# Solving 2×2 Games by Hand: Best Responses, Pure Equilibria, and Mixing > Learn to solve two-by-two strategic-form games without prior game-theory training. The lecture builds payoff matrices from familiar stories, marks each player's best responses to locate pure Nash equilibria, and uses Matching Pennies to explain why predictable pure choices can be exploited. Expected-payoff graphs then derive mixed strategies through indifference. An asymmetric inspection game exposes the cross-player probability rule, and a final guided example leads to a practical six-step hand procedure. - Canonical watch page: [Solving 2×2 Games by Hand: Best Responses, Pure Equilibria, and Mixing](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Economics - Published: 2026-08-28T19:07:13.510Z - Updated: 2026-08-28T19:07:13.510Z - Duration: PT1402S (23 minutes 22 seconds) - Chapters: 5 - Views: 1 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TXXS3DT7ZDH2RQ7RMREF3/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TXXS3DT7ZDH2RQ7RMREF3/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TXXS3DT7ZDH2RQ7RMREF3/0/dark/poster.jpg) ## Description Solve 2×2 games by hand using payoff matrices, best-response marks, pure Nash equilibria, expected payoffs, and mixed strategies. ## Chapters - [00:00–05:26.89 · Payoffs, Best Responses, and Pure Equilibria](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=0) - [05:26.89–08:37.793 · Matching Pennies and Exploitation](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=326.89004166666666) - [08:37.793–12:53.318 · Mixing and Indifference](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=517.7931458333334) - [12:53.318–17:59.728 · The Asymmetric Inspection Game](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=773.3182916666667) - [17:59.728–23:22 · Guided Practice and the Hand Procedure](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1079.7282916666668) ## Transcript ### [00:00 · Payoffs, Best Responses, and Pure Equilibria](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=0) You already know the Prisoner's Dilemma as a story. Our goal is to turn that kind of story into a calculation you can carry out by hand. Two people choose actions, their choices select one outcome, and we want every outcome where neither person benefits by changing alone. The method has four stages. Read the payoff pairs, mark each player's best replies, keep every cell with both marks, and if no cell survives, ask whether randomized choices can create an equilibrium. Start with the object the whole method uses: a payoff matrix. Row has two actions, Top and Bottom. Column has two actions, Left and Right. One action from each player selects exactly one of the four inner cells. Each inner cell contains an ordered pair of numbers. The first number is Row's payoff, and the second number is Column's payoff. In the upper-left cell, Row receives four and Column receives three. A payoff can mean money, points, votes, years of freedom, or simply a ranking. The scale depends on the story. For solving the game, each player compares only that player's own numbers, and a larger number means a preferred outcome. The players choose without first observing the other's current choice. Still, to analyse incentives we ask a conditional question. If I knew which action the other player had chosen, which of my actions would give me the largest payoff? That conditional answer is called a best response. Begin with Row and suppose Column chooses Left. Compare Row's first numbers in the Left column: four from Top and two from Bottom. Four is larger, so Top is Row's best response to Left. Now suppose Column chooses Right. Row compares zero from Top with three from Bottom. Three is larger, so Bottom is Row's best response to Right. We mark that first payoff in red as well. Now change viewpoints. Hold Row fixed on Top and compare Column's second numbers across that row. Column gets three from Left and one from Right. Left is better, so the second payoff three receives Column's blue mark. Hold Row fixed on Bottom. Column compares zero from Left with four from Right. Right is better, so the four receives the second blue mark. Row's comparisons ran down columns; Column's comparisons ran across rows. Now inspect whole cells. The upper-left cell carries Row's red mark and Column's blue mark. The lower-right cell also carries both. At either outcome, each player's chosen action is a best response to the other player's chosen action. A cell with both marks is a pure-strategy Nash equilibrium. Pure means that each player chooses one action with certainty. Nash equilibrium means that, holding the other player's action fixed, neither player gains by changing alone. Check the upper-left outcome directly. If Column stays Left, Row falls from four to two by switching. If Row stays Top, Column falls from three to one by switching. Neither unilateral change helps. The same check works at the lower-right outcome. Row would fall from three to zero by switching, and Column would fall from four to zero. A game can therefore have more than one pure equilibrium, and we must keep every double-marked cell. There is one small rule about ties. If two available actions give the same maximum payoff against an opponent's action, both are best responses. Mark both. Never break a payoff tie merely to force one answer. Now put the familiar Prisoner's Dilemma into this formal language. Each prisoner chooses Cooperate or Defect. Mutual cooperation gives two each. A lone defector receives three while the cooperator receives zero, and mutual defection gives one each. Mark Row's best responses first. Against Column's cooperation, Row prefers three from defecting to two from cooperating. Against Column's defection, Row prefers one from defecting to zero from cooperating. Both red marks land in the Defect row. Now mark Column's best responses using the second numbers. Against Row's cooperation, Column receives three by defecting. Against Row's defection, Column receives one by defecting. Both blue marks land in the Defect column. Only the lower-right cell has both marks, so mutual defection is the unique pure Nash equilibrium. Mutual cooperation gives both players more, but either player can gain individually by defecting while the other cooperates. That distinction matters. Equilibrium is a claim about incentives against one-player deviations, not a claim that the outcome is fair, cooperative, or jointly best. We now have a mechanical test for pure equilibria. Next we need a game where that test leaves no cell at all. ### [05:26.89 · Matching Pennies and Exploitation](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=326.89004166666666) Matching Pennies is the smallest game in which the pure-equilibrium search fails completely. Each player secretly chooses Heads or Tails. Row earns one when the faces match, Column earns one when they differ, and the loser receives minus one. Apply the same marking procedure. If Column chooses Heads, Row wants Heads. If Column chooses Tails, Row wants Tails. Row's best responses are the two matching cells, so mark Row's payoff in each of those cells red. Column wants exactly the opposite pattern. If Row chooses Heads, Column wants Tails. If Row chooses Tails, Column wants Heads. Column's two best responses are the mismatching cells, so mark those second payoffs blue. Inspect all four cells. Every cell carries one player's mark, but no cell carries both. At every deterministic outcome, one player is losing and can reverse the result by switching. Therefore Matching Pennies has no pure Nash equilibrium. The incentives form a cycle. Start at Heads, Heads. Row wins there, so Column wants to switch to Tails. That change carries the outcome to Heads, Tails. Now Row is losing, so Row switches to Tails. At Tails, Tails, Column is losing and switches to Heads. Then Row is losing and switches back to Heads. We return to the starting outcome without ever reaching a cell where both players want to stay. This cycle is another way to check the matrix. A pure equilibrium would be a stopping point with no profitable arrow leaving it. Here every one of the four outcomes has an escape for exactly one player. The practical problem is predictability. Suppose Row always chooses Heads, or follows a pattern Column has learned. Column chooses Tails, forces a mismatch, and wins every round. The same vulnerability runs in the other direction. If Column always chooses Heads, Row copies Heads and wins every round. In this game, any deterministic pattern that an opponent can predict reveals the pure reply that defeats it. Randomizing does not mean alternating according to a visible schedule. Heads, Tails, Heads, Tails is deterministic and therefore exploitable once noticed. A mixed strategy assigns probabilities and uses genuine random choice so that past actions do not reveal the next one. The next question is precise. Can we choose probabilities that leave the opponent indifferent between Heads and Tails? If both pure replies give the same expected payoff, the opponent has no profitable way to exploit one of them. That indifference condition is the key to a mixed-strategy equilibrium. We will write each pure action's expected payoff as a function of the opponent's probability, plot the two functions, and find exactly where they cross. ### [08:37.793 · Mixing and Indifference](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=517.7931458333334) Let q be the probability that Column chooses Heads. We are choosing Column's probability, but the equations we compare are Row's payoffs. Column's mix must remove Row's preference between Row's Heads and Row's Tails. If Row chooses Heads, Row earns one when Column chooses Heads and minus one when Column chooses Tails. Weight those two payoffs by q and one minus q. Simplifying gives two q minus one. The red line on the plot rises with q because Heads becomes more attractive as Column chooses Heads more often. If Row instead chooses Tails, matching Column's Heads loses one and differing from Column's Tails wins one. The weighted payoff is q times minus one plus one minus q times one. That simplifies to one minus two q. The blue line falls as q rises because Tails becomes less attractive when Column chooses Heads more often. At q equals zero, Column always chooses Tails. Row strongly prefers Tails, so the blue payoff is one and the red payoff is minus one. Move q upward and that advantage shrinks. Indifference occurs where the two payoff lines cross. Set two q minus one equal to one minus two q. Adding two q and adding one gives four q equals two, so q equals one half. At the yellow crossing, Row's Heads and Tails both have expected payoff zero. This calculation found Column's probability from Row's payoffs. That cross-player direction is not a trick. A player's mix is chosen to control the opponent's incentives. Now let p be the probability that Row chooses Heads. To find p, compare Column's two pure actions using Column's payoffs. Row's probability must make Column indifferent. If Column chooses Heads, Column loses one when Row also chooses Heads and wins one when Row chooses Tails. The expected payoff is p times minus one plus one minus p times one. That simplifies to one minus two p. If Column chooses Tails, Column wins against Row's Heads and loses against Row's Tails, giving two p minus one. Set the two expressions equal. One minus two p equals two p minus one, so p equals one half. The symmetry of Matching Pennies produced the same half-and-half probability for both players. That symmetry is special. The method, comparing the opponent's pure-action payoffs, is the part that generalizes. The mixed-strategy equilibrium has each player choose Heads with probability one half and Tails with probability one half. Each of the four outcome cells then occurs with probability one quarter. Verify Row first. Against Column's half-and-half mix, Heads wins one half the time and loses one half, so its expected payoff is zero. Tails has the same calculation and also gives zero. Verify Column in the same way. Either pure action wins half the time and loses half, so both give zero. Because neither player has a better pure reply, neither can improve by changing to any other mixture either. Notice what equilibrium does not require. The realized actions can differ from round to round, and after any particular round one player may wish the coin had landed differently. Equilibrium says the probability rule itself cannot be profitably replaced while the opponent keeps the equilibrium mix. We now know the general shape of a mixing calculation. Assign a probability to one player's first action, compute the other player's two pure-action payoffs, set them equal, and solve. The asymmetric example next will show why remembering whose payoffs to use is essential. ### [12:53.318 · The Asymmetric Inspection Game](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=773.3182916666667) Now take a genuinely asymmetric game. A worker chooses Work or Shirk, while an inspector chooses Inspect or Do Not Inspect. The roles differ, the available actions differ, and the first payoff in each cell belongs to the worker. Read the worker's incentives. If inspection occurs, Work gives two while Shirk gives minus two, so Work is better. Without inspection, Shirk gives three while Work gives two, so Shirk is better. Now read the inspector's incentives using the second payoffs. If the worker Works, avoiding the inspection cost gives zero instead of minus one. If the worker Shirks, inspection gives one instead of minus two. No cell carries both marks. Work makes Do Not Inspect attractive, which then makes Shirk attractive. Shirk makes Inspect attractive, which then makes Work attractive. The pure incentives cycle, so there is no pure equilibrium. A natural guess says that if the worker Works most of the time, the inspector must also Inspect most of the time. Perhaps inspection needs probability one half or more. That guess confuses the frequency of an action with the strength of the incentive created by it. Let q be the probability that the inspector inspects. To find q, use the worker's payoffs. The inspector must choose q so that the worker is indifferent between Work and Shirk. Work pays the worker two if inspected and two if not inspected. Its expected payoff is two q plus two times one minus q, which is simply two. The blue line is flat. Shirk pays minus two if inspected and three if not inspected. Its expected payoff is minus two q plus three times one minus q. Simplifying gives three minus five q. The red line slopes downward because more inspection makes shirking less attractive. At an inspection probability of zero, Shirk pays three and beats Work's two. Move q to one tenth and Shirk still pays two point five, so the worker still prefers Shirk. Set the two worker payoffs equal. Two equals three minus five q. Solving gives q equals one fifth, or twenty percent. At exactly twenty percent inspection, Work and Shirk both give the worker an expected payoff of two. Below that crossing, Shirk is better. Above it, Work is better. The mixing point is the boundary between those strict preferences. Now let p be the probability that the worker Works. To find p, switch to the inspector's payoffs. The worker must choose p so that Inspect and Do Not Inspect give the inspector the same expected payoff. Inspect gives the inspector minus one against Work and one against Shirk. Its expected payoff is minus p plus one minus p, which simplifies to one minus two p. Do Not Inspect gives zero against Work and minus two against Shirk. Its expected payoff is zero times p minus two times one minus p, which simplifies to minus two plus two p. Set the two inspector payoffs equal. One minus two p equals minus two plus two p. Rearranging gives three equals four p, so p equals three quarters. Move to p equals zero point seven five. The two payoff lines cross at minus one half. Inspect and Do Not Inspect are equally good, so the inspector can genuinely mix. The equilibrium has the worker Work with probability three quarters and the inspector Inspect with probability one fifth. Frequent work is supported by infrequent inspection because being caught while shirking is costly. Verify the worker. At q equals one fifth, Work gives two. Shirk gives three minus five times one fifth, also two. Verify the inspector. At p equals three quarters, both Inspect and Do Not Inspect give minus one half. Here is the rule worth carrying away. The worker's payoff numbers determined the inspector's mixing probability. The inspector's payoff numbers determined the worker's mixing probability. Your probability is chosen to erase the opponent's strict preference. That is why intuition based only on how often an action appears can be misleading. Mixed equilibrium probabilities are not direct measures of effort, importance, or virtue. They are the probabilities that balance the other player's expected payoffs. ### [17:59.728 · Guided Practice and the Hand Procedure](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1079.7282916666668) Let's finish by solving a new game from beginning to end. Row chooses Up or Down. Column chooses Left or Right. The question asks for every equilibrium, so we must check pure cells first and then ask whether a mixed equilibrium also exists. Pause at the matrix and begin with Row. Against Column's Left, compare Row's first payoffs three and zero. Up is the unique best response, so mark the three red. Against Column's Right, Row compares zero with one. Down is the best response, so mark the one in the lower-right cell red. Now analyse Column using the second numbers. Against Row's Up, Column compares two from Left with zero from Right. Left is better, so mark the two blue. Against Row's Down, Column compares zero from Left with three from Right. Right is better, so mark the three blue. Inspect complete cells. Up, Left carries both marks, and Down, Right carries both marks. Those are two pure-strategy Nash equilibria. Do not stop merely because pure equilibria exist. Some two-by-two games, including this one, also have a mixed equilibrium. We find it with exactly the same indifference method used before. Let p be the probability that Row chooses Up, and q the probability that Column chooses Left. Remember the cross-player rule. Use Row's payoffs to find q, because q must make Row indifferent. If Row chooses Up, the payoff is three against Left and zero against Right. The expected payoff is therefore three q. If Row chooses Down, the payoff is zero against Left and one against Right. Its expected payoff is one minus q. Set Row's two payoffs equal. Three q equals one minus q, so four q equals one and q equals one quarter. Column chooses Left with probability one quarter to keep Row willing to mix. Now use Column's payoffs to find p. If Column chooses Left, the expected payoff is two p. If Column chooses Right, the expected payoff is three times one minus p. Set those equal. Two p equals three minus three p, so five p equals three and p equals three fifths. Row chooses Up with probability three fifths to keep Column indifferent. Notice that the probabilities are not mirror images of the most attractive payoffs. Column's one-quarter probability came from Row's payoff comparison, while Row's three-fifths probability came from Column's payoff comparison. Always verify a mixed result before accepting it. At q equals one quarter, Row's Up payoff is three times one quarter, or three quarters. Row's Down payoff is one minus one quarter, also three quarters. At p equals three fifths, Column's Left payoff is two times three fifths, or six fifths. Column's Right payoff is three times two fifths, also six fifths. Both indifference conditions hold. A directional check catches sign errors. If q rises above one quarter, Up becomes better for Row; if q falls below one quarter, Down becomes better. The crossing is exactly where Row changes preferred actions. Likewise, if p rises above three fifths, Left becomes better for Column; below three fifths, Right becomes better. At the crossing, each player is willing to use either pure action. This game therefore has three equilibria: the two double-marked pure cells and the interior mixed equilibrium we just verified. Finding one equilibrium is not permission to stop when the question asks for all of them. Here is the complete hand method in six steps. First, label the payoff order. Second, for each column mark every largest Row payoff. Third, for each row mark every largest Column payoff. Fourth, every cell with both marks is a pure equilibrium. Fifth, if mixing is relevant, introduce probabilities and use each probability to make the opponent indifferent. Your mix controls the opponent's incentives. Sixth, verify. Probabilities must lie between zero and one. Pure actions used with positive probability must give equal expected payoffs, and any unused action must not give more. The shortest memory aid is this: your mixing probability is solved from the opponent's payoff comparison. It is chosen to make the opponent indifferent, not to make your own two payoffs equal directly. With that discipline, a two-by-two game becomes a small sequence of comparisons and two linear equations. Read the pairs, mark the best responses, keep every double mark, balance the opponent when mixing, and check the result. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TXXS3DT7ZDH2RQ7RMREF3/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Payoffs, Best Responses, and Pure Equilibria](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=0) Span: 00:00–05:26.89 (0s–326.89004166666666s). #### Objects - cell\_reading: a Math \[text\] that says "$(4, 3): quad 4 thin upright("for Row"), quad 3 thin upright("for Column")$" - column\_rule: a Text \[text\] that says "For Column: hold one row fixed, compare the second numbers, and mark every maximum." - coordination: a Table \[text\] that says "Column: Left Column: Right Row: Top $(4, 3)$ $(0, 1)$ Row: Bottom $(2, 0)$ $(3, 4)$" (rows=(('', 'Column: Left', 'Column: Right'), ('Row: Top', '$(4, 3)$'…, header=True) - double\_rule: a Text \[text\] that says "A cell carrying both players' marks is a pure-strategy Nash equilibrium." - head\_matrix: a Heading that says "How to Read a Payoff Matrix" - head\_pd: a Heading that says "The Prisoner's Dilemma" - head\_pure: a Heading that says "Two Pure Equilibria" - head\_responses: a Heading that says "Mark the Best Responses" - matrix\_prompt: a Text \[text\] that says "Row chooses Top or Bottom. Column chooses Left or Right. Larger own payoffs are preferred." - opening: a Panel that says "Given two decision makers, two actions each, and four possible outcomes, how can we find every outcome where neither person wants to change alone?" - payoff\_order: a Panel that says "The first number in every pair belongs to Row. The second belongs to Column." - pd\_answer: a Math \[text\] that says "$(upright("Defect"), upright("Defect"))$" - pd\_matrix: a Table \[text\] that says "Column: Cooperate Column: Defect Row: Cooperate $(2, 2)$ $(0, 3)$ Row: Defect $(3, 0)$ $(1, 1)$" (rows=(('', 'Column: Cooperate', 'Column: Defect'), ('Row: Cooperate'…, header=True) - pd\_method: a Text \[text\] that says "Compare Row's first numbers down columns, then Column's second numbers across rows." - pd\_warning: a Math \[text\] that says "$upright("equilibrium") eq.not upright("jointly best")$" - pure\_answer: a Math \[text\] that says "$(T,L) quad upright("and") quad (B,R)$" - pure\_definition: a Panel that says "An action pair in which each player's action is a best response to the other player's action." - route: a Block \[text\] that says "Read each ordered payoff pair. Mark each player's best responses. Find cells carrying both marks. If no cell carries both marks, consider mixing." - row\_rule: a Text \[text\] that says "For Row: hold one column fixed, compare the first numbers, and mark every maximum." - tie\_rule: a Math \[text\] that says "$upright("tie") arrow.r upright("mark both")$" #### Beats ##### [00:00](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=0) Narration: You already know the Prisoner's Dilemma as a story. Our goal is to turn that kind of story into a calculation you can carry out by hand. Two people choose actions, their choices select one outcome, and we want every outcome where neither person benefits by changing alone. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=0): opening is shown on the screen, written out. ##### [00:16.669](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=16.6685) Narration: The method has four stages. Read the payoff pairs, mark each player's best replies, keep every cell with both marks, and if no cell survives, ask whether randomized choices can create an equilibrium. Board: opening — a Panel that says "Given two decision makers, two actions each, and four possible outcomes, how can we find every outcome where neither person wants to change alone?" Actions: - [00:17.679](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=17.679000000000002): route is shown on the screen, written out. - [00:30.45](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=30.4495): opening is hidden from the screen — left the board. - [00:30.45](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=30.4495): route is hidden from the screen — left the board. ##### [00:31.65](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=31.649500000000003) Narration: Start with the object the whole method uses: a payoff matrix. Row has two actions, Top and Bottom. Column has two actions, Left and Right. One action from each player selects exactly one of the four inner cells. Board: Empty. Actions: - [00:31.65](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=31.649500000000003): head\_matrix is shown on the screen, written out. - [00:35.655](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=35.65500000000001): coordination is shown on the screen, written out. - [00:36.99](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=36.99): matrix\_prompt is shown on the screen, written out. - [00:38.558](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=38.55800000000001): coordination is shown on the screen, written out. - [00:38.987](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=38.987): coordination is shown on the screen, written out. ##### [00:49.119](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=49.118500000000004) Narration: Each inner cell contains an ordered pair of numbers. The first number is Row's payoff, and the second number is Column's payoff. In the upper-left cell, Row receives four and Column receives three. Board: matrix\_prompt — a Text \[text\] that says "Row chooses Top or Bottom. Column chooses Left or Right. Larger own payoffs are preferred."; head\_matrix — a Heading that says "How to Read a Payoff Matrix" Actions: - [00:53.612](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=53.61200000000001): payoff\_order is shown on the screen, written out. - [00:58.604](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=58.604000000000006): coordination (the "$(4, 3)$" part) is indicated — a transient flash. - [01:0.52](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=60.52000000000001): cell\_reading is shown on the screen, written out. ##### [01:3.663](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=63.66250000000001) Narration: A payoff can mean money, points, votes, years of freedom, or simply a ranking. The scale depends on the story. For solving the game, each player compares only that player's own numbers, and a larger number means a preferred outcome. Board: matrix\_prompt — a Text \[text\] that says "Row chooses Top or Bottom. Column chooses Left or Right. Larger own payoffs are preferred."; payoff\_order — a Panel that says "The first number in every pair belongs to Row. The second belongs to Column."; cell\_reading — a Math \[text\] that says "$(4, 3): quad 4 thin upright("for Row"), quad 3 thin upright("for Column")$"; head\_matrix — a Heading that says "How to Read a Payoff Matrix" Actions: - [01:16.026](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=76.02600000000001): coordination (the "4" part) is indicated — a transient flash. - [01:17.338](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=77.338): coordination (the "3" part) is indicated — a transient flash. ##### [01:20.307](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=80.30700000000002) Narration: The players choose without first observing the other's current choice. Still, to analyse incentives we ask a conditional question. If I knew which action the other player had chosen, which of my actions would give me the largest payoff? Board: Unchanged from the preceding beat in this scene. Actions: - [01:34.576](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=94.57550000000002): cell\_reading is hidden from the screen — left the board. - [01:34.576](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=94.57550000000002): head\_matrix is hidden from the screen — left the board. - [01:34.576](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=94.57550000000002): matrix\_prompt is hidden from the screen — left the board. - [01:34.576](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=94.57550000000002): payoff\_order is hidden from the screen — left the board. ##### [01:35.776](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=95.77550000000002) Narration: That conditional answer is called a best response. Begin with Row and suppose Column chooses Left. Compare Row's first numbers in the Left column: four from Top and two from Bottom. Four is larger, so Top is Row's best response to Left. Board: Empty. Actions: - [01:35.776](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=95.77550000000002): head\_responses is shown on the screen, written out. - [01:37.889](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=97.88900000000002): row\_rule is shown on the screen, written out. - [01:46.004](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=106.00400000000002): coordination (the "4" part) is emphasized. ##### [01:53.837](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=113.83700000000002) Narration: Now suppose Column chooses Right. Row compares zero from Top with three from Bottom. Three is larger, so Bottom is Row's best response to Right. We mark that first payoff in red as well. Board: row\_rule — a Text \[text\] that says "For Row: hold one column fixed, compare the first numbers, and mark every maximum."; head\_responses — a Heading that says "Mark the Best Responses" Actions: - [01:59.038](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=119.03800000000001): coordination (the "3" part) is emphasized. ##### [02:8.358](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=128.35750000000002) Narration: Now change viewpoints. Hold Row fixed on Top and compare Column's second numbers across that row. Column gets three from Left and one from Right. Left is better, so the second payoff three receives Column's blue mark. Board: Unchanged from the preceding beat in this scene. Actions: - [02:15.347](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=135.34700000000004): column\_rule is shown on the screen, written out. - [02:15.974](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=135.974): coordination (the "3" part) is emphasized. ##### [02:23.319](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=143.31900000000002) Narration: Hold Row fixed on Bottom. Column compares zero from Left with four from Right. Right is better, so the four receives the second blue mark. Row's comparisons ran down columns; Column's comparisons ran across rows. Board: row\_rule — a Text \[text\] that says "For Row: hold one column fixed, compare the first numbers, and mark every maximum."; column\_rule — a Text \[text\] that says "For Column: hold one row fixed, compare the second numbers, and mark every maximum."; head\_responses — a Heading that says "Mark the Best Responses" Actions: - [02:28.253](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=148.25300000000004): coordination (the "4" part) is emphasized. - [02:34.673](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=154.67300000000003): row\_rule (the "one column" part) is indicated — a transient flash. - [02:37.03](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=157.03000000000003): column\_rule (the "one row" part) is indicated — a transient flash. ##### [02:38.826](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=158.82600000000002) Narration: Now inspect whole cells. The upper-left cell carries Row's red mark and Column's blue mark. The lower-right cell also carries both. At either outcome, each player's chosen action is a best response to the other player's chosen action. Board: Unchanged from the preceding beat in this scene. Actions: - [02:40.01](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=160.01000000000002): double\_rule is shown on the screen, written out. - [02:41.694](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=161.69400000000002): coordination (the "$(4, 3)$" part) is indicated — a transient flash. - [02:46.175](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=166.175): coordination (the "$(3, 4)$" part) is indicated — a transient flash. - [02:54.801](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=174.80100000000002): column\_rule is hidden from the screen — left the board. - [02:54.801](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=174.80100000000002): double\_rule is hidden from the screen — left the board. - [02:54.801](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=174.80100000000002): head\_responses is hidden from the screen — left the board. - [02:54.801](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=174.80100000000002): row\_rule is hidden from the screen — left the board. ##### [02:56.001](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=176.00100000000003) Narration: A cell with both marks is a pure-strategy Nash equilibrium. Pure means that each player chooses one action with certainty. Nash equilibrium means that, holding the other player's action fixed, neither player gains by changing alone. Board: Empty. Actions: - [02:56.001](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=176.00100000000003): head\_pure is shown on the screen, written out. - [02:58.462](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=178.46200000000005): pure\_definition is shown on the screen, written out. - [02:59.762](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=179.76200000000006): pure\_answer is shown on the screen, written out. ##### [03:12.797](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=192.79650000000004) Narration: Check the upper-left outcome directly. If Column stays Left, Row falls from four to two by switching. If Row stays Top, Column falls from three to one by switching. Neither unilateral change helps. Board: pure\_definition — a Panel that says "An action pair in which each player's action is a best response to the other player's action."; pure\_answer — a Math \[text\] that says "$(T,L) quad upright("and") quad (B,R)$"; head\_pure — a Heading that says "Two Pure Equilibria" Actions: - [03:13.632](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=193.63200000000003): coordination (the "$(4, 3)$" part) is indicated — a transient flash. ##### [03:28.293](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=208.29250000000005) Narration: The same check works at the lower-right outcome. Row would fall from three to zero by switching, and Column would fall from four to zero. A game can therefore have more than one pure equilibrium, and we must keep every double-marked cell. Board: Unchanged from the preceding beat in this scene. Actions: - [03:29.86](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=209.86000000000007): coordination (the "$(3, 4)$" part) is indicated — a transient flash. - [03:38.95](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=218.95000000000007): pure\_answer is indicated — a transient flash. ##### [03:44.113](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=224.11250000000004) Narration: There is one small rule about ties. If two available actions give the same maximum payoff against an opponent's action, both are best responses. Mark both. Never break a payoff tie merely to force one answer. Board: Unchanged from the preceding beat in this scene. Actions: - [03:46.121](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=226.12100000000007): tie\_rule is shown on the screen, written out. - [03:58.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=238.61400000000003): coordination is hidden from the screen — left the board. - [03:58.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=238.61400000000003): head\_pure is hidden from the screen — left the board. - [03:58.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=238.61400000000003): pure\_answer is hidden from the screen — left the board. - [03:58.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=238.61400000000003): pure\_definition is hidden from the screen — left the board. - [03:58.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=238.61400000000003): tie\_rule is hidden from the screen — left the board. ##### [03:59.214](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=239.21400000000003) Narration: Now put the familiar Prisoner's Dilemma into this formal language. Each prisoner chooses Cooperate or Defect. Mutual cooperation gives two each. A lone defector receives three while the cooperator receives zero, and mutual defection gives one each. Board: Empty. Actions: - [03:59.214](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=239.21400000000003): head\_pd is shown on the screen, written out. - [04:3.626](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=243.62600000000006): pd\_matrix is shown on the screen, written out. - [04:7.607](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=247.60700000000006): pd\_matrix is shown on the screen, written out. - [04:15.142](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=255.14200000000005): pd\_matrix is shown on the screen, written out. ##### [04:17.507](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=257.507) Narration: Mark Row's best responses first. Against Column's cooperation, Row prefers three from defecting to two from cooperating. Against Column's defection, Row prefers one from defecting to zero from cooperating. Both red marks land in the Defect row. Board: head\_pd — a Heading that says "The Prisoner's Dilemma" Actions: - [04:17.855](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=257.855): pd\_method is shown on the screen, written out. - [04:23.707](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=263.707): pd\_matrix (the "3" part) is emphasized. - [04:29.187](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=269.187): pd\_matrix (the "1" part) is emphasized. ##### [04:35.603](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=275.6035) Narration: Now mark Column's best responses using the second numbers. Against Row's cooperation, Column receives three by defecting. Against Row's defection, Column receives one by defecting. Both blue marks land in the Defect column. Board: pd\_method — a Text \[text\] that says "Compare Row's first numbers down columns, then Column's second numbers across rows."; head\_pd — a Heading that says "The Prisoner's Dilemma" Actions: - [04:42.639](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=282.639): pd\_matrix (the "3" part) is emphasized. - [04:46.876](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=286.87600000000003): pd\_matrix (the "1#2" part) is emphasized. ##### [04:52.156](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=292.1555) Narration: Only the lower-right cell has both marks, so mutual defection is the unique pure Nash equilibrium. Mutual cooperation gives both players more, but either player can gain individually by defecting while the other cooperates. Board: Unchanged from the preceding beat in this scene. Actions: - [04:52.98](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=292.9800000000001): pd\_answer is shown on the screen, written out. - [04:54.152](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=294.15200000000004): pd\_matrix (the "$(1, 1)$" part) is indicated — a transient flash. ##### [05:7.43](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=307.4305) Narration: That distinction matters. Equilibrium is a claim about incentives against one-player deviations, not a claim that the outcome is fair, cooperative, or jointly best. We now have a mechanical test for pure equilibria. Next we need a game where that test leaves no cell at all. Board: pd\_method — a Text \[text\] that says "Compare Row's first numbers down columns, then Column's second numbers across rows."; pd\_answer — a Math \[text\] that says "$(upright("Defect"), upright("Defect"))$"; head\_pd — a Heading that says "The Prisoner's Dilemma" Actions: - [05:8.232](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=308.232): pd\_warning is shown on the screen, written out. - [05:25.848](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=325.848375): head\_pd is hidden from the screen — left the board. - [05:25.848](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=325.848375): pd\_answer is hidden from the screen — left the board. - [05:25.848](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=325.848375): pd\_matrix is hidden from the screen — left the board. - [05:25.848](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=325.848375): pd\_method is hidden from the screen — left the board. - [05:25.848](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=325.848375): pd\_warning is hidden from the screen — left the board. ### Scene 2: [Matching Pennies and Exploitation](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=326.89004166666666) Span: 05:26.89–08:37.793 (326.89004166666666s–517.7931458333334s). #### Objects - column\_explanation: a Text \[text\] that says "If Column's next face is predictable, Row copies that face and wins." - column\_label: a Tex \[text\] that says "Column is predictable" - column\_move\_one: a CurvedArrow \[blue\] labelled "C" drawn in cycle (start=(1.0, 3.0), end=(3.0, 3.0), bend=0.18) - column\_move\_two: a CurvedArrow \[blue\] labelled "C" drawn in cycle (start=(3.0, 1.0), end=(1.0, 1.0), bend=0.18) - column\_predictable: a Math \[text\] that says "$P(H\_C)=1 quad arrow.r quad R upright(" chooses H")$" - cycle: a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)) - cycle\_key: a Math \[text\] that says "$R: upright("Row switches"), quad C: upright("Column switches")$" - head\_cycle: a Heading that says "Every Pure Outcome Has an Escape" - head\_matrix: a Heading that says "Matching Pennies" - head\_predict: a Heading that says "Predictability Can Be Exploited" - hh: a Point \[green\] labelled "(H,H)" drawn in cycle (location=(1.0, 3.0)) - ht: a Point \[blue\] labelled "(H,T)" drawn in cycle (location=(3.0, 3.0)) - marking: a Text \[text\] that says "Red marks Row's best responses. Blue marks Column's best responses." - mixing\_question: a Panel that says "Can each player randomize so that the opponent has no better pure reply?" - no\_pure: a Math \[text\] that says "$upright("no cell has both marks")$" - payoffs: a Table \[text\] that says "Column: Heads Column: Tails Row: Heads $(1, -1)$ $(-1, 1)$ Row: Tails $(-1, 1)$ $(1, -1)$" (rows=(('', 'Column: Heads', 'Column: Tails'), ('Row: Heads', '$(1, -…, header=True) - row\_explanation: a Text \[text\] that says "If Row's next face is predictable, Column chooses the opposite face and wins." - row\_label: a Tex \[text\] that says "Row is predictable" - row\_move\_one: a CurvedArrow \[red\] labelled "R" drawn in cycle (start=(3.0, 3.0), end=(3.0, 1.0), bend=0.18) - row\_move\_two: a CurvedArrow \[red\] labelled "R" drawn in cycle (start=(1.0, 1.0), end=(1.0, 3.0), bend=0.18) - row\_predictable: a Math \[text\] that says "$P(H\_R)=1 quad arrow.r quad C upright(" chooses T")$" - rules: a Panel that says "Each player chooses Heads or Tails. Row wins when the choices match. Column wins when they differ." - th: a Point \[blue\] labelled "(T,H)" drawn in cycle (location=(1.0, 1.0)) - tt: a Point \[green\] labelled "(T,T)" drawn in cycle (location=(3.0, 1.0)) #### Beats ##### [05:26.89](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=326.89004166666666) Narration: Matching Pennies is the smallest game in which the pure-equilibrium search fails completely. Each player secretly chooses Heads or Tails. Row earns one when the faces match, Column earns one when they differ, and the loser receives minus one. Board: Empty. Actions: - [05:26.89](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=326.89004166666666): head\_matrix is shown on the screen, written out. - [05:33.519](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=333.5190416666667): rules is shown on the screen, written out. - [05:34.495](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=334.4950416666667): payoffs is shown on the screen, written out. - [05:35.911](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=335.9110416666667): payoffs is shown on the screen, written out. - [05:38.419](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=338.41904166666666): payoffs is shown on the screen, written out. ##### [05:43.141](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=343.14054166666665) Narration: Apply the same marking procedure. If Column chooses Heads, Row wants Heads. If Column chooses Tails, Row wants Tails. Row's best responses are the two matching cells, so mark Row's payoff in each of those cells red. Board: rules — a Panel that says "Each player chooses Heads or Tails. Row wins when the choices match. Column wins when they differ."; head\_matrix — a Heading that says "Matching Pennies" Actions: - [05:44.046](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=344.04604166666667): marking is shown on the screen, written out. - [05:46.414](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=346.41404166666666): payoffs (the "1" part) is emphasized. - [05:49.804](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=349.80404166666665): payoffs (the "1" part) is emphasized. ##### [05:58.915](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=358.91504166666664) Narration: Column wants exactly the opposite pattern. If Row chooses Heads, Column wants Tails. If Row chooses Tails, Column wants Heads. Column's two best responses are the mismatching cells, so mark those second payoffs blue. Board: rules — a Panel that says "Each player chooses Heads or Tails. Row wins when the choices match. Column wins when they differ."; marking — a Text \[text\] that says "Red marks Row's best responses. Blue marks Column's best responses."; head\_matrix — a Heading that says "Matching Pennies" Actions: - [06:3.176](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=363.17604166666666): payoffs (the "1#2" part) is emphasized. - [06:4.488](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=364.4880416666667): payoffs (the "1#2" part) is emphasized. ##### [06:15.467](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=375.46654166666667) Narration: Inspect all four cells. Every cell carries one player's mark, but no cell carries both. At every deterministic outcome, one player is losing and can reverse the result by switching. Therefore Matching Pennies has no pure Nash equilibrium. Board: Unchanged from the preceding beat in this scene. Actions: - [06:20.598](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=380.59804166666663): no\_pure is shown on the screen, written out. - [06:32.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=392.5215416666666): head\_matrix is hidden from the screen — left the board. - [06:32.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=392.5215416666666): marking is hidden from the screen — left the board. - [06:32.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=392.5215416666666): no\_pure is hidden from the screen — left the board. - [06:32.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=392.5215416666666): payoffs is hidden from the screen — left the board. - [06:32.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=392.5215416666666): rules is hidden from the screen — left the board. ##### [06:33.722](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=393.72154166666667) Narration: The incentives form a cycle. Start at Heads, Heads. Row wins there, so Column wants to switch to Tails. That change carries the outcome to Heads, Tails. Board: Empty. Actions: - [06:33.722](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=393.72154166666667): head\_cycle is shown on the screen, written out. - [06:33.722](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=393.72154166666667): cycle is shown on the screen, written out. - [06:36.821](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=396.8210416666667): hh is shown on the screen, written out. - [06:37.021](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=397.0210416666667): ht is shown on the screen, written out. - [06:37.421](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=397.42104166666667): tt is shown on the screen, written out. - [06:38.021](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=398.0210416666667): th is shown on the screen, written out. - [06:40.769](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=400.7690416666667): column\_move\_one is shown on the screen, written out. ##### [06:46.083](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=406.08254166666666) Narration: Now Row is losing, so Row switches to Tails. At Tails, Tails, Column is losing and switches to Heads. Then Row is losing and switches back to Heads. We return to the starting outcome without ever reaching a cell where both players want to stay. Board: cycle — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); head\_cycle — a Heading that says "Every Pure Outcome Has an Escape"; hh — a Point \[green\] labelled "(H,H)" drawn in cycle (location=(1.0, 3.0)); ht — a Point \[blue\] labelled "(H,T)" drawn in cycle (location=(3.0, 3.0)); tt — a Point \[green\] labelled "(T,T)" drawn in cycle (location=(3.0, 1.0)); th — a Point \[blue\] labelled "(T,H)" drawn in cycle (location=(1.0, 1.0)); column\_move\_one — a CurvedArrow \[blue\] labelled "C" drawn in cycle (start=(1.0, 3.0), end=(3.0, 3.0), bend=0.18) Actions: - [06:48.021](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=408.02104166666663): row\_move\_one is shown on the screen, written out. - [06:51.388](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=411.38804166666665): column\_move\_two is shown on the screen, written out. - [06:55.777](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=415.77704166666666): row\_move\_two is shown on the screen, written out. - [06:58.761](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=418.76104166666664): cycle\_key is shown on the screen, written out. ##### [07:3.773](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=423.77304166666664) Narration: This cycle is another way to check the matrix. A pure equilibrium would be a stopping point with no profitable arrow leaving it. Here every one of the four outcomes has an escape for exactly one player. Board: cycle — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); cycle\_key — a Math \[text\] that says "$R: upright("Row switches"), quad C: upright("Column switches")$"; head\_cycle — a Heading that says "Every Pure Outcome Has an Escape"; hh — a Point \[green\] labelled "(H,H)" drawn in cycle (location=(1.0, 3.0)); ht — a Point \[blue\] labelled "(H,T)" drawn in cycle (location=(3.0, 3.0)); tt — a Point \[green\] labelled "(T,T)" drawn in cycle (location=(3.0, 1.0)); th — a Point \[blue\] labelled "(T,H)" drawn in cycle (location=(1.0, 1.0)); column\_move\_one — a CurvedArrow \[blue\] labelled "C" drawn in cycle (start=(1.0, 3.0), end=(3.0, 3.0), bend=0.18); row\_move\_one — a CurvedArrow \[red\] labelled "R" drawn in cycle (start=(3.0, 3.0), end=(3.0, 1.0), bend=0.18); column\_move\_two — a CurvedArrow \[blue\] labelled "C" drawn in cycle (start=(3.0, 1.0), end=(1.0, 1.0), bend=0.18); row\_move\_two — a CurvedArrow \[red\] labelled "R" drawn in cycle (start=(1.0, 1.0), end=(1.0, 3.0), bend=0.18) Actions: - [07:13.548](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=433.5480416666667): hh is indicated — a transient flash. - [07:13.748](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=433.74804166666667): ht is indicated — a transient flash. - [07:14.148](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=434.14804166666664): tt is indicated — a transient flash. - [07:14.748](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=434.74804166666667): th is indicated — a transient flash. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): cycle is hidden from the screen — left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): hh is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): ht is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): tt is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): th is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): column\_move\_one is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): row\_move\_one is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): column\_move\_two is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): row\_move\_two is hidden from the screen — cycle left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): cycle\_key is hidden from the screen — left the board. - [07:16.868](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=436.86804166666667): head\_cycle is hidden from the screen — left the board. ##### [07:18.068](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=438.06804166666666) Narration: The practical problem is predictability. Suppose Row always chooses Heads, or follows a pattern Column has learned. Column chooses Tails, forces a mismatch, and wins every round. Board: Empty. Actions: - [07:18.068](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=438.06804166666666): head\_predict is shown on the screen, written out. - [07:21.818](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=441.81804166666666): row\_label is shown on the screen, written out. - [07:22.143](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=442.14304166666665): row\_predictable is shown on the screen, written out. - [07:25.278](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=445.27804166666664): row\_explanation is shown on the screen, written out. ##### [07:31.66](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=451.66004166666664) Narration: The same vulnerability runs in the other direction. If Column always chooses Heads, Row copies Heads and wins every round. In this game, any deterministic pattern that an opponent can predict reveals the pure reply that defeats it. Board: row\_label — a Tex \[text\] that says "Row is predictable"; row\_predictable — a Math \[text\] that says "$P(H\_R)=1 quad arrow.r quad C upright(" chooses T")$"; row\_explanation — a Text \[text\] that says "If Row's next face is predictable, Column chooses the opposite face and wins."; head\_predict — a Heading that says "Predictability Can Be Exploited" Actions: - [07:35.364](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=455.36404166666665): column\_label is shown on the screen, written out. - [07:35.735](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=455.7350416666667): column\_predictable is shown on the screen, written out. - [07:43.943](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=463.94304166666666): column\_explanation is shown on the screen, written out. ##### [07:47.933](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=467.93304166666667) Narration: Randomizing does not mean alternating according to a visible schedule. Heads, Tails, Heads, Tails is deterministic and therefore exploitable once noticed. A mixed strategy assigns probabilities and uses genuine random choice so that past actions do not reveal the next one. Board: row\_label — a Tex \[text\] that says "Row is predictable"; row\_predictable — a Math \[text\] that says "$P(H\_R)=1 quad arrow.r quad C upright(" chooses T")$"; row\_explanation — a Text \[text\] that says "If Row's next face is predictable, Column chooses the opposite face and wins."; column\_label — a Tex \[text\] that says "Column is predictable"; column\_predictable — a Math \[text\] that says "$P(H\_C)=1 quad arrow.r quad R upright(" chooses H")$"; column\_explanation — a Text \[text\] that says "If Column's next face is predictable, Row copies that face and wins."; head\_predict — a Heading that says "Predictability Can Be Exploited" Actions: - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): column\_explanation is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): column\_label is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): column\_predictable is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): head\_predict is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): row\_explanation is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): row\_label is hidden from the screen — left the board. - [08:7.241](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=487.2405416666667): row\_predictable is hidden from the screen — left the board. ##### [08:8.441](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=488.44054166666666) Narration: The next question is precise. Can we choose probabilities that leave the opponent indifferent between Heads and Tails? If both pure replies give the same expected payoff, the opponent has no profitable way to exploit one of them. Board: Empty. Actions: - [08:9.427](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=489.4270416666667): mixing\_question is shown on the screen, written out. ##### [08:23.205](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=503.20454166666667) Narration: That indifference condition is the key to a mixed-strategy equilibrium. We will write each pure action's expected payoff as a function of the opponent's probability, plot the two functions, and find exactly where they cross. Board: mixing\_question — a Panel that says "Can each player randomize so that the opponent has no better pure reply?" Actions: - [08:36.751](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=516.7514791666667): mixing\_question is hidden from the screen — left the board. ### Scene 3: [Mixing and Indifference](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=517.7931458333334) Span: 08:37.793–12:53.318 (517.7931458333334s–773.3182916666667s). #### Objects - column\_h\_curve: a FunctionPlot \[red\] labelled "upright("Column Heads")" drawn in p\_plot (function=\) - column\_h\_probe: a PlotPoint \[red\] drawn in p\_plot (target='column\_h\_curve', x=\) - column\_t\_curve: a FunctionPlot \[blue\] labelled "upright("Column Tails")" drawn in p\_plot (function=\) - column\_t\_probe: a PlotPoint \[blue\] drawn in p\_plot (target='column\_t\_curve', x=\) - equilibrium: a Math \[text\] that says "$P(R upright(" Heads"))=frac(1,2), quad P(C upright(" Heads"))=frac(1,2)$" - head\_p: a Heading that says "Choose $p$ to Make Column Indifferent" - head\_q: a Heading that says "Choose $q$ to Make Row Indifferent" - head\_result: a Heading that says "The Mixed-Strategy Equilibrium" - mixed\_definition: a Panel that says "A probability distribution over pure actions. In equilibrium, every pure action used with positive probability gives the same expected payoff." - outcome\_probability: a Math \[text\] that says "$P(H,H)=P(H,T)=P(T,H)=P(T,T)=frac(1,4)$" - p\_answer: a Math \[text\] that says "$P(R upright(" chooses Heads"))=frac(1,2)$" - p\_crossing: a Point \[yellow\] labelled "p=frac(1,2)" drawn in p\_plot (location=(0.5, 0.0)) - p\_definition: a Math \[text\] that says "$p=P(R upright(" chooses Heads"))$" - p\_live: a VariableNumber (format\_spec='.2f') - p\_plot: an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5) - p\_work: a Derivation \[text\] that says "$U\_C(H \| p) &= p(-1)+(1-p)(1) \\ &=1-2p \\ U\_C(T \| p) &= p(1)+(1-p)(-1) \\ &=2p-1 \\ 1-2p &= 2p-1 \\ p &= frac(1,2)$" - q\_answer: a Math \[text\] that says "$P(C upright(" chooses Heads"))=frac(1,2)$" - q\_crossing: a Point \[yellow\] labelled "q=frac(1,2)" drawn in q\_plot (location=(0.5, 0.0)) - q\_definition: a Math \[text\] that says "$q=P(C upright(" chooses Heads"))$" - q\_live: a VariableNumber (format\_spec='.2f') - q\_plot: an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5) - q\_work: a Derivation \[text\] that says "$U\_R(H \| q) &= q(1)+(1-q)(-1) \\ &=2q-1 \\ U\_R(T \| q) &= q(-1)+(1-q)(1) \\ &=1-2q \\ 2q-1 &= 1-2q \\ q &= frac(1,2)$" - row\_h\_curve: a FunctionPlot \[red\] labelled "upright("Row Heads")" drawn in q\_plot (function=\) - row\_h\_probe: a PlotPoint \[red\] drawn in q\_plot (target='row\_h\_curve', x=\) - row\_t\_curve: a FunctionPlot \[blue\] labelled "upright("Row Tails")" drawn in q\_plot (function=\) - row\_t\_probe: a PlotPoint \[blue\] drawn in q\_plot (target='row\_t\_curve', x=\) - verification: a Math \[text\] that says "$U\_R(H)=U\_R(T)=0, quad U\_C(H)=U\_C(T)=0$" #### Beats ##### [08:37.793](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=517.7931458333334) Narration: Let q be the probability that Column chooses Heads. We are choosing Column's probability, but the equations we compare are Row's payoffs. Column's mix must remove Row's preference between Row's Heads and Row's Tails. Board: Empty. Actions: - [08:37.793](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=517.7931458333334): head\_q is shown on the screen, written out. - [08:38.095](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=518.0951458333334): q\_definition is shown on the screen, written out. - [08:38.582](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=518.5821458333334): q\_plot is shown on the screen, written out. - [08:44.643](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=524.6431458333334): row\_h\_probe is shown on the screen, written out. - [08:44.643](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=524.6431458333334): row\_t\_probe is shown on the screen, written out. - [08:50.018](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=530.0181458333334): row\_h\_curve is shown on the screen, drawn. - [08:51.098](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=531.0981458333333): row\_t\_curve is shown on the screen, drawn. ##### [08:52.894](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=532.8941458333334) Narration: If Row chooses Heads, Row earns one when Column chooses Heads and minus one when Column chooses Tails. Weight those two payoffs by q and one minus q. Board: q\_definition — a Math \[text\] that says "$q=P(C upright(" chooses Heads"))$"; q\_plot — an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5); head\_q — a Heading that says "Choose $q$ to Make Row Indifferent"; row\_h\_curve — a FunctionPlot \[red\] labelled "upright("Row Heads")" drawn in q\_plot (function=\); row\_t\_curve — a FunctionPlot \[blue\] labelled "upright("Row Tails")" drawn in q\_plot (function=\); row\_h\_probe — a PlotPoint \[red\] drawn in q\_plot (target='row\_h\_curve', x=\); row\_t\_probe — a PlotPoint \[blue\] drawn in q\_plot (target='row\_t\_curve', x=\) Actions: - [08:55.506](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=535.5061458333333): q\_work is shown on the screen, written out. - [09:0.626](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=540.6261458333333): q\_work (the "q(1)" part) is emphasized. - [09:2.855](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=542.8551458333334): q\_work (the "(1-q)(-1)" part) is emphasized. - [09:2.855](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=542.8551458333334): q\_work (the "q(1)" part) is no longer emphasized. - [09:4.19](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=544.1901458333334): q\_work (the "(1-q)(-1)" part) is no longer emphasized. ##### [09:4.79](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=544.7901458333333) Narration: Simplifying gives two q minus one. The red line on the plot rises with q because Heads becomes more attractive as Column chooses Heads more often. Board: Unchanged from the preceding beat in this scene. Actions: - [09:5.092](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=545.0921458333333): q\_work is shown on the screen, written out. - [09:8.436](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=548.4361458333334): row\_h\_curve is indicated — a transient flash. ##### [09:15.932](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=555.9321458333334) Narration: If Row instead chooses Tails, matching Column's Heads loses one and differing from Column's Tails wins one. The weighted payoff is q times minus one plus one minus q times one. Board: Unchanged from the preceding beat in this scene. Actions: - [09:16.849](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=556.8491458333334): q\_work is shown on the screen, written out. ##### [09:30.116](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=570.1156458333334) Narration: That simplifies to one minus two q. The blue line falls as q rises because Tails becomes less attractive when Column chooses Heads more often. Board: Unchanged from the preceding beat in this scene. Actions: - [09:30.638](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=570.6381458333334): q\_work is shown on the screen, written out. - [09:33.587](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=573.5871458333334): row\_t\_curve is indicated — a transient flash. ##### [09:41.014](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=581.0136458333334) Narration: At q equals zero, Column always chooses Tails. Row strongly prefers Tails, so the blue payoff is one and the red payoff is minus one. Move q upward and that advantage shrinks. Board: Unchanged from the preceding beat in this scene. Actions: - [09:51.729](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=591.7291458333334): row\_h\_probe is redrawn as the numbers it depends on change. - [09:51.729](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=591.7291458333334): row\_t\_probe is redrawn as the numbers it depends on change. - [09:51.729](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=591.7291458333334): q\_live ticks to 0.25. ##### [09:55.215](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=595.2146458333334) Narration: Indifference occurs where the two payoff lines cross. Set two q minus one equal to one minus two q. Board: Unchanged from the preceding beat in this scene. Actions: - [09:58.03](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=598.0301458333333): row\_h\_probe is redrawn as the numbers it depends on change. - [09:58.03](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=598.0301458333333): row\_t\_probe is redrawn as the numbers it depends on change. - [09:58.03](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=598.0301458333333): q\_live ticks to 0.5. - [09:59.342](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=599.3421458333333): q\_work is shown on the screen, written out. ##### [10:4.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=604.5216458333334) Narration: Adding two q and adding one gives four q equals two, so q equals one half. At the yellow crossing, Row's Heads and Tails both have expected payoff zero. Board: Unchanged from the preceding beat in this scene. Actions: - [10:9.949](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=609.9491458333334): q\_answer is shown on the screen, written out. - [10:9.949](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=609.9491458333334): q\_work is shown on the screen, written out. - [10:11.946](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=611.9461458333334): q\_crossing is shown on the screen, written out. - [10:16.729](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=616.7291458333334): A box is drawn around q\_answer. ##### [10:17.329](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=617.3291458333334) Narration: This calculation found Column's probability from Row's payoffs. That cross-player direction is not a trick. A player's mix is chosen to control the opponent's incentives. Board: q\_definition — a Math \[text\] that says "$q=P(C upright(" chooses Heads"))$"; q\_answer — a Math \[text\] that says "$P(C upright(" chooses Heads"))=frac(1,2)$"; q\_plot — an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5); head\_q — a Heading that says "Choose $q$ to Make Row Indifferent"; row\_h\_curve — a FunctionPlot \[red\] labelled "upright("Row Heads")" drawn in q\_plot (function=\); row\_t\_curve — a FunctionPlot \[blue\] labelled "upright("Row Tails")" drawn in q\_plot (function=\); row\_h\_probe — a PlotPoint \[red\] drawn in q\_plot (target='row\_h\_curve', x=\); row\_t\_probe — a PlotPoint \[blue\] drawn in q\_plot (target='row\_t\_curve', x=\); q\_crossing — a Point \[yellow\] labelled "q=frac(1,2)" drawn in q\_plot (location=(0.5, 0.0)) Actions: - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): head\_q is hidden from the screen — left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): q\_answer is hidden from the screen — left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): q\_definition is hidden from the screen — left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): q\_plot is hidden from the screen — left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): row\_h\_curve is hidden from the screen — q\_plot left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): row\_t\_curve is hidden from the screen — q\_plot left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): row\_h\_probe is hidden from the screen — q\_plot left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): row\_t\_probe is hidden from the screen — q\_plot left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): q\_crossing is hidden from the screen — q\_plot left the board. - [10:28.313](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=628.3126458333334): q\_work is hidden from the screen — left the board. ##### [10:29.513](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=629.5126458333334) Narration: Now let p be the probability that Row chooses Heads. To find p, compare Column's two pure actions using Column's payoffs. Row's probability must make Column indifferent. Board: Empty. Actions: - [10:29.513](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=629.5126458333334): head\_p is shown on the screen, written out. - [10:30.267](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=630.2671458333334): p\_definition is shown on the screen, written out. - [10:30.801](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=630.8011458333334): p\_plot is shown on the screen, written out. - [10:34.818](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=634.8181458333333): column\_h\_probe is shown on the screen, written out. - [10:34.818](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=634.8181458333333): column\_t\_probe is shown on the screen, written out. - [10:35.26](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=635.2601458333334): column\_h\_curve is shown on the screen, drawn. - [10:36.049](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=636.0491458333333): column\_t\_curve is shown on the screen, drawn. ##### [10:42.658](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=642.6576458333334) Narration: If Column chooses Heads, Column loses one when Row also chooses Heads and wins one when Row chooses Tails. The expected payoff is p times minus one plus one minus p times one. Board: p\_definition — a Math \[text\] that says "$p=P(R upright(" chooses Heads"))$"; p\_plot — an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5); head\_p — a Heading that says "Choose $p$ to Make Column Indifferent"; column\_h\_curve — a FunctionPlot \[red\] labelled "upright("Column Heads")" drawn in p\_plot (function=\); column\_t\_curve — a FunctionPlot \[blue\] labelled "upright("Column Tails")" drawn in p\_plot (function=\); column\_h\_probe — a PlotPoint \[red\] drawn in p\_plot (target='column\_h\_curve', x=\); column\_t\_probe — a PlotPoint \[blue\] drawn in p\_plot (target='column\_t\_curve', x=\) Actions: - [10:45.043](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=645.0431458333334): p\_work is shown on the screen, written out. ##### [10:55.942](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=655.9416458333334) Narration: That simplifies to one minus two p. If Column chooses Tails, Column wins against Row's Heads and loses against Row's Tails, giving two p minus one. Board: Unchanged from the preceding beat in this scene. Actions: - [10:56.371](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=656.3711458333335): p\_work is shown on the screen, written out. - [10:57.741](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=657.7411458333333): p\_work is shown on the screen, written out. - [11:0.017](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=660.0171458333334): p\_work is shown on the screen, written out. ##### [11:6.91](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=666.9096458333333) Narration: Set the two expressions equal. One minus two p equals two p minus one, so p equals one half. Board: Unchanged from the preceding beat in this scene. Actions: - [11:8.419](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=668.4191458333335): p\_work is shown on the screen, written out. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): column\_h\_probe is redrawn as the numbers it depends on change. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): column\_t\_probe is redrawn as the numbers it depends on change. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): p\_crossing is shown on the screen, written out. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): p\_answer is shown on the screen, written out. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): p\_work is shown on the screen, written out. - [11:13.992](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=673.9921458333333): p\_live ticks to 0.5. - [11:15.014](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=675.0136458333334): A box is drawn around p\_answer. ##### [11:15.614](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=675.6136458333334) Narration: The symmetry of Matching Pennies produced the same half-and-half probability for both players. That symmetry is special. The method, comparing the opponent's pure-action payoffs, is the part that generalizes. Board: p\_definition — a Math \[text\] that says "$p=P(R upright(" chooses Heads"))$"; p\_answer — a Math \[text\] that says "$P(R upright(" chooses Heads"))=frac(1,2)$"; p\_plot — an Axes (y\_range=(-1.2, 1.2), x\_ticks\_every=0.25, y\_ticks\_every=0.5); head\_p — a Heading that says "Choose $p$ to Make Column Indifferent"; column\_h\_curve — a FunctionPlot \[red\] labelled "upright("Column Heads")" drawn in p\_plot (function=\); column\_t\_curve — a FunctionPlot \[blue\] labelled "upright("Column Tails")" drawn in p\_plot (function=\); column\_h\_probe — a PlotPoint \[red\] drawn in p\_plot (target='column\_h\_curve', x=\); column\_t\_probe — a PlotPoint \[blue\] drawn in p\_plot (target='column\_t\_curve', x=\); p\_crossing — a Point \[yellow\] labelled "p=frac(1,2)" drawn in p\_plot (location=(0.5, 0.0)) Actions: - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): head\_p is hidden from the screen — left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): p\_answer is hidden from the screen — left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): p\_definition is hidden from the screen — left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): p\_plot is hidden from the screen — left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): column\_h\_curve is hidden from the screen — p\_plot left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): column\_t\_curve is hidden from the screen — p\_plot left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): column\_h\_probe is hidden from the screen — p\_plot left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): column\_t\_probe is hidden from the screen — p\_plot left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): p\_crossing is hidden from the screen — p\_plot left the board. - [11:30.242](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=690.2421458333333): p\_work is hidden from the screen — left the board. ##### [11:31.442](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=691.4421458333334) Narration: The mixed-strategy equilibrium has each player choose Heads with probability one half and Tails with probability one half. Each of the four outcome cells then occurs with probability one quarter. Board: Empty. Actions: - [11:31.442](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=691.4421458333334): head\_result is shown on the screen, written out. - [11:31.988](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=691.9881458333334): mixed\_definition is shown on the screen, written out. - [11:36.121](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=696.1211458333333): equilibrium is shown on the screen, written out. - [11:43.075](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=703.0751458333334): outcome\_probability is shown on the screen, written out. ##### [11:44.999](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=704.9991458333334) Narration: Verify Row first. Against Column's half-and-half mix, Heads wins one half the time and loses one half, so its expected payoff is zero. Tails has the same calculation and also gives zero. Board: equilibrium — a Math \[text\] that says "$P(R upright(" Heads"))=frac(1,2), quad P(C upright(" Heads"))=frac(1,2)$"; mixed\_definition — a Panel that says "A probability distribution over pure actions. In equilibrium, every pure action used with positive probability gives the same expected payoff."; outcome\_probability — a Math \[text\] that says "$P(H,H)=P(H,T)=P(T,H)=P(T,T)=frac(1,4)$"; head\_result — a Heading that says "The Mixed-Strategy Equilibrium" Actions: - [11:54.553](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=714.5531458333334): verification is shown on the screen, written out. - [11:56.527](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=716.5271458333334): verification (the "U\_R(H)=U\_R(T)=0" part) is emphasized. ##### [11:59.96](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=719.9601458333334) Narration: Verify Column in the same way. Either pure action wins half the time and loses half, so both give zero. Because neither player has a better pure reply, neither can improve by changing to any other mixture either. Board: equilibrium — a Math \[text\] that says "$P(R upright(" Heads"))=frac(1,2), quad P(C upright(" Heads"))=frac(1,2)$"; mixed\_definition — a Panel that says "A probability distribution over pure actions. In equilibrium, every pure action used with positive probability gives the same expected payoff."; outcome\_probability — a Math \[text\] that says "$P(H,H)=P(H,T)=P(T,H)=P(T,T)=frac(1,4)$"; verification — a Math \[text\] that says "$U\_R(H)=U\_R(T)=0, quad U\_C(H)=U\_C(T)=0$"; head\_result — a Heading that says "The Mixed-Strategy Equilibrium" Actions: - [12:0.877](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=720.8771458333334): verification (the "U\_C(H)=U\_C(T)=0" part) is emphasized. - [12:0.877](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=720.8771458333334): verification (the "U\_R(H)=U\_R(T)=0" part) is no longer emphasized. - [12:14.438](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=734.4376458333334): verification (the "U\_C(H)=U\_C(T)=0" part) is no longer emphasized. ##### [12:15.038](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=735.0376458333334) Narration: Notice what equilibrium does not require. The realized actions can differ from round to round, and after any particular round one player may wish the coin had landed differently. Equilibrium says the probability rule itself cannot be profitably replaced while the opponent keeps the equilibrium mix. Board: Unchanged from the preceding beat in this scene. Actions: - [12:27.854](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=747.8541458333334): equilibrium is indicated — a transient flash. ##### [12:34.434](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=754.4336458333333) Narration: We now know the general shape of a mixing calculation. Assign a probability to one player's first action, compute the other player's two pure-action payoffs, set them equal, and solve. The asymmetric example next will show why remembering whose payoffs to use is essential. Board: Unchanged from the preceding beat in this scene. Actions: - [12:52.277](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=772.276625): equilibrium is hidden from the screen — left the board. - [12:52.277](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=772.276625): head\_result is hidden from the screen — left the board. - [12:52.277](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=772.276625): mixed\_definition is hidden from the screen — left the board. - [12:52.277](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=772.276625): outcome\_probability is hidden from the screen — left the board. - [12:52.277](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=772.276625): verification is hidden from the screen — left the board. ### Scene 4: [The Asymmetric Inspection Game](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=773.3182916666667) Span: 12:53.318–17:59.728 (773.3182916666667s–1079.7282916666668s). #### Objects - cross\_player: a Panel that says "Worker payoffs determine the inspector's probability. Inspector payoffs determine the worker's probability." - equilibrium: a Math \[text\] that says "$P(upright("Work"))=frac(3,4), quad P(upright("Inspect"))=frac(1,5)$" - head\_game: a Heading that says "An Inspection Game" - head\_p: a Heading that says "How Often Does the Worker Work?" - head\_q: a Heading that says "How Often Must the Inspector Inspect?" - head\_result: a Heading that says "The Counterintuitive Result" - inspect\_curve: a FunctionPlot \[blue\] labelled "upright("Inspect")" drawn in p\_plot (function=\) - inspect\_probe: a PlotPoint \[blue\] drawn in p\_plot (target='inspect\_curve', x=\) - inspection: a Table \[text\] that says "Inspect Do Not Inspect Work $(2, -1)$ $(2, 0)$ Shirk $(-2, 1)$ $(3, -2)$" (rows=(('', 'Inspect', 'Do Not Inspect'), ('Work', '$(2, -1)$', '$(2,…, header=True) - inspector\_working: a Derivation \[text\] that says "$U\_I(I \| p) &= -p+(1-p) \\ &=1-2p \\ U\_I(N \| p) &= 0p-2(1-p) \\ &=-2+2p \\ 1-2p &= -2+2p \\ p &= frac(3,4)$" - intuition: a Text \[text\] that says "Tempting guess: frequent work must require frequent inspection." - no\_inspect\_curve: a FunctionPlot \[red\] labelled "upright("Do Not Inspect")" drawn in p\_plot (function=\) - no\_inspect\_probe: a PlotPoint \[red\] drawn in p\_plot (target='no\_inspect\_curve', x=\) - no\_pure: a Math \[text\] that says "$upright("no pure equilibrium")$" - p\_answer: a Math \[text\] that says "$P(upright("Work"))=frac(3,4)=0.75$" - p\_crossing: a Point \[yellow\] labelled "p=frac(3,4)" drawn in p\_plot (location=(0.75, -0.5)) - p\_live: a VariableNumber (format\_spec='.2f') - p\_plot: an Axes (y\_range=(-2.5, 1.5), x\_ticks\_every=0.25, y\_ticks\_every=0.5) - q\_answer: a Math \[text\] that says "$P(upright("Inspect"))=frac(1,5)=0.20$" - q\_crossing: a Point \[yellow\] labelled "q=frac(1,5)" drawn in q\_plot (location=(0.2, 2.0)) - q\_live: a VariableNumber (format\_spec='.2f') - q\_plot: an Axes (y\_range=(-2.5, 3.5), x\_ticks\_every=0.2, y\_ticks\_every=1.0) - shirk\_curve: a FunctionPlot \[red\] labelled "upright("Shirk")" drawn in q\_plot (function=\) - shirk\_probe: a PlotPoint \[red\] drawn in q\_plot (target='shirk\_curve', x=\) - story: a Panel that says "Worker chooses Work or Shirk. Inspector chooses Inspect or Do Not Inspect. The first payoff belongs to Worker." - verification: a Table \[text\] that says "Player Pure actions at equilibrium Expected payoff Worker Work and Shirk $2$ Inspector Inspect and Do Not Inspect $-1/2$" (rows=(('Player', 'Pure actions at equilibrium', 'Expected payoff'), …, header=True) - work\_curve: a FunctionPlot \[blue\] labelled "upright("Work")" drawn in q\_plot (function=\) - work\_probe: a PlotPoint \[blue\] drawn in q\_plot (target='work\_curve', x=\) - worker\_working: a Derivation \[text\] that says "$U\_W(W \| q) &= 2q+2(1-q) \\ &=2 \\ U\_W(S \| q) &= -2q+3(1-q) \\ &=3-5q \\ 2 &= 3-5q \\ q &= frac(1,5)$" #### Beats ##### [12:53.318](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=773.3182916666667) Narration: Now take a genuinely asymmetric game. A worker chooses Work or Shirk, while an inspector chooses Inspect or Do Not Inspect. The roles differ, the available actions differ, and the first payoff in each cell belongs to the worker. Board: Empty. Actions: - [12:53.318](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=773.3182916666667): head\_game is shown on the screen, written out. - [12:56.604](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=776.6042916666667): story is shown on the screen, written out. - [12:57.532](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=777.5322916666668): inspection is shown on the screen, written out. - [12:58.171](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=778.1712916666667): inspection is shown on the screen, written out. - [13:5.183](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=785.1832916666667): inspection is shown on the screen, written out. ##### [13:10.52](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=790.5202916666667) Narration: Read the worker's incentives. If inspection occurs, Work gives two while Shirk gives minus two, so Work is better. Without inspection, Shirk gives three while Work gives two, so Shirk is better. Board: story — a Panel that says "Worker chooses Work or Shirk. Inspector chooses Inspect or Do Not Inspect. The first payoff belongs to Worker."; head\_game — a Heading that says "An Inspection Game" Actions: - [13:15.942](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=795.9422916666667): inspection (the "2" part) is emphasized. - [13:21.782](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=801.7822916666668): inspection (the "3" part) is emphasized. ##### [13:26.051](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=806.0507916666667) Narration: Now read the inspector's incentives using the second payoffs. If the worker Works, avoiding the inspection cost gives zero instead of minus one. If the worker Shirks, inspection gives one instead of minus two. Board: Unchanged from the preceding beat in this scene. Actions: - [13:34.05](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=814.0502916666667): inspection (the "0" part) is emphasized. - [13:35.432](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=815.4322916666667): inspection (the "1" part) is emphasized. ##### [13:41.813](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=821.8132916666667) Narration: No cell carries both marks. Work makes Do Not Inspect attractive, which then makes Shirk attractive. Shirk makes Inspect attractive, which then makes Work attractive. The pure incentives cycle, so there is no pure equilibrium. Board: Unchanged from the preceding beat in this scene. Actions: - [13:42.289](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=822.2892916666667): no\_pure is shown on the screen, written out. ##### [14:0.154](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=840.1537916666667) Narration: A natural guess says that if the worker Works most of the time, the inspector must also Inspect most of the time. Perhaps inspection needs probability one half or more. That guess confuses the frequency of an action with the strength of the incentive created by it. Board: story — a Panel that says "Worker chooses Work or Shirk. Inspector chooses Inspect or Do Not Inspect. The first payoff belongs to Worker."; no\_pure — a Math \[text\] that says "$upright("no pure equilibrium")$"; head\_game — a Heading that says "An Inspection Game" Actions: - [14:0.757](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=840.7572916666667): intuition is shown on the screen, written out. - [14:16.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=856.6507916666667): head\_game is hidden from the screen — left the board. - [14:16.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=856.6507916666667): inspection is hidden from the screen — left the board. - [14:16.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=856.6507916666667): intuition is hidden from the screen — left the board. - [14:16.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=856.6507916666667): no\_pure is hidden from the screen — left the board. - [14:16.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=856.6507916666667): story is hidden from the screen — left the board. ##### [14:17.851](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=857.8507916666667) Narration: Let q be the probability that the inspector inspects. To find q, use the worker's payoffs. The inspector must choose q so that the worker is indifferent between Work and Shirk. Board: Empty. Actions: - [14:17.851](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=857.8507916666667): head\_q is shown on the screen, written out. - [14:18.547](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=858.5472916666668): q\_plot is shown on the screen, written out. - [14:28.427](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=868.4272916666667): work\_probe is shown on the screen, written out. - [14:28.427](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=868.4272916666667): shirk\_probe is shown on the screen, written out. - [14:29.843](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=869.8432916666667): work\_curve is shown on the screen, drawn. - [14:30.494](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=870.4942916666668): shirk\_curve is shown on the screen, drawn. ##### [14:32.011](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=872.0107916666667) Narration: Work pays the worker two if inspected and two if not inspected. Its expected payoff is two q plus two times one minus q, which is simply two. The blue line is flat. Board: q\_plot — an Axes (y\_range=(-2.5, 3.5), x\_ticks\_every=0.2, y\_ticks\_every=1.0); head\_q — a Heading that says "How Often Must the Inspector Inspect?"; work\_curve — a FunctionPlot \[blue\] labelled "upright("Work")" drawn in q\_plot (function=\); shirk\_curve — a FunctionPlot \[red\] labelled "upright("Shirk")" drawn in q\_plot (function=\); work\_probe — a PlotPoint \[blue\] drawn in q\_plot (target='work\_curve', x=\); shirk\_probe — a PlotPoint \[red\] drawn in q\_plot (target='shirk\_curve', x=\) Actions: - [14:33.474](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=873.4742916666668): q\_plot moves to a new place on the board. - [14:33.474](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=873.4742916666668): worker\_working is shown on the screen, written out. - [14:42.239](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=882.2392916666668): worker\_working is shown on the screen, written out. - [14:43.714](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=883.7142916666667): work\_curve is indicated — a transient flash. ##### [14:45.655](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=885.6547916666667) Narration: Shirk pays minus two if inspected and three if not inspected. Its expected payoff is minus two q plus three times one minus q. Board: Unchanged from the preceding beat in this scene. Actions: - [14:46.787](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=886.7872916666668): worker\_working is shown on the screen, written out. ##### [14:55.915](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=895.9147916666667) Narration: Simplifying gives three minus five q. The red line slopes downward because more inspection makes shirking less attractive. Board: Unchanged from the preceding beat in this scene. Actions: - [14:56.234](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=896.2342916666668): worker\_working is shown on the screen, written out. - [14:59.682](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=899.6822916666667): shirk\_curve is indicated — a transient flash. ##### [15:4.799](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=904.7987916666667) Narration: At an inspection probability of zero, Shirk pays three and beats Work's two. Move q to one tenth and Shirk still pays two point five, so the worker still prefers Shirk. Board: Unchanged from the preceding beat in this scene. Actions: - [15:11.579](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=911.5792916666667): work\_probe is redrawn as the numbers it depends on change. - [15:11.579](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=911.5792916666667): shirk\_probe is redrawn as the numbers it depends on change. - [15:11.579](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=911.5792916666667): q\_live ticks to 0.1. ##### [15:17.577](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=917.5772916666667) Narration: Set the two worker payoffs equal. Two equals three minus five q. Solving gives q equals one fifth, or twenty percent. Board: Unchanged from the preceding beat in this scene. Actions: - [15:19.319](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=919.3192916666667): worker\_working is shown on the screen, written out. - [15:24.555](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=924.5552916666668): worker\_working is shown on the screen, written out. - [15:25.716](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=925.7162916666667): work\_probe is redrawn as the numbers it depends on change. - [15:25.716](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=925.7162916666667): shirk\_probe is redrawn as the numbers it depends on change. - [15:25.716](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=925.7162916666667): q\_crossing is shown on the screen, written out. - [15:25.716](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=925.7162916666667): q\_answer is shown on the screen, written out. - [15:25.716](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=925.7162916666667): q\_live ticks to 0.2. ##### [15:27.593](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=927.5932916666667) Narration: At exactly twenty percent inspection, Work and Shirk both give the worker an expected payoff of two. Below that crossing, Shirk is better. Above it, Work is better. The mixing point is the boundary between those strict preferences. Board: q\_answer — a Math \[text\] that says "$P(upright("Inspect"))=frac(1,5)=0.20$"; q\_plot — an Axes (y\_range=(-2.5, 3.5), x\_ticks\_every=0.2, y\_ticks\_every=1.0); head\_q — a Heading that says "How Often Must the Inspector Inspect?"; work\_curve — a FunctionPlot \[blue\] labelled "upright("Work")" drawn in q\_plot (function=\); shirk\_curve — a FunctionPlot \[red\] labelled "upright("Shirk")" drawn in q\_plot (function=\); work\_probe — a PlotPoint \[blue\] drawn in q\_plot (target='work\_curve', x=\); shirk\_probe — a PlotPoint \[red\] drawn in q\_plot (target='shirk\_curve', x=\); q\_crossing — a Point \[yellow\] labelled "q=frac(1,5)" drawn in q\_plot (location=(0.2, 2.0)) Actions: - [15:28.777](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=928.7772916666668): A box is drawn around q\_answer. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): head\_q is hidden from the screen — left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): q\_answer is hidden from the screen — left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): q\_plot is hidden from the screen — left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): work\_curve is hidden from the screen — q\_plot left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): shirk\_curve is hidden from the screen — q\_plot left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): work\_probe is hidden from the screen — q\_plot left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): shirk\_probe is hidden from the screen — q\_plot left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): q\_crossing is hidden from the screen — q\_plot left the board. - [15:43.697](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=943.6967916666667): worker\_working is hidden from the screen — left the board. ##### [15:44.897](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=944.8967916666668) Narration: Now let p be the probability that the worker Works. To find p, switch to the inspector's payoffs. The worker must choose p so that Inspect and Do Not Inspect give the inspector the same expected payoff. Board: Empty. Actions: - [15:44.897](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=944.8967916666668): head\_p is shown on the screen, written out. - [15:45.709](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=945.7092916666668): p\_plot is shown on the screen, written out. - [15:54.846](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=954.8462916666667): inspect\_curve is shown on the screen, drawn. - [15:55.531](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=955.5312916666667): no\_inspect\_curve is shown on the screen, drawn. - [15:57.017](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=957.0172916666668): inspect\_probe is shown on the screen, written out. - [15:57.017](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=957.0172916666668): no\_inspect\_probe is shown on the screen, written out. ##### [16:0.091](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=960.0907916666667) Narration: Inspect gives the inspector minus one against Work and one against Shirk. Its expected payoff is minus p plus one minus p, which simplifies to one minus two p. Board: p\_plot — an Axes (y\_range=(-2.5, 1.5), x\_ticks\_every=0.25, y\_ticks\_every=0.5); head\_p — a Heading that says "How Often Does the Worker Work?"; inspect\_curve — a FunctionPlot \[blue\] labelled "upright("Inspect")" drawn in p\_plot (function=\); no\_inspect\_curve — a FunctionPlot \[red\] labelled "upright("Do Not Inspect")" drawn in p\_plot (function=\); inspect\_probe — a PlotPoint \[blue\] drawn in p\_plot (target='inspect\_curve', x=\); no\_inspect\_probe — a PlotPoint \[red\] drawn in p\_plot (target='no\_inspect\_curve', x=\) Actions: - [16:1.913](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=961.9132916666667): p\_plot moves to a new place on the board. - [16:1.913](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=961.9132916666667): inspector\_working is shown on the screen, written out. - [16:8.391](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=968.3912916666667): inspect\_curve is indicated — a transient flash. - [16:9.854](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=969.8542916666668): inspector\_working is shown on the screen, written out. ##### [16:12.8](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=972.7997916666667) Narration: Do Not Inspect gives zero against Work and minus two against Shirk. Its expected payoff is zero times p minus two times one minus p, which simplifies to minus two plus two p. Board: Unchanged from the preceding beat in this scene. Actions: - [16:14.413](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=974.4132916666667): inspector\_working is shown on the screen, written out. - [16:15.737](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=975.7372916666667): no\_inspect\_curve is indicated — a transient flash. - [16:23.515](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=983.5152916666667): inspector\_working is shown on the screen, written out. ##### [16:27.07](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=987.0697916666668) Narration: Set the two inspector payoffs equal. One minus two p equals minus two plus two p. Rearranging gives three equals four p, so p equals three quarters. Board: Unchanged from the preceding beat in this scene. Actions: - [16:28.945](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=988.9452916666667): inspector\_working is shown on the screen, written out. - [16:37.246](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=997.2462916666667): inspector\_working is shown on the screen, written out. ##### [16:38.984](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=998.9837916666668) Narration: Move to p equals zero point seven five. The two payoff lines cross at minus one half. Inspect and Do Not Inspect are equally good, so the inspector can genuinely mix. Board: Unchanged from the preceding beat in this scene. Actions: - [16:40.621](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1000.6212916666667): inspect\_probe is redrawn as the numbers it depends on change. - [16:40.621](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1000.6212916666667): no\_inspect\_probe is redrawn as the numbers it depends on change. - [16:40.621](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1000.6212916666667): p\_answer is shown on the screen, written out. - [16:40.621](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1000.6212916666667): p\_live ticks to 0.75. - [16:44.023](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1004.0232916666668): p\_crossing is shown on the screen, written out. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): head\_p is hidden from the screen — left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): inspector\_working is hidden from the screen — left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): p\_answer is hidden from the screen — left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): p\_plot is hidden from the screen — left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): inspect\_curve is hidden from the screen — p\_plot left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): no\_inspect\_curve is hidden from the screen — p\_plot left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): inspect\_probe is hidden from the screen — p\_plot left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): no\_inspect\_probe is hidden from the screen — p\_plot left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): p\_crossing is hidden from the screen — p\_plot left the board. - [16:52.022](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.0222916666667): A box is drawn around p\_answer. ##### [16:52.622](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.6222916666667) Narration: The equilibrium has the worker Work with probability three quarters and the inspector Inspect with probability one fifth. Frequent work is supported by infrequent inspection because being caught while shirking is costly. Board: Empty. Actions: - [16:52.622](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1012.6222916666667): head\_result is shown on the screen, written out. - [16:53.191](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1013.1912916666668): equilibrium is shown on the screen, written out. - [17:2.514](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1022.5142916666667): equilibrium is indicated — a transient flash. ##### [17:6.806](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1026.8057916666667) Narration: Verify the worker. At q equals one fifth, Work gives two. Shirk gives three minus five times one fifth, also two. Verify the inspector. At p equals three quarters, both Inspect and Do Not Inspect give minus one half. Board: equilibrium — a Math \[text\] that says "$P(upright("Work"))=frac(3,4), quad P(upright("Inspect"))=frac(1,5)$"; head\_result — a Heading that says "The Counterintuitive Result" Actions: - [17:7.154](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1027.1542916666667): verification is shown on the screen, written out. - [17:7.758](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1027.7582916666668): verification is shown on the screen, written out. - [17:18.311](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1038.3112916666666): verification is shown on the screen, written out. ##### [17:26.295](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1046.2952916666668) Narration: Here is the rule worth carrying away. The worker's payoff numbers determined the inspector's mixing probability. The inspector's payoff numbers determined the worker's mixing probability. Your probability is chosen to erase the opponent's strict preference. Board: Unchanged from the preceding beat in this scene. Actions: - [17:27.142](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1047.1422916666668): cross\_player is shown on the screen, written out. - [17:29.441](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1049.4412916666668): cross\_player (the "Worker payoffs" part) is emphasized. - [17:31.02](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1051.0202916666667): cross\_player (the "Inspector payoffs" part) is emphasized. - [17:31.02](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1051.0202916666667): cross\_player (the "Worker payoffs" part) is no longer emphasized. - [17:42.212](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1062.2117916666666): cross\_player (the "Inspector payoffs" part) is no longer emphasized. ##### [17:42.812](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1062.8117916666667) Narration: That is why intuition based only on how often an action appears can be misleading. Mixed equilibrium probabilities are not direct measures of effort, importance, or virtue. They are the probabilities that balance the other player's expected payoffs. Board: equilibrium — a Math \[text\] that says "$P(upright("Work"))=frac(3,4), quad P(upright("Inspect"))=frac(1,5)$"; cross\_player — a Panel that says "Worker payoffs determine the inspector's probability. Inspector payoffs determine the worker's probability."; head\_result — a Heading that says "The Counterintuitive Result" Actions: - [17:49.232](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1069.2322916666667): equilibrium is indicated — a transient flash. - [17:58.687](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1078.686625): cross\_player is hidden from the screen — left the board. - [17:58.687](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1078.686625): equilibrium is hidden from the screen — left the board. - [17:58.687](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1078.686625): head\_result is hidden from the screen — left the board. - [17:58.687](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1078.686625): verification is hidden from the screen — left the board. ### Scene 5: [Guided Practice and the Hand Procedure](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1079.7282916666668) Span: 17:59.728–23:22.095 (1079.7282916666668s–1402.0950625000003s). #### Objects - all\_equilibria: a Math \[text\] that says "$(U,L), quad (D,R), quad upright("and the mixed equilibrium")$" - column\_work: a Derivation \[text\] that says "$U\_C(L \| p) &= 2p \\ U\_C(R \| p) &= 3(1-p) \\ 2p &= 3-3p \\ p &= frac(3,5)$" - definitions: a Math \[text\] that says "$p=P(R upright(" chooses Up")), quad q=P(C upright(" chooses Left"))$" - final\_formula: a Math \[text\] that says "$upright("your mix") arrow.r upright("opponent indifferent")$" - final\_note: a Panel that says "A probability must lie between zero and one. A solution outside that interval is not an interior mixed strategy." - head\_check: a Heading that says "Verify Before You Accept" - head\_method: a Heading that says "The Six-Step Hand Method" - head\_mix\_column: a Heading that says "Choose $p$ to Make Column Indifferent" - head\_mix\_row: a Heading that says "Choose $q$ to Make Row Indifferent" - head\_practice: a Heading that says "Try a New Game" - heading: a Heading that says "The Idea to Remember" - mixed\_result: a Math \[text\] that says "$P(R upright(" Up"))=frac(3,5), quad P(C upright(" Left"))=frac(1,4)$" - pause\_note: a Panel that says "Before calculating, mark Row's first payoffs down each column and Column's second payoffs across each row." - practice: a Table \[text\] that says "Column: Left Column: Right Row: Up $(3, 2)$ $(0, 0)$ Row: Down $(0, 0)$ $(1, 3)$" (rows=(('', 'Column: Left', 'Column: Right'), ('Row: Up', '$(3, 2)$',…, header=True) - practice\_prompt: a Text \[text\] that says "Row chooses Up or Down. Column chooses Left or Right. Find every equilibrium." - preference\_check: a Block \[text\] that says "If $q\>1/4$, Row prefers Up; if $q\<1/4$, Row prefers Down. If $p\>3/5$, Column prefers Left; if $p\<3/5$, Column prefers Right. At the crossing values, each player is willing to randomize." - procedure: a Block \[text\] that says "Label whose payoff is first and whose is second. For each column, mark every largest Row payoff. For each row, mark every largest Column payoff. Every double-marked cell is a pure equilibrium. For mixing, make each player indifferent using…" - pure\_result: a Math \[text\] that says "$(U,L) quad upright("and") quad (D,R)$" - row\_work: a Derivation \[text\] that says "$U\_R(U \| q) &= 3q \\ U\_R(D \| q) &= 1-q \\ 3q &= 1-q \\ q &= frac(1,4)$" - verification: a Table \[text\] that says "Check Calculation Result Row $3(1/4)$ and $1-1/4$ $3/4=3/4$ Column $2(3/5)$ and $3(1-3/5)$ $6/5=6/5$" (rows=(('Check', 'Calculation', 'Result'), ('Row', '$3(1/4)$ and $1-1…, header=True) #### Beats ##### [17:59.728](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1079.7282916666668) Narration: Let's finish by solving a new game from beginning to end. Row chooses Up or Down. Column chooses Left or Right. The question asks for every equilibrium, so we must check pure cells first and then ask whether a mixed equilibrium also exists. Board: Empty. Actions: - [17:59.728](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1079.7282916666668): head\_practice is shown on the screen, written out. - [18:1.295](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1081.2952916666668): practice is shown on the screen, written out. - [18:3.35](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1083.3502916666669): practice\_prompt is shown on the screen, written out. - [18:4.256](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1084.2562916666668): practice is shown on the screen, written out. - [18:4.744](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1084.7442916666669): practice is shown on the screen, written out. ##### [18:16.501](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1096.5012916666667) Narration: Pause at the matrix and begin with Row. Against Column's Left, compare Row's first payoffs three and zero. Up is the unique best response, so mark the three red. Board: practice\_prompt — a Text \[text\] that says "Row chooses Up or Down. Column chooses Left or Right. Find every equilibrium."; head\_practice — a Heading that says "Try a New Game" Actions: - [18:18.452](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1098.4522916666667): pause\_note is shown on the screen, written out. - [18:23.235](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1103.2352916666669): practice (the "3" part) is emphasized. ##### [18:29.408](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1109.4077916666668) Narration: Against Column's Right, Row compares zero with one. Down is the best response, so mark the one in the lower-right cell red. Board: practice\_prompt — a Text \[text\] that says "Row chooses Up or Down. Column chooses Left or Right. Find every equilibrium."; pause\_note — a Panel that says "Before calculating, mark Row's first payoffs down each column and Column's second payoffs across each row."; head\_practice — a Heading that says "Try a New Game" Actions: - [18:32.798](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1112.7982916666667): practice (the "1" part) is emphasized. ##### [18:39.203](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1119.2027916666668) Narration: Now analyse Column using the second numbers. Against Row's Up, Column compares two from Left with zero from Right. Left is better, so mark the two blue. Board: Unchanged from the preceding beat in this scene. Actions: - [18:44.822](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1124.8222916666668): practice (the "2" part) is emphasized. ##### [18:50.785](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1130.7852916666668) Narration: Against Row's Down, Column compares zero from Left with three from Right. Right is better, so mark the three blue. Board: Unchanged from the preceding beat in this scene. Actions: - [18:54.559](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1134.5592916666667): practice (the "3" part) is emphasized. ##### [18:59.35](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1139.3497916666668) Narration: Inspect complete cells. Up, Left carries both marks, and Down, Right carries both marks. Those are two pure-strategy Nash equilibria. Board: Unchanged from the preceding beat in this scene. Actions: - [19:1.916](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1141.916291666667): practice (the "$(3, 2)$" part) is indicated — a transient flash. - [19:4.354](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1144.3542916666668): practice (the "$(1, 3)$" part) is indicated — a transient flash. - [19:7.651](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1147.6512916666668): pure\_result is shown on the screen, written out. ##### [19:11.131](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1151.1307916666667) Narration: Do not stop merely because pure equilibria exist. Some two-by-two games, including this one, also have a mixed equilibrium. We find it with exactly the same indifference method used before. Board: practice\_prompt — a Text \[text\] that says "Row chooses Up or Down. Column chooses Left or Right. Find every equilibrium."; pause\_note — a Panel that says "Before calculating, mark Row's first payoffs down each column and Column's second payoffs across each row."; pure\_result — a Math \[text\] that says "$(U,L) quad upright("and") quad (D,R)$"; head\_practice — a Heading that says "Try a New Game" Actions: - [19:23.925](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1163.9247916666668): practice moves to a new place on the board. - [19:23.925](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1163.9247916666668): head\_practice is hidden from the screen — left the board. - [19:23.925](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1163.9247916666668): pause\_note is hidden from the screen — left the board. - [19:23.925](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1163.9247916666668): practice\_prompt is hidden from the screen — left the board. - [19:23.925](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1163.9247916666668): pure\_result is hidden from the screen — left the board. ##### [19:25.125](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1165.1247916666669) Narration: Let p be the probability that Row chooses Up, and q the probability that Column chooses Left. Remember the cross-player rule. Use Row's payoffs to find q, because q must make Row indifferent. Board: Empty. Actions: - [19:25.125](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1165.1247916666669): head\_mix\_row is shown on the screen, written out. - [19:25.728](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1165.7282916666668): definitions is shown on the screen, written out. ##### [19:39.704](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1179.7037916666668) Narration: If Row chooses Up, the payoff is three against Left and zero against Right. The expected payoff is therefore three q. Board: definitions — a Math \[text\] that says "$p=P(R upright(" chooses Up")), quad q=P(C upright(" chooses Left"))$"; head\_mix\_row — a Heading that says "Choose $q$ to Make Row Indifferent" Actions: - [19:42.211](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1182.2112916666667): row\_work is shown on the screen, written out. - [19:42.85](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1182.8502916666669): practice (the "$(3, 2)$" part) is indicated — a transient flash. ##### [19:48.571](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1188.5707916666668) Narration: If Row chooses Down, the payoff is zero against Left and one against Right. Its expected payoff is one minus q. Board: Unchanged from the preceding beat in this scene. Actions: - [19:52.831](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1192.8312916666669): practice (the "$(1, 3)$" part) is indicated — a transient flash. - [19:55.211](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1195.2112916666667): row\_work is shown on the screen, written out. ##### [19:57.402](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1197.401791666667) Narration: Set Row's two payoffs equal. Three q equals one minus q, so four q equals one and q equals one quarter. Column chooses Left with probability one quarter to keep Row willing to mix. Board: Unchanged from the preceding beat in this scene. Actions: - [19:58.946](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1198.9462916666669): row\_work is shown on the screen, written out. - [20:4.588](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1204.588291666667): row\_work is shown on the screen, written out. - [20:10.812](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1210.8117916666667): definitions moves to a new place on the board. - [20:10.812](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1210.8117916666667): head\_mix\_row is hidden from the screen — left the board. - [20:10.812](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1210.8117916666667): row\_work is hidden from the screen — left the board. ##### [20:12.012](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1212.0117916666668) Narration: Now use Column's payoffs to find p. If Column chooses Left, the expected payoff is two p. If Column chooses Right, the expected payoff is three times one minus p. Board: definitions — a Math \[text\] that says "$p=P(R upright(" chooses Up")), quad q=P(C upright(" chooses Left"))$" Actions: - [20:12.012](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1212.0117916666668): head\_mix\_column is shown on the screen, written out. - [20:18.663](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1218.6632916666667): column\_work is shown on the screen, written out. - [20:23.04](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1223.0402916666667): column\_work is shown on the screen, written out. ##### [20:25.8](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1225.800291666667) Narration: Set those equal. Two p equals three minus three p, so five p equals three and p equals three fifths. Row chooses Up with probability three fifths to keep Column indifferent. Board: definitions — a Math \[text\] that says "$p=P(R upright(" chooses Up")), quad q=P(C upright(" chooses Left"))$"; head\_mix\_column — a Heading that says "Choose $p$ to Make Column Indifferent" Actions: - [20:26.659](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1226.6592916666668): column\_work is shown on the screen, written out. - [20:33.149](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1233.1492916666668): mixed\_result is shown on the screen, written out. - [20:33.149](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1233.1492916666668): column\_work is shown on the screen, written out. ##### [20:40.471](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1240.4707916666669) Narration: Notice that the probabilities are not mirror images of the most attractive payoffs. Column's one-quarter probability came from Row's payoff comparison, while Row's three-fifths probability came from Column's payoff comparison. Board: definitions — a Math \[text\] that says "$p=P(R upright(" chooses Up")), quad q=P(C upright(" chooses Left"))$"; mixed\_result — a Math \[text\] that says "$P(R upright(" Up"))=frac(3,5), quad P(C upright(" Left"))=frac(1,4)$"; head\_mix\_column — a Heading that says "Choose $p$ to Make Column Indifferent" Actions: - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): column\_work is hidden from the screen — left the board. - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): definitions is hidden from the screen — left the board. - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): head\_mix\_column is hidden from the screen — left the board. - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): mixed\_result is hidden from the screen — left the board. - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): practice is hidden from the screen — left the board. - [20:54.322](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1254.3217916666667): A box is drawn around mixed\_result. ##### [20:55.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1255.5217916666668) Narration: Always verify a mixed result before accepting it. At q equals one quarter, Row's Up payoff is three times one quarter, or three quarters. Row's Down payoff is one minus one quarter, also three quarters. Board: Empty. Actions: - [20:55.522](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1255.5217916666668): head\_check is shown on the screen, written out. - [20:56.369](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1256.3692916666669): verification is shown on the screen, written out. - [21:1.304](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1261.3042916666668): verification is shown on the screen, written out. ##### [21:10.704](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1270.704291666667) Narration: At p equals three fifths, Column's Left payoff is two times three fifths, or six fifths. Column's Right payoff is three times two fifths, also six fifths. Both indifference conditions hold. Board: head\_check — a Heading that says "Verify Before You Accept" Actions: - [21:13.223](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1273.2232916666667): verification is shown on the screen, written out. ##### [21:25.097](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1285.0967916666668) Narration: A directional check catches sign errors. If q rises above one quarter, Up becomes better for Row; if q falls below one quarter, Down becomes better. The crossing is exactly where Row changes preferred actions. Board: Unchanged from the preceding beat in this scene. Actions: - [21:25.596](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1285.5962916666667): preference\_check is shown on the screen, written out. - [21:29.346](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1289.3462916666667): preference\_check (the "q\>1/4" part) is emphasized. - [21:33.259](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1293.2592916666667): preference\_check (the "q\>1/4" part) is no longer emphasized. ##### [21:40.837](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1300.8367916666668) Narration: Likewise, if p rises above three fifths, Left becomes better for Column; below three fifths, Right becomes better. At the crossing, each player is willing to use either pure action. Board: preference\_check — a Block \[text\] that says "If $q\>1/4$, Row prefers Up; if $q\<1/4$, Row prefers Down. If $p\>3/5$, Column prefers Left; if $p\<3/5$, Column prefers Right. At the crossing values, each player is willing to randomize."; head\_check — a Heading that says "Verify Before You Accept" Actions: - [21:42.624](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1302.6242916666667): preference\_check (the "p\>3/5" part) is emphasized. - [21:53.631](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1313.6312916666668): preference\_check (the "p\>3/5" part) is no longer emphasized. ##### [21:54.231](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1314.2312916666667) Narration: This game therefore has three equilibria: the two double-marked pure cells and the interior mixed equilibrium we just verified. Finding one equilibrium is not permission to stop when the question asks for all of them. Board: Unchanged from the preceding beat in this scene. Actions: - [21:55.775](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1315.7752916666668): all\_equilibria is shown on the screen, written out. - [22:7.374](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1327.3737916666669): all\_equilibria is hidden from the screen — left the board. - [22:7.374](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1327.3737916666669): head\_check is hidden from the screen — left the board. - [22:7.374](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1327.3737916666669): preference\_check is hidden from the screen — left the board. - [22:7.374](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1327.3737916666669): verification is hidden from the screen — left the board. ##### [22:8.574](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1328.5737916666667) Narration: Here is the complete hand method in six steps. First, label the payoff order. Second, for each column mark every largest Row payoff. Third, for each row mark every largest Column payoff. Board: Empty. Actions: - [22:8.574](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1328.5737916666667): head\_method is shown on the screen, written out. - [22:10.605](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1330.6052916666667): procedure is shown on the screen, written out. - [22:12.114](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1332.1142916666668): procedure (the "Label whose payoff is first" part) is emphasized. - [22:15.017](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1335.0172916666668): procedure (the "For each column" part) is emphasized. - [22:15.017](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1335.0172916666668): procedure (the "Label whose payoff is first" part) is no longer emphasized. - [22:19.255](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1339.2552916666668): procedure (the "For each column" part) is no longer emphasized. - [22:19.255](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1339.2552916666668): procedure (the "For each row" part) is emphasized. ##### [22:23.686](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1343.6862916666669) Narration: Fourth, every cell with both marks is a pure equilibrium. Fifth, if mixing is relevant, introduce probabilities and use each probability to make the opponent indifferent. Your mix controls the opponent's incentives. Board: procedure — a Block \[text\] that says "Label whose payoff is first and whose is second. For each column, mark every largest Row payoff. For each row, mark every largest Column payoff. Every double-marked cell is a pure equilibrium. For mixing, make each player indifferent using…"; head\_method — a Heading that says "The Six-Step Hand Method" Actions: - [22:24.162](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1344.1622916666668): procedure (the "Every double-marked cell" part) is emphasized. - [22:24.162](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1344.1622916666668): procedure (the "For each row" part) is no longer emphasized. - [22:28.539](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1348.539291666667): procedure (the "Every double-marked cell" part) is no longer emphasized. - [22:28.539](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1348.539291666667): procedure (the "For mixing" part) is emphasized. ##### [22:38.938](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1358.9382916666668) Narration: Sixth, verify. Probabilities must lie between zero and one. Pure actions used with positive probability must give equal expected payoffs, and any unused action must not give more. Board: Unchanged from the preceding beat in this scene. Actions: - [22:39.286](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1359.2862916666668): procedure (the "Check probabilities" part) is emphasized. - [22:39.286](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1359.2862916666668): procedure (the "For mixing" part) is no longer emphasized. - [22:52.429](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1372.4287916666667): head\_method is hidden from the screen — left the board. - [22:52.429](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1372.4287916666667): procedure is hidden from the screen — left the board. - [22:52.429](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1372.4287916666667): procedure (the "Check probabilities" part) is no longer emphasized. ##### [22:53.029](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1373.0287916666668) Narration: The shortest memory aid is this: your mixing probability is solved from the opponent's payoff comparison. It is chosen to make the opponent indifferent, not to make your own two payoffs equal directly. Board: Empty. Actions: - [22:55.861](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1375.8612916666668): final\_formula is shown on the screen, written out. - [22:56.465](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1376.4652916666669): final\_note is shown on the screen, written out. ##### [23:6.388](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1386.3882916666669) Narration: With that discipline, a two-by-two game becomes a small sequence of comparisons and two linear equations. Read the pairs, mark the best responses, keep every double mark, balance the opponent when mixing, and check the result. Board: final\_formula — a Math \[text\] that says "$upright("your mix") arrow.r upright("opponent indifferent")$"; final\_note — a Panel that says "A probability must lie between zero and one. A solution outside that interval is not an interior mixed strategy." Actions: - [23:7.305](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1387.305291666667): A box is drawn around final\_formula. - [23:21.053](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1401.0533958333335): final\_formula is hidden from the screen — left the board. - [23:21.053](https://academa.ai/lectures/solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria?t=1401.0533958333335): final\_note is hidden from the screen — left the board.