{"version":1,"lectureId":"01M14TXXS3DT7ZDH2RQ7RMREF3","attempt":0,"publication":{"slug":"solving-2-2-games-best-responses-pure-and-mixed-nash-equilibria","title":"Solving 2×2 Games by Hand: Best Responses, Pure Equilibria, and Mixing","subject":"economics","summary":"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.","metaDescription":"Solve 2×2 games by hand using payoff matrices, best-response marks, pure Nash equilibria, expected payoffs, and mixed strategies.","transcript":"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. 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. 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. 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. 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.","watch":{"version":1,"scenes":[{"title":"Payoffs, Best Responses, and Pure Equilibria","start":0,"end":326.89004166666666,"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. 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One action from each player selects exactly one of the four inner cells.","live":[],"does":[[31.649500000000003,"head_matrix is shown on the screen, written out."],[35.65500000000001,"coordination is shown on the screen, written out."],[36.99,"matrix_prompt is shown on the screen, written out."],[38.55800000000001,"coordination is shown on the screen, written out."],[38.987,"coordination is shown on the screen, written out."]]},{"start":49.118500000000004,"say":"Each inner cell contains an ordered pair of numbers. The first number is Row's payoff, and the second number is Column's payoff. 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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?","live":null,"does":[[94.57550000000002,"cell_reading is hidden from the screen — left the board."],[94.57550000000002,"head_matrix is hidden from the screen — left the board."],[94.57550000000002,"matrix_prompt is hidden from the screen — left the board."],[94.57550000000002,"payoff_order is hidden from the screen — left the board."]]},{"start":95.77550000000002,"say":"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.","live":[],"does":[[95.77550000000002,"head_responses is shown on the screen, written out."],[97.88900000000002,"row_rule is shown on the screen, written out."],[106.00400000000002,"coordination (the \"4\" part) is emphasized."]]},{"start":113.83700000000002,"say":"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.","live":["row_rule","head_responses"],"does":[[119.03800000000001,"coordination (the \"3\" part) is emphasized."]]},{"start":128.35750000000002,"say":"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.","live":null,"does":[[135.34700000000004,"column_rule is shown on the screen, written out."],[135.974,"coordination (the \"3\" part) is emphasized."]]},{"start":143.31900000000002,"say":"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.","live":["row_rule","column_rule","head_responses"],"does":[[148.25300000000004,"coordination (the \"4\" part) is emphasized."],[154.67300000000003,"row_rule (the \"one column\" part) is indicated — a transient flash."],[157.03000000000003,"column_rule (the \"one row\" part) is indicated — a transient flash."]]},{"start":158.82600000000002,"say":"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.","live":null,"does":[[160.01000000000002,"double_rule is shown on the screen, written out."],[161.69400000000002,"coordination (the \"$(4, 3)$\" part) is indicated — a transient flash."],[166.175,"coordination (the \"$(3, 4)$\" part) is indicated — a transient flash."],[174.80100000000002,"column_rule is hidden from the screen — left the board."],[174.80100000000002,"double_rule is hidden from the screen — left the board."],[174.80100000000002,"head_responses is hidden from the screen — left the board."],[174.80100000000002,"row_rule is hidden from the screen — left the board."]]},{"start":176.00100000000003,"say":"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.","live":[],"does":[[176.00100000000003,"head_pure is shown on the screen, written out."],[178.46200000000005,"pure_definition is shown on the screen, written out."],[179.76200000000006,"pure_answer is shown on the screen, written out."]]},{"start":192.79650000000004,"say":"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.","live":["pure_definition","pure_answer","head_pure"],"does":[[193.63200000000003,"coordination (the \"$(4, 3)$\" part) is indicated — a transient flash."]]},{"start":208.29250000000005,"say":"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. 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Never break a payoff tie merely to force one answer.","live":null,"does":[[226.12100000000007,"tie_rule is shown on the screen, written out."],[238.61400000000003,"coordination is hidden from the screen — left the board."],[238.61400000000003,"head_pure is hidden from the screen — left the board."],[238.61400000000003,"pure_answer is hidden from the screen — left the board."],[238.61400000000003,"pure_definition is hidden from the screen — left the board."],[238.61400000000003,"tie_rule is hidden from the screen — left the board."]]},{"start":239.21400000000003,"say":"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.","live":[],"does":[[239.21400000000003,"head_pd is shown on the screen, written out."],[243.62600000000006,"pd_matrix is shown on the screen, written out."],[247.60700000000006,"pd_matrix is shown on the screen, written out."],[255.14200000000005,"pd_matrix is shown on the screen, written out."]]},{"start":257.507,"say":"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.","live":["head_pd"],"does":[[257.855,"pd_method is shown on the screen, written out."],[263.707,"pd_matrix (the \"3\" part) is emphasized."],[269.187,"pd_matrix (the \"1\" part) is emphasized."]]},{"start":275.6035,"say":"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.","live":["pd_method","head_pd"],"does":[[282.639,"pd_matrix (the \"3\" part) is emphasized."],[286.87600000000003,"pd_matrix (the \"1#2\" part) is emphasized."]]},{"start":292.1555,"say":"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.","live":null,"does":[[292.9800000000001,"pd_answer is shown on the screen, written out."],[294.15200000000004,"pd_matrix (the \"$(1, 1)$\" part) is indicated — a transient flash."]]},{"start":307.4305,"say":"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.","live":["pd_method","pd_answer","head_pd"],"does":[[308.232,"pd_warning is shown on the screen, written out."],[325.848375,"head_pd is hidden from the screen — left the board."],[325.848375,"pd_answer is hidden from the screen — left the board."],[325.848375,"pd_matrix is hidden from the screen — left the board."],[325.848375,"pd_method is hidden from the screen — left the board."],[325.848375,"pd_warning is hidden from the screen — left the board."]]}]},{"title":"Matching Pennies and Exploitation","start":326.89004166666666,"end":517.7931458333334,"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":[{"start":326.89004166666666,"say":"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.","live":[],"does":[[326.89004166666666,"head_matrix is shown on the screen, written out."],[333.5190416666667,"rules is shown on the screen, written out."],[334.4950416666667,"payoffs is shown on the screen, written out."],[335.9110416666667,"payoffs is shown on the screen, written out."],[338.41904166666666,"payoffs is shown on the screen, written out."]]},{"start":343.14054166666665,"say":"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.","live":["rules","head_matrix"],"does":[[344.04604166666667,"marking is shown on the screen, written out."],[346.41404166666666,"payoffs (the \"1\" part) is emphasized."],[349.80404166666665,"payoffs (the \"1\" part) is emphasized."]]},{"start":358.91504166666664,"say":"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.","live":["rules","marking","head_matrix"],"does":[[363.17604166666666,"payoffs (the \"1#2\" part) is emphasized."],[364.4880416666667,"payoffs (the \"1#2\" part) is emphasized."]]},{"start":375.46654166666667,"say":"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.","live":null,"does":[[380.59804166666663,"no_pure is shown on the screen, written out."],[392.5215416666666,"head_matrix is hidden from the screen — left the board."],[392.5215416666666,"marking is hidden from the screen — left the board."],[392.5215416666666,"no_pure is hidden from the screen — left the board."],[392.5215416666666,"payoffs is hidden from the screen — left the board."],[392.5215416666666,"rules is hidden from the screen — left the board."]]},{"start":393.72154166666667,"say":"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.","live":[],"does":[[393.72154166666667,"head_cycle is shown on the screen, written out."],[393.72154166666667,"cycle is shown on the screen, written out."],[396.8210416666667,"hh is shown on the screen, written out."],[397.0210416666667,"ht is shown on the screen, written out."],[397.42104166666667,"tt is shown on the screen, written out."],[398.0210416666667,"th is shown on the screen, written out."],[400.7690416666667,"column_move_one is shown on the screen, written out."]]},{"start":406.08254166666666,"say":"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.","live":["cycle","head_cycle","hh","ht","tt","th","column_move_one"],"does":[[408.02104166666663,"row_move_one is shown on the screen, written out."],[411.38804166666665,"column_move_two is shown on the screen, written out."],[415.77704166666666,"row_move_two is shown on the screen, written out."],[418.76104166666664,"cycle_key is shown on the screen, written out."]]},{"start":423.77304166666664,"say":"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.","live":["cycle","cycle_key","head_cycle","hh","ht","tt","th","column_move_one","row_move_one","column_move_two","row_move_two"],"does":[[433.5480416666667,"hh is indicated — a transient flash."],[433.74804166666667,"ht is indicated — a transient flash."],[434.14804166666664,"tt is indicated — a transient flash."],[434.74804166666667,"th is indicated — a transient flash."],[436.86804166666667,"cycle is hidden from the screen — left the board."],[436.86804166666667,"hh is hidden from the screen — cycle left the board."],[436.86804166666667,"ht is hidden from the screen — cycle left the board."],[436.86804166666667,"tt is hidden from the screen — cycle left the board."],[436.86804166666667,"th is hidden from the screen — cycle left the board."],[436.86804166666667,"column_move_one is hidden from the screen — cycle left the board."],[436.86804166666667,"row_move_one is hidden from the screen — cycle left the board."],[436.86804166666667,"column_move_two is hidden from the screen — cycle left the board."],[436.86804166666667,"row_move_two is hidden from the screen — cycle left the board."],[436.86804166666667,"cycle_key is hidden from the screen — left the board."],[436.86804166666667,"head_cycle is hidden from the screen — left the board."]]},{"start":438.06804166666666,"say":"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.","live":[],"does":[[438.06804166666666,"head_predict is shown on the screen, written out."],[441.81804166666666,"row_label is shown on the screen, written out."],[442.14304166666665,"row_predictable is shown on the screen, written out."],[445.27804166666664,"row_explanation is shown on the screen, written out."]]},{"start":451.66004166666664,"say":"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.","live":["row_label","row_predictable","row_explanation","head_predict"],"does":[[455.36404166666665,"column_label is shown on the screen, written out."],[455.7350416666667,"column_predictable is shown on the screen, written out."],[463.94304166666666,"column_explanation is shown on the screen, written out."]]},{"start":467.93304166666667,"say":"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.","live":["row_label","row_predictable","row_explanation","column_label","column_predictable","column_explanation","head_predict"],"does":[[487.2405416666667,"column_explanation is hidden from the screen — left the board."],[487.2405416666667,"column_label is hidden from the screen — left the board."],[487.2405416666667,"column_predictable is hidden from the screen — left the board."],[487.2405416666667,"head_predict is hidden from the screen — left the board."],[487.2405416666667,"row_explanation is hidden from the screen — left the board."],[487.2405416666667,"row_label is hidden from the screen — left the board."],[487.2405416666667,"row_predictable is hidden from the screen — left the board."]]},{"start":488.44054166666666,"say":"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.","live":[],"does":[[489.4270416666667,"mixing_question is shown on the screen, written out."]]},{"start":503.20454166666667,"say":"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.","live":["mixing_question"],"does":[[516.7514791666667,"mixing_question is hidden from the screen — left the board."]]}]},{"title":"Mixing and Indifference","start":517.7931458333334,"end":773.3182916666667,"objects":{"column_h_curve":"a FunctionPlot [red] labelled \"upright(\"Column Heads\")\" drawn in p_plot (function=<function>)","column_h_probe":"a PlotPoint [red] drawn in p_plot (target='column_h_curve', x=<VariableNumber p_live = 0.5>)","column_t_curve":"a FunctionPlot [blue] labelled \"upright(\"Column Tails\")\" drawn in p_plot (function=<function>)","column_t_probe":"a PlotPoint [blue] drawn in p_plot (target='column_t_curve', x=<VariableNumber p_live = 0.5>)","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=<function>)","row_h_probe":"a PlotPoint [red] drawn in q_plot (target='row_h_curve', x=<VariableNumber q_live = 0.5>)","row_t_curve":"a FunctionPlot [blue] labelled \"upright(\"Row Tails\")\" drawn in q_plot (function=<function>)","row_t_probe":"a PlotPoint [blue] drawn in q_plot (target='row_t_curve', x=<VariableNumber q_live = 0.5>)","verification":"a Math [text] that says \"$U_R(H)=U_R(T)=0, quad U_C(H)=U_C(T)=0$\""},"beats":[{"start":517.7931458333334,"say":"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.","live":[],"does":[[517.7931458333334,"head_q is shown on the screen, written out."],[518.0951458333334,"q_definition is shown on the screen, written out."],[518.5821458333334,"q_plot is shown on the screen, written out."],[524.6431458333334,"row_h_probe is shown on the screen, written out."],[524.6431458333334,"row_t_probe is shown on the screen, written out."],[530.0181458333334,"row_h_curve is shown on the screen, drawn."],[531.0981458333333,"row_t_curve is shown on the screen, drawn."]]},{"start":532.8941458333334,"say":"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.","live":["q_definition","q_plot","head_q","row_h_curve","row_t_curve","row_h_probe","row_t_probe"],"does":[[535.5061458333333,"q_work is shown on the screen, written out."],[540.6261458333333,"q_work (the \"q(1)\" part) is emphasized."],[542.8551458333334,"q_work (the \"(1-q)(-1)\" part) is emphasized."],[542.8551458333334,"q_work (the \"q(1)\" part) is no longer emphasized."],[544.1901458333334,"q_work (the \"(1-q)(-1)\" part) is no longer emphasized."]]},{"start":544.7901458333333,"say":"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.","live":null,"does":[[545.0921458333333,"q_work is shown on the screen, written out."],[548.4361458333334,"row_h_curve is indicated — a transient flash."]]},{"start":555.9321458333334,"say":"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.","live":null,"does":[[556.8491458333334,"q_work is shown on the screen, written out."]]},{"start":570.1156458333334,"say":"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.","live":null,"does":[[570.6381458333334,"q_work is shown on the screen, written out."],[573.5871458333334,"row_t_curve is indicated — a transient flash."]]},{"start":581.0136458333334,"say":"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.","live":null,"does":[[591.7291458333334,"row_h_probe is redrawn as the numbers it depends on change."],[591.7291458333334,"row_t_probe is redrawn as the numbers it depends on change."],[591.7291458333334,"q_live ticks to 0.25."]]},{"start":595.2146458333334,"say":"Indifference occurs where the two payoff lines cross. Set two q minus one equal to one minus two q.","live":null,"does":[[598.0301458333333,"row_h_probe is redrawn as the numbers it depends on change."],[598.0301458333333,"row_t_probe is redrawn as the numbers it depends on change."],[598.0301458333333,"q_live ticks to 0.5."],[599.3421458333333,"q_work is shown on the screen, written out."]]},{"start":604.5216458333334,"say":"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.","live":null,"does":[[609.9491458333334,"q_answer is shown on the screen, written out."],[609.9491458333334,"q_work is shown on the screen, written out."],[611.9461458333334,"q_crossing is shown on the screen, written out."],[616.7291458333334,"A box is drawn around q_answer."]]},{"start":617.3291458333334,"say":"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.","live":["q_definition","q_answer","q_plot","head_q","row_h_curve","row_t_curve","row_h_probe","row_t_probe","q_crossing"],"does":[[628.3126458333334,"head_q is hidden from the screen — left the board."],[628.3126458333334,"q_answer is hidden from the screen — left the board."],[628.3126458333334,"q_definition is hidden from the screen — left the board."],[628.3126458333334,"q_plot is hidden from the screen — left the board."],[628.3126458333334,"row_h_curve is hidden from the screen — q_plot left the board."],[628.3126458333334,"row_t_curve is hidden from the screen — q_plot left the board."],[628.3126458333334,"row_h_probe is hidden from the screen — q_plot left the board."],[628.3126458333334,"row_t_probe is hidden from the screen — q_plot left the board."],[628.3126458333334,"q_crossing is hidden from the screen — q_plot left the board."],[628.3126458333334,"q_work is hidden from the screen — left the board."]]},{"start":629.5126458333334,"say":"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.","live":[],"does":[[629.5126458333334,"head_p is shown on the screen, written out."],[630.2671458333334,"p_definition is shown on the screen, written out."],[630.8011458333334,"p_plot is shown on the screen, written out."],[634.8181458333333,"column_h_probe is shown on the screen, written out."],[634.8181458333333,"column_t_probe is shown on the screen, written out."],[635.2601458333334,"column_h_curve is shown on the screen, drawn."],[636.0491458333333,"column_t_curve is shown on the screen, drawn."]]},{"start":642.6576458333334,"say":"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.","live":["p_definition","p_plot","head_p","column_h_curve","column_t_curve","column_h_probe","column_t_probe"],"does":[[645.0431458333334,"p_work is shown on the screen, written out."]]},{"start":655.9416458333334,"say":"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.","live":null,"does":[[656.3711458333335,"p_work is shown on the screen, written out."],[657.7411458333333,"p_work is shown on the screen, written out."],[660.0171458333334,"p_work is shown on the screen, written out."]]},{"start":666.9096458333333,"say":"Set the two expressions equal. One minus two p equals two p minus one, so p equals one half.","live":null,"does":[[668.4191458333335,"p_work is shown on the screen, written out."],[673.9921458333333,"column_h_probe is redrawn as the numbers it depends on change."],[673.9921458333333,"column_t_probe is redrawn as the numbers it depends on change."],[673.9921458333333,"p_crossing is shown on the screen, written out."],[673.9921458333333,"p_answer is shown on the screen, written out."],[673.9921458333333,"p_work is shown on the screen, written out."],[673.9921458333333,"p_live ticks to 0.5."],[675.0136458333334,"A box is drawn around p_answer."]]},{"start":675.6136458333334,"say":"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.","live":["p_definition","p_answer","p_plot","head_p","column_h_curve","column_t_curve","column_h_probe","column_t_probe","p_crossing"],"does":[[690.2421458333333,"head_p is hidden from the screen — left the board."],[690.2421458333333,"p_answer is hidden from the screen — left the board."],[690.2421458333333,"p_definition is hidden from the screen — left the board."],[690.2421458333333,"p_plot is hidden from the screen — left the board."],[690.2421458333333,"column_h_curve is hidden from the screen — p_plot left the board."],[690.2421458333333,"column_t_curve is hidden from the screen — p_plot left the board."],[690.2421458333333,"column_h_probe is hidden from the screen — p_plot left the board."],[690.2421458333333,"column_t_probe is hidden from the screen — p_plot left the board."],[690.2421458333333,"p_crossing is hidden from the screen — p_plot left the board."],[690.2421458333333,"p_work is hidden from the screen — left the board."]]},{"start":691.4421458333334,"say":"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.","live":[],"does":[[691.4421458333334,"head_result is shown on the screen, written out."],[691.9881458333334,"mixed_definition is shown on the screen, written out."],[696.1211458333333,"equilibrium is shown on the screen, written out."],[703.0751458333334,"outcome_probability is shown on the screen, written out."]]},{"start":704.9991458333334,"say":"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.","live":["equilibrium","mixed_definition","outcome_probability","head_result"],"does":[[714.5531458333334,"verification is shown on the screen, written out."],[716.5271458333334,"verification (the \"U_R(H)=U_R(T)=0\" part) is emphasized."]]},{"start":719.9601458333334,"say":"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.","live":["equilibrium","mixed_definition","outcome_probability","verification","head_result"],"does":[[720.8771458333334,"verification (the \"U_C(H)=U_C(T)=0\" part) is emphasized."],[720.8771458333334,"verification (the \"U_R(H)=U_R(T)=0\" part) is no longer emphasized."],[734.4376458333334,"verification (the \"U_C(H)=U_C(T)=0\" part) is no longer emphasized."]]},{"start":735.0376458333334,"say":"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.","live":null,"does":[[747.8541458333334,"equilibrium is indicated — a transient flash."]]},{"start":754.4336458333333,"say":"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.","live":null,"does":[[772.276625,"equilibrium is hidden from the screen — left the board."],[772.276625,"head_result is hidden from the screen — left the board."],[772.276625,"mixed_definition is hidden from the screen — left the board."],[772.276625,"outcome_probability is hidden from the screen — left the board."],[772.276625,"verification is hidden from the screen — left the board."]]}]},{"title":"The Asymmetric Inspection Game","start":773.3182916666667,"end":1079.7282916666668,"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=<function>)","inspect_probe":"a PlotPoint [blue] drawn in p_plot (target='inspect_curve', x=<VariableNumber p_live = 0.75>)","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)$\"","intuition":"a Text [text] that says \"Tempting guess: frequent work must require frequent inspection.\"","no_inspect_curve":"a FunctionPlot [red] labelled \"upright(\"Do Not Inspect\")\" drawn in p_plot (function=<function>)","no_inspect_probe":"a PlotPoint [red] drawn in p_plot (target='no_inspect_curve', x=<VariableNumber p_live = 0.75>)","no_pure":"a Math [text] that says \"$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=<function>)","shirk_probe":"a PlotPoint [red] drawn in q_plot (target='shirk_curve', x=<VariableNumber q_live = 0.2>)","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=<function>)","work_probe":"a PlotPoint [blue] drawn in q_plot (target='work_curve', x=<VariableNumber q_live = 0.2>)","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":[{"start":773.3182916666667,"say":"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.","live":[],"does":[[773.3182916666667,"head_game is shown on the screen, written out."],[776.6042916666667,"story is shown on the screen, written out."],[777.5322916666668,"inspection is shown on the screen, written out."],[778.1712916666667,"inspection is shown on the screen, written out."],[785.1832916666667,"inspection is shown on the screen, written out."]]},{"start":790.5202916666667,"say":"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.","live":["story","head_game"],"does":[[795.9422916666667,"inspection (the \"2\" part) is emphasized."],[801.7822916666668,"inspection (the \"3\" part) is emphasized."]]},{"start":806.0507916666667,"say":"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.","live":null,"does":[[814.0502916666667,"inspection (the \"0\" part) is emphasized."],[815.4322916666667,"inspection (the \"1\" part) is emphasized."]]},{"start":821.8132916666667,"say":"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.","live":null,"does":[[822.2892916666667,"no_pure is shown on the screen, written out."]]},{"start":840.1537916666667,"say":"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.","live":["story","no_pure","head_game"],"does":[[840.7572916666667,"intuition is shown on the screen, written out."],[856.6507916666667,"head_game is hidden from the screen — left the board."],[856.6507916666667,"inspection is hidden from the screen — left the board."],[856.6507916666667,"intuition is hidden from the screen — left the board."],[856.6507916666667,"no_pure is hidden from the screen — left the board."],[856.6507916666667,"story is hidden from the screen — left the board."]]},{"start":857.8507916666667,"say":"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.","live":[],"does":[[857.8507916666667,"head_q is shown on the screen, written out."],[858.5472916666668,"q_plot is shown on the screen, written out."],[868.4272916666667,"work_probe is shown on the screen, written out."],[868.4272916666667,"shirk_probe is shown on the screen, written out."],[869.8432916666667,"work_curve is shown on the screen, drawn."],[870.4942916666668,"shirk_curve is shown on the screen, drawn."]]},{"start":872.0107916666667,"say":"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.","live":["q_plot","head_q","work_curve","shirk_curve","work_probe","shirk_probe"],"does":[[873.4742916666668,"q_plot moves to a new place on the board."],[873.4742916666668,"worker_working is shown on the screen, written out."],[882.2392916666668,"worker_working is shown on the screen, written out."],[883.7142916666667,"work_curve is indicated — a transient flash."]]},{"start":885.6547916666667,"say":"Shirk pays minus two if inspected and three if not inspected. Its expected payoff is minus two q plus three times one minus q.","live":null,"does":[[886.7872916666668,"worker_working is shown on the screen, written out."]]},{"start":895.9147916666667,"say":"Simplifying gives three minus five q. The red line slopes downward because more inspection makes shirking less attractive.","live":null,"does":[[896.2342916666668,"worker_working is shown on the screen, written out."],[899.6822916666667,"shirk_curve is indicated — a transient flash."]]},{"start":904.7987916666667,"say":"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.","live":null,"does":[[911.5792916666667,"work_probe is redrawn as the numbers it depends on change."],[911.5792916666667,"shirk_probe is redrawn as the numbers it depends on change."],[911.5792916666667,"q_live ticks to 0.1."]]},{"start":917.5772916666667,"say":"Set the two worker payoffs equal. Two equals three minus five q. Solving gives q equals one fifth, or twenty percent.","live":null,"does":[[919.3192916666667,"worker_working is shown on the screen, written out."],[924.5552916666668,"worker_working is shown on the screen, written out."],[925.7162916666667,"work_probe is redrawn as the numbers it depends on change."],[925.7162916666667,"shirk_probe is redrawn as the numbers it depends on change."],[925.7162916666667,"q_crossing is shown on the screen, written out."],[925.7162916666667,"q_answer is shown on the screen, written out."],[925.7162916666667,"q_live ticks to 0.2."]]},{"start":927.5932916666667,"say":"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.","live":["q_answer","q_plot","head_q","work_curve","shirk_curve","work_probe","shirk_probe","q_crossing"],"does":[[928.7772916666668,"A box is drawn around q_answer."],[943.6967916666667,"head_q is hidden from the screen — left the board."],[943.6967916666667,"q_answer is hidden from the screen — left the board."],[943.6967916666667,"q_plot is hidden from the screen — left the board."],[943.6967916666667,"work_curve is hidden from the screen — q_plot left the board."],[943.6967916666667,"shirk_curve is hidden from the screen — q_plot left the board."],[943.6967916666667,"work_probe is hidden from the screen — q_plot left the board."],[943.6967916666667,"shirk_probe is hidden from the screen — q_plot left the board."],[943.6967916666667,"q_crossing is hidden from the screen — q_plot left the board."],[943.6967916666667,"worker_working is hidden from the screen — left the board."]]},{"start":944.8967916666668,"say":"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.","live":[],"does":[[944.8967916666668,"head_p is shown on the screen, written out."],[945.7092916666668,"p_plot is shown on the screen, written out."],[954.8462916666667,"inspect_curve is shown on the screen, drawn."],[955.5312916666667,"no_inspect_curve is shown on the screen, drawn."],[957.0172916666668,"inspect_probe is shown on the screen, written out."],[957.0172916666668,"no_inspect_probe is shown on the screen, written out."]]},{"start":960.0907916666667,"say":"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.","live":["p_plot","head_p","inspect_curve","no_inspect_curve","inspect_probe","no_inspect_probe"],"does":[[961.9132916666667,"p_plot moves to a new place on the board."],[961.9132916666667,"inspector_working is shown on the screen, written out."],[968.3912916666667,"inspect_curve is indicated — a transient flash."],[969.8542916666668,"inspector_working is shown on the screen, written out."]]},{"start":972.7997916666667,"say":"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.","live":null,"does":[[974.4132916666667,"inspector_working is shown on the screen, written out."],[975.7372916666667,"no_inspect_curve is indicated — a transient flash."],[983.5152916666667,"inspector_working is shown on the screen, written out."]]},{"start":987.0697916666668,"say":"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.","live":null,"does":[[988.9452916666667,"inspector_working is shown on the screen, written out."],[997.2462916666667,"inspector_working is shown on the screen, written out."]]},{"start":998.9837916666668,"say":"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.","live":null,"does":[[1000.6212916666667,"inspect_probe is redrawn as the numbers it depends on change."],[1000.6212916666667,"no_inspect_probe is redrawn as the numbers it depends on change."],[1000.6212916666667,"p_answer is shown on the screen, written out."],[1000.6212916666667,"p_live ticks to 0.75."],[1004.0232916666668,"p_crossing is shown on the screen, written out."],[1012.0222916666667,"head_p is hidden from the screen — left the board."],[1012.0222916666667,"inspector_working is hidden from the screen — left the board."],[1012.0222916666667,"p_answer is hidden from the screen — left the board."],[1012.0222916666667,"p_plot is hidden from the screen — left the board."],[1012.0222916666667,"inspect_curve is hidden from the screen — p_plot left the board."],[1012.0222916666667,"no_inspect_curve is hidden from the screen — p_plot left the board."],[1012.0222916666667,"inspect_probe is hidden from the screen — p_plot left the board."],[1012.0222916666667,"no_inspect_probe is hidden from the screen — p_plot left the board."],[1012.0222916666667,"p_crossing is hidden from the screen — p_plot left the board."],[1012.0222916666667,"A box is drawn around p_answer."]]},{"start":1012.6222916666667,"say":"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.","live":[],"does":[[1012.6222916666667,"head_result is shown on the screen, written out."],[1013.1912916666668,"equilibrium is shown on the screen, written out."],[1022.5142916666667,"equilibrium is indicated — a transient flash."]]},{"start":1026.8057916666667,"say":"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.","live":["equilibrium","head_result"],"does":[[1027.1542916666667,"verification is shown on the screen, written out."],[1027.7582916666668,"verification is shown on the screen, written out."],[1038.3112916666666,"verification is shown on the screen, written out."]]},{"start":1046.2952916666668,"say":"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.","live":null,"does":[[1047.1422916666668,"cross_player is shown on the screen, written out."],[1049.4412916666668,"cross_player (the \"Worker payoffs\" part) is emphasized."],[1051.0202916666667,"cross_player (the \"Inspector payoffs\" part) is emphasized."],[1051.0202916666667,"cross_player (the \"Worker payoffs\" part) is no longer emphasized."],[1062.2117916666666,"cross_player (the \"Inspector payoffs\" part) is no longer emphasized."]]},{"start":1062.8117916666667,"say":"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.","live":["equilibrium","cross_player","head_result"],"does":[[1069.2322916666667,"equilibrium is indicated — a transient flash."],[1078.686625,"cross_player is hidden from the screen — left the board."],[1078.686625,"equilibrium is hidden from the screen — left the board."],[1078.686625,"head_result is hidden from the screen — left the board."],[1078.686625,"verification is hidden from the screen — left the board."]]}]},{"title":"Guided Practice and the Hand Procedure","start":1079.7282916666668,"end":1402.0950625000003,"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":[{"start":1079.7282916666668,"say":"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.","live":[],"does":[[1079.7282916666668,"head_practice is shown on the screen, written out."],[1081.2952916666668,"practice is shown on the screen, written out."],[1083.3502916666669,"practice_prompt is shown on the screen, written out."],[1084.2562916666668,"practice is shown on the screen, written out."],[1084.7442916666669,"practice is shown on the screen, written out."]]},{"start":1096.5012916666667,"say":"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.","live":["practice_prompt","head_practice"],"does":[[1098.4522916666667,"pause_note is shown on the screen, written out."],[1103.2352916666669,"practice (the \"3\" part) is emphasized."]]},{"start":1109.4077916666668,"say":"Against Column's Right, Row compares zero with one. Down is the best response, so mark the one in the lower-right cell red.","live":["practice_prompt","pause_note","head_practice"],"does":[[1112.7982916666667,"practice (the \"1\" part) is emphasized."]]},{"start":1119.2027916666668,"say":"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.","live":null,"does":[[1124.8222916666668,"practice (the \"2\" part) is emphasized."]]},{"start":1130.7852916666668,"say":"Against Row's Down, Column compares zero from Left with three from Right. Right is better, so mark the three blue.","live":null,"does":[[1134.5592916666667,"practice (the \"3\" part) is emphasized."]]},{"start":1139.3497916666668,"say":"Inspect complete cells. Up, Left carries both marks, and Down, Right carries both marks. Those are two pure-strategy Nash equilibria.","live":null,"does":[[1141.916291666667,"practice (the \"$(3, 2)$\" part) is indicated — a transient flash."],[1144.3542916666668,"practice (the \"$(1, 3)$\" part) is indicated — a transient flash."],[1147.6512916666668,"pure_result is shown on the screen, written out."]]},{"start":1151.1307916666667,"say":"Do not stop merely because pure equilibria exist. Some two-by-two games, including this one, also have a mixed equilibrium. We find it with exactly the same indifference method used before.","live":["practice_prompt","pause_note","pure_result","head_practice"],"does":[[1163.9247916666668,"practice moves to a new place on the board."],[1163.9247916666668,"head_practice is hidden from the screen — left the board."],[1163.9247916666668,"pause_note is hidden from the screen — left the board."],[1163.9247916666668,"practice_prompt is hidden from the screen — left the board."],[1163.9247916666668,"pure_result is hidden from the screen — left the board."]]},{"start":1165.1247916666669,"say":"Let p be the probability that Row chooses Up, and q the probability that Column chooses Left. Remember the cross-player rule. Use Row's payoffs to find q, because q must make Row indifferent.","live":[],"does":[[1165.1247916666669,"head_mix_row is shown on the screen, written out."],[1165.7282916666668,"definitions is shown on the screen, written out."]]},{"start":1179.7037916666668,"say":"If Row chooses Up, the payoff is three against Left and zero against Right. The expected payoff is therefore three q.","live":["definitions","head_mix_row"],"does":[[1182.2112916666667,"row_work is shown on the screen, written out."],[1182.8502916666669,"practice (the \"$(3, 2)$\" part) is indicated — a transient flash."]]},{"start":1188.5707916666668,"say":"If Row chooses Down, the payoff is zero against Left and one against Right. Its expected payoff is one minus q.","live":null,"does":[[1192.8312916666669,"practice (the \"$(1, 3)$\" part) is indicated — a transient flash."],[1195.2112916666667,"row_work is shown on the screen, written out."]]},{"start":1197.401791666667,"say":"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.","live":null,"does":[[1198.9462916666669,"row_work is shown on the screen, written out."],[1204.588291666667,"row_work is shown on the screen, written out."],[1210.8117916666667,"definitions moves to a new place on the board."],[1210.8117916666667,"head_mix_row is hidden from the screen — left the board."],[1210.8117916666667,"row_work is hidden from the screen — left the board."]]},{"start":1212.0117916666668,"say":"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.","live":["definitions"],"does":[[1212.0117916666668,"head_mix_column is shown on the screen, written out."],[1218.6632916666667,"column_work is shown on the screen, written out."],[1223.0402916666667,"column_work is shown on the screen, written out."]]},{"start":1225.800291666667,"say":"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.","live":["definitions","head_mix_column"],"does":[[1226.6592916666668,"column_work is shown on the screen, written out."],[1233.1492916666668,"mixed_result is shown on the screen, written out."],[1233.1492916666668,"column_work is shown on the screen, written out."]]},{"start":1240.4707916666669,"say":"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.","live":["definitions","mixed_result","head_mix_column"],"does":[[1254.3217916666667,"column_work is hidden from the screen — left the board."],[1254.3217916666667,"definitions is hidden from the screen — left the board."],[1254.3217916666667,"head_mix_column is hidden from the screen — left the board."],[1254.3217916666667,"mixed_result is hidden from the screen — left the board."],[1254.3217916666667,"practice is hidden from the screen — left the board."],[1254.3217916666667,"A box is drawn around mixed_result."]]},{"start":1255.5217916666668,"say":"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.","live":[],"does":[[1255.5217916666668,"head_check is shown on the screen, written out."],[1256.3692916666669,"verification is shown on the screen, written out."],[1261.3042916666668,"verification is shown on the screen, written out."]]},{"start":1270.704291666667,"say":"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.","live":["head_check"],"does":[[1273.2232916666667,"verification is shown on the screen, written out."]]},{"start":1285.0967916666668,"say":"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.","live":null,"does":[[1285.5962916666667,"preference_check is shown on the screen, written out."],[1289.3462916666667,"preference_check (the \"q>1/4\" part) is emphasized."],[1293.2592916666667,"preference_check (the \"q>1/4\" part) is no longer emphasized."]]},{"start":1300.8367916666668,"say":"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.","live":["preference_check","head_check"],"does":[[1302.6242916666667,"preference_check (the \"p>3/5\" part) is emphasized."],[1313.6312916666668,"preference_check (the \"p>3/5\" part) is no longer emphasized."]]},{"start":1314.2312916666667,"say":"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.","live":null,"does":[[1315.7752916666668,"all_equilibria is shown on the screen, written out."],[1327.3737916666669,"all_equilibria is hidden from the screen — left the board."],[1327.3737916666669,"head_check is hidden from the screen — left the board."],[1327.3737916666669,"preference_check is hidden from the screen — left the board."],[1327.3737916666669,"verification is hidden from the screen — left the board."]]},{"start":1328.5737916666667,"say":"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.","live":[],"does":[[1328.5737916666667,"head_method is shown on the screen, written out."],[1330.6052916666667,"procedure is shown on the screen, written out."],[1332.1142916666668,"procedure (the \"Label whose payoff is first\" part) is emphasized."],[1335.0172916666668,"procedure (the \"For each column\" part) is emphasized."],[1335.0172916666668,"procedure (the \"Label whose payoff is first\" part) is no longer emphasized."],[1339.2552916666668,"procedure (the \"For each column\" part) is no longer emphasized."],[1339.2552916666668,"procedure (the \"For each row\" part) is emphasized."]]},{"start":1343.6862916666669,"say":"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.","live":["procedure","head_method"],"does":[[1344.1622916666668,"procedure (the \"Every double-marked cell\" part) is emphasized."],[1344.1622916666668,"procedure (the \"For each row\" part) is no longer emphasized."],[1348.539291666667,"procedure (the \"Every double-marked cell\" part) is no longer emphasized."],[1348.539291666667,"procedure (the \"For mixing\" part) is emphasized."]]},{"start":1358.9382916666668,"say":"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.","live":null,"does":[[1359.2862916666668,"procedure (the \"Check probabilities\" part) is emphasized."],[1359.2862916666668,"procedure (the \"For mixing\" part) is no longer emphasized."],[1372.4287916666667,"head_method is hidden from the screen — left the board."],[1372.4287916666667,"procedure is hidden from the screen — left the board."],[1372.4287916666667,"procedure (the \"Check probabilities\" part) is no longer emphasized."]]},{"start":1373.0287916666668,"say":"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.","live":[],"does":[[1375.8612916666668,"final_formula is shown on the screen, written out."],[1376.4652916666669,"final_note is shown on the screen, written out."]]},{"start":1386.3882916666669,"say":"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.","live":["final_formula","final_note"],"does":[[1387.305291666667,"A box is drawn around final_formula."],[1401.0533958333335,"final_formula is hidden from the screen — left the board."],[1401.0533958333335,"final_note is hidden from the screen — left the board."]]}]}]},"durationSeconds":1402,"chapters":[{"title":"Payoffs, Best Responses, and Pure Equilibria","startSeconds":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. 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."},{"title":"Matching Pennies and Exploitation","startSeconds":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. 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."},{"title":"Mixing and Indifference","startSeconds":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. 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."},{"title":"The Asymmetric Inspection Game","startSeconds":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. 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."},{"title":"Guided Practice and the Hand Procedure","startSeconds":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. 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."}]}}