# Why Truthful Bidding Works—and Why First-Price Bidders Shade > A case-by-case explanation of auction incentives for viewers familiar with online bidding but new to economics. Holding one bidder's value fixed, the lecture proves why bids above or below that value can only tie or worsen the outcome in a second-price auction. It then changes the payment rule, derives first-price bid shading from the tradeoff between winning and profit, and states revenue equivalence with its assumptions and real-world limits. - Canonical watch page: [Why Truthful Bidding Works—and Why First-Price Bidders Shade](https://academa.ai/lectures/second-price-auctions-and-honest-bidding) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Economics - Published: 2026-08-28T18:55:54.598Z - Updated: 2026-08-28T18:55:54.598Z - Duration: PT769S (12 minutes 49 seconds) - Chapters: 4 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TXWNP8RS8MQP0KDSRPNQJ/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TXWNP8RS8MQP0KDSRPNQJ/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TXWNP8RS8MQP0KDSRPNQJ/0/dark/poster.jpg) ## Description See truthful bidding proved by cases, first-price bid shading derived from incentives, and revenue equivalence stated with its assumptions. ## Chapters - [00:00–02:40.158 · If You Bid Above Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0) - [02:40.158–05:26.349 · If You Bid Below Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664) - [05:26.349–08:44.798 · Why First-Price Bidders Shade](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=326.34902083333327) - [08:44.798–12:49 · What Revenue Equivalence Really Says](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666) ## Transcript ### [00:00 · If You Bid Above Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0) Suppose an online auction offers a lamp that is worth exactly one hundred dollars to you. That number is personal: owning the lamp rather than keeping one hundred dollars leaves you indifferent. Should you bid more? Here is the rule. Everyone submits one sealed number. The highest number wins, but the winner does not pay their own bid. They pay the highest bid among everybody else. Your payoff is simple. If you win at price p, you receive something worth one hundred and give up p, so your gain is one hundred minus p. If you lose, your gain is zero. Notice the division of labor. Your bid helps decide whether you win. Somebody else's bid determines what you pay. Your bid is a gate, not the number printed on the bill. Now hold your value fixed at one hundred and inflate your bid to one hundred thirty. The green point is what the lamp is worth to you. The yellow point is what you wrote. Every possible best rival bid falls into one of three ranges. We will walk all three, so there is nowhere for a hidden advantage to escape. First, suppose the best rival bid is above one hundred thirty. Let it be one hundred forty-five. Your inflated bid loses, and your truthful bid of one hundred would also lose. Nothing changes. Second, suppose the best rival bid is below your value. Move it to sixty. With a bid of one hundred you win and pay sixty. With a bid of one hundred thirty you also win and pay sixty. Your payoff is forty dollars either way. Raising your bid did not raise the price, but it did not improve anything either. This entire range is another tie. Third, put the rival between your value and your inflated bid. Say one hundred fifteen. With the truthful bid of one hundred, you lose. That loss is good news. The rival is willing to pay one hundred fifteen for a lamp worth only one hundred to you, so you should let them have it. But the inflated bid of one hundred thirty wins. The second price is the rival's one hundred fifteen, so you pay one hundred fifteen for something worth one hundred. Your payoff is minus fifteen. Only this interval changes the winner, and every change inside it is bad for you. The higher bid can turn a sensible loss into an unwanted purchase, but it can never create a profitable purchase. So bidding above your value is never better than bidding your value. In two ranges it changes nothing. In the remaining range it makes you strictly worse off. ### [02:40.158 · If You Bid Below Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664) Now try the opposite deviation. The lamp is still worth one hundred dollars to you, but you shade your bid down to eighty. Again, every rival bid lies in one of three ranges. First, suppose the best rival bid is above your value. Put it at one hundred twenty. A bid of eighty loses, and a truthful bid of one hundred also loses. Nothing changes. Second, suppose the best rival bid is below eighty. Move it to fifty-five. Your low bid wins and pays fifty-five. Your truthful bid would also win and pay exactly the same fifty-five. The payoff is forty-five dollars under either bid. Once more, changing your bid changes neither the allocation nor the price. The only interesting range lies between your low bid and your value. Put the best rival bid at ninety. With a truthful bid of one hundred, you win and pay ninety. The lamp is worth one hundred to you, so that purchase produces ten dollars of profit. With the shaded bid of eighty, you lose. You do not hand over money, but you throw away a purchase that would have made you ten dollars better off. So bidding below your value has the same logical shape as bidding above. It ties the truthful bid in two ranges. In the only range where it changes the result, it changes the result against you. Combine the two halves. A bid above value risks buying at a loss. A bid below value risks missing a gain. The truthful bid avoids both errors. This is why economists call truthful bidding weakly dominant in a second-price auction. Whatever the rival bids turn out to be, truth is at least as good as the deviation. Weakly matters. Often a different bid produces exactly the same winner, price, and payoff. Dominance does not mean truth wins strictly in every case. It means no alternative can do better in any case. The case proof treats your one hundred dollars as a private value. It is your own payoff from the lamp, and it does not rise or fall merely because somebody else values the lamp differently. It also takes the auction rule seriously: the highest bid wins and the winner pays the highest rival bid, apart from a fixed tie rule. The proof does not require you to predict the rivals. Risk neutrality, symmetry, and independent value draws are not needed for this elementary dominance comparison. They will matter when we predict first-price bids and compare the seller's expected revenue. ### [05:26.349 · Why First-Price Bidders Shade](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=326.34902083333327) Now keep the same lamp, the same one-hundred-dollar value, and the same sealed bids. Change one rule: the highest bidder still wins, but the winner pays their own bid. Try the truthful bid of one hundred. If you lose, your payoff is zero. If you win, you pay one hundred for something worth one hundred, so your payoff is still zero. Truth now destroys every dollar of profit conditional on winning. To make a positive gain, you must bid below your value. But moving down also makes it easier for a rival to beat you. Let us make that tension numerical. There are two bidders. Your value is one hundred, and the rival's value is equally likely to be anywhere from zero to one hundred. Assume the rival follows the symmetric strategy we are testing. With two uniform bidders, that strategy will turn out to be bidding half of value. Equivalently, the rival's bid is uniform from zero to fifty. It is convenient to rescale the horizontal calculation. If b denotes a candidate cutoff from zero to one hundred, the chance of beating a uniform rival cutoff is b over one hundred. Your expected gain is the chance of winning times the profit if you win. The curve shows that product for every candidate bid. Start at one hundred. You win for sure, but keep zero, so the average gain is zero. Shade to seventy. You win about seven times in ten and keep thirty dollars when you win. Seven tenths of thirty gives an average gain of twenty-one. Shade to fifty. Now the chance and the profit are both one half of their scales. You win half the time, keep fifty when you win, and average twenty-five. Shade farther to thirty. The profit after a win rises to seventy, but the win chance falls to three tenths. The average drops back to twenty-one. So shading is not free money. Too little shading leaves almost no profit. Too much shading throws away too many wins. The best bid balances those two effects. For this two-bidder uniform example, that balance occurs at half your value. A bidder worth one hundred bids fifty. A bidder worth eighty bids forty. The higher value still submits the higher bid. With n symmetric risk-neutral bidders whose private values are independent and uniform, the same calculation gives this equilibrium rule. The bid is n minus one over n times value. With two bidders that is one half. With more rivals the fraction rises, because stronger competition makes aggressive bidding more valuable. This formula is not a universal law of first-price auctions. It uses uniform independent values, symmetric bidders, and risk neutrality. Change those assumptions and the amount of shading changes. The durable conclusion is the incentive, not this particular fraction. Because your own bid becomes your price, you trade a greater chance of winning against a smaller profit when you win. ### [08:44.798 · What Revenue Equivalence Really Says](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666) We have reached the seller's question. Bidders behave differently under the two rules, so does one format reliably collect more money? Use the same two bidders, with independent values uniformly distributed from zero to one hundred. In second price they bid their values. In first price the symmetric equilibrium bid is half of value. Take one draw: values ninety and twenty. Second price charges the lower value, twenty. First price receives half of the higher value, forty-five. The blue first-price result is larger in this particular auction. Revenue equivalence does not deny that. Take values eighty and forty. Second price collects forty. The high-value bidder shades eighty to forty in first price, so that format also collects forty. Take values fifty and thirty. Second price collects thirty. First price collects twenty-five. This time second price earns more. Sale by sale, either format can lead. The theorem concerns the average across all possible value draws, with bidders using equilibrium strategies. Average over every pair. Both markers land at thirty-three dollars and a third. That equality of expectations is the claim. Here is the calculation behind that landing. In second price, seller revenue is the lower of the two values. Its expected value is one hundred over three. In first price, the winner has the higher value but bids half of it. The expected maximum of two uniform values is two hundred over three. Half of two hundred over three is again one hundred over three. Different bidding behavior and different payment rules produce the same expected seller revenue. That numerical match is one instance of a broader theorem, but the theorem has assumptions. First, values are private and independently drawn. Second, bidders are symmetric: their values come from the same distribution and they face the same opportunities. Third, bidders are risk neutral. They compare strategies by expected money payoff, without an extra preference for safer outcomes. Fourth, the allocation is efficient: the highest-value bidder receives the object. The formats also need matched participation and the same expected payoff for the lowest possible type. Under those conditions, efficient auction formats with the same payoff for the lowest type generate the same expected payment from every bidder type. That is the precise content of revenue equivalence. Real auctions can violate the conditions. Values may be correlated or partly common, bidders may differ, and people may care about risk. Budgets, entry costs, reserve prices, platform fees, imperfect learning, and collusion can also change participation, allocation, strategies, and revenue. The theorem is a benchmark, not a promise about every marketplace. So what should the seller hear? Under the standard assumptions and equilibrium play, first-price and second-price auctions deliver the same expected revenue. They need not collect the same amount in a particular sale. They need not expose the seller to the same variation, and the conclusion can fail when the assumptions fail. For bidders, the contrast remains sharp. Second price separates winning from payment, so truthful bidding is weakly dominant. First price makes your bid your payment, so equilibrium bids are shaded. For the seller, revenue equivalence says something narrower and more careful: after averaging over value draws, under the ideal conditions, neither standard format has an expected-revenue advantage. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TXWNP8RS8MQP0KDSRPNQJ/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [If You Bid Above Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0) Span: 00:00–02:40.158 (0s–160.15779166666664s). #### Objects - bid\_dot: a Point \[yellow\] labelled "130" drawn in line (location=(\, 0.0)) - heading: a Heading that says "If You Bid Above Your Value" - line: a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0) - payoff: a Derivation \[text\] that says "$upright("win") &arrow.r 100 - p \\ upright("lose") &arrow.r 0$" - point: a Point \[yellow\] drawn in line (location=(100.0, 0.0)) - question: a Panel that says "A lamp is worth $100$ to you. In a sealed-bid second-price auction, should you bid more than $100$?" - rival: a VariableNumber (initial\_value=145.0, format\_spec='.0f') - rival\_dot: a Point \[red\] labelled "145" drawn in line (location=(\, 0.0)) - rule\_panel: a Panel that says "Everyone submits one sealed bid. The highest bidder wins, but pays the highest losing bid." - table\_high: a Table \[text\] that says "Best rival bid Bid 100 Bid 130 Effect above 130 lose lose none below 100 win, pay rival win, pay rival none 100 to 130 lose win, overpay worse" (rows=(('Best rival bid', 'Bid 100', 'Bid 130', 'Effect'), ('above 13…, header=True) - value\_dot: a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)) - verdict: a Math \[text\] that says "$upright("bid above value") arrow.r upright("never better")$" #### Beats ##### [00:00](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0) Narration: Suppose an online auction offers a lamp that is worth exactly one hundred dollars to you. That number is personal: owning the lamp rather than keeping one hundred dollars leaves you indifferent. Should you bid more? Board: Empty. Actions: - [00:00](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0): question is shown on the screen, written out. - [00:00](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=0): line is shown on the screen, written out. - [00:3.286](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=3.286): value\_dot is shown on the screen, written out. ##### [00:12.953](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=12.953) Narration: Here is the rule. Everyone submits one sealed number. The highest number wins, but the winner does not pay their own bid. They pay the highest bid among everybody else. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0); question — a Panel that says "A lamp is worth $100$ to you. In a sealed-bid second-price auction, should you bid more than $100$?"; value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)) Actions: - [00:13.696](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=13.696): line moves to a new place on the board. - [00:13.696](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=13.696): rule\_panel is shown on the screen, written out. ##### [00:24.502](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=24.5015) Narration: Your payoff is simple. If you win at price p, you receive something worth one hundred and give up p, so your gain is one hundred minus p. If you lose, your gain is zero. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0); rule\_panel — a Panel that says "Everyone submits one sealed bid. The highest bidder wins, but pays the highest losing bid."; question — a Panel that says "A lamp is worth $100$ to you. In a sealed-bid second-price auction, should you bid more than $100$?"; value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)) Actions: - [00:26.719](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=26.718999999999998): payoff is shown on the screen, written out. - [00:33.511](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=33.510999999999996): payoff is shown on the screen, written out. ##### [00:36.085](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=36.085) Narration: Notice the division of labor. Your bid helps decide whether you win. Somebody else's bid determines what you pay. Your bid is a gate, not the number printed on the bill. Board: Unchanged from the preceding beat in this scene. Actions: - [00:43.364](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=43.36400000000001): payoff (the "100 - p" part) is emphasized. - [00:46.836](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=46.836000000000006): payoff (the "100 - p" part) is no longer emphasized. - [00:47.568](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=47.5675): line moves to a new place on the board. - [00:47.568](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=47.5675): payoff is hidden from the screen — left the board. - [00:47.568](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=47.5675): question is hidden from the screen — left the board. - [00:47.568](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=47.5675): rule\_panel is hidden from the screen — left the board. ##### [00:48.168](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=48.167500000000004) Narration: Now hold your value fixed at one hundred and inflate your bid to one hundred thirty. The green point is what the lamp is worth to you. The yellow point is what you wrote. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0); value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)) Actions: - [00:48.168](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=48.167500000000004): heading is shown on the screen, written out. - [00:48.806](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=48.80600000000001): table\_high is shown on the screen, written out. - [00:57.072](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=57.07200000000001): bid\_dot is shown on the screen, written out. ##### [00:59.483](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=59.483000000000004) Narration: Every possible best rival bid falls into one of three ranges. We will walk all three, so there is nowhere for a hidden advantage to escape. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0); value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)); heading — a Heading that says "If You Bid Above Your Value"; bid\_dot — a Point \[yellow\] labelled "130" drawn in line (location=(\, 0.0)) Actions: - None. ##### [01:9.453](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=69.453) Narration: First, suppose the best rival bid is above one hundred thirty. Let it be one hundred forty-five. Your inflated bid loses, and your truthful bid of one hundred would also lose. Nothing changes. Board: Unchanged from the preceding beat in this scene. Actions: - [01:14.225](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=74.22500000000001): rival\_dot is shown on the screen, written out. - [01:20.714](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=80.71400000000001): table\_high is shown on the screen, written out. - [01:21.051](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=81.051): table\_high (the "row=2" part) is indicated — a transient flash. ##### [01:22.615](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=82.6145) Narration: Second, suppose the best rival bid is below your value. Move it to sixty. With a bid of one hundred you win and pay sixty. With a bid of one hundred thirty you also win and pay sixty. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 170.0), include\_numbers=True, ticks\_every=25.0); value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)); heading — a Heading that says "If You Bid Above Your Value"; bid\_dot — a Point \[yellow\] labelled "130" drawn in line (location=(\, 0.0)); rival\_dot — a Point \[red\] labelled "145" drawn in line (location=(\, 0.0)) Actions: - [01:26.875](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=86.875): rival\_dot is redrawn as the numbers it depends on change. - [01:26.875](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=86.875): rival ticks to 60.0. - [01:32.495](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=92.495): table\_high is shown on the screen, written out. ##### [01:35.208](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=95.208) Narration: Your payoff is forty dollars either way. Raising your bid did not raise the price, but it did not improve anything either. This entire range is another tie. Board: Unchanged from the preceding beat in this scene. Actions: - [01:36.265](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=96.265): The segment (60.0, 0.0) to (100.0, 0.0) in line is lit up. - [01:43.8](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=103.8): line: retire a lit segment (unemphasize\_line). ##### [01:45.201](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=105.201) Narration: Third, put the rival between your value and your inflated bid. Say one hundred fifteen. With the truthful bid of one hundred, you lose. Board: Unchanged from the preceding beat in this scene. Actions: - [01:49.996](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=109.99599999999998): rival\_dot is redrawn as the numbers it depends on change. - [01:49.996](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=109.99599999999998): rival ticks to 115.0. - [01:52.318](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=112.31799999999998): point is shown on the screen, grown. - [01:54.318](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=114.31799999999998): point is hidden from the screen. ##### [01:55.681](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=115.681) Narration: That loss is good news. The rival is willing to pay one hundred fifteen for a lamp worth only one hundred to you, so you should let them have it. Board: Unchanged from the preceding beat in this scene. Actions: - None. ##### [02:4.663](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=124.663) Narration: But the inflated bid of one hundred thirty wins. The second price is the rival's one hundred fifteen, so you pay one hundred fifteen for something worth one hundred. Your payoff is minus fifteen. Board: Unchanged from the preceding beat in this scene. Actions: - [02:9.226](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=129.226): rival\_dot is indicated — a transient flash. - [02:15.101](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=135.10099999999997): table\_high is shown on the screen, written out. ##### [02:16.885](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=136.885) Narration: Only this interval changes the winner, and every change inside it is bad for you. The higher bid can turn a sensible loss into an unwanted purchase, but it can never create a profitable purchase. Board: Unchanged from the preceding beat in this scene. Actions: - [02:17.767](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=137.76699999999997): The segment (100.0, 0.0) to (130.0, 0.0) in line is lit up. - [02:27.044](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=147.04399999999998): line: retire a lit segment (unemphasize\_line). ##### [02:29.037](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=149.03699999999998) Narration: So bidding above your value is never better than bidding your value. In two ranges it changes nothing. In the remaining range it makes you strictly worse off. Board: Unchanged from the preceding beat in this scene. Actions: - [02:30.918](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=150.91799999999995): verdict is shown on the screen, written out. - [02:37.989](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=157.98899999999995): A box is drawn around verdict. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): heading is hidden from the screen — left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): line is hidden from the screen — left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): value\_dot is hidden from the screen — line left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): bid\_dot is hidden from the screen — line left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): rival\_dot is hidden from the screen — line left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): table\_high is hidden from the screen — left the board. - [02:39.116](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=159.11612499999998): verdict is hidden from the screen — left the board. ### Scene 2: [If You Bid Below Your Value](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664) Span: 02:40.158–05:26.349 (160.15779166666664s–326.34902083333327s). #### Objects - bid\_dot: a Point \[yellow\] labelled "80" drawn in line (location=(\, 0.0)) - dominance: a Math \[text\] that says "$upright("your bid") = upright("your value")$" - heading: a Heading that says "If You Bid Below Your Value" - heading\_2: a Heading that says "What This Proof Assumes" - line: a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 150.0), include\_numbers=True, ticks\_every=25.0) - payoff: a Derivation \[text\] that says "$upright("win") &arrow.r 100 - p \\ upright("lose") &arrow.r 0$" - point: a Point \[yellow\] drawn in line (location=(100.0, 0.0)) - private\_note: a Panel that says "Your value is the most you would willingly pay, and it does not change when you learn who else wins or what their values are." - rival: a VariableNumber (initial\_value=120.0, format\_spec='.0f') - rival\_dot: a Point \[red\] labelled "120" drawn in line (location=(\, 0.0)) - table\_low: a Table \[text\] that says "Best rival bid Bid 100 Bid 80 Effect above 100 lose lose none below 80 win, pay rival win, pay rival none 80 to 100 win profitably lose worse" (rows=(('Best rival bid', 'Bid 100', 'Bid 80', 'Effect'), ('above 100…, header=True) - truth\_heading: a Heading that says "The Truthful Bid" - truth\_note: a Panel that says "For every possible set of rival bids, bidding your value gives an outcome at least as good as any other bid. For some rival bids it is strictly better." - value\_dot: a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)) - verdict: a Math \[text\] that says "$upright("bid below value") arrow.r upright("never better")$" #### Beats ##### [02:40.158](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664) Narration: Now try the opposite deviation. The lamp is still worth one hundred dollars to you, but you shade your bid down to eighty. Again, every rival bid lies in one of three ranges. Board: Empty. Actions: - [02:40.158](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664): heading is shown on the screen, written out. - [02:40.158](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=160.15779166666664): line is shown on the screen, written out. - [02:43.537](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=163.53679166666663): value\_dot is shown on the screen, written out. - [02:46.265](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=166.26479166666664): bid\_dot is shown on the screen, written out. - [02:50.026](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=170.02579166666663): line moves to a new place on the board. - [02:50.026](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=170.02579166666663): table\_low is shown on the screen, written out. ##### [02:51.613](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=171.61329166666664) Narration: First, suppose the best rival bid is above your value. Put it at one hundred twenty. A bid of eighty loses, and a truthful bid of one hundred also loses. Nothing changes. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 150.0), include\_numbers=True, ticks\_every=25.0); heading — a Heading that says "If You Bid Below Your Value"; value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)); bid\_dot — a Point \[yellow\] labelled "80" drawn in line (location=(\, 0.0)) Actions: - [02:56.211](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=176.21079166666664): rival\_dot is shown on the screen, written out. - [03:2.585](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=182.58479166666663): table\_low is shown on the screen, written out. ##### [03:4.496](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=184.49629166666665) Narration: Second, suppose the best rival bid is below eighty. Move it to fifty-five. Your low bid wins and pays fifty-five. Your truthful bid would also win and pay exactly the same fifty-five. Board: line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 150.0), include\_numbers=True, ticks\_every=25.0); heading — a Heading that says "If You Bid Below Your Value"; value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)); bid\_dot — a Point \[yellow\] labelled "80" drawn in line (location=(\, 0.0)); rival\_dot — a Point \[red\] labelled "120" drawn in line (location=(\, 0.0)) Actions: - [03:9.222](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=189.22179166666663): rival\_dot is redrawn as the numbers it depends on change. - [03:9.222](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=189.22179166666663): rival ticks to 55.0. - [03:16.095](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=196.09479166666665): table\_low is shown on the screen, written out. ##### [03:18.1](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=198.09979166666665) Narration: The payoff is forty-five dollars under either bid. Once more, changing your bid changes neither the allocation nor the price. Board: Unchanged from the preceding beat in this scene. Actions: - [03:19.342](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=199.34179166666664): The segment (55.0, 0.0) to (100.0, 0.0) in line is lit up. - [03:25.379](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=205.37879166666664): line: retire a lit segment (unemphasize\_line). ##### [03:26.769](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=206.76879166666663) Narration: The only interesting range lies between your low bid and your value. Put the best rival bid at ninety. Board: Unchanged from the preceding beat in this scene. Actions: - [03:28.673](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=208.67279166666663): The segment (80.0, 0.0) to (100.0, 0.0) in line is lit up. - [03:32.899](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=212.89879166666663): rival\_dot is redrawn as the numbers it depends on change. - [03:32.899](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=212.89879166666663): rival ticks to 90.0. ##### [03:34.358](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=214.35829166666662) Narration: With a truthful bid of one hundred, you win and pay ninety. The lamp is worth one hundred to you, so that purchase produces ten dollars of profit. Board: Unchanged from the preceding beat in this scene. Actions: - [03:34.997](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=214.99679166666664): point is shown on the screen, grown. - [03:36.997](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=216.99679166666664): point is hidden from the screen. - [03:37.551](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=217.55079166666664): rival\_dot is indicated — a transient flash. ##### [03:44.455](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=224.45529166666662) Narration: With the shaded bid of eighty, you lose. You do not hand over money, but you throw away a purchase that would have made you ten dollars better off. Board: Unchanged from the preceding beat in this scene. Actions: - [03:50.481](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=230.48079166666662): table\_low is shown on the screen, written out. ##### [03:54.692](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=234.69179166666663) Narration: So bidding below your value has the same logical shape as bidding above. It ties the truthful bid in two ranges. In the only range where it changes the result, it changes the result against you. Board: Unchanged from the preceding beat in this scene. Actions: - [04:6.858](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=246.85779166666663): verdict is shown on the screen, written out. - [04:6.858](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=246.85779166666663): line: retire a lit segment (unemphasize\_line). ##### [04:8.468](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=248.46829166666663) Narration: Combine the two halves. A bid above value risks buying at a loss. A bid below value risks missing a gain. The truthful bid avoids both errors. Board: verdict — a Math \[text\] that says "$upright("bid below value") arrow.r upright("never better")$"; line — a NumberLine labelled "upright("dollars")" (x\_range=(0.0, 150.0), include\_numbers=True, ticks\_every=25.0); heading — a Heading that says "If You Bid Below Your Value"; value\_dot — a Point \[green\] labelled "upright("your value")" drawn in line (location=(100.0, 0.0)); bid\_dot — a Point \[yellow\] labelled "80" drawn in line (location=(\, 0.0)); rival\_dot — a Point \[red\] labelled "120" drawn in line (location=(\, 0.0)) Actions: - [04:19.428](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=259.4277916666666): A box is drawn around verdict. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): heading is hidden from the screen — left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): line is hidden from the screen — left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): value\_dot is hidden from the screen — line left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): bid\_dot is hidden from the screen — line left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): rival\_dot is hidden from the screen — line left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): table\_low is hidden from the screen — left the board. - [04:20.369](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=260.36879166666665): verdict is hidden from the screen — left the board. ##### [04:21.569](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=261.56879166666664) Narration: This is why economists call truthful bidding weakly dominant in a second-price auction. Whatever the rival bids turn out to be, truth is at least as good as the deviation. Board: Empty. Actions: - [04:21.569](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=261.56879166666664): truth\_heading is shown on the screen, written out. - [04:23.566](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=263.5657916666666): dominance is shown on the screen, written out. - [04:24.378](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=264.3777916666666): truth\_note is shown on the screen, written out. ##### [04:33.326](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=273.32579166666665) Narration: Weakly matters. Often a different bid produces exactly the same winner, price, and payoff. Dominance does not mean truth wins strictly in every case. It means no alternative can do better in any case. Board: dominance — a Math \[text\] that says "$upright("your bid") = upright("your value")$"; truth\_note — a Panel that says "For every possible set of rival bids, bidding your value gives an outcome at least as good as any other bid. For some rival bids it is strictly better."; truth\_heading — a Heading that says "The Truthful Bid" Actions: - [04:42.08](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=282.0797916666666): dominance is indicated — a transient flash. - [04:47.803](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=287.8032916666666): dominance moves to a new place on the board. - [04:47.803](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=287.8032916666666): truth\_heading is hidden from the screen — left the board. - [04:47.803](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=287.8032916666666): truth\_note is hidden from the screen — left the board. ##### [04:48.403](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=288.40329166666663) Narration: The case proof treats your one hundred dollars as a private value. It is your own payoff from the lamp, and it does not rise or fall merely because somebody else values the lamp differently. Board: dominance — a Math \[text\] that says "$upright("your bid") = upright("your value")$" Actions: - [04:51.247](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=291.24679166666664): private\_note is shown on the screen, written out. ##### [05:0.16](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=300.1597916666666) Narration: It also takes the auction rule seriously: the highest bid wins and the winner pays the highest rival bid, apart from a fixed tie rule. The proof does not require you to predict the rivals. Board: dominance — a Math \[text\] that says "$upright("your bid") = upright("your value")$"; private\_note — a Panel that says "Your value is the most you would willingly pay, and it does not change when you learn who else wins or what their values are." Actions: - None. ##### [05:13.113](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=313.1127916666666) Narration: Risk neutrality, symmetry, and independent value draws are not needed for this elementary dominance comparison. They will matter when we predict first-price bids and compare the seller's expected revenue. Board: Unchanged from the preceding beat in this scene. Actions: - [05:18.732](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=318.7317916666666): dominance is indicated — a transient flash. - [05:25.307](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=325.30735416666664): dominance is hidden from the screen — left the board. - [05:25.307](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=325.30735416666664): private\_note is hidden from the screen — left the board. ### Scene 3: [Why First-Price Bidders Shade](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=326.34902083333327) Span: 05:26.349–08:44.798 (326.34902083333327s–524.7982291666666s). #### Objects - best\_mark: a Line \[green\] drawn in curve\_axes (start=(50.0, 0.0), end=(50.0, 25.0), dashed=True) - bid: a VariableNumber (initial\_value=100.0, format\_spec='.0f') - curve: a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)) - curve\_axes: an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0) - equilibrium: a Math \[text\] that says "$b(v) = frac(n-1, n) v$" - gain: a VariableNumber (format\_spec='.0f') - gain\_rule: a Math \[text\] that says "$E(pi(b)) = P(upright("win with") thin b) dot.op (100-b)$" - head\_equilibrium: a Heading that says "The Symmetric Equilibrium" - head\_first: a Heading that says "Change One Rule" - head\_tradeoff: a Heading that says "Probability Against Profit" - ledger: a Table \[text\] that says "Your bid Win chance Profit if win Average 100 1.00 0 0 70 0.70 30 21 50 0.50 50 25 30 0.30 70 21" (rows=(('Your bid', 'Win chance', 'Profit if win', 'Average'), ('100'…, header=True) - qualification: a Text \[text\] that says "For $n$ risk-neutral bidders with independent values uniformly distributed on the same interval." - rule\_panel: a Panel that says "Everyone submits one sealed bid and the highest bidder wins. The winner now pays their own bid." - spot: a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\) - tradeoff: a Panel that says "A higher bid wins more often but leaves less profit when it wins. A lower bid leaves more profit but loses more often." - value\_mark: a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True) - zero\_profit: a Math \[text\] that says "$upright("bid") thin 100 arrow.r upright("profit if you win") thin 0$" #### Beats ##### [05:26.349](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=326.34902083333327) Narration: Now keep the same lamp, the same one-hundred-dollar value, and the same sealed bids. Change one rule: the highest bidder still wins, but the winner pays their own bid. Board: Empty. Actions: - [05:26.349](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=326.34902083333327): head\_first is shown on the screen, written out. - [05:32.781](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=332.7810208333333): rule\_panel is shown on the screen, written out. ##### [05:38.257](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=338.2570208333333) Narration: Try the truthful bid of one hundred. If you lose, your payoff is zero. If you win, you pay one hundred for something worth one hundred, so your payoff is still zero. Board: rule\_panel — a Panel that says "Everyone submits one sealed bid and the highest bidder wins. The winner now pays their own bid."; head\_first — a Heading that says "Change One Rule" Actions: - [05:48.323](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=348.32302083333326): zero\_profit is shown on the screen, written out. ##### [05:49.771](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=349.77052083333325) Narration: Truth now destroys every dollar of profit conditional on winning. To make a positive gain, you must bid below your value. But moving down also makes it easier for a rival to beat you. Board: rule\_panel — a Panel that says "Everyone submits one sealed bid and the highest bidder wins. The winner now pays their own bid."; zero\_profit — a Math \[text\] that says "$upright("bid") thin 100 arrow.r upright("profit if you win") thin 0$"; head\_first — a Heading that says "Change One Rule" Actions: - [06:1.659](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=361.6590208333333): head\_first is hidden from the screen — left the board. - [06:1.659](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=361.6590208333333): rule\_panel is hidden from the screen — left the board. - [06:1.659](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=361.6590208333333): zero\_profit is hidden from the screen — left the board. ##### [06:2.859](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=362.85902083333326) Narration: Let us make that tension numerical. There are two bidders. Your value is one hundred, and the rival's value is equally likely to be anywhere from zero to one hundred. Board: Empty. Actions: - [06:2.859](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=362.85902083333326): head\_tradeoff is shown on the screen, written out. - [06:4.554](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=364.55402083333325): curve\_axes is shown on the screen, written out. - [06:8.954](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=368.9540208333333): value\_mark is shown on the screen, written out. ##### [06:15.755](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=375.7545208333333) Narration: Assume the rival follows the symmetric strategy we are testing. With two uniform bidders, that strategy will turn out to be bidding half of value. Equivalently, the rival's bid is uniform from zero to fifty. Board: curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); head\_tradeoff — a Heading that says "Probability Against Profit"; value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True) Actions: - None. ##### [06:29.172](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=389.17152083333326) Narration: It is convenient to rescale the horizontal calculation. If b denotes a candidate cutoff from zero to one hundred, the chance of beating a uniform rival cutoff is b over one hundred. Board: Unchanged from the preceding beat in this scene. Actions: - [06:36.695](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=396.6950208333333): curve\_axes moves to a new place on the board. - [06:36.695](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=396.6950208333333): gain\_rule is shown on the screen, written out. - [06:39.215](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=399.21502083333326): gain\_rule becomes "$E(pi(b)) = frac(b, 100) (100-b)$". ##### [06:41.394](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=401.39352083333324) Narration: Your expected gain is the chance of winning times the profit if you win. The curve shows that product for every candidate bid. Board: gain\_rule — a Math \[text\] that says "$E(pi(b)) = P(upright("win with") thin b) dot.op (100-b)$"; curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); head\_tradeoff — a Heading that says "Probability Against Profit"; value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True) Actions: - [06:46.014](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=406.0140208333333): curve is shown on the screen, written out. - [06:47.779](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=407.7790208333333): ledger is shown on the screen, written out. ##### [06:49.493](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=409.4930208333333) Narration: Start at one hundred. You win for sure, but keep zero, so the average gain is zero. Board: gain\_rule — a Math \[text\] that says "$E(pi(b)) = P(upright("win with") thin b) dot.op (100-b)$"; curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); head\_tradeoff — a Heading that says "Probability Against Profit"; value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)) Actions: - [06:52.419](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=412.41902083333326): gain\_rule (the "(100-b)" part) is emphasized. - [06:52.663](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=412.6630208333333): spot is shown on the screen, written out. - [06:52.663](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=412.6630208333333): ledger is shown on the screen, written out. - [06:55.577](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=415.5770208333333): gain\_rule (the "(100-b)" part) is no longer emphasized. ##### [06:56.177](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=416.17702083333324) Narration: Shade to seventy. You win about seven times in ten and keep thirty dollars when you win. Seven tenths of thirty gives an average gain of twenty-one. Board: gain\_rule — a Math \[text\] that says "$E(pi(b)) = P(upright("win with") thin b) dot.op (100-b)$"; curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); head\_tradeoff — a Heading that says "Probability Against Profit"; value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)); spot — a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\) Actions: - [06:57.14](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=417.14002083333327): spot is redrawn as the numbers it depends on change. - [06:57.14](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=417.14002083333327): bid ticks to 70.0. - [06:57.14](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=417.14002083333327): gain ticks to 21.0. - [07:4.943](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=424.94302083333326): ledger is shown on the screen, written out. ##### [07:6.518](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=426.51802083333325) Narration: Shade to fifty. Now the chance and the profit are both one half of their scales. You win half the time, keep fifty when you win, and average twenty-five. Board: Unchanged from the preceding beat in this scene. Actions: - [07:7.331](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=427.33102083333324): spot is redrawn as the numbers it depends on change. - [07:7.331](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=427.33102083333324): bid ticks to 50.0. - [07:7.331](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=427.33102083333324): gain ticks to 25.0. - [07:15.678](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=435.6780208333333): best\_mark is shown on the screen, written out. - [07:15.678](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=435.6780208333333): ledger is shown on the screen, written out. ##### [07:17.428](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=437.42752083333323) Narration: Shade farther to thirty. The profit after a win rises to seventy, but the win chance falls to three tenths. The average drops back to twenty-one. Board: gain\_rule — a Math \[text\] that says "$E(pi(b)) = P(upright("win with") thin b) dot.op (100-b)$"; curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); head\_tradeoff — a Heading that says "Probability Against Profit"; value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)); spot — a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\); best\_mark — a Line \[green\] drawn in curve\_axes (start=(50.0, 0.0), end=(50.0, 25.0), dashed=True) Actions: - [07:18.588](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=438.58802083333325): spot is redrawn as the numbers it depends on change. - [07:18.588](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=438.58802083333325): bid ticks to 30.0. - [07:18.588](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=438.58802083333325): gain ticks to 21.0. - [07:26.239](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=446.23902083333326): ledger is shown on the screen, written out. ##### [07:28.001](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=448.00052083333327) Narration: So shading is not free money. Too little shading leaves almost no profit. Too much shading throws away too many wins. The best bid balances those two effects. Board: Unchanged from the preceding beat in this scene. Actions: - [07:37.765](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=457.76502083333327): curve is indicated — a transient flash. ##### [07:40.385](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=460.3845208333333) Narration: For this two-bidder uniform example, that balance occurs at half your value. A bidder worth one hundred bids fifty. A bidder worth eighty bids forty. The higher value still submits the higher bid. Board: Unchanged from the preceding beat in this scene. Actions: - [07:44.146](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=464.1460208333333): spot is redrawn as the numbers it depends on change. - [07:44.146](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=464.1460208333333): bid ticks to 50.0. - [07:44.146](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=464.1460208333333): gain ticks to 25.0. - [07:53.783](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=473.78252083333325): gain\_rule is hidden from the screen — left the board. - [07:53.783](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=473.78252083333325): head\_tradeoff is hidden from the screen — left the board. - [07:53.783](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=473.78252083333325): ledger is hidden from the screen — left the board. ##### [07:54.983](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=474.9825208333333) Narration: With n symmetric risk-neutral bidders whose private values are independent and uniform, the same calculation gives this equilibrium rule. Board: curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)); spot — a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\); best\_mark — a Line \[green\] drawn in curve\_axes (start=(50.0, 0.0), end=(50.0, 25.0), dashed=True) Actions: - [07:54.983](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=474.9825208333333): head\_equilibrium is shown on the screen, written out. - [08:1.345](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=481.34502083333325): tradeoff is shown on the screen, written out. - [08:3.249](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=483.24902083333325): equilibrium is shown on the screen, written out. ##### [08:4.615](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=484.6150208333333) Narration: The bid is n minus one over n times value. With two bidders that is one half. With more rivals the fraction rises, because stronger competition makes aggressive bidding more valuable. Board: curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)); spot — a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\); best\_mark — a Line \[green\] drawn in curve\_axes (start=(50.0, 0.0), end=(50.0, 25.0), dashed=True); tradeoff — a Panel that says "A higher bid wins more often but leaves less profit when it wins. A lower bid leaves more profit but loses more often."; equilibrium — a Math \[text\] that says "$b(v) = frac(n-1, n) v$"; head\_equilibrium — a Heading that says "The Symmetric Equilibrium" Actions: - [08:12.138](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=492.13802083333326): equilibrium (the "frac(n-1, n)" part) is emphasized. - [08:17.189](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=497.1885208333333): equilibrium (the "frac(n-1, n)" part) is no longer emphasized. ##### [08:17.789](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=497.7885208333333) Narration: This formula is not a universal law of first-price auctions. It uses uniform independent values, symmetric bidders, and risk neutrality. Change those assumptions and the amount of shading changes. Board: Unchanged from the preceding beat in this scene. Actions: - [08:23.002](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=503.00202083333323): qualification is shown on the screen, written out. ##### [08:32.077](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=512.0770208333333) Narration: The durable conclusion is the incentive, not this particular fraction. Because your own bid becomes your price, you trade a greater chance of winning against a smaller profit when you win. Board: curve\_axes — an Axes (x\_range=(0.0, 108.0), y\_range=(-2.0, 30.0), x\_ticks\_every=25.0); value\_mark — a Line \[red\] labelled "upright("value")" drawn in curve\_axes (start=(100.0, 0.0), end=(100.0, 27.0), dashed=True); curve — a FunctionPlot \[blue\] drawn in curve\_axes (function=\, x\_range=(0.0, 100.0)); spot — a PlotPoint \[yellow\] labelled "0" drawn in curve\_axes (target='curve', x=\); best\_mark — a Line \[green\] drawn in curve\_axes (start=(50.0, 0.0), end=(50.0, 25.0), dashed=True); tradeoff — a Panel that says "A higher bid wins more often but leaves less profit when it wins. A lower bid leaves more profit but loses more often."; equilibrium — a Math \[text\] that says "$b(v) = frac(n-1, n) v$"; qualification — a Text \[text\] that says "For $n$ risk-neutral bidders with independent values uniformly distributed on the same interval."; head\_equilibrium — a Heading that says "The Symmetric Equilibrium" Actions: - [08:39.949](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=519.9490208333332): tradeoff (the "A higher bid wins more often" part) is indicated — a transient flash. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): curve\_axes is hidden from the screen — left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): value\_mark is hidden from the screen — curve\_axes left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): curve is hidden from the screen — curve\_axes left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): spot is hidden from the screen — curve\_axes left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): best\_mark is hidden from the screen — curve\_axes left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): equilibrium is hidden from the screen — left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): head\_equilibrium is hidden from the screen — left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): qualification is hidden from the screen — left the board. - [08:43.757](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=523.7565625): tradeoff is hidden from the screen — left the board. ### Scene 4: [What Revenue Equivalence Really Says](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666) Span: 08:44.798–12:49.22 (524.7982291666666s–769.2204999999999s). #### Objects - assumptions: a Block \[text\] that says "Independent private values: each bidder knows their own value, and draws are independent. Symmetric bidders: values come from the same distribution and bidders face the same rules. Risk neutrality: bidders maximize expected monetary payoff…" - draws: a Table \[text\] that says "Two values Second price First price 90 and 20 20 45 80 and 40 40 40 50 and 30 30 25 average over all draws 33.3 33.3" (rows=(('Two values', 'Second price', 'First price'), ('90 and 20', '…, header=True) - equal\_result: a Math \[text\] that says "$E(R\_1) = E(R\_2)$" - expectation\_work: a Derivation \[text\] that says "$E(R\_2) &= E(min(V\_1,V\_2)) = frac(100, 3) \\ E(R\_1) &= E(frac(1, 2) max(V\_1,V\_2)) \\ &= frac(1, 2) dot.op frac(200, 3) = frac(100, 3)$" - first\_dot: a Point \[blue\] labelled "45.0" drawn in take\_line (location=(\, 0.0)) - first\_take: a VariableNumber (initial\_value=45.0, format\_spec='.1f') - head\_assumptions: a Heading that says "The Standard Assumptions" - head\_average: a Heading that says "Expected Revenue" - head\_close: a Heading that says "What the Seller Should Hear" - head\_draws: a Heading that says "Auction by Auction" - head\_limits: a Heading that says "The Theorem and the Real World" - ideal: a Panel that says "Any two standard auction formats that allocate efficiently and give the lowest type the same expected payoff generate the same expected payment for every bidder type." - limits: a Panel that says "Risk aversion, correlated or common values, asymmetric information, budgets, entry costs, reserve prices, fees, learning, and collusion can all break the comparison." - second\_dot: a Point \[green\] labelled "20.0" drawn in take\_line (location=(\, 0.0)) - second\_take: a VariableNumber (initial\_value=20.0, format\_spec='.1f') - seller\_claim: a Math \[text\] that says "$E(upright("seller revenue in first price")) = E(upright("seller revenue in second price"))$" - take\_line: a NumberLine labelled "upright("seller revenue")" (x\_range=(0.0, 62.0), include\_numbers=True, ticks\_every=10.0) - takeaway: a Panel that says "The theorem compares expected revenue under its assumptions. It does not say the two auctions collect the same amount in each sale, have the same risk, or work equally well outside the model." - tex: a Tex \[text\] that says "Idealized theorem" - tex\_2: a Tex \[text\] that says "Real-world complications" #### Beats ##### [08:44.798](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666) Narration: We have reached the seller's question. Bidders behave differently under the two rules, so does one format reliably collect more money? Board: Empty. Actions: - [08:44.798](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666): head\_draws is shown on the screen, written out. - [08:44.798](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=524.7982291666666): take\_line is shown on the screen, written out. - [08:51.915](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=531.9152291666666): take\_line moves to a new place on the board. - [08:51.915](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=531.9152291666666): draws is shown on the screen, written out. ##### [08:53.49](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=533.4902291666666) Narration: Use the same two bidders, with independent values uniformly distributed from zero to one hundred. In second price they bid their values. In first price the symmetric equilibrium bid is half of value. Board: take\_line — a NumberLine labelled "upright("seller revenue")" (x\_range=(0.0, 62.0), include\_numbers=True, ticks\_every=10.0); head\_draws — a Heading that says "Auction by Auction" Actions: - None. ##### [09:7.825](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=547.8247291666667) Narration: Take one draw: values ninety and twenty. Second price charges the lower value, twenty. First price receives half of the higher value, forty-five. Board: Unchanged from the preceding beat in this scene. Actions: - [09:8.695](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=548.6952291666666): draws is shown on the screen, written out. - [09:10.379](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=550.3792291666666): second\_dot is shown on the screen, written out. - [09:17.543](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=557.5432291666666): first\_dot is shown on the screen, written out. ##### [09:19.35](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=559.3502291666666) Narration: The blue first-price result is larger in this particular auction. Revenue equivalence does not deny that. Board: take\_line — a NumberLine labelled "upright("seller revenue")" (x\_range=(0.0, 62.0), include\_numbers=True, ticks\_every=10.0); head\_draws — a Heading that says "Auction by Auction"; second\_dot — a Point \[green\] labelled "20.0" drawn in take\_line (location=(\, 0.0)); first\_dot — a Point \[blue\] labelled "45.0" drawn in take\_line (location=(\, 0.0)) Actions: - [09:21.382](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=561.3822291666665): first\_dot is indicated — a transient flash. ##### [09:26.696](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=566.6957291666666) Narration: Take values eighty and forty. Second price collects forty. The high-value bidder shades eighty to forty in first price, so that format also collects forty. Board: Unchanged from the preceding beat in this scene. Actions: - [09:28.251](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=568.2512291666666): second\_dot is redrawn as the numbers it depends on change. - [09:28.251](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=568.2512291666666): first\_dot is redrawn as the numbers it depends on change. - [09:28.251](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=568.2512291666666): second\_take ticks to 40.0. - [09:28.251](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=568.2512291666666): first\_take ticks to 40.0. - [09:35.833](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=575.8332291666666): draws is shown on the screen, written out. ##### [09:38.094](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=578.0937291666667) Narration: Take values fifty and thirty. Second price collects thirty. First price collects twenty-five. This time second price earns more. Board: Unchanged from the preceding beat in this scene. Actions: - [09:39.928](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=579.9282291666666): second\_dot is redrawn as the numbers it depends on change. - [09:39.928](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=579.9282291666666): second\_take ticks to 30.0. - [09:44.827](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=584.8272291666666): first\_dot is redrawn as the numbers it depends on change. - [09:44.827](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=584.8272291666666): first\_take ticks to 25.0. - [09:48.194](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=588.1942291666666): draws is shown on the screen, written out. ##### [09:49.653](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=589.6532291666666) Narration: Sale by sale, either format can lead. The theorem concerns the average across all possible value draws, with bidders using equilibrium strategies. Board: Unchanged from the preceding beat in this scene. Actions: - [09:51.348](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=591.3482291666667): draws (the "column=2" part) is emphasized. - [09:51.348](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=591.3482291666667): draws (the "column=3" part) is emphasized. - [09:54.866](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=594.8662291666666): draws (the "column=2" part) is no longer emphasized. - [09:54.866](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=594.8662291666666): draws (the "column=3" part) is no longer emphasized. ##### [10:0.737](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=600.7372291666666) Narration: Average over every pair. Both markers land at thirty-three dollars and a third. That equality of expectations is the claim. Board: Unchanged from the preceding beat in this scene. Actions: - [10:4.58](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=604.5802291666666): second\_dot is redrawn as the numbers it depends on change. - [10:4.58](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=604.5802291666666): first\_dot is redrawn as the numbers it depends on change. - [10:4.58](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=604.5802291666666): second\_take ticks to 33.333333333333336. - [10:4.58](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=604.5802291666666): first\_take ticks to 33.333333333333336. - [10:7.064](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=607.0642291666666): draws is shown on the screen, written out. - [10:9.817](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=609.8167291666666): draws is hidden from the screen — left the board. - [10:9.817](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=609.8167291666666): head\_draws is hidden from the screen — left the board. ##### [10:10.417](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=610.4167291666666) Narration: Here is the calculation behind that landing. In second price, seller revenue is the lower of the two values. Its expected value is one hundred over three. Board: take\_line — a NumberLine labelled "upright("seller revenue")" (x\_range=(0.0, 62.0), include\_numbers=True, ticks\_every=10.0); second\_dot — a Point \[green\] labelled "20.0" drawn in take\_line (location=(\, 0.0)); first\_dot — a Point \[blue\] labelled "45.0" drawn in take\_line (location=(\, 0.0)) Actions: - [10:10.417](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=610.4167291666666): head\_average is shown on the screen, written out. - [10:11.264](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=611.2642291666666): expectation\_work is shown on the screen, written out. - [10:15.85](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=615.8502291666666): expectation\_work (the "min(V\_1,V\_2)" part) is emphasized. - [10:20.761](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=620.7612291666666): expectation\_work (the "min(V\_1,V\_2)" part) is no longer emphasized. ##### [10:21.361](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=621.3612291666666) Narration: In first price, the winner has the higher value but bids half of it. The expected maximum of two uniform values is two hundred over three. Board: take\_line — a NumberLine labelled "upright("seller revenue")" (x\_range=(0.0, 62.0), include\_numbers=True, ticks\_every=10.0); second\_dot — a Point \[green\] labelled "20.0" drawn in take\_line (location=(\, 0.0)); first\_dot — a Point \[blue\] labelled "45.0" drawn in take\_line (location=(\, 0.0)); head\_average — a Heading that says "Expected Revenue" Actions: - [10:23.3](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=623.3002291666667): expectation\_work (the "max(V\_1,V\_2)" part) is emphasized. - [10:24.531](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=624.5312291666667): expectation\_work is shown on the screen, written out. - [10:30.545](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=630.5447291666666): expectation\_work (the "max(V\_1,V\_2)" part) is no longer emphasized. ##### [10:31.145](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=631.1447291666666) Narration: Half of two hundred over three is again one hundred over three. Different bidding behavior and different payment rules produce the same expected seller revenue. Board: Unchanged from the preceding beat in this scene. Actions: - [10:31.4](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=631.4002291666666): expectation\_work is shown on the screen, written out. - [10:38.296](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=638.2962291666666): equal\_result is shown on the screen, written out. - [10:39.539](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=639.5392291666666): A box is drawn around equal\_result. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): equal\_result is hidden from the screen — left the board. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): expectation\_work is hidden from the screen — left the board. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): head\_average is hidden from the screen — left the board. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): take\_line is hidden from the screen — left the board. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): second\_dot is hidden from the screen — take\_line left the board. - [10:40.572](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=640.5722291666666): first\_dot is hidden from the screen — take\_line left the board. ##### [10:41.172](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=641.1722291666666) Narration: That numerical match is one instance of a broader theorem, but the theorem has assumptions. First, values are private and independently drawn. Board: Empty. Actions: - [10:41.172](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=641.1722291666666): head\_assumptions is shown on the screen, written out. - [10:41.172](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=641.1722291666666): assumptions is shown on the screen, written out. - [10:48.544](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=648.5442291666666): assumptions (the "Independent private values" part) is indicated — a transient flash. ##### [10:51.281](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=651.2807291666666) Narration: Second, bidders are symmetric: their values come from the same distribution and they face the same opportunities. Board: assumptions — a Block \[text\] that says "Independent private values: each bidder knows their own value, and draws are independent. Symmetric bidders: values come from the same distribution and bidders face the same rules. Risk neutrality: bidders maximize expected monetary payoff…"; head\_assumptions — a Heading that says "The Standard Assumptions" Actions: - [10:52.941](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=652.9412291666666): assumptions (the "Symmetric bidders" part) is indicated — a transient flash. ##### [10:59.381](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=659.3807291666666) Narration: Third, bidders are risk neutral. They compare strategies by expected money payoff, without an extra preference for safer outcomes. Board: Unchanged from the preceding beat in this scene. Actions: - [11:1.099](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=661.0992291666666): assumptions (the "Risk neutrality" part) is indicated — a transient flash. ##### [11:8.723](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=668.7227291666666) Narration: Fourth, the allocation is efficient: the highest-value bidder receives the object. The formats also need matched participation and the same expected payoff for the lowest possible type. Board: Unchanged from the preceding beat in this scene. Actions: - [11:10.847](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=670.8472291666667): assumptions (the "Efficient allocation" part) is indicated — a transient flash. - [11:16.396](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=676.3962291666667): assumptions (the "Matched participation" part) is indicated — a transient flash. - [11:21.389](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=681.3887291666666): assumptions is hidden from the screen — left the board. - [11:21.389](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=681.3887291666666): head\_assumptions is hidden from the screen — left the board. ##### [11:21.989](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=681.9887291666666) Narration: Under those conditions, efficient auction formats with the same payoff for the lowest type generate the same expected payment from every bidder type. That is the precise content of revenue equivalence. Board: Empty. Actions: - [11:21.989](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=681.9887291666666): head\_limits is shown on the screen, written out. - [11:22.894](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=682.8942291666667): ideal is shown on the screen, written out. ##### [11:34.733](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=694.7332291666667) Narration: Real auctions can violate the conditions. Values may be correlated or partly common, bidders may differ, and people may care about risk. Board: ideal — a Panel that says "Any two standard auction formats that allocate efficiently and give the lowest type the same expected payoff generate the same expected payment for every bidder type."; head\_limits — a Heading that says "The Theorem and the Real World" Actions: - [11:35.139](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=695.1392291666666): limits is shown on the screen, written out. ##### [11:44.505](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=704.5047291666666) Narration: Budgets, entry costs, reserve prices, platform fees, imperfect learning, and collusion can also change participation, allocation, strategies, and revenue. The theorem is a benchmark, not a promise about every marketplace. Board: ideal — a Panel that says "Any two standard auction formats that allocate efficiently and give the lowest type the same expected payoff generate the same expected payment for every bidder type."; limits — a Panel that says "Risk aversion, correlated or common values, asymmetric information, budgets, entry costs, reserve prices, fees, learning, and collusion can all break the comparison."; head\_limits — a Heading that says "The Theorem and the Real World" Actions: - [11:57.612](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=717.6122291666667): limits (the "can all break the comparison" part) is indicated — a transient flash. - [12:0.794](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=720.7937291666666): head\_limits is hidden from the screen — left the board. - [12:0.794](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=720.7937291666666): ideal is hidden from the screen — left the board. - [12:0.794](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=720.7937291666666): limits is hidden from the screen — left the board. ##### [12:1.394](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=721.3937291666666) Narration: So what should the seller hear? Under the standard assumptions and equilibrium play, first-price and second-price auctions deliver the same expected revenue. Board: Empty. Actions: - [12:1.394](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=721.3937291666666): head\_close is shown on the screen, written out. - [12:9.764](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=729.7642291666666): seller\_claim is shown on the screen, written out. ##### [12:12.129](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=732.1292291666666) Narration: They need not collect the same amount in a particular sale. They need not expose the seller to the same variation, and the conclusion can fail when the assumptions fail. Board: seller\_claim — a Math \[text\] that says "$E(upright("seller revenue in first price")) = E(upright("seller revenue in second price"))$"; head\_close — a Heading that says "What the Seller Should Hear" Actions: - [12:14.253](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=734.2532291666666): takeaway is shown on the screen, written out. ##### [12:22.597](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=742.5972291666666) Narration: For bidders, the contrast remains sharp. Second price separates winning from payment, so truthful bidding is weakly dominant. First price makes your bid your payment, so equilibrium bids are shaded. Board: seller\_claim — a Math \[text\] that says "$E(upright("seller revenue in first price")) = E(upright("seller revenue in second price"))$"; takeaway — a Panel that says "The theorem compares expected revenue under its assumptions. It does not say the two auctions collect the same amount in each sale, have the same risk, or work equally well outside the model."; head\_close — a Heading that says "What the Seller Should Hear" Actions: - None. ##### [12:36.108](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=756.1082291666667) Narration: For the seller, revenue equivalence says something narrower and more careful: after averaging over value draws, under the ideal conditions, neither standard format has an expected-revenue advantage. Board: Unchanged from the preceding beat in this scene. Actions: - [12:38.918](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=758.9182291666666): A box is drawn around seller\_claim. - [12:48.179](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=768.1788333333333): head\_close is hidden from the screen — left the board. - [12:48.179](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=768.1788333333333): seller\_claim is hidden from the screen — left the board. - [12:48.179](https://academa.ai/lectures/second-price-auctions-and-honest-bidding?t=768.1788333333333): takeaway is hidden from the screen — left the board.