In a first-price sealed-bid auction with 5 symmetric bidders drawing valuations uniformly from [0, $100], a bidder with true valuation $80 submits a bid of $80. Is this optimal?
AYes — bidding true value maximizes win probability, which is the primary goal
BYes — like a second-price auction, truthful bidding is the dominant strategy in any sealed-bid format
CNo — bidding $80 guarantees zero surplus even if she wins; the equilibrium bid is $64, shading by (1/N) to balance win probability against surplus
DNo — she should bid above $80 to outcompete rivals with higher valuations
In a first-price auction, winners pay their own bid, so bidding true value yields zero profit even when winning. The equilibrium strategy is b(v) = ((N-1)/N)v — with N=5, she should bid (4/5)×$80 = $64. This shading trades a lower win probability for positive surplus when she does win. Option B is the critical misconception: truthful bidding is dominant only in second-price auctions, where the winner pays the second-highest bid regardless of her own bid.
Question 2 Multiple Choice
A seller must choose between a first-price and second-price auction. A consultant argues: 'The second-price format will generate more revenue because bidders reveal their true values instead of shading.' Under standard assumptions (symmetric, independent private values, risk-neutral bidders), is this correct?
AYes — truthful bidding always extracts full valuation surplus for the seller
BNo — revenue equivalence holds: both formats yield the same expected seller revenue because equilibrium bid shading in the first-price format exactly offsets the higher nominal bids in the second-price format
CYes — bid shading in first-price auctions necessarily hurts the seller relative to second-price
DNo — first-price always generates more revenue because the winner pays more than the second-highest value
Revenue equivalence is the key theorem: under symmetric independent private values with risk-neutral bidders, first-price and second-price auctions generate identical expected seller revenue. In second-price, the winner bids truthfully but pays only the second-highest value. In first-price, the winner shades to (N-1)/N of her value and pays that shaded amount — which turns out to equal the expected second-highest valuation given she has the highest. The expected payments are identical. Revenue equivalence breaks down only when bidders are risk-averse, valuations are asymmetric, or there is a common-value component.
Question 3 True / False
In a first-price sealed-bid auction, as the number of symmetric bidders increases without limit, equilibrium bids converge toward each bidder's true valuation.
TTrue
FFalse
Answer: True
The equilibrium bid formula b(v) = ((N-1)/N)v approaches v as N → ∞. With many competitors, the expected gap between the highest and second-highest valuation shrinks — you cannot afford to shade aggressively without risking losing to a rival with a nearly identical valuation. In the limit, competition does the seller's work of extracting true valuations, analogously to how perfect competition drives price to marginal cost.
Question 4 True / False
In a first-price sealed-bid auction, bidding your true valuation is the dominant strategy, just as it is in a second-price (Vickrey) auction.
TTrue
FFalse
Answer: False
In a second-price auction, the winner pays the second-highest bid, so your own bid determines only whether you win, not what you pay — making truthful bidding dominant regardless of others' bids. In a first-price auction, the winner pays their own bid, so winning at your true value yields exactly zero surplus. The optimal strategy is to shade below true value, with b(v) = ((N-1)/N)v in the symmetric uniform-values model. Truthful bidding is never optimal in a first-price auction with positive competition.
Question 5 Short Answer
Why does the revenue equivalence theorem hold between first-price and second-price auctions, despite the fact that bidders shade their bids in first-price but bid truthfully in second-price?
Think about your answer, then reveal below.
Model answer: In a second-price auction, winners bid truthfully but pay only the second-highest bid — retaining surplus equal to (own value − second-highest value). In a first-price auction, winners shade their bids to (N-1)/N of their value and pay that shaded amount. The equilibrium shaded bid turns out to equal the expected value of the second-order statistic from the valuation distribution, given that the bidder has the highest valuation. So in expectation, the winner pays the same amount in both formats, and the seller receives the same expected revenue. What the seller gains from truthful bidding in the second-price format is exactly offset by the shading in the first-price format.
The theorem is counterintuitive because the mechanisms feel so different. The insight is that rational agents adjust their strategies to equalize expected payoffs, and in doing so equate expected payments to the seller. Revenue equivalence breaks down when these equilibrating adjustments cannot fully compensate — for instance, when risk-averse bidders shade less in first-price (because the certain win at a shaded bid is preferred over a risky bet), raising first-price revenue above the second-price benchmark.