Questions: Auction Design: First-Price and Second-Price Sealed-Bid Auctions

Question 1

Your true value for an item is $400. In a second-price sealed-bid auction, what bid maximizes your expected payoff?

• A. $400 — bidding your true value is the dominant strategy regardless of how many bidders there are
• B. Less than $400 — you should shade your bid to reduce what you pay if you win
• C. More than $400 — overbidding increases your chance of winning without changing what you pay
• D. It depends on the number of other bidders and your beliefs about their values

Answer: (A)

In a second-price auction, the winner pays the second-highest bid, not their own. This decouples your bid (which determines whether you win) from what you pay (determined by others). Bidding above $400 risks winning at a price above your value — a loss. Bidding below $400 risks losing to someone who bid $350 when you could have won with positive surplus. Bidding exactly $400 is the only bid that cannot hurt you. This is true regardless of competition — it is a dominant strategy, not a best response to particular beliefs about others.

Question 2

In a first-price sealed-bid auction with two bidders drawing values independently from a uniform distribution on [0, 1], the symmetric equilibrium strategy is:

• A. Bid your true value — underbidding only risks losing to a close competitor
• B. Bid half your true value — the equilibrium formula is (n-1)/n times your value, with n=2
• C. Bid zero — in a one-shot game, credible commitment is impossible
• D. Bid your value minus a constant depending on the number of bidders

Answer: (B)

In a first-price auction with n bidders and values uniform on [0,1], the symmetric Bayesian Nash equilibrium bid is (n-1)/n times your value. With n=2, each bidder bids half their value. Bidding your true value guarantees zero surplus if you win, which is irrational when you could shade down and earn positive surplus. The formula shows that bid-shading decreases as competition increases: with 10 bidders you bid 90% of your value, because the risk of losing to a close rival outweighs the benefit of shading further.

Question 3

In a first-price auction, bidding your true value maximizes your chance of winning and is therefore the dominant strategy.

• T. True
• F. False

Answer: False

Bidding your true value in a first-price auction maximizes your probability of winning but yields zero surplus if you win — you pay exactly what the item is worth to you. The rational strategy is bid-shading: bidding below your true value to preserve positive surplus upon winning. Truth-telling is the dominant strategy in second-price auctions, where your bid determines only whether you win, not what you pay. In first-price auctions, your bid determines both, making bid-shading the equilibrium response.

Question 4

The Revenue Equivalence Theorem states that under independent private values with risk-neutral bidders, a seller's expected revenue is the same in first-price and second-price auctions.

• T. True
• F. False

Answer: True

This is the theorem's central claim. In second-price auctions, winners bid their true values but pay less (the second-highest value). In first-price auctions, winners shade their bids but pay their own (shaded) bid. These effects exactly offset under independent private values with risk-neutral bidders, yielding identical expected revenue. The theorem breaks down with risk aversion (first-price generates more revenue because bidders shade less), correlated values, or asymmetric bidders — which is why real auction design requires attention to the specific environment.

Question 5

Why is truth-telling a dominant strategy in a second-price auction but not in a first-price auction? Explain the key difference in how each payment rule affects bidding incentives.

Model answer: In a second-price auction, your bid determines only whether you win — not what you pay if you win. Payment equals the second-highest bid, which is independent of your own bid. This decoupling means shading your bid can only hurt you (by sometimes losing when you could have won), while overbidding cannot help (you still pay the second-highest bid). Bidding your true value is safe in all cases — a dominant strategy. In a first-price auction, your bid determines both whether you win and exactly what you pay. Bidding your true value guarantees zero surplus upon winning, so every rational bidder shades their bid below their value to trade a lower win probability for positive surplus when they do win.

The distinction hinges on whether the bid is pivotal for winning only or for both winning and paying. Second-price auctions decouple these, creating a dominant strategy. First-price auctions couple them, requiring strategic calculation about competitors' values and producing a Bayesian Nash equilibrium rather than a dominant strategy.