Questions: Auction Formats and Revenue Equivalence

Question 1

A bidder values an item at $100. In which auction format is bidding exactly $100 a weakly dominant strategy — meaning it is never worse than any other bid, regardless of what others do?

• A. Sealed-bid first-price auction, because winning with a lower bid wastes surplus
• B. Dutch auction, because claiming the item early avoids the risk of losing
• C. Sealed-bid second-price (Vickrey) auction, because winning means you pay the second-highest bid, not your own
• D. English auction only when fewer than three competitors are present

Answer: (C)

In a Vickrey (second-price) auction, truthful bidding is a weakly dominant strategy: if you win, you pay someone else's bid, not yours, so bidding your true value cannot hurt you. Bidding above your value risks winning and overpaying; bidding below risks losing when you could have won profitably. In first-price and Dutch auctions, you pay your own bid, so the optimal strategy is bid shading — bidding below true value — making truthful bidding suboptimal there.

Question 2

A seller knows her bidders are strongly risk-averse (they dislike uncertainty about whether they will win). Compared to a second-price auction, a first-price auction will generate:

• A. The same revenue, because the revenue equivalence theorem holds regardless of bidder risk preferences
• B. Lower revenue, because risk-averse bidders shade their bids further below value to avoid overpaying
• C. Higher revenue, because risk-averse bidders shade their bids less (they bid more aggressively to reduce the chance of losing)
• D. Revenue that depends only on the number of bidders, not on their risk attitudes

Answer: (C)

Revenue equivalence breaks down when bidders are risk-averse. In a first-price auction, where losing means getting nothing, risk-averse bidders shade their bids *less* aggressively than risk-neutral bidders would — they accept a smaller surplus to increase their probability of winning. This higher bidding translates into higher expected revenue for the seller. Second-price auctions do not produce this effect because the winner's payment is independent of their own bid.

Question 3

A Dutch auction (descending-price) and a sealed-bid first-price auction are strategically equivalent: the optimal bidding strategy and the distribution of outcomes are identical in both.

• T. True
• F. False

Answer: True

In both formats, a bidder must commit to a price before knowing others' bids, and the winner pays exactly their own bid. The strategic problem is identical: how much to shade below your true value, trading off a higher probability of winning (bid high) against a larger surplus if you do win (bid low). Because the information structure and payoff structure are the same, both formats induce the same equilibrium bids and the same expected outcomes.

Question 4

The revenue equivalence theorem implies that bidders behave the same way across most four standard auction formats.

• T. True
• F. False

Answer: False

Revenue equivalence says expected *revenues* are equal under its assumptions — not that bidding behavior is identical. In fact, bidding behavior is strikingly different: in English and Vickrey auctions, truthful bidding is dominant; in Dutch and first-price auctions, the equilibrium involves bid shading below true value. The theorem's insight is that these very different strategies happen to produce the same expected revenue for the seller. Behavior diverges; outcomes (in expectation) do not.

Question 5

Explain the core logic of the revenue equivalence theorem: why should four auction formats with such different rules produce the same expected revenue?

Model answer: Under the theorem's assumptions (risk-neutral, symmetric bidders; highest-value bidder wins; bidder with lowest value pays zero), what ultimately determines revenue is the underlying distribution of bidder values and the number of bidders. Strategic differences across formats (truthful bidding vs. shading) exactly offset each other: in formats where bidders shade down, they win at lower prices but more often in the right situations, leaving the seller's expected take unchanged. The information rents captured by each bidder are the same in all formats because the winner's expected payment is pinned down by the second-highest value — regardless of how that payment is realized.

The deep intuition is that a seller cannot extract more than the 'information rent' a bidder has by virtue of having a higher value than competitors. Any format that awards the item to the highest-value bidder and leaves the lowest-value bidder with zero surplus must generate the same expected revenue — the format's rules determine how revenue is distributed across outcomes, but the expectation is invariant. Revenue equivalence fails when one of these conditions breaks down: asymmetric bidders have different information rents; correlated values (winner's curse) change incentives; risk aversion changes the bid-shading calculus.