Game Theory: Auction Analysis

William Vickrey's 1961 discovery is one of the cleanest results in economics: in a second-price sealed-bid auction, bidding your true value is a dominant strategy — the best move regardless of what others bid. The mechanism works because you pay the second-highest bid, not your own. Overbidding your true value doesn't help you (you might win but pay more than the item is worth); underbidding doesn't help you either (you might lose an item worth more than you'd have paid). So you bid your true value and let the second-highest bid determine the price. Vickrey received the Nobel Prize in 1996 for this result and related work.

First-price auctions are strategically different: you pay what you bid, so optimal play requires shading your bid below your true value. The optimal shade depends on the number of competitors (shade more with more competitors) and the distribution of their valuations (shade more when competition is intense). In equilibrium, first-price and second-price auctions generate the same expected revenue — the revenue equivalence theorem — under standard conditions.

The winner's curse is the most common failure mode in common-value auctions (where the item has an underlying objective value everyone is trying to estimate, rather than a private personal value). Winning means you bid highest, which means your estimate was the most optimistic among all bidders. In expectation, if you bid your unconditional estimate and win, you've overpaid — because winning reveals that you were the most optimistic, not the most accurate. The correct bid is your estimate conditional on winning, which is lower than your unconditional estimate.

Paul Milgrom and Robert Wilson (Nobel 2020) developed the modern theory of auction design, including the simultaneous ascending auction used in FCC spectrum allocation — showing how auction design directly affects both revenue and efficient allocation.

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