EXAMPLE 9 The Vickrey Auction

Suppose that Anneliese, Binh, and Charlie bid for a stamp in a Vickrey auction. Anneliese, Binh, and Charlie value the stamp at $300, $200, and $100, respectively. To see that it is always at least as good for Anneliese to bid her true valuation of the stamp, consider the two possible results of bidding $300. To simplify the analysis, we will assume that each bidder’s bid is distinct, so no two bidders bid the same amount.

  • If Anneliese’s bid of $300 is the highest bid, then she wins the item but pays the second-highest bid $, which is strictly less than $300. To represent the Vickrey auction as a game, Anneliese’s payoff in this case is given by , which is a positive number because Anneliese was able to buy the stamp for less than she valued it. If she had decided to bid more than $300, then she would still be the highest bidder and still pay the same amount $. However, if she decides to bid less than $300, then it is possible that her bid will be less than $, in which case she loses the item and has a payoff of $0. Anneliese has no incentive to raise or to lower her bid because she can do no better than a payoff of , but she can do worse, receiving a payoff of $0 if she bids too low.
  • If Anneliese’s bid of $300 is not the highest bid, then she does not receive the item, and naturally does not have to pay for it—she receives a payoff of $0. Let $ be the highest bid. If she had decided to bid more than $300, say, more than $, then she becomes the highest bidder and would have had to pay $, which is more than $300—the value she placed on the stamp. This means that she would have overpaid for the stamp and her payoff would be the negative amount of . If she bids less than $300, she would still fail to have the highest bid and would still not have to pay anything. As before, Anneliese has no incentive to raise or to lower her bid.

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Anneliese’s strategy of bidding $300 weakly dominates any other strategy she could use. A similar analysis holds for Binh and Charlie. Collectively, it is a Nash equilibrium for Anneliese, Binh, and Charlie to each bid his or her true valuation for the stamp. To see this, if Anneliese, Binh, and Charlie bid $300, $200, and $100, respectively, for the stamp, then Anneliese will be the highest bidder and pay $200 for the stamp. The payoffs for Anneliese, Binh, and Charlie are given by the triplet . Neither Binh nor Charlie could change his bid to receive a positive payoff. Anneliese’s bid cannot affect the price she pays; it can determine only whether or not she is the highest bidder. Likewise, she cannot change her bid to do better and could only do worse by bidding too little and losing the stamp to Binh.