How much should you bid?
How much should you bid? Since bidding 5—which is in excess of what the piece is worth to you—is strictly dominated by bidding 4, then, clearly, you don’t want to bid that much. Also, you probably don’t want to bid 4—which is your valuation of the item—since that is weakly dominated by any lower bid. If you bid your valuation, you’re assured of a zero payoff, regardless of whether you win. Thus, it would be better to bid lower and have a chance of getting a positive payoff. You can then rule out bidding at or above your valu- ation. The minimum bid of 1 is also weakly dominated. We’ve then eliminated bids 1, 4, and 5 because they are either strictly or weakly dominated. Can we say more? Unfortunately, no. Either a bid of 2 or 3 may be best, depending on what the other bidder submits. If you think she’ll bid 1, then you want to bid 2. If you think she’ll bid 3, then you’ll want to match that bid. You want to shade your bid below your valuation, but how much to shade depends on what you think the other bidder will bid.
That you’d want to shade your bid is not that surprising, since, at a first-price auction, you want to bid just high enough to win the item (all the while making sure that the bid is below your valuation). Which bid achieves that objective de- pends on the other bids submitted. However, a slight modification in the auction format results in the surprising finding that a bidder’s optimal bid is independent of how others bid. Devised by Nobel Laureate William Vickrey, the modified for- mat is like the first-price auction in that the bidder with the highest bid wins, but it differs from the first-price auction in that the winner pays, not his bid, but the second-highest bid. Thus, this format is known as the second-price auction.
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FIGURE 3.7 First-Price Auction
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The strategic form game for the second price auction is illustrated in FIGURE 3.8. For example, if you bid 3 and she bids 2, then you win the item and pay a price of 2, because that is the second-highest bid. In that case, your payoff is 2 (� 4 � 2). In fact, your payoff is 2 whether you bid 3, 4, or 5, because the price you pay is not your bid, but the other bidder’s bid. Your bid only influences whether or not you win. If you were to bid 1, then, since she is bidding 2, your payoff would be affected—it is now zero—since your low bid causes you to lose the auction.
Inspection of Figure 3.8 reveals that a bid of 4 weakly dominates every other bid for you. It would then make sense for you to bid 4, regardless of how you think your former girl friend will bid. As for her, a bid of 3 weakly
dominates every other one of her bids. Note that for each of you, the weakly dominant bid equals your valuation. This is not coincidental: in every second- price auction, bidding your valuation weakly dominates every other bid!
In the first-price auction, the motivation for shading your bid below your valuation is to lower the price you pay in the event that you win. That strat- egy doesn’t work in the second-price auction, since the price you pay is not what you bid, but what someone else bid. Bidding below your valuation only reduces your chances of winning at a price below your valuation, and that’s a bad deal.
Figuring out your optimal bid at a second-price auction is a piece of cake. A bidder just needs to determine what the item is worth and bid that value. There is no need for a certified psychologist to help you evaluate the psyche of other bidders! There is no need for the services of a well-trained game theorist to tell you how to bid! You just need to know yourself.