Bidding at an Auction
Bidding at an Auction It’s been 20 years since you’ve graduated from college, and you’ve just sold your Internet company for a cool $50 million. With all this cash on hand, you decide to indulge your passion for modern art. An auction house is selling an Andy Warhol piece that you’ve been coveting for some time. The rules are that all interested parties must submit a written bid by this Friday at 5 P.M. Whoever submits the highest bid wins the Warhol piece and pays a price equal to the bid—a format known as the first-price auction.
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64 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
*A more formal mathematical definition is provided in Section 3.5, which is an appendix to this chapter.
FIGURE 3.6 Revised Tosca Game
2,2 4,2
1,4 3,3 Tosca
Scarpia
Stab
Consent
Real Blanks
3.2 Solving a Game when Players Are Rational 65
The Warhol piece is worth $400,000 to you. If you win the item, your pay- off equals $400,000 less the price you paid, while if you don’t win, your payoff is zero. Hence, if you end up paying $400,000, you’re no better off, while you’re better (worse) off if you get it for less (more) than $400,000.
You’ve just learned that there is only one other bidder: your old college girl friend, who has recently cashed in stock options after being CEO for a bio- medical company. You know that she values the piece at $300,000, and fur- thermore, she knows that you value it at $400,000. (In Chapter 10, we’ll ex- plore the more realistic case when each bidder’s valuation is known only to him or her.)
The auctioneer announces that bids must be in increments of $100,000 and that the minimum bid is $100,000. We’ll also assume that the maxi- mum bid is $500,000. If the bids are equal, the auctioneer flips a coin to determine the winner. The strategic form of the first-price auction is shown in FIGURE 3.7, where strategies and payoffs are in hundreds of thousands of dollars. For ex- ample, if you bid 3 and she bids 1, then you win the auction, pay a price of 3, and receive a payoff of 1 (� 4 � 3). If you both bid 1, then there is a 50% chance that you’re declared the winner—in which case your payoff is 3 (from paying a price of 1)—and a 50% chance that you’re not the win- ner—in which case your payoff is zero; the ex- pected payoff is then . (We’ll explain more about expected payoffs in Chapter 7, so if you don’t understand, trust me.)