SITUATION: THE PROXY BID PARADOX AT eBAY
Have you ever bid for an item at eBay? Since an auction typically take days, eBay was smart enough to provide a mechanism that doesn’t require a bidder to hang out online 24/7; instead, you can enter a proxy bid, which works as fol- lows: As long as the highest bid of the other bidders is below your proxy bid, you’ll be the top bidder, with a bid equal to the highest bid of the other bid- ders plus the minimum bid increment. As soon as the highest bid of the other bidders exceeds your proxy bid, you drop out of the bidding (although you can always return with a higher proxy bid).
So what should your proxy bid be? Note that if, at the end of the auction, you submitted the highest proxy bid, then you win the item and pay a price equal to the second-highest proxy bid (plus the minimum bid increment). In this way, the eBay auction has the property of a second-price auction, and accordingly, you should submit a proxy bid equal to your valuation. Furthermore, once you’ve submitted such a bid, you can just return to the auc- tion site at its completion to find out whether you’ve won. How simple!
This argument for setting your proxy bid equal to your valuation is hit with a full body slam when it gets in the ring with reality. Contrary to what the the- ory prescribes, people frequently change their proxy bid over the course of an
66 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
FIGURE 3.8 Second-Price Auction
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3.2 Solving a Game when Players Are Rational 67
eBay auction. For example, say Dave enters a proxy bid of $150 for a ticket to a Rolling Stones concert and, coming back a day later, sees that the highest bid has reached $180 (so that it exceeds his proxy bid). Dave then changes his proxy bid to $200. But if it was originally worth $200 to Dave to see Mick Jagger and his buddies, why didn’t he just submit a proxy bid of $200 at the start? Why mess around with this lower proxy bid?
Because the phenomenon of bidders changing their proxy bids happens fairly often, we cannot summarily dismiss it as “stupid bidding.” The phe- nomenon represents systematic behavior, and as social scientists, our ob- jective is to understand it, not judge it. If we accept the idea that the bid- ders are doing exactly what they intend to do, it’s the theory that’s stupid— or, to say it more eloquently, our auction model is missing some relevant factors.
What could be missing? Several possibilities have been identified, but we have space to discuss only one. A potentially significant departure between the model and reality is that eBay runs multiple auctions for the same item. Think about how this could alter someone’s bidding strategy. Perhaps the Stones concert is worth $200 to you, which means that you would prefer to pay anything less than $200 than not get a ticket. But if you’re bidding for a ticket at an eBay auction, losing the auction doesn’t necessarily mean not getting a ticket; you might instead participate in another auction for a Stones ticket.
To see what difference this new information makes, imagine that you’re participating at an auction that ends two weeks prior to the concert. If you win the auction at a price of $199, your payoff is then $1, which is your val- uation less the price. Although $1 is higher than your payoff from not having a ticket, which is zero, it may not be higher than your expected payoff from participating in another auction. You might prefer not to win at $199 in order to have the option of winning at a lower price in a later auction.
It isn’t hard to see how this scenario could cause you to change your proxy bid over time. Suppose that you are currently watching two auctions and auction I ends to- morrow and auction II ends in two days. Suppose also that you have a proxy bid in auction II, but you’re keeping track of the price in auction I. As just argued, your proxy bid is not your valuation, but instead something that depends on what kind of price you think you would need to pay to win at another auction. If auction I closes at a higher price than you expected, you may conclude that the remaining tickets will go for higher prices, and this con- clusion could cause you to raise your proxy bid at auction II. Your optimal proxy bid changes over time as you learn what these tickets are selling for at other eBay auctions.
The gap between the theory’s prediction and actual be- havior at eBay auctions indicates a problem, not with game theory, but rather with the particular game-theoretic model. Game theory is immensely flexible and, when combined with an observant and clever mind, can offer cogent explanations of many social phenomena.
For the game in FIGURE 3.9, find the strategies that are strictly dominated and those which are weakly dominated.
FIGURE 3.9
3.1 CHECK YOUR UNDERSTANDING*
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*Recall that all answers to Check Your Understanding are at the back of the book.
3.3 Solving a Game when Players Are Rational and Players Know That Players Are Rational
[Saddam Hussein] starts out with a very menacing image. It sets you back a bit. I remember looking at my hands, and I was sweating. I was conscious that he knew what his reputation was. And he knew that I knew his reputa- tion. —BILL RICHARDSON
IN A SECOND-PRICE AUCTION, a player’s optimal bid could be determined without figuring out what bids others would submit. However, in a first-price auction, how much a bid should be shaded below your valuation depends on how ag- gressively you think other bidders will bid. Games commonly require the kind of thinking that goes into bidding in a first-price auction, in that a player must prognosticate what others will do.
In this section, we begin our journey into solving that problem by consid- ering some games for which it’s not enough to assume players are rational. However, if we assume just a bit more—such as the assumption that each player knows that the other players are rational—then reasonable conclusions can be drawn about how players will behave, at least in some situations.
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