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All-Pay Auctions and Contests

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All-Pay Auctions and Contests All-Pay Auctions and Contests 1 Plan • Today: contests • Thursday: recap • Thursday night: HW5 due • Online course evaluations are open till Friday • Final exam next Thursday, 7:45 am, Soc Sci 6102 • Any questions before we get started? 398 2 Before we start: a common mistake on HW4 • Before we start, I wanted to mention a common mistake that the grader flagged on HW4 • On question 1.e, you've already verified that in the two-bidder all-pay auction with valuations 2 vi uniform on [0; 30], it's an equilibrium for both bidders to bid bi = 60 • Now you're asked to calculate expected revenue, which is the expected sum of the payments from both bidders, or v2 v2 E 1 + 2 60 60 • We can think of these as bidder 1 and bidder 2, or we can think of them as the higher-value and the lower-value bidder, and rewrite this as v2 v2 1 1 E max + min = E v2 + E v2 60 60 60 max 60 min where vmax and vmin are maxfv1; v2g and minfv1; v2g, or whichever valuation turns out to be higher and whichever turns out to be lower, respectively • Up to here, everything's fine; but at this point, several people noted that Evmax = 20 and Evmin = 10, 202+102 plugged those in, and mis-calculated revenue as 60 , which gave the wrong answer • Why can't you do that? Because if x is a random variable, E(x2) 6= (E(x))2 2 2 • (In fact, E(x ) − (E(x)) is the variance of x, so unless vmax and vmin have zero variance, this is guaranteed to give the wrong answer) • More generally, for a general function g, the only time that E(g(x)) = g(E(x)) { the only time you can plug in the mean of x to evaluate the mean of g(x){ is when g is linear • In all other cases, you have to calculate the expected value of a function by integrating 399 3 All-Pay Auctions • This segues nicely into today's starting point: all-pay auctions • When I first mentioned the all-pay auction in class, I mentioned that these are rarely if ever literally used to sell things, but that they're sometimes used as an analogy for certain types of winner-take-all competition • A paper from 19931 uses an all-pay auction as a model for politicians and lobbying • Suppose a corrupt politician gets to make a policy choice { say, the location of a military base { and would happily sell the outcome to whichever side is willing to pay more • However, that's illegal, so instead, the different sides \wine and dine" the politician with campaign contributions, fancy meals, and so on, and then the politician makes a decision, but all the lobbyists have \paid" their bribes { nobody gets their campaign contribution back because they're unhappy with the policy the politician chose • The structure of payoffs is exactly the same as in an all-pay auction { each lobbyist gets a certain value from winning, minus the bribe they pay, and the politician awards the outcome based on the biggest bribe, but keeps all the bribes, not just the winner's 1Baye, Kovenock and de Vries (1993), \Rigging the Lobbying Process: An Application of the All-Pay Auction," American Economic Review 83.1 400 • So, an all-pay auction might be a pretty reasonable, natural way to think about certain legislative/lobbying settings • However, when all-pay auctions are used to model situations like this, there's one key difference from the way we've been thinking about auctions • We've been thinking of settings where bidders' valuations are private information; in lobbying models, it's assumed that while different lobbyists may value winning differently, everyone knows everyone else's valuation • That is, if the politician is considering putting a military base in Bethesda, or Portland, or Newport News, everyone knows how much value Bethesda would get from being chosen, and how much value Portland would get, and so on • Modeling auctions with complete information is messier than with private information • When bidders' valuations are private information, each \type" of bidder can have a unique equilibrium bid, and uncertainty about which valuations your opponent has creates a tradeoff between price and likelihood of winning • That is, perhaps in equilibrium, if I have valuation 70, you know I'll bid 35, and if I have valuation 80, you know I'll bid 40, but since you don't know what my valuation turned out to be, you don't know how I'll bid, which creates a trade-off for you between how much you pay and how likely you are to win, which is what makes the equilibrium \work" • But when bidders know each others' valuations, this no longer works • There can't be an equilibrium in pure strategies { an equilibrium where each bidder makes a predictable bid { or someone would always have an incentive to change their bid 401 • For example, consider an all-pay setting with two bidders, where their valuations for winning are v1 = 1 and v2 = 2 • Of course, bidder 1 can't bid more than 1 in equilibrium, or he'd get a negative payoff even if he won; so bidder 2 would have to win in any equilibrium • If bidder 1 was bidding more than 0 and losing, he'd want to lower his bid; but if bidder 1 was bidding 0, bidder 2 would want to win as cheaply as possible; and then bidder 1 would want to outbid him • So in an all-pay auction with complete information, there's no equilibrium in the sense we've been considering so far { for any combination of bids, some bidder will always have an incentive to change their bid • So how do we proceed? • Let's put aside the all-pay auction for a few minutes, and think about a game you might have more intuition for: Scissors-Paper-Rock 402 3.1 Scissors-Paper-Rock and Mixed Strategies • Suppose you and I decide to bet a dollar on a single game of scissors-paper-rock • If you play scissors and I play rock, rock smashes scissors and I win your dollar • If you play scissors and I play paper, scissors cuts paper and you win my dollar • If we both play paper, we tie, and nobody wins any money • There's clearly no equilibrium where we both pick a single action to play • Remember, the logic of equilibrium is that in equilibrium, I correctly anticipate how you're going to play, and best-respond to that, and you correctly anticipate how I'm going to play, and best-respond to that • But if I'm going to play rock, your best-response is easy { play paper { but then if you're going to play paper, I don't want to play rock, I want to play scissors • So how do we deal with this? • Instead of thinking of each of us choosing a single action to play, think of each of us choosing a probability distribution over actions { a probability of playing rock, a probability of playing paper, and a probability of playing scissors { and then at the very last second before we play, choosing randomly according to those probabilities • So, maybe I'll play rock half the time, scissors a quarter of the time, and paper a quarter of the time { and maybe you know that in equilibrium, but you don't know which one I'll choose this particular time 403 • Thinking about strategies in this way, scissors-paper-rock has a unique equilibrium: I play each action with probability exactly one-third, and you play each action with probability one-third • If I'm playing each action with probability one-third, then no matter what you do, your expected outcome is the same: you'll win a third of the time, lose a third of the time, and tie a third of the time • So any action, or any combination of actions, is a best-response, because each one is just as good as the others • And if you're playing each action one-third of the time, then each action is just as good for me, so any combination is a best-response • And that's an equilibrium • A little more formally, a mixed-strategy equilibrium is when... { each player chooses a probability distribution over actions, and { given the other players' strategies, each action I play in equilibrium gives me the same expected payoff, which is at least as high as anything else I could do 404 3.2 Back to the complete-information all-pay auction... • So let's go back to the all-pay auction, where there's no private information, v1 = 1, and v2 = 2 • The mixed-strategy equilibrium turns out to look like this: bidder 2 always bids, and mixes uniformly over the interval (0; 1]; and bidder 1 only bids half the time, and mixes uniformly over (0; 1] when he bids • Why is this an equilibrium? • First, consider bidder 1's problem, if bidder 2 is following this strategy • If bidder 1 bids b1, her expected payoff is v1 Pr(winjb1) − b1 = Pr(b2 < b1) − b1 = b1 − b1 = 0 which is the same for every b1 2 [0; 1], and she can't do better than that by bidding more than 1, so any bid in [0; 1] { or any mixture of those bids { is a best-response • What about bidder 2? • If bidder 2 bids b2 > 0, his expected payoff is 1 1 v Pr(winjb ) − b = 2 + b − b = 1 + b − b = 1 2 2 2 2 2 2 2 2 2 for any b2 2 (0; 1] • (If bidder 2 bids exactly 0, then when b1 = 0, they tie, and he only wins half the time, 1 1 which would make his expected payoff 4 2 − 0 = 2 ) • So any bid in (0; 1] is equally good, giving payoff 1, and is beter than bidding 0 or strictly above 1; so any bid in (0; 1], or any mix of such bids, is a best-response • so given these strategies, both bidders are best-responding to each other, and so this is the mixed-strategy equilibrium 405 • So, many papers consider the complete-information all-pay auction, as an analogy for lobbying • Baye, Kovenock and de Vries show, interestingly, that if there are more than two lobbyists, the politician (seler) may increase revenue by excluding some competitors { and may want to exclude the one who values the outcome the most! • (They note that one example of excluding some competitors is announcing a small field of “finalists”) • If one competitor is much stronger than the others, the others may not want to compete much knowing they'll probably lose, and so the strong one may not have to compete much either; • If the politician excludes the strong competitor, the weaker ones have a better shot at winning, and will be willing to expend more resources) • In the example above, suppose we started with three lobbyists, with vi = 1, 1, and 2 • If you exclude one of the weak bidders..
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