When we did independent private values and revenue equivalence, one of the auction types we mentioned was an all-pay auction

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1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 15 October 28, 2008 When we did independent private values and revenue equivalence, one of the auction types we mentioned was an all-pay auction All-pay auctions are generally not used in actual auction settings, but are used to model lobbying efforts, or research and development races, or other winner-take-all scenarios involving sunk costs Another variation is an all-pay version of an ascending auction, called a war of attrition These are sometimes used to model industry shakeouts lots of firms trying to compete in an industry (or market) that is a natural monopoly or will only support a small number of firms The idea is that, while the war of attrition is going on, firms operate at a loss; over time, firms give up and drop out, and eventually the remaining firms are able to operate profitably In entry scenarios like this, firms which drop out stop incurring costs at the time they drop out Bulow and Klemperer ( The Generalized War of Attrition ) give another example of a war of attrition, this time in a political setting. The 1993 budget passed by the Clinton administration was hugely unpopular politically; Democrats did not want to see the new Democratic president lose the fight, but also did not want to support the bill if they could avoid it. The bill passed the House, , and the Senate, with the tiebreaker, both after long delays as one member after another reluctantly fell into line (NY Times 7/7/93). Here, the prize the congressmen were competing over was the ability to not vote for the bill, knowing that the game could not end until sufficiently many gave up on the prize (agreed to support it). Bulow and Klemperer point out that wars of attrition are also useful in modeling situations where firms which give up continue to incur costs until the game ends The example they use for this a competition to decide on a standard. Suppose there are three firms with similar technologies, competing to establish a standard the whole industry will use. One firm may give up on trying to push through their own standard; but as long as the other two firms are still battling, none of them can start selling products based on the standard; so even the firm that has dropped out still incurs costs until the war is over Bulow and Klemperer offer a model that nests both of these, and prove some nice results 1

2 Model There are N + K players competing for N prizes, so K players must drop out for the game to end Each bidder has a private value v i for winning a prize (the prizes are identical), independently drawn from a distribution F with support [ V, V ], with V > 0 Bidders incur a cost of 1 per unit time as long as they remain in the competition, and then a cost c > 0 per unit time after dropping out until the game ends (The c = 0 case is the usual entry game; they consider the limit as c 0, since when c = 0, the game does not have a symmetric equilibrium in pure strategies. They generally interpret c as being less than 1, but allow for it to be greater.) Dropouts are visible, so strategies can condition on how many other players are still active They focus on symmetric perfect Bayesian equilibrium, which they show is unique Results Like in an ascending auction, a strategy is a plan for when to drop out in a given stage of the game Since there are private values, a continuation game does not depend on the exact history of when players have dropped out, only the number of players left and the lowest type that could still be in the game given history (They show that each continuation game has a unique symmetric PBE) Their first lemma is that at any point in the game, higher types plan to wait longer before dropping out than lower types, and therefore that the allocation of winners is efficient: at any point in the game, the probability you win is the probability you are among the highest N types remaining. (The logic is the same single-crossing differences intuition we had in auctions.) The proof that equilibrium is unique is by induction on the number of players left. Basically, given a number N + k of players remaining, and a knowledge of the unique equilibrium (and therefore payoffs) in a continuation game with N +k 1 players, expected payoffs are uniquely determined by the Envelope Theorem, and therefore uniquely determine expected payments in this stage, which lead to a unique determination of strategies. This leads to a simple characterization of the unique symmetric equilibrium of the game. In the stage with N + k players left, when v is the lowest possible type still in the game given equilibrium strategies, a player with type v plans to drop out (if nobody else has) in time v T (v, v, k) = c k 1 Nx f(x) 1 F (x) dx 2 v

3 Note that for c < 1, this means dropouts occur more rapidly while there are still lots of players left in the game, then slow down as there are fewer and fewer players left. The intuition for T : first consider the game when k = 1 (the next player to drop out ends the game). Consider a player with type v, who s supposed to drop out at time T (v, v, 1), but who decides to wait till time T (v + dv, v, 1) instead. Conditional on the game not ending before T (v), and for dv small, he pays T (v)dv more, and in return, wins a prize worth v with additional probability Nf(v)dv/(1 F (v)), the probability that one of the other N players has a type between v and v + dv. T being a best-response requires these to be equal; integrating up to v gives the result for k = 1. For k > 1, the argument is a bit different. Conditional on reaching time T (v) with k > 1 too many players left, waiting till T (v + dv) for dv small still doesn t give you any chance of winning. What quitting a later gains you is this: we said that dropouts happen faster when there are more players in the game. So if you stay in a little bit longer, it speeds up the game for the other players, so it reduces how long you ll have to pay the cost c for. They show that for indifference to hold, T (v, v, k) = ct (v, v, k 1). (The LHS is what you pay by staying in a little bit longer; the RHS is what you gain by shortening the game.) Repeated k times, this gives T (v, v, k) = c k 1 T (v, v, 1) = c k 1 Nf(v)dv/(1 F (v)). Again, for c < 1, dropouts happen fast early in the game (when k is large), then slow down as k falls. (The paper gives some corollaries about this.) One interesting case is the limit as c 0. The first k 1 dropouts occur immediately; all that takes time is the reduction from N + 1 to N. To see why, suppose not: that is, suppose that c 0 but early dropouts take time. I m type v, there are k too many players left, and we reach time T (v ɛ). By sticking around till my equilibrium dropout point, my chance of winning is of order ɛ k 1, since it depends on k 1 other players having types between v ɛ and v. But by dropping out early, I save ɛt in costs. By making ɛ small relative to T, I always want to drop out earlier, and the equilibrium unravels. They also spend some time on the c = 1 case, which they describe as a standards battle. (We all want to sell HD-DVD players; whichever firm s technology is chosen as the industry standard also makes more money from intellectual property licensing; but all firms incur costs (delayed sales) until a standard is chosen, even the firms that have given up on winning the standards battle.) They point out that the c = 0 and c = 1 games take the same length of time to reduce from N + 1 to N, but the c = 1 game takes far longer to get from N + K to N + 1. (Another example they give for c = 1: the chair of the econ department calls a faculty meeting, and asks for five volunteers to sit on a committee. The meeting will not end nobody can 3

4 leave until five have volunteered. So the costs of the ongoing meeting are borne by all, even those who have conceded the prize of avoiding committee work. The c = 0 case would correspond to a world where faculty who agreed to serve could leave immediately.) (So they point out that once four people have already volunteered, the two games are the same, and take the same amount of time to resolve; but that starting at the beginning, the game where people who have already volunteered get to leave immediately goes by much quicker: four people volunteer almost immediately and leave. In the game where everyone stays until all five volunteer, the early rounds take time.) (They also mention a couple of examples of natural monopoly -type settings which very quickly devolved into two-horse races. The examples they give are a little dated, but we can think of PC and Mac, the two competing high-def DVD standards, and so on.) 4

5 Siegel, All-Pay Contests Another paper I like on a similar topic is by a friend of mine from Stanford, Ron Siegel (now at Northwestern): All-Pay Contests The basic setting is a sealed-bid all-pay auction with full information, that is, everyone knows everyone else s type This means, of course, there is no pure strategy equilibrium: if there was, the losers would all be bidding 0, which means the winners would be bidding very close to 0, which means the losers would deviate to higher bids and win So any equilibrium (for nontrivial cases) is in mixed strategies Besides differences in how they value the prize, though, Ron allows for another type of asymmetry across bidders Consider a lobbying example. The legislature is going to pass one of three alternative bills. There is one lobbyist favoring each bill. The bill chosen will be based on which lobbyist has the most influence. There is one lobbyist who is well-connected by lazy. It costs him very little to achieve a certain (moderate) level of influence; but he has a high marginal cost beyond that point. (DRAW IT.) The other two lobbyists are energetic but not as well-connected. Their marginal costs are everywhere positive, but less so at the extreme. This is also a good model for settings like procurement auctions, where bids can vary in many dimensions. (A bid for a contract can specify both the price that will be charged, and lots of specifics that affect the value of fulfillment to the government.) This model basically compresses all these dimensions down to a single one how attractive the bid is and compresses all firm heterogeneity down to a cost function how costly it is to fulfill a contract of a given attractiveness. (If the auctioneer s preferences are known, this is basically WLOG.) Ron terms the attractiveness dimension score, and claims the model allows for different production technologies, costs of capital, prior investments, conditional and unconditional investments, and attitudes towards risk, among others. Once we allow heterogeneous cost functions, we no longer need variability in how the players value the prize: since games are unaffected by, say, linear scaling of a player s payoff function, we can normalize every player s value of the prize to 1. That is, rather than saying a player s payoff is v i Pr(win) c i (b i ), we can rescale this to Pr(win) c i(b i ) v i. 5

6 (In fact, Ron allows for more generality than additive preferences v i c i (s i ) that is, he allows the cost function to vary depending on whether or not you win, as in cases where some of the investments are sunk and some are not. For today, we ll stick to the simpler case.) The main result in Ron s paper is this: all equilibria of this game are payoff-equivalent, and the expected payoffs of each player admit a simple, easily-interpreted closed form. Since it s a job market paper, he then goes much further in exploring: he shows there are special cases where equilibrium is unique and can be explicitly calculated; he calculates total expenditures for special cases; he examines the comparative statics of adding players, adding prizes, and increasing the value of prizes. But to me, the cool part of the paper is the payoff result. Model and Notation We ll use a special case of Ron s model. There are n players, and m < n prizes. Each prize is worth 1 to all players, and each bidder i has a cost function c i specifying a cost c i (x) of submitting a bid with score x. The m bidders whose bids have the highest scores win; all bidders pay the cost of their bids. Assume that c i (0) = 0 (just a normalization), and that c i is continuous and weakly increasing. We ll require it to be strictly increasing in a particular region, but it can be flat in some places. (For instance, a lobbyist with an established reputation can be thought of as having zero cost to get up to a score of a i > 0.) Define player i s reach as the highest score at which he can win with non-negative profits, that is, r i solves c i (r i ) = 1. Reindex the players by decreasing reach, so r 1 r 2... r n. The player with the m + 1 st -highest reach is the marginal player. The reach of the marginal player is the threshold T = r m+1. Player i s power is his profit from winning at the threshold: w i = 1 c i (T ). By definition, the marginal player has power 0; players 1, 2,..., m have nonnegative power, and m+2,..., n have nonpositive power. The regularity conditions he imposes are that there is a unique player with power 0, and that his cost function c m+1 is strictly increasing at T 6

7 Results Theorem 1. In any equilibrium of a contest meeting these conditions, the expected payoff of each players is the max of his power and 0. Ron defines players 1 through m as winners, and m + 1 through n as losers, based on having positive versus 0 expected payoffs. Losers do sometimes win a prize, they just have expected profits of 0. To prove the theorem, first note that no player will ever bid higher than his reach: even if he won, he d get negative payoff, so this is strictly dominated by bidding 0. So the n m losers, who all have reaches less than or equal to T, will all never bid more than T. So players 1 through m can guarantee payoffs arbitrarily close to their power by bidding T +ɛ for small enough ɛ, since this is guaranteed to win. So player i m can guarantee a payoff of 1 c i (T + ɛ) w i. So for players 1 through m, expected payoffs are bounded below by w i. To show that losers get 0 expected payoffs, Ron proves a lemma that at least n m players must sometimes play strategies which win with probability close to zero. Since players playing mixed strategies are indifferent among the strategies they play, this means these players have nonpositive expected payoffs. Since they can get 0 profit by bidding 0, their expected payoffs in equilibrium must be 0. To prove the lemma, suppose m+1 bidders never played strategies that always lost. Take the union of the supports of their equilibrium strategies; someone must be playing an equilibrium arbitrarily close to the infimum of the set, which means it always loses to the other m bidders. (He uses a separate lemma to rule out ties screwing up the result.) So we know losers, including the marginal player, get 0 expected profits. Finally, we need to show that winners can t do better than their power in expectation. He does this by showing that winners must all bid arbitrarily close to the threshold, T, with positive probability. Bidders m + 2,..., n never bid higher than their reach, which is strictly lower than T. If one of the winners never bid close to T, then there would be some ɛ where by bidding T ɛ, the marginal player could always win, giving him strictly positive payoffs. But we know he gets 0. So every winner must sometimes bid right near the threshold, which gives them payoff less than or equal to 1 c i (T ) = w i. Since players are indifferent among the strategies they mix among, and since we know winners get at least w i, they must get exactly w i. (Again, don t be confused by the terms winners and losers. We know that the marginal player bids close to T some of the time; to compensate for his costs, he must win an item 7

8 some of the time. Winners are so named because they have positive expected payoffs, not because they are guaranteed to get an object.) Mapping Siegal to a Second-Price Auction One of the things I like about the model is how nicely it maps to a regular (non-all-pay) auction Consider the special case of a private-value auction for one item: valuations for winning, and all have the same c i (b) = b players have different (We can rescale the preferences to be 1 1 v i c i (b) = 1 1 v i b to fit with his setting, but for now, stick to the unscaled version.) Each player s reach, then, is v i the amount he could pay and break even Obviously, with one prize, the marginal player is player 2, so the threshold is v 2 This means that in Ron s all-pay setting, the expected payoff to the winner is his power, which is v 1 v 2, and the expected payoffs to everyone else are 0 These, of course, are the exact payoffs they d get in a second-price (non-all-pay) auction But here, we re getting them in the mixed-strategy equilibrium of an all-pay auction with full information However, there is a difference: to the seller. Since the all-pay auction gives the prize to the wrong guy with positive probability, it is inefficient; and since the bidders get the same payoffs as in a second-price auction, the seller must do worse. This last point is a tricky one. Ron characterizes expected payoffs to each player. But if the value of winning is different to each, the overall surplus generated by the game is much harder to pin down, and so the aggregate expenditure (seller s revenue if the money is going to the seller) is not always pinned down. As I said, Ron uses a more general formulation, where a player s cost function can be different depending on whether he wins or loses. He looks at participation (how many players actively participate), gives conditions under which only the winners and the marginal player participate, but also gives an algorithm to generate games with any number of participants. He looks at a bunch of comparative statics (the effects of adding a player, adding a prize, increasing the value of the prizes), and shows conditions under which equilibrium is unique and constructible. In the case of two players and one prize, if cost functions are strictly increasing everywhere, he constructs the unique equilibrium. Some of these additional results are very tricky to prove, but what I like about the paper is how clean and intuitive the payoff characterization is. (Practice job talk.) 8