INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION DAVID P. MYATT Department of Economics, Oxford University This version: October 27, 2003. Correspondence: St. Catherine’s College, Oxford OX1 3UJ, United Kingdom david.myatt@economics.ox.ac.uk Abstract. Two players engage in a costly fight. A conceding player yields a prize of privately-known value to her opponent. As is well known, this classic game exhibits multiple equilibria. Perturbing the war of attrition—for instance, by imposing a time limit or allowing for the small probability of players who are restricted to fight forever—yields a unique equilibrium. In this paper, I examine the properties of this unique equilibrium in the limit, as such perturbations are allowed to vanish, hence selecting an equilibrium from the classic war of attrition. I employ a measure of stochastic strength to order the distributions from which players’ prize valuations are drawn. This measure is based on the relative-hazard rates of the distributions in their upper tails. In the selected equilibrium, the stochastically weaker player always exits at the beginning of the game, even though her realized prize valuation may well be greater. Thus a war of attrition is resolved immediately, in favor of a player who is merely perceived to be stronger ex ante , rather than actually stronger ex post 1. Wars of Attrition In a classic two-player war of attrition, the combatants compete to win an indivisible prize. At a player’s disposal is a single weapon: Her time of exit. The first player to quit concedes the prize to her opponent. Fighting is costly, and hence is worthwhile only if a player expects her opponent to quit in the near future. These features ensure that the war of attrition serves as a stylized representation of many important economic phenomena, including labor-market negotiations, the voluntary provision of public goods, macroeconomic stabilization, the adop- tion of technological standards, and political lobbying. 1 An analysis of these scenarios must ask two simple questions: Who will win? When will the war of attrition end? Early answers to these questions suffered from the presence of multiple equilibria. 2 A number of authors, most notably Fudenberg and Tirole (1986), Kornhauser, Rubinstein, and Wilson (Acknowledgements are omitted from this review copy. This paper is based on Myatt (1999).) 1 I give a fuller account of such applications, as well as the related literature, in Sections 2 and 7. 2 Consider the simplest complete-information war of attrition: The costs incurred by the players are directly proportional to the length of the war, and their valuations for the prize are commonly known. In his study of this game, Maynard Smith (1974) examined a symmetric mixed-strategy equilibrium, in which each player exits with a constant hazard rate. There are, however, many other equilibria. For instance, it is an equilibrium for one player to fight forever, while her opponent quits at the beginning. 2 DAVID P. MYATT (1989), Amann and Leininger (1996), and Riley (1999), successfully addressed this issue by “perturbing” the game in different ways. In the first two of these papers, the players are restricted to fight forever with positive probability. 3 In the other two papers, the winner’s costs respond positively to her own planned exit time as well as the exit time of the loser. These perturbations (and indeed others) yield a game with a unique equilibrium, and hence an opportunity to “select” an equilibrium from the classic war of attrition by allowing the perturbation to vanish. So, for this selected equilibrium, who wins and when? Kornauser et al (1989) and Riley (1999) provided answers when the players’ prize valuations are commonly known: In an asymmetric game, the player with the lowest prize valuation (the “weaker” player) concedes immediately 4 This is efficient, since the player with the highest prize valuation (the “stronger” player) receives the prize at zero cost. 5 In an incomplete-information world, however, a player’s prize valuation is the privately observed realization of a random variable. A player is stronger ex post if her realized prize valuation is highest. She may be weaker ex ante , however, if her valuation is expected to be lower. An open question, therefore, is whether the outcome of a war of attrition is determined by a player’s real strength (her ex post prize valuation) or her perceived strength (derived from an ex ante ranking of prize-valuation distributions). 6 In this paper, I offer an answer. I argue that a player who is merely perceived to be weaker ex ante will exit immediately The implications of this claim are immediate. First, players’ true prize-valuations (their real strengths) play no role in determining the outcome of a war of attrition. Second, the allocation of the prize may, therefore, be dramatically inefficient. Third, since the game ends immediately, the classic war of attrition cannot, by itself, explain the existence of delay in concessionary environments. Fourth, any perceived asymmetries ex ante may be critical to a player’s likely success. Fifth, players will have an incentive, therefore, to engage in activities that enhance their perceived valuation of a prize, rather than the value itself. The “instant exit” claim requires a formal definition of perceived strength, and an appro- priate equilibrium-selection mechanism. Ex ante , I rank the distributions from which prize 3 More accurately, Fudenberg and Tirole (1986) assume that a player’s cost of fighting is sometimes negative, while Kornhauser et al (1989) assume that a player is sometimes “irrational.” See Sections 2 and 3. 4 If the players share the same valuation for the prize, then the symmetric equilibrium is selected. The players are equally likely to win the prize, and the length of the war of attrition is random. 5 Notice that the winning player is stronger ex post , in that her valuation is the highest. She is also stronger ex ante , in that it is commonly known that this is the case at the start of the game. 6 Establishing uniqueness is not a contribution of this paper—Fudenberg and Tirole (1986), Ponsati and S ́ akovics (1995), and Amann and Leininger (1996) tie down unique equilibria in incomplete-information wars of attrition. With ex ante symmetry, so that valuations are independently drawn from the same distribution, the unique equilibrium is symmetric. The player with the highest valuation ex post will fight for longest, and hence win the war. These authors did not, however, fully explore the properties of equilibria in asymmetric wars of attrition. Hence they were unable to assess the relative importance of ex ante and ex post strength. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 3 valuations are drawn in three different ways. First, a player stochastically dominates her op- ponent if her valuation is more likely to exceed any benchmark level. Second, she hazard-rate dominates if this likelihood ratio is increasing in the benchmark. Third, she is stochastically stronger if the likelihood ratio increases without bound as the benchmark grows large. 7 To obtain a unique equilibrium, I use three mechanisms to “perturb” the classic war of attri- tion. First, each player suffers exit failure and is forced to fight forever with some exogenous probability. Second, I employ a hybrid all-pay auction in which a winner’s costs respond positively to her planned stopping time. Third, I impose a finite time limit , after which the prize is awarded at random. For each mechanism, allowing the perturbation to vanish (for instance, when exit failure occurs very rarely), a unique equilibrium to the classic war of attrition is selected. Although I offer results for each of the ex ante strength measures described above, the most dramatic is this: A stochastically weaker player exits immediately. Of course, the results are subject to qualifications. Most importantly, the “instant exit” claim may be sensitive to the exact size of the perturbation used to select a unique equilibrium. To assess such sensitivity, I examine a simple example in which the prize valuations are drawn from uniform distributions. I find that the perturbation must become very small indeed before dramatic “instant exit” is observed. A similar finding emerges from a second example in which prize valuations are exponentially distributed. Hence, if a non-negligible perturbation is present, then the ex ante weaker player’s exit is rapid, rather than instant. A review of the related literature provides further motivation in Section 2. I describe the classic war of attrition and the different selection mechanisms in Section 3. Analysis proceeds in two steps. In Section 4 I determine the basic properties of any equilibrium, and in Section 5 I present the substantive results of the paper, which are subject to the aforementioned sensitivity analysis in Section 6. I conclude in Section 7. All proofs are appendicized. 2. Related Literature 2.1. Concession Games and their Applications. Wars of attrition are important, sim- ply because they are common. Workers and employers may prolong a costly strike in order to obtain a preferred resolution (Kennan and Wilson 1989). Potential providers of a public good may delay their private contributions in an effort to free ride on others (Bliss and Nalebuff 1984, Bilodeau and Slivinski 1996). Oligopolists in a declining industry may incur losses in anticipation of profitability following the exit of a competitor (Fudenberg and Ti- role 1986, Ghemawat and Nalebuff 1985, 1990). Socioeconomic interest groups may delay macroeconomic stabilization in order to bias the burden of an agreement towards others 7 Thus stochastic dominance is the textbook first-order ranking, hazard-rate dominance corresponds to the conditional stochastic-dominance ranking of Maskin and Riley (2000), and stochastic strength is related to the unbounded likelihood-ratio property required for the implementation of Mirrlees (1999) contracts in moral-hazard problems. These conditions hold for a wide range of specifications. 4 DAVID P. MYATT (Alesina and Drazen 1991, Casella and Eichengreen 1996). The sponsor of a technological standard may continue its costly promotion in the hope that a competitor will abandon her own standard (Farrell and Saloner 1988, David and Monroe 1994, Farrell 1996). In these settings a participant is free to concede at any time. For instance, a firm may choose to adopt an opponent’s technology standard. Similarly, a striking union may choose to accept an employer’s current wage offer. Nevertheless, by “holding out” for a moment longer the player might be rewarded with a concession from her opponent. 8 When stopping times are labelled as bids, a war of attrition may be interpreted as an “all-pay” auction (Klemperer 1999): The loser, despite forgoing the prize, pays her bid. The winner, who may cease fighting following a concession, pays the loser’s bid. 9 All-pay models find a rich variety of applications. Political actors may expend irrecoverable lobbying costs in exchange for political influence (Hillman and Samet 1987, Hillman and Riley 1989). Individuals may queue to obtain a scarce resource (Holt and Sherman 1982). Models of races are also related, including those of innovative firms chasing a research and development goal (Harris and Vickers 1985, Fudenberg, Gilbert, Stiglitz, and Tirole 1983) or considering the adoption of a process innovation (Reinagum 1981a,b). 2.2. Rent Dissipation and Multiple Equilibria. The applications described above sug- gest that wars of attrition are important phenomena, justifying the careful studies that are present in the literature. Tracing a selection of theoretical contributions helps me to explain the role of this paper. In the first such study Maynard Smith (1974) distinguished between biological “tournaments” and “displays.” He described a tournament as a “fight” between competing animals, with victory enjoyed by the stronger individual. In contrast, he defined a display as a contest in which “no physical contact takes place” but where delay is costly and “the winner is the contestant who continues for longer, and the loser the one who first gives way.” For this second scenario, he identified a symmetric mixed-strategy equilibrium in which each player concedes with a constant hazard rate. This rate is chosen so that, at each point in time, a player is just indifferent between conceding the prize and fighting for a little longer. The additional costs of remaining in the war are exactly balanced by the possibility that an opponent yields. Hence, in equilibrium, a player is indifferent between fighting for any length of time and conceding at the start of the war. In expectation, the war of attrition yields no benefit to either player, and in economic parlance there is complete rent-dissipation (Posner 1975, Fudenberg and Tirole 1987). Maynard Smith (1974) described this quirky feature as an apparent “absurdity.” 8 In this way, a war of attrition is a bargaining game in which proposals are fixed, but agreement requires approval from both of the impatient participants, and hence the acquiescence of one (Osborne 1985, Ordover and Rubinstein 1986, Chatterjee and Samuelson 1987, Abreu and Gul 2000, Kambe 1999). 9 This is a “second-price” all-pay auction. Other formats include “first-price all-pay” auctions, where all participants pay their chosen bids (Baye, Kovenock and de Vries, 1993, 1996). In contrast to the second- price format, a player finds it costly to raise her bid even if her existing bid is already the highest. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 5 Mixed-strategy equilibria in asymmetric wars of attrition are even more quirky: A stronger (higher valuation) player exits more quickly (with a higher hazard rate) than her weaker (lower valuation) opponent. The reasoning is straightforward: The weaker opponent needs a stronger incentive to fight in order to make her indifferent between quitting and continuing. Kornhauser, Rubinstein, and Wilson (1989) found such equilibria to be troubling. They claimed that intuition “suggests that the weaker player [. . . ] should concede immediately.” Of course, such counter-intuitive equilibria arise at the discretion of the analyst, since the classic war of attrition exhibits multiple equilibria. In addition to the mixed equilibrium described above, it is also an equilibrium for one player to concede immediately in the face of a promise to fight forever from her opponent, so that the “intuitive” outcome described by Kornhauser et al (1989) is, in fact, consistent with equilibrium behavior. 10 Thus, any prediction of play must solve an equilibrium-selection problem. Furthermore, the introduction of incomplete information does not necessarily eliminate the problem. Authors such as Bishop, Cannings, and Maynard Smith (1978) studied games in which a player’s prize valuation is the privately observed realization of a random variable. As noted by Riley (1980) and others, however, such games also exhibit multiple equilibria. 11 2.3. Perturbed Wars of Attrition. Fortunately, a number of authors have addressed the equilibrium-selection problem. Ponsati and S ́ akovics (1995) offered the following succinct summary: “There are a continuum of equilibria characterized by a system of ordinary dif- ferential equations. Uniqueness may be achieved by perturbing the game, imposing that for a positive measure of types it is a dominant strategy not to concede.” For instance, Fudenberg and Tirole (1986) allowed for players with negative fighting costs. Such a player enjoys fighting and hence has a dominant strategy to stay in the war forever. A similar approach was taken by Kornhauser, Rubinstein, and Wilson (1989). They began with a complete-information war of attrition and, following Kreps and Wilson (1982a) and Mil- grom and Roberts (1982), added the possibility that a player is “irrationally” committed to playing a fixed strategy. Both procedures pin down a unique equilibrium. Uniqueness may 10 Furthermore, it is also an equilibrium for either player to concede with arbitrary positive probability at the beginning of the game, prior to continuous attrition in the usual fashion. Hence, in both symmetric and asymmetric settings, there are infinitely many equilibria. 11 These equilibria may involve the rapid exit of either player. Alternatively, when players are symmetric ex ante (so that valuations are drawn from the same distribution) there is a symmetric equilibrium in which both players use the same stopping rule. In such a symmetric equilibrium, a player’s concession time is increasing in her prize valuation, so that in such a symmetric equilibrium the player with the highest prize valuation (the ex post stronger player) wins—an efficient allocation. Many theorists have studied games in which players are symmetric ex ante , and hence have focused upon symmetric equilibria (Bulow and Klemperer 1999, Krishna and Morgan 1997, for example). Whereas symmetric equilibria might be described as “focal,” most applications will involve some (perhaps small) asymmetry between the contestants. With such asymmetries in place, there is no symmetric equilibrium on which to focus. It is unclear (at least to me) why a selected equilibrium in an asymmetric game should, necessarily, be particularly symmetric. 6 DAVID P. MYATT be achieved by other means. Amann and Leininger (1996) and Riley (1999) studied two- player hybrid all-pay auctions in which the loser pays her own bid, whereas the winner pays a convex combination of the two bids. So long as the winner’s payment positively responds to her own bid, the equilibrium is unique. These different perturbations are subject to a common interpretation: They force threats of great aggression to be credible . To explore this further, consider an unperturbed game. Take a player who, in equilibrium, is prepared to choose an extremely large exit time. Such an aggressor will, almost always, win the war of attrition. The end of the war will then be determined by her opponent’s exit time. But this means that an increase in the aggressor’s own exit time will have a negligible effect on her expected costs. Of course, this argument fails when, following Fudenberg and Tirole (1986), the aggressor’s opponent fights forever with positive probability: Any expansion in exit time must be paid for with this, non-negligible, probability. When, following Amann and Leininger (1996), the winner of an all-pay auction pays a convex combination of her own and her opponent’s bids, she will always find it costly to increase her bid. Thus, perturbation devices employed by these authors also ensure that any threats made by a player are always costly, and hence must be credible. 2.4. Equilibrium Selection and Instant Exit. For each of the perturbations described above, the unique equilibrium may be examined in the limit as the perturbation is allowed to vanish. This procedure “selects” an equilibrium of the unperturbed game. A number of authors have followed this procedure for the complete-information war of attrition. For in- stance, Kornhauser et al (1989) allowed the probability of irrationality to vanish to zero. For his hybrid all-pay auction model, Riley (1999) considered the limiting case, where a player’s cost depend only on upon the second-highest bid. In their studies of concessionary bargain- ing, Abreu and Gul (2000) and Kambe (1999) assumed that players are “stubborn” with some small probability, and considered equilibria as the probability of stubborness vanishes to zero. These procedures all select an equilibrium with the following characteristic: The player with the lowest prize-valuation (the “weaker” player) concedes immediately Thus, fighting never takes place, and the stronger player wins the prize at zero cost. All of these authors restricted to a classic war of attrition with complete information: The prize valuations of the players are commonly known. This means that, at the start of the game, it is common knowledge that one player is stronger than her opponent. The move to an incomplete-information war of attrition, in which a player’s prize valuation is private information, is important since it permits a separation of ex ante and ex post strength. Fudenberg and Tirole (1986), Ponsati and S ́ akovics (1995), and Amann and Leininger (1996), all offered characterizations of the unique equilibrium to a perturbed war of attrition. They did not, however, select an equilibrium from the classic war of attrition by taking the limit as the relevant perturbation is allowed to vanish. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 7 This paper, therefore, offers the following contributions. First, I obtain an “instant exit” result even when players’ prize valuations are private information. In doing so, I obtain a measure of ex ante strength (namely, the “stochastic strength” of Definition 3) that deter- mines the winner of a war of attrition. In this sense, the contribution of the paper is to complete a branch of literature that analyzes equilibrium behavior in the war of attrition. Second, the results offer a contribution that is of wider interest to economists. The results imply that the outcome of a war of attrition is due to the ex ante perceptions of players, rather than their ex post valuations. Moreover, the effect of ex ante strength is so strong that the war of attrition ends immediately. This means that the classic war of attrition model cannot explain the delays that are observed in economic applications of the game. 3. Modelling the War of Attrition 3.1. The Classic War of Attrition. In a classic war of attrition two players i ∈ { 1 , 2 } each choose a stopping time t i ∈ R + ∪ {∞} , which may be revised at any time t ≤ t i Player i is characterized by a fighting cost of c i > 0 and a valuation u i for a prize, where u i ∈ ( u i , u i ) ⊆ R and u i > u i . Only the ratio u i /c i will be of interest to Player i , and hence it is without loss of generality to adopt the normalization c i = 1. Realized payoffs are π i ( t i , t j ) = u i [ I { t i > t j } + I { t i = t j } 2 ] − min { t i , t j } , where I is the indicator function. 12 Following Maynard Smith (1974), the costs incurred by a player are directly proportional to the time elapsed. Other formulations are possible, in which “leader” and “follower” payoffs are general functions of time (Bishop and Cannings 1978, Hendricks, Weiss, and Wilson 1988). 13 They lead to similar insights. The “linear costs” approach is convenient in that the war of attrition may be interpreted as an ascending-price all-pay auction: The price t rises until a player concedes, and both players pay the exit price. Following Bishop, Cannings, and Maynard Smith (1978) and Riley (1979, 1980) information is incomplete: Players observe only their own valuations. It is commonly known that u i is 12 The prize is awarded at random if both players exit simultaneously ( t i = t j ). Other tie-break rules may be employed while retaining most of the results, so long as no player wins a tie with probability one. 13 A more general “leader-follower” game might be formulated in which the exiting player receives a payoff L i ( t ) and the follower F i ( t ). Insisting that L ′ i ( t ) < 0 and F ′ i ( t ) < 0 ensures that a player would, other things equal, rather quit sooner than later. When F i ( t ) > L i ( t ), however, a player is willing to wait for the anticipated exit of her opponent. A special case of this is when the “fighting costs” correspond to the delay before the award of a second prize following the exit of the losing player. For A > B > 0 this might be implemented via L i ( t ) = Be − δ i t and F i ( t ) = Ae − δ i t , where players differ in their patience δ i . Alternatively, as in Ponsati and S ́ akovics (1995), it may be implemented via L i ( t ) = B i e − t and F i ( t ) = e − t where players differ in their reservation payoff B i 8 DAVID P. MYATT drawn from the distribution F i ( u ) with strictly-positive continuous density f i ( u ). Further- more, when u i < ∞ I insist that lim u → u i f i ( u ) ≡ f i ( u i ) exists and that f i ( u i ) ∈ (0 , ∞ ). 14 Under an auction interpretation, the players have independent private values. The classic war of attrition exhibits multiple equilibria. For instance, when min { u 1 , u 2 } > 0 it is an equilibrium for one player to be infinitely aggressive and fight forever ( t i = ∞ ) and her opponent to quit immediately ( t j = 0). 15 This extreme equilibrium is, however, particularly sensitive to the exact specification, in that Player i ’s own payoff does not respond to her exact choice of t i , and hence she is happy to wait forever: The threat of limitless aggression is costless. Removing this possibility ties down a unique equilibrium. In the context of this paper, I do this by either ensuring that aggression is always costly (Sections 3.2–3.3) or by imposing an upper limit to the length of the war of attrition (Section 3.4). 3.2. Exit Failure. In the classic war of attrition a player successfully exits at her chosen time. In a first change to the basic specification I assume that each player fails to exit with commonly known probability ξ > 0. Exit failures are independent events for the two players and are independent of the players’ prize valuations. 16 If exit failure occurs, then a player is forced to fight forever. This possibility (of forced infinite aggression) means that a player never intentionally fights forever. If Player i were to choose t i = ∞ , then with probability ξ > 0 she would face infinite costs that exceed her prize valuation u i . Thus the possibility of exit failure will automatically eliminate the “extreme” equilibria described in Section 3.1. Other interpretations of this specification, other than exit failure, are possible. For instance, with probability ξ a player might be “crazy” and insist on fighting forever. This corresponds to the approach taken by Kornhauser, Rubinstein, and Wilson (1989), who, following Kreps and Wilson (1982a), Milgrom and Roberts (1982) and Kreps, Milgrom, Roberts, and Wilson (1982), introduced “irrationality” into a complete-information model. 17 3.3. All-Pay Auctions. As noted in Section 1, the war of attrition may be interpreted as an ascending-price all-pay auction. This is similar to a second-price sealed-bid all-pay 14 This technical restriction is required since the support ( u i , u i ) is an open set. If the support were compact, so that u i ∈ [ u i , u i ], then this requirement could be dropped. 15 Behavior “off the equilibrium path” must also be specified: Player i always stays in forever and, off the equilibrium path, Player j quits whenever she can. Players retain their prior beliefs over opposing valuations. 16 The model and analysis may be extended to incorporate asymmetric exit failure probabilities ξ 1 and ξ 2 . I impose the assumption ξ 1 = ξ 2 in order to simplify the statement of the results. 17 A player does not necessarily need to be insane or irrational in order to fight forever. It might be that a player has good reason to do so. This latter approach was taken by Fudenberg and Tirole (1986) and Ponsati and S ́ akovics (1995), among others. In their models, players’ valuations are fixed. Instead, the fighting cost of a player is unknown to her opponent. Crucially, a player’s fighting cost is sometimes negative. Fudenberg and Tirole’s (1986) justification for this assumption was based upon their model’s application. They considered exit from a declining industry, where a firm’s characteristics might enable it to remain profitable forever, even when facing a competitor. These interpretations of exit failure lead to an equilibrium in which, as in Abreu and Gul (2000) and Kambe (1999), a player’s “reputation” is important. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 9 auction, where the stopping times t 1 and t 2 are interpreted as sealed bids. 18 The highest bidder wins the prize, and both bidders pay the lowest (i.e. second-price) bid. The class of all-pay auctions extends beyond the war of attrition. In a first-price all-pay auction, each player pays their bid, with the prize once again awarded to the highest bidder: π i ( t i , t j ) = u i [ I { t i > t j } + I { t i = t j } 2 ] − t i The first- and second-price all-pay auctions may be combined to yield a “hybrid” auction format. This mechanism, employed by G ̈ uth and van Damme (1986), Amann and Leininger (1996) and Riley (1999), works as follows. The prize is awarded to the highest bidder. The loser pays her bid. The winner pays a convex combination of her own bid and that of the loser. Specifically, for some parameter β ∈ [0 , 1], when t 1 > t 2 , Player 1 wins the object and pays a price βt 1 + (1 − β ) t 2 . Clearly, the first-price all-pay auction is obtained when β = 1, and the war of attrition is obtained when β = 0. Formally π i ( t i , t j ) = u i [ I { t i > t j } + I { t i = t j } 2 ] − [min { t i , t j } + max { β ( t i − t j ) , 0 } ] 19 Setting β > 0 ensures that a finite-valuation player will never choose t i = ∞ 3.4. Time Limits. The modifications presented in Sections 3.2–3.3 ensure that a player’s payoff always responds to a change in her chosen exit time. Not only is fighting always costly but the marginal cost of fighting a little longer is bounded away from zero. In the exit-failure case it exceeds ξ > 0, and in the hybrid-auction case it exceeds β > 0. Thus, so long as max { ξ, β } > 0, a player always faces a strict incentive to tame her aggression. The imposition of a time limit directly tames the aggression of players and, following Can- nings and Whittaker (1995), yields a finite-horizon war of attrition. 20 To implement this idea, suppose that players may only fight up to time T 21 If they both fight until this time, 18 In the sealed-bid interpretation players choose their stopping times simultaneously at the beginning of the game, rather than revising them as the game progresses. Assuming that a player pays the second highest bid is equivalent to assuming that, following the exit of her opponent, she revises her bid and stops immediately. 19 This hybrid auction may continue to be supported by a war of attrition interpretation. Hendricks, Weiss, and Wilson (1988) and Pitchik (1982) refer to wars of attrition as “noisy” games of timing. The idea is that each player can “hear” the exit of her opponent, and hence respond by following her exit. In contrast, in a “silent” game of timing an player does not observe her opponent’s exit—yielding an all-pay auction. The hybrid model described here can be interpreted as one in which a player fails to observe her opponent’s exit with probability β , and in this case must continue to her original exit time. Alternatively, it may take some time for a player to “brake” following the exit of her opponent, and hence the parameter β might be interpreted as a measure of “braking distance.” 20 Bilodeau and Slivinski (1996) follow Bliss and Nalebuff (1984) in modelling the voluntary provision of a public good as a war of attrition. They impose a time limit to the game, and obtain a unique equilibrium. Their mechanism is subtly different to the one considered here. A player’s (commonly known) valuation for the prize is the present-discounted value of the prize’s flow utility until the time limit. Hence, as the time limit is reached, lower valuation players have a dominant strategy to fight until the end. 21 A player suffering exit failure (Section 3.2) is assumed to exit at time T 10 DAVID P. MYATT then the prize is allocated randomly. So long as T is not too large, a player will fight until the time limit with positive probability: By doing so, Player i guarantees a payoff of at least ( u i / 2) − T To ensure that T is neither too large nor too small to have an effect, I make two simplifying assumptions. First, whenever T < ∞ , so that an effective limit is in place, I assume that u 1 = u 2 = ∞ . This ensures that there is always positive probability that a player has a dominant strategy to fight until the time limit, and that this remains true as T → ∞ 22 , 23 Second, I assume that a player does not always have a dominant strategy to fight until the limit, so that T > max { u 1 , u 2 } / 2. 3.5. Equilibrium Selection and Stochastic Strength. Exit failures, hybrid all-pay auc- tions and time limits all represent perturbations to the classic war of attrition. Allowing ξ → 0, β → 0, or T → ∞ eliminates the perturbation. Away from the limit, however, the incomplete information “perturbed” war of attrition exhibits a unique equilibrium (Sec- tion 5). This is not a new finding (Fudenberg and Tirole 1986, Amann and Leininger 1996). Instead, the contribution of this paper stems from an examination of this unique equilibrium in the limit as the perturbation vanishes (following Kohlberg and Mertens (1986) and others) and hence the characteristics of a “selected” equilibrium of the classic war of attrition. 24 When the players share common expectations over their valuations, so that F 1 ( · ) ≡ F 2 ( · ), the game is symmetric, and so a unique equilibrium must also be symmetric. This applies in the limit and hence the perturbation procedures described above will select a symmet- ric equilibrium, and the player with the highest valuation will win the war. 25 Symmetry, however, is a particularly strong assumption. The ex post realized valuations of the two players—their real strengths —will almost always be different. Importantly, however, ex ante expectations of players’ valuations may also be different—these valuations may be drawn from different distributions, so that F 1 ( · ) 6 = F 2 ( · ). To aid analysis of the asymmetric case I must order the two players ex ante and hence formalize the idea of perceived strength . I consider three different partial orderings, beginning with the following. Definition 1. Player 1 stochastically dominates Player 2 , denoted F 1 FSD F 2 , if F 1 first- order stochastically dominates F 2 . Formally F 1 ( u ) < F 2 ( u ) for all u ∈ ( u 1 , u 1 ) ∩ ( u 2 , u 2 ) 22 Player i has a dominant strategy to set t i = T whenever u i / 2 > T 23 Hence, whenever I impose a time limit T < ∞ I am implicitly restriction attention to valuation distributions with unbounded support. 24 The perturbations considered here are not exhaustive. Other approaches might involve the addition of noise to the actions taken by players. Anderson, Goeree and Holt (1998a, 1998b), for instance, studied boundedly-rational versions of the all pay auction and war of attrition, respectively. In their models, players’ decisions are determined by a logic probabilistic choice rule. They identified logit equilibria, in the sense of McKelvey and Palfrey (1995, 1996), which exhibit “sensible” comparative statics. 25 Furthermore, the revenue-equivalence theorem applies (Vickrey 1961, Myerson 1981, Riley and Samuelson 1981), and hence, in its auction interpretation, the war of attrition will raise the same expected revenue as other standard auction formats. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 11 When this familiar dominance condition holds, for any increasing function h ( u ), the expec- tation E[ h ( u )] is higher under F 1 than under F 2 A trivial corollary is that E[ u 1 ] > E[ u 2 ] so that Player 1 has a higher expected valuation ex ante : Absent any other information or feasible allocation mechanism a social planner might choose to give the prize to Player 1. More stringent criteria are also used by a number of authors to characterize a “stronger” player. Angeles de Frutos (2000), for instance, analyzes auction procedures for allocating the assets of a dissolving partnership. She uses a ranking of the hazard rates of competing distributions to order competing bidders. This is hazard-rate dominance Definition 2. Player 1 hazard-rate dominates Player 2 , denoted F 1 HRD F 2 , if u 1 ≥ u 2 , u 1 ≥ u 2 , and for all u satisfying u ≥ 0 and u 2 > u > u 1 , the hazard rates satisfy f 1 ( u ) 1 − F 1 ( u ) < f 2 ( u ) 1 − F 2 ( u ) Hazard-rate dominance implies first-order stochastic dominance. An equivalent condition is ∂ ∂u { 1 − F 1 ( u ) 1 − F 2 ( u ) } > 0 (1) To interpret this inequality, note that first-order stochastic dominance (Definition 1) says that 1 − F 1 ( u ) > 1 − F 2 ( u ), so that the event u 1 > u is more likely than u 2 > u . Hazard-rate dominance (Definition 2) goes further, saying that the odds of u 1 > u versus u 2 > u are strictly increasing in u 26 Furthermore, this means that, conditional on u 1 > ̃ u and u 2 > ̃ u , the conditional distribution of u 1 continues to stochastically dominate that of u 2 27 For this reason Maskin and Riley (2000), in their study of asymmetric auctions, refer to such a ranking as conditional stochastic dominance . They compare first- and second-price sealed- bid auctions in an independent-private-value setting. Departing from many of the classic auction studies, they allow two bidders to be asymmetric. A “strong” bidder in their model conditionally stochastically dominates her opponent. For the purposes of some results presented here, however, I will not need the hazard rates of the distributions to be ranked everywhere. Instead, I need the hazard rates to be ranked for high valuations. I call this measure stochastic strength 26 Hence hazard-rate dominance is related to Milgrom’s (1981) notions of good news and bad news. For instance, suppose that an observer wishes to assess the probability that a particular individual is in fact Player i , and observes the valuation u . An increase in u increases the odds that Player i is the individual in question, and hence is “good news” for the hypothesis. For more on the relationship between hazard rates and monotone-likelihood-ratio dominance, see Shaked and Shanthikumar (1994). 27 To see this formally, observe that: Pr[ u 1 > u | u 1 > ̃ u ] Pr[ u 2 > u | u 2 > ̃ u ] = [1 − F 1 ( u )] / [1 − F 1 ( ̃ u )] [1 − F 2 ( u )] / [1 − F 2 ( ̃ u )] > 1 ⇔ 1 − F 1 ( u ) 1 − F 2 ( u ) > 1 − F 1 ( ̃ u ) 1 − F 2 ( ̃ u ) 12 DAVID P. MYATT Definition 3. Player 1 is stochastically stronger than Player 2 if u 1 > u 2 (the upper bound to the support of u 1 is greater than that of u 2 ), or u 1 = u 2 = ∞ (unbounded support) and lim inf u →∞ [ f 2 ( u ) 1 − F 2 ( u ) − f 1 ( u ) 1 − F 1 ( u ) ] > 0 (2) This is sometimes denoted F 1 AHRD F 2 , indicating asymptotic hazard-rate dominance. Notice that Equation 2 is automatically true when u 1 > u 2 , but that Definition 3 cannot be applied when u 1 = u 2 < ∞ 28 Furthermore, it does not apply when the hazard rates of the two distributions converge in the upper tails, or when the ranking of the hazard rates switches repeatedly. A stochastically stronger player is far more likely (in a relative sense) to experience very high valuations. In fact, when Player 1 is stochastically stronger than Player 2 (Definition 3) lim u → u 2 1 − F 1 ( u ) 1 − F 2 ( u ) = ∞ (3) This helps to clarify the relationship between the different competing notions of ex ante strength considered here. If F 1 FSD F 2 , then u 1 > u is more likely than u 2 > u ; if F 1 HRD F 2 , then the relatively likelihood of u 1 > u versus u 2 > u is strictly increasing in u ; and finally, if F 1 AHRD F 2 then this likelihood ratio is unbounded for large u . Of course, for F 1 AHRD F 2 , there is no restriction placed on the hazard rates of the distributions for smaller u As shown by example below, the distributions may be ranked by second-order stochastic dominance (Rothschild and Stiglitz 1970), so that F 1 is “riskier” than F 2 Although only a partial ordering, stochastic strength may be used as criterion for a wide range of distributional specifications. A simple example is the use of uniform distributions. Example 1 (Uniform dist