Instant Exit from the Asymmetric War of Attrition - David Peter Myatt

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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 Sákovics (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 Sákovics (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 Sákovics (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 ti ∈ R+ ∪ {∞}, which may be revised at any time t ≤ ti . Player i is characterized by a fighting cost of ci > 0 and a valuation ui for a prize, where ui ∈ (ui , ui ) ⊆ R and ui > ui . Only the ratio ui /ci will be of interest to Player i, and hence it is without loss of generality to adopt the normalization ci = 1. Realized payoffs are   I{ti = tj } πi (ti , tj ) = ui I{ti > tj } + − min{ti , tj }, 2 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 ui is 12The prize is awarded at random if both players exit simultaneously (ti = tj ). 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 Li (t) and the follower Fi (t). Insisting that L0i (t) < 0 and Fi0 (t) < 0 ensures that a player would, other things equal, rather quit sooner than later. When Fi (t) > Li (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 Li (t) = Be−δi t and Fi (t) = Ae−δi t , where players differ in their patience δi . Alternatively, as in Ponsati and Sákovics (1995), it may be implemented via Li (t) = Bi e−t and Fi (t) = e−t where players differ in their reservation payoff Bi . 8 DAVID P. MYATT drawn from the distribution Fi (u) with strictly-positive continuous density fi (u). Further- more, when ui < ∞ I insist that limu→ui fi (u) ≡ fi (ui ) exists and that fi (ui ) ∈ (0, ∞).14 Under an auction interpretation, the players have independent private values. The classic war of attrition exhibits multiple equilibria. For instance, when min{u1 , u2 } > 0 it is an equilibrium for one player to be infinitely aggressive and fight forever (ti = ∞) and her opponent to quit immediately (tj = 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 ti , 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 ti = ∞, then with probability ξ > 0 she would face infinite costs that exceed her prize valuation ui . 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 14This technical restriction is required since the support (ui , ui ) is an open set. If the support were compact, so that ui ∈ [ui , ui ], then this requirement could be dropped. 15Behavior “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. 16The model and analysis may be extended to incorporate asymmetric exit failure probabilities ξ and ξ . I 1 2 impose the assumption ξ1 = ξ2 in order to simplify the statement of the results. 17A 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 Sákovics (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 t1 and t2 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:   I{ti = tj } πi (ti , tj ) = ui I{ti > tj } + − ti . 2 The first- and second-price all-pay auctions may be combined to yield a “hybrid” auction format. This mechanism, employed by Güth 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 t1 > t2 , Player 1 wins the object and pays a price βt1 + (1 − β)t2 . Clearly, the first-price all-pay auction is obtained when β = 1, and the war of attrition is obtained when β = 0. Formally   I{ti = tj } πi (ti , tj ) = ui I{ti > tj } + − [min{ti , tj } + max{β(ti − tj ), 0}] .19 2 Setting β > 0 ensures that a finite-valuation player will never choose ti = ∞. 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, 18In 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. 19This 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.” 20Bilodeau 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. 21A 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 (ui /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 u1 = u2 = ∞. 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{u1 , u2 }/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 F1 (·) ≡ F2 (·), 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 F1 (·) 6= F2 (·). 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 F1 FSD F2 , if F1 first- order stochastically dominates F2 . Formally F1 (u) < F2 (u) for all u ∈ (u1 , u1 ) ∩ (u2 , u2 ). 22Player i has a dominant strategy to set ti = T whenever ui /2 > T . 23Hence, whenever I impose a time limit T < ∞ I am implicitly restriction attention to valuation distributions with unbounded support. 24The 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. 25Furthermore, 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 F1 than under F2 . A trivial corollary is that E[u1 ] > E[u2 ] 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 F1 HRD F2 , if u1 ≥ u2 , u1 ≥ u2 , and for all u satisfying u ≥ 0 and u2 > u > u1 , the hazard rates satisfy f1 (u) f2 (u) < . 1 − F1 (u) 1 − F2 (u) Hazard-rate dominance implies first-order stochastic dominance. An equivalent condition is   ∂ 1 − F1 (u) > 0. (1) ∂u 1 − F2 (u) To interpret this inequality, note that first-order stochastic dominance (Definition 1) says that 1 − F1 (u) > 1 − F2 (u), so that the event u1 > u is more likely than u2 > u. Hazard-rate dominance (Definition 2) goes further, saying that the odds of u1 > u versus u2 > u are strictly increasing in u.26 Furthermore, this means that, conditional on u1 > ũ and u2 > ũ, the conditional distribution of u1 continues to stochastically dominate that of u2 .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. 26Hence 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). 27To see this formally, observe that: Pr[u1 > u | u1 > ũ] [1 − F1 (u)]/[1 − F1 (ũ)] 1 − F1 (u) 1 − F1 (ũ) = >1 ⇔ > . Pr[u2 > u | u2 > ũ] [1 − F2 (u)]/[1 − F2 (ũ)] 1 − F2 (u) 1 − F2 (ũ) 12 DAVID P. MYATT Definition 3. Player 1 is stochastically stronger than Player 2 if u1 > u2 (the upper bound to the support of u1 is greater than that of u2 ), or u1 = u2 = ∞ (unbounded support) and   f2 (u) f1 (u) lim inf − > 0. (2) u→∞ 1 − F2 (u) 1 − F1 (u) This is sometimes denoted F1 AHRD F2 , indicating asymptotic hazard-rate dominance. Notice that Equation 2 is automatically true when u1 > u2 , but that Definition 3 cannot be applied when u1 = u2 < ∞.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) 1 − F1 (u) lim = ∞. (3) u→u2 1 − F2 (u) This helps to clarify the relationship between the different competing notions of ex ante strength considered here. If F1 FSD F2 , then u1 > u is more likely than u2 > u; if F1 HRD F2 , then the relatively likelihood of u1 > u versus u2 > u is strictly increasing in u; and finally, if F1 AHRD F2 then this likelihood ratio is unbounded for large u. Of course, for F1 AHRD F2 , 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 F1 is “riskier” than F2 . 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 distributions). ui ∼ U (ui , ui ), so that Fi (ui ) = (ui − ui )/(ui − ui ) Here, Player 1 is stochastically stronger whenever u1 > u2 , so that the distributions have “shifted support.” If, in addition, u1 ≥ u2 then Player 1 is hazard-rate dominant (F1 HRD F2 ) and stochastically dominant (F1 FSD F2 ). I study this example in greater depth in Section 6, since closed form results are available. For u1 = u2 = 0, it is equivalent to Maskin and Riley’s (2000, p. 416) Example 2. F1 is, in their parlance, a “stretched” version of F2 . For a second example I consider a distribution with unbounded support. Example 2 (Exponential distributions). fi (u)/(1 − Fi (u)) = λi for all u ∈ (0, ∞). For Example 2, Player 1 is stochastically stronger whenever λ2 > λ1 . The same inequality ensures that she is hazard-rate dominant and stochastically dominant. 28Suppose that u2 < u1 . As u → u2 the hazard rate of F1 remains finite while the hazard rate of F2 diverges to ∞, since f2 (u) → f2 (u2 ) ∈ (0, ∞) and 1 − F2 (u) → 0. Suppose instead that u1 = u2 < ∞. Then Equation 3 fails, following an application of l’Hôptital’s rule. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 13 Fixing the bottom of the support at u1 = u2 = 0, changes in u1 and u2 for Example 1 change the relative means and variances of the two distributions in the same direction. Similarly, changes in λ1 and λ2 for Example 2 have a similar effect. Employing normal distributions, the means and variances of the competing distributions may be changed independently. Example 3 (Normal distributions). ui ∼ N (µi , σi2 ), so that Fi (u) = Φ((u − µi )/σi ). Here, Φ(z) represents the distribution function of the standard normal and φ(z) the corre- sponding density. The asymptotic linearity of the normal distribution’s hazard rate yields     f2 (u) f1 (u) u − µ2 u − µ1 − −→ − as u → ∞. 1 − F2 (u) 1 − F1 (u) σ22 σ12 By inspection, when σ1 = σ2 , Player 1 is stochastically stronger if and only if µ1 > µ2 . Thus an upward shift in a valuation distribution, perhaps unsurprisingly, creates a stochastic strength advantage. However, if σ1 > σ2 then Player 1 is stochastically stronger even when µ1 < µ2 . This reveals that variability in a player’s valuation distribution is a key determinant of stochastic strength: An increase in the variance increases the probability that a player will experience very high valuations, and hence enhances her stochastic strength. I conclude this section with a note of caution. Stochastic dominance (Definition 1) and hazard-rate dominance (Definition 2) are properties that depend on a support-wide ranking of the two distributions. In contrast, stochastic strength (Definition 3) depends exclusively on the “upper tail” behavior of the distributions. It would be expected, therefore, to be particularly sensitive to the exact assumptions of the model.29 4. Existence of a Unique Equilibrium Here I confirm the existence of a unique pure-strategy Bayesian-Nash equilibrium for the perturbed war of attrition, and describe its basic features. The results in this section are mild modifications of similar results presented by Fudenberg and Tirole (1986), Amann and Leininger (1996), Ponsati and Sákovics (1995), and others—I describe the substantive results of the paper in Section 5. Throughout, I assume that either max{ξ, β} > 0 or T < ∞. 4.1. Basic Properties. A pure strategy is a stopping rule ti (ui ), mapping a player’s valua- tion into the extended real line, subject to any time limit. For a pure-strategy Bayesian-Nash equilibrium (Harsanyi 1967–68), stopping rules need to be mutually optimal.30 Equilibrium stopping rules must exhibit certain basic properties. First, the expected fighting costs of a player are strictly increasing in her stopping time, and hence equilibrium stopping 29Thisis analogous to the use Mirrlees (1999) contracts in moral-hazard problems—see Section 5. 30When ξ > 0 or T < ∞ all information sets are reached with positive probability. In that case, any Bayesian Nash equilibrium profile will yield a Kreps-Wilson (1982b) sequential equilibrium. 14 DAVID P. MYATT rules are monotonic: A higher valuation is necessary to justify fighting for longer.31 Second, a player will never exit with positive probability at any time t ∈ (0, T ): An “atom” at time t makes a player predictable. Third, players must begin exiting at time zero, and will stop exiting at the same time: If not, then there are opportunities for a player to reduce her exit time without harming her chance of winning. Finally, when a time limit is in place, there can be a single discontinuity in a player’s stopping rule: At a critical time, a player finds it worthwhile to stay in until the time limit, hence guaranteeing an expected benefit of ui /2. Proposition 1. In equilibrium, a player’s stopping rule is weakly increasing in her valuation. There exist u∗i and u∗i satisfying ui ≤ u∗i < u∗i ≤ ui for each i ∈ {1, 2}, and t > 0, such that (1) ti (u) = 0 for all u ∈ (ui , u∗i ) and ti (u) = T for all u ∈ (u∗i , ui ), (2) ti (u) is strictly increasing and continuous for u ∈ (u∗i , u∗i ), (3) limu↓u∗i ti (u) = 0 and limu↑u∗i ti (u) = t, (4) if T = ∞ then u∗i = ui , and if T < ∞ then u∗i < ui . Hence players exit continuously over (0, t), with possible atoms at t = 0 and t = T < ∞. Proposition 1 reveals that a player’s stopping rule ti (u) is strictly increasing and continuous for u ∈ (u∗i , u∗i ), or equivalently t ∈ (0, t). It follows that its inverse is well defined for such valuations or, equivalently, at such points in time. I use v(t) and w(t) to denote the inverses for Player 1 and Player 2 respectively. Thus t = t1 (v(t)) = t2 (w(t)). Following from Proposition 1, v(0) ≡ limt↓0 v(t) and v(t) ≡ limt↑t v(t) are both well defined, and satisfy v(0) = u∗1 and v(t) = u∗1 . Similarly, w(0) = u∗2 and w(t) = u∗2 . 4.2. First-Order Conditions. For ti (u) ∈ (0, t), and so long as the inverse-stopping rule of her opponent is differentiable, first-order conditions must be satisfied by a player’s exit time. I use Gi (t) ≡ Pr[ti ≤ t] to denote the cumulative distribution function of Player i’s exit time ti ∈ (0, t), and gi (t) for its corresponding density, whenever this density exists. Hence G1 (t) = (1 − ξ)F1 (v(t)) ⇒ g1 (t) = v 0 (t)(1 − ξ)f1 (v(t)), and G2 (t) = (1 − ξ)F2 (w(t)) ⇒ g2 (t) = w0 (t)(1 − ξ)f2 (w(t)). Suppose that Player 2 is considering exiting at time t, and that v(t) is differentiable at this point. By waiting for some small amount of time dt, she increases the probability that Player 1 exits before her by g1 (t), yielding an expected benefit of g1 (t)w(t) dt. She also increases her fighting costs. With probability G1 (t) she is the winner, yielding a cost increase of β dt. With probability 1 − G1 (t) she is the loser, generating additional costs of dt. Hence, by 31 This is not necessarily true in the classic war of attrition. For instance, when ui < ∞ Player i could choose ti = 0 for all valuations while her opponent chooses any particular stopping time greater than ui . INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 15 waiting a little longer her expected costs rise by [1 − (1 − β)G1 (t)] dt. She will be indifferent to exit at time t when these costs and benefits precisely balance: 1 − (1 − β)G1 (t) g1 (t)w(t) dt = [1 − (1 − β)G1 (t)] dt ⇒ 1= . g1 (t)w(t) A similar indifference condition may be obtained for Player 1. Substituting for {Gi (t), gi (t)}: Proposition 2. The inverse stopping rules v(t) and w(t) are differentiable for t ∈ (0, t), 1 − (1 − β)(1 − ξ)F1 (v(t)) 1 − (1 − β)(1 − ξ)F2 (w(t)) v 0 (t) = and w0 (t) = . (4) (1 − ξ)f1 (v(t))w(t) (1 − ξ)f2 (w(t))v(t) Take two stopping times 0 < tL < tH < t. The inverse stopping-rules satisfy Z v(tH ) Z w(tH ) (1 − β)(1 − ξ)f1 (x) (1 − β)(1 − ξ)f2 (x) dx = dx. (5) v(tL ) x(1 − (1 − β)(1 − ξ)F1 (x)) w(tL ) x(1 − (1 − β)(1 − ξ)F2 (x)) Furthermore, for any t ∈ (0, t), v(t) and w(t) satisfy Λ1 (v(t)) = Λ2 (w(t)) where Z u∗i (1 − β)(1 − ξ)fi (x) Λi (u) ≡ dx. (6) u x(1 − (1 − β)(1 − ξ)Fi (x)) The final part of Proposition 2 is obtained by setting tL = t and tH → t in Equation (5). The integral on the right-hand side of Equation (6) exists only because either max{ξ, β} > 0 or T < ∞, so that the game is perturbed; the final part of the proposition would fail for the classic war of attrition. I return to this point just below. 4.3. Uniqueness. The pair of differential equations (Equations 4) solve to yield a family of potential equilibria to the war of attrition. For a unique solution, boundary conditions are needed. Such boundary conditions may be obtained by considering behavior at the beginning of the game (t = 0) or the end (t = t). Consider behavior at t = 0, and suppose that prize valuations are bounded away from zero (min{u1 , u2 } > 0). Neither player has a dominant strategy to exit at the beginning of the game. In this case, a single boundary condition is available. If a player exits with positive probability at the beginning of the game, then her opponent will always find it profitable to fight for some period of time: There cannot be “instant exit” by both players. Formally, this means that either v(0) = u∗1 = u1 or w(0) = u∗2 = u2 . This argument leads, therefore, to a single boundary condition, and this is not enough to tie down a unique equilibrium. Next, suppose suppose that prize valuations extend below zero, so that max{u1 , u2 } < 0. This ensures that, with probabilities F1 (0) and F2 (0) respectively, the players do not wish to win the prize: A player with a valuation ui ≤ 0 has a strictly dominant strategy to exit at time t = 0. Thus there is (trivially) instant exit from players with negative valuations. This means, in turn, that players with strictly positive valuations will always fight for some period of time. Formally, v(0) = u∗1 = 0 or w(0) = u∗2 = 0. 16 DAVID P. MYATT This second case yields a pair of boundary conditions, and would appear to suggest a unique solution. Unfortunately, this is false since (Fudenberg and Tirole 1986) the differentiable equations are not Lipschitz continuous at t = 0.32 In both of the cases considered here, an additional boundary condition is needed—other cases (e.g. u1 > 0 > u2 ) yield similar conclusions. The search for a further boundary condition leads to an examination of behavior as t → t. As is well known, this attempt fails for the classic war of attrition. To see why, set ξ = β = 0 and T = ∞. For simplicity suppose that the hazard rate of F1 is increasing.33 Further suppose that u1 = ∞. Then Z v(tH ) Z v(tH ) f1 (x) f (v(tL )) 1 dx ≥ dx v(tL ) x(1 − F1 (x)) (1 − F (v(tL )) v(tL ) x   f (v(tL )) v(tH ) = log −→ ∞ as tH → t. (1 − F (v(tL )) v(tL ) This follows since v(tH ) → ∞ as t → t. The same is true, following a slightly different argument, when u1 < ∞.34 Similarly, the right hand side of Equation 5 also diverges. The source of divergence is the fact that 1 − F1 (v(t)) → 0 as t grows large. In other words, the probability that Player 1 fights beyond t becomes vanishingly small. When either β > 0, ξ > 0, or T < ∞, so that the marginal cost of increased aggression is bounded away from zero, a consideration of t → t is successful. Consider first the case where T = ∞, and for notational simplicity set ξ = 0 but β > 0. Allowing tH → t, Z v(tH ) Z v(tH ) (1 − β)f1 (x) 1 (1 − β)f1 (x) dx ≤ dx v(tL ) x(1 − (1 − β)F1 (x)) v(tL ) v(tL ) 1 − (1 − β)F1 (x)     1 1 − (1 − β)F1 (v(tL )) 1 1 − (1 − β)F1 (v(tL )) = log −→ log . v(tL ) 1 − (1 − β)F1 (v(tH )) v(tL ) β Similarly, the right hand side of Equation 5 also converges as tH → t. When T < ∞, v(t) → u∗i < ui as t → t, and once again there is convergence. Such convergence yields a terminal boundary condition, as described in the final part of Proposition 2. This condition is obtained by “integrating up” the first-order conditions from time t until the last exit time t. Equivalently, it may be obtained by integrating backwards from t until t. In essence, therefore, it is a consequence of backward induction. Utilizing boundary conditions at both t = 0 (as in Section 4) and as t → t ties down a unique equilibrium. The only exception is 32ExaminingEquation 4, notice that the denominators of both expressions tend to zero as t → 0. 33It would be sufficient to assume that the hazard rate is bounded away from zero for large valuations. 34When Player 1’s valuation is bounded above Z v(tH ) Z v(tH )   f1 (x) 1 f1 (x) 1 1 − F1 (v(tL )) dx ≥ dx = log → ∞, v(tL ) x(1 − F1 (x)) u1 v(tL ) 1 − F1 (x) u1 1 − F1 (v(tH )) since 1 − F1 (v(tH )) → 0 as tH → t. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 17 that, when T < ∞, a player with valuation u∗i is exactly indifferent between exiting at t and fighting until the time limit T . Thus, I describe the equilibrium as “essentially unique.” Proposition 3. There is an essentially-unique pure-strategy Bayesian-Nash equilibrium. Propositions 1–3 confirm the existence of a pure-strategy Bayesian-Nash equilibrium, and characterize its basic properties. As noted previously, these results are not novel, but are a reformulation of similar results offered by Fudenberg and Tirole (1986) and Amann and Leininger (1996). Nevertheless, they offer the potential for comparative-static analysis, and in particular an examination of the unique equilibrium as max{ξ, β} → 0 and T → ∞, so that the vanishing perturbations select an equilibrium from the multiplicity present in the classic war of attrition.35 This permits an analysis of the relative exit times of the two players for the selected equilibrium. Such an analysis is contained in the next section, and is (to the best of my knowledge) a new contribution with a surprising message. 5. Equilibrium Selection and Instant Exit Here I present the substantive results of the paper. I assume that either max{ξ, β} > 0 or T < ∞ (but not both), yielding a perturbed war of attrition with a unique equilibrium (Proposition 3); the case when both inequalities hold is omitted for simplicity. I first derive some basic properties of this equilibrium (Propositions 4 and 5) before moving to the main result of the paper (Proposition 6), in which I allow the perturbations to vanish, hence selecting an equilibrium from the multiplicity present in the classic war of attrition. I begin by considering min{u1 , u2 } > 0, so that the valuations of the players are bounded away from zero, and hence at most one of the players exits at the beginning of the game. Two immediate questions arise: Will one of the players exit at the beginning with positive probability? If so, which player? The final part of Proposition 2 may be used to answer these questions, by determining the time-zero exit probabilities of the players. When a player is perceived to be weaker in the sense of stochastic dominance, she will be the one who engages in instant concession. Formally: Proposition 4. If min{u1 , u2 } > 0 and F1 FSD F2 then Pr[t2 = 0] > 0 and Pr[t1 = 0] = 0. 35Setting ξ = 0, β = 1 and T = ∞, the game reduces to a simple first-price all-pay auction with two bidders. This is, in fact, a game studied by Amann and Leininger (1996). The final part of Proposition 2 reduces to R u1 R u2 v(t) (1/x)f1 (x) dx = w(t) (1/x)f2 (x) dx, or equivalently: Pr[u1 > v(t)]E[(1/u1 ) | u1 > v(t)] = Pr[u2 > w(t)]E[(1/u2 ) | u2 > w(t)] Notice that E[(1/ui )] is the expectation of the inverse of the prize valuation. Recall that I fixed the fighting cost at 1 per unit of time, allowing the valuation to vary. Alternatively, for a fixed positive valuation the fighting cost of a player could be allowed to vary. Thus E[(1/ui )] is the expectation of Player i’s fighting cost, relative to a prize with unit value. 18 DAVID P. MYATT A (very rough) intuition for this result is as follows. Begin by fixing a stopping rule that is adopted by both players. Suppose that both players update their strategies by myopic best-response. It is relatively less likely that Player 2 reaches any particular stopping time (her valuation tends to be lower), and hence Player 1 finds it relatively less costly to increase her own stopping time. This pushes Player 1’s stopping rule up relative to that of Player 2. Updating by best response once more, this effect is reinforced. In equilibrium, therefore, Player 1 will adopt a relatively aggressive stance. This means that, when her valuation is low, Player 2 will prefer not to fight—in other words, there will be instant exit. A more precise explanation may be obtained by examining cases in which Player 1 is per- ceived to be stronger than Player 2 in the sense of hazard-rate dominance (Defintion 2). (Of course, F1 HRD F2 implies that F1 FSD F2 , and Proposition 4 continues to hold.) Suppose that T < ∞, so that a time limit is in place, but that ξ = β = 0. Before proceeding, I must consider the values taken by u∗1 and u∗2 . Recall that limu↑u∗i ti (u) = t, but that ti (u) = T for all u ∈ (u∗i , ui ). Hence, at a valuation u = u∗i , Player i’s exit time jumps from t to T . Thus, with this valuation, the player must be just indifferent between exiting at t and remaining until the time limit T . Continued fighting from t until T involves additional costs of T − t. When the war is over at T , the player i receives the prize with probability 1/2, and hence u∗i /2 = T − t. This means that u∗1 = u∗2 = u∗ = 2(T − t). Effectively, the game ends at time t, and at this time the lower bound to the players’ valuations is this same. Equivalently, the inverse stopping rules cross: v(t) = w(t) = u∗ . By examining the first-order conditions from Proposition 2, I may establish an ordering of the stopping rules for the two players. Taking the ratio of v 0 (t) and w0 (t) from Equation 4: t01 (u∗ ) w0 (t) 1 − F2 (u∗ ) f1 (u∗ ) = = × < 1, t02 (u∗ ) v 0 (t) f2 (u∗ ) 1 − F1 (u∗ ) following from F1 HRD F2 . Hence t1 (u) is steeper than t2 (u) as u → u∗ , and thus t2 (u) < t1 (u) for u just below u∗ . In fact, the same argument may be employed at any crossing point of the stopping rules. This means that the crossing rules cross only once, and hence t2 (u) lies below t1 (u). This observation is sufficient to yield the conclusion of Proposition 4. Using this last part of Proposition 2, Λ1 (u∗1 ) = Λ2 (u∗2 ), and hence Z u∗ Z u∗ f1 (x) f2 (x) dx = dx. u∗1 x(1 − F1 (x)) u∗2 x(1 − F2 (x)) Given F1 HRD F2 , and for each x, the integrand on the right hand side exceeds the integrand on the left. Hence, if the equality is to be maintained, the range of integration on the left must be larger: u∗1 < u∗2 . But, since u1 ≥ u2 (a requirement of Definition 2), this implies that u∗2 > u2 : Exactly the instant-exit predicted by Proposition 4. The formal proof of the proposition shows that first-order stochastic dominance is sufficient for the result to obtain, and that the result holds for T = ∞ with max{ξ, β} > 0. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 19 Of course, Proposition 4 is limited in two ways. First, it applies only when valuations are bounded away from zero. Second, it establishes that instant exit occurs with some probability, but does not show that a weaker player will almost always give up immediately. To address the first limitation, turn to the case max{u1 , u2 } < 0. Trivially, both players “instantly exit” when their valuations are negative, and a player with a positive valuation is always willing to fight for some time; there is no instant exit from a player who values the prize. Nevertheless, it is still possible to order the behavior of the players. So long as the Player 1 is stronger than Player 2 in the sense of hazard-rate dominance (F1 HRD F2 ), the argument given above ensures that t1 (u) > t2 (u) whenever a time limit is in place (T < ∞). In fact, a similar argument may be employed when the classic war of attrition is subject to the other perturbations. For instance, set u1 = u2 = ∞, ξ = 0 and T = ∞. Thus the only perturbation is the “first-price effect” β > 0. For this configuration, u∗1 = u∗2 = ∞, and Proposition 2 yields Z ∞ Z ∞ 1 f1 (x) 1 f2 (x) dx = dx. v(t) x 1 − (1 − β)F1 (x) w(t) x 1 − (1 − β)F2 (x) For any x, and β sufficiently small, the integrand of the right hand expression exceeds that of the left, following from F1 HRD F2 , and hence, once again, v(t) < w(t). The only qualification here is that the perturbation must be sufficiently small. Formally: Proposition 5. Suppose that u1 = u2 = ∞ and F1 HRD F2 . For u > max{u1 , 0}, if either T < ∞ and u < u∗ , or T = ∞ and for max{ξ, β} sufficiently small, then t1 (u) > t2 (u). Hence the ex ante stronger player is willing to fight for longer given any particular prize valuation. This addresses the first limitation described above. Turning to the second limiti- ation, I may draw upon the notion of stochastic strength to obtain a much stronger result, despite the fact that the stochastic strength ordering limits only the upper tail behavior of the two distributions. For ease of exposition, suppose that Proposition 4 holds, so that u∗2 > u∗1 . Then, when T < ∞, Λ1 (u∗1 ) = Λ2 (u∗2 ) implies that Z u∗2 Z u∗   f1 (x) 1 f2 (x) f1 (x) dx = − dx. (7) u∗1 x(1 − F1 (x)) u∗2 x 1 − F2 (x) 1 − F1 (x) It is straightforward to show (as part of the appendicized proofs) that u∗ → ∞ as T → ∞. This means that, if the hazard rates of F1 and F2 are bounded apart in the upper tails, then the right-hand side of Equation 7 tends to ∞ as T → ∞. To offset this effect, and maintain the equality, u∗2 must satisfy u∗2 → ∞ as T → ∞. In the limit, Player 2 always exits at the beginning of the game, and concedes the prize to her (stochastically stronger) opponent. Proposition 6. If Player 1 is stochastically stronger than Player 2 and T < ∞, then (1) if min{u1 , u2 } > 0 then Pr[t1 = 0] = 0 for large T and Pr[t2 = 0] → 1 as T → ∞, 20 DAVID P. MYATT (2) if max{u1 , u2 } < 0 then Pr[t2 < t1 | min{u1 , u2 } > 0] → 1 as T → ∞, and (3) for all u ∈ (u2 , u2 ), t2 (u) → 0 as T → ∞. Hence the stochastically weaker player almost always loses the war of attrition. If, instead, T = ∞ and max{ξ, β} > 0, then the same results hold as max{ξ, β} → 0 instead of T → ∞. When Player 1 is stochastically stronger, she is able, in equilibrium, to threaten to fight for longer periods of time with greater credibility. To see this, note that, since threats to fight for long periods of time are always costly, a player must have a sufficiently high valuation with sufficiently high probability in order to make them. As the threat grows, the probability that the valuation is sufficiently high shrinks to zero. However, the likelihood ratio that Player 1 is able to execute a threat versus Player 2 diverges. In other words, it is many times more likely that Player 1 has a very high valuation than Player 2—this is the content of Equation 3.36 Notice that, as a corollary of Proposition 6, and in the limit, the war of attrition will end at time t = 0. Thus the result implies that the selected equilibrium in a classic war of attrition cannot explain the existence of delay in concessionary games. 6. Sensitivity Analysis The analysis of Section 5 employs two procedures. First, the classic war of attrition is perturbed to yield a unique equilibrium. The selected equilibrium sometimes involves instant exit (Proposition 4) and sometimes a bias towards one of the players (Proposition 5). Second, the perturbation is allowed to vanish to zero, yielding the “almost-always instant-exit” result of Proposition 6. This second procedure relies upon the stochastic strength of one of the players, and an arbitrarily small perturbation. In this section, I use Examples 1 and 2 to ascertain how sensitive the results are to these requirements. 6.1. Uniform Distributions (Example 1). Turning to Example 1, the simplicity of the uniform distribution allows me to obtain closed-form solutions, and hence provide easy nu- merical illustrations of the results. I set T = ∞ (since u1 < ∞ and u2 < ∞) and, for simplicity, ξ = 0. Hence the only perturbation is the “first-price effect” β > 0. 36There are parallels between Proposition 6 and the “unpleasant theorem” of Mirrlees (1999). He considered the design of an incentive contract under moral hazard. Roughly speaking, a special case of his model is as follows: A principal wishes to induce an agent to choose action i = 1 rather than i = 2, and may base compensation on y ∼ Fi (y). There are two critical assumptions: First, the agent’s utility is unbounded below—hence infinite punishments are possible. Second, F1 (y)/F2 (y) → 0 as y → −∞. This unbounded likelihood ratio means that the principal can threaten the agent with a very large punishment if y ≤ y ∗ . With y ∗ suitably chosen, the punishment takes place with arbitrarily low probability under i = 1, but is (although still very unlikely) infinitely more likely under i = 2. Analogs to both of these assumptions are present in the current model: The unbounded likelihood ratio stems from stochastic strength, and infinite punishments are possible since the war of attrition can go on forever, at least when T = ∞. When T → ∞, the “maximum punishment” is allowed to become arbitrarily large. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 21 Proposition 7. If u1 > u2 , then as β → 0 Player 1 almost always wins almost immediately. If u1 = u2 = u and u1 > u2 with u1 > 0, then β → 0 yields an equilibrium in which w(t) v(t) u − u1 u1 = × , and hence w(0) = u − w(t) u − v(t) u − u2 1 + (u1 − u2 )/u Hence Player 2 exits immediately with positive probability, but w(t) < v(t). The first part of Proposition 7 follows straightforwardly from Proposition 6. For the second claim, however, u1 = u2 . Thus Player 1 is not stochastically stronger than Player 2, and Proposition 6 does not apply. Nevertheless, it is straightforward to observe that F1 FSD F2 and hence Proposition 4 does apply. Thus Player 2 exits at time t = 0 with positive probability. Interestingly, however, for this case w(t) < v(t). This means that, valuation for valuation, Player 1 is less likely to win the war of attrition—it is biased against her. To see why, return to the environment in which ξ > 0, and set β = 0. (Proposition 7 continues to apply, with ξ → 0 replacing β → 0.) Player 2 instantly exits at time zero with positive probability. Just after t = 0, the probability that Player 1 suffers exit failure is close to ξ. The probability that Player 2 suffers exit failure, conditional on this time being reached, is greater than ξ. If exit failure is interpreted as “craziness,” then the initial instant-exit of Player 2 enhances her reputation for such craziness whenever she remains in the game. As a counterpoint to Proposition 6, the u1 = u2 case establishes that fighting does occur.37 It might be argued, however, that this is a knife-edge case. Hence, if u2 ≤ u1 < ∞, one might normally expect u2 < u1 and thus the stochastic-strength advantage of Player 1. Of course, when the players are ranked by stochastic strength, the immediate exit of the stochastically weaker player only happens in the limit as β → 0 (or, equivalently, ξ → 0). When β > 0 is fixed, taking away any asymmetry reverses the result. I wish to calculate, therefore, the degree to which the “instant exit” results are sensitive to the exact choice of β. To do this, I turn once again to Example 1. For simplicity, I set u1 = u2 = 0. In Appendix A.3, I solve for v(t)/u1 in terms of w(t). Of course, v(t)/u1 = Pr[u1 ≤ v(t)]. This is the probability that Player 1 loses the war of attrition, given that Player 2 has a valuation u2 = w(t). Equivalently, it is the probability Player 2 wins the war of attrition given that she has a valuation u2 = w(t). Fixing u2 , this probability vanishes as β → 0. Away from the limit, however, I may calculate this probability for a range of different parameter values. I plot the results from such a numerical exercise in Figure 1. When u2 is moderately large, Player 2 retains a “fighting chance” of winning the war until β vanishes completely (notice the use of a log scale for β). The basic lesson, therefore, is that the perturbation to the war of attrition has to be very small indeed for the instant-exit result to have bite. Hence, when 37Propositions 4–6 fail to apply when the players are symmetric. Indeed, the uniqueness result of Propo- sition 3 ensures that, with symmetric players, the unique equilibrium will be symmetric and the ex post stronger player will win. 22 DAVID P. MYATT 0.6 1.0 ......................................... ...... . .... ... ... ... ... .................................. . ..... ..... ...... . .... .. ........... .... .... . . . 0.5 .... ..................... u2 = 1 ........ .... .... . Pr[t1 (u1 ) < t2 (u2 ) | u2 ] Pr[t2 (u2 ) < t1 (u1 ) | u1 ] ... 2 0.8 . . . . ... .. ... ..... . .. . . ... . ... .. 0.4 ... ... ..... ..... .... u2 = 7 ... . .. . .. . .. .... ... 20 0.6 . ... .. ... ... ... .. .. . .. ... ... . . . . . u2 = 2 . 0.3 ... . . . . ..................... ... . . ... 5 . ... . β = 10−6 ... .... 0.4 ... .. . . .. 0.2 ... .... . .. ... .... .. .. ..... ..... .... β = 10−3 .. .. .. . .. . .. .......... ... .. . . . .. .. 0.2 .... .. 0.1 .. ... .. ........ ... .. . . . . .... ..... ............................ ....... ..... β = 1 . . . . . . ....... ....... ................................... ........... 0.0 . . . ..... 0.0 . 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 − log10 β u1 = F1 (u1 ) Figure 1. Uniform Distributions (Example 1, u1 = 1 and u2 = 54 ) 0.7 1.0 ........................................................................................................................... ... ......... .. ... .. . .. ..... ... 0.6 ... ... .. . . ... .. . . .. .. Pr[t1 (u1 ) < t2 (u2 ) | u2 ] Pr[t2 (u2 ) < t1 (u1 ) | u1 ] ... ... 0.8 . ... ... 0.5 .... ..................... u2 = 1 ... .. .. . . .... . .. .. ... .... .... 0.6 ... .. .. . 0.4 ... .... ..... ..... .... u2 = 2 . . .. ... .... 3 ... ... . . ... ..... .. .. ..................... T = 10000 0.3 .... ..... ...... . . . . . u2 = 1 3 0.4 ... ... .. . .... ...... .. . .. .... .... ... .. .... 0.2 .. ... .... . .............. .. ... ..... ..... .... T = 100 . .. ..... .... ................... 0.2 ... .... .. ..... .... 0.1 . . . . . . .... ..... ..... ......................................... .. ..... T = 1 . . . . . . . . . . . ....... ..... ... ... 0.0 0.0 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 log10 T F1 (u1 ) = 1 − e−λ1 u1 Figure 2. Exponential Distributions (Example 2, λ1 = 1 and λ2 = 1 12 ) the perturbation is taken seriously—small, but not taken to the limit of zero—then the exit of the stochastically weaker player is relatively rapid, rather than almost always instant. 6.2. Exponential Distributions (Example 2). To investigate this point further, I con- sider Example 2. I set ξ = β = 0, but impose a time limit T < ∞ on the game. Algebraically, this example is relatively straightforward, since Λi (u) = λi log[u∗ /u]. For the illustrations of Figure 2, I set λ1 = 1 and λ2 = 1 21 , so that E[u1 ] = 1 and E[u2 ] = 23 . This means that Pr[u1 > u2 ] = 53 , and Player 1 is stronger ex ante in the sense of stochastic dominance, hazard-rate dominance, and stochastic strength (Definitions 1–3). INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 23 I plot the results for this second example in Figure 2. Once again, the perturbation to the classic war of attrition must be very small before the instant-exit results begin to bite. In this case, a small perturbation corresponds to a large time limit T . For instance, when u2 takes its median value of u2 = log 2/λ2 ≈ 0.46 and T = 10, Player 2 retains a 15% chance of winning the war of attrition, compared to a 37% chance of having the highest valuation. Taken together, these sensitivity analyses suggest that the selection of the “instant exit” equilibrium is an extreme case. In any application of the war of attrition, therefore, the analyst may wish to take seriously the different perturbations that may be present, and hence assess the exact speed of the “rapid exit” that may occur. 7. Discussion As noted in Section 1, the war of attrition lends itself to a wide range of social-scientific applications. Here, I discuss the implications of the results for a number of these. 7.1. Rent Dissipation. In a classic paper, Posner (1975) argued that the social costs of monopoly should include monopoly profits.38 His idea was that such profits would be dis- sipated by rent-seeking activities. Wars of attrition may be used to model rent-seeking. For instance, Fudenberg and Tirole (1987) used “game theory to shed light on a particular issue—the hypothesis of monopoly rent dissipation.” In their complete-information model, the symmetric equilibrium succeeds in dissipating all rents. Posner’s (1975) claim is subject to a number of critiques. Fisher (1985), for instance, argued that the rents attributed to an initial advantage should not be counted as a social cost. An interpretation of this is that asymmetries between the participants may prevent the complete dissipation of the prize. Others have considered the actual size of rent-seeking costs. Tullock (1967), for instance, recognized that political actors will expend resources in order to obtain political rents. Later (Tullock 1980), however, he noted that the costs incurred by participants appear to be relatively low compared to the prizes involved. Examining this “Tullock paradox,” Riley (1999) demonstrated that asymmetries in a complete-information war of attrition may dramatically reduce rent-seeking costs. The present paper, therefore, extends Riley’s (1999) explanation to the incomplete-information (or random rewards) case: Merely the perception of an initial advantage may dramatically reduce Tullock costs. 7.2. All-Pay Auctions. Wars of attrition have attracted the attention of leading auction theorists. Bulow and Klemperer (1999), for instance, considered a generalized war of attri- tion. Specifically, suppose that N + K symmetric bidders compete for N privately valued 38Posner (1975) went on to combine monopoly profits with the usual Harberger (1954) deadweight loss triangles, and turned to appopriate data to obtain a relatively large estimate of the social costs of monopoly. Wenders (1987) went further by adding “rent-defending” to the “rent-seeking” equation. 24 DAVID P. MYATT prizes. As is well known (Haigh and Cannings 1989), when K > 1 such a game has no symmetric equilibrium; K − 1 players must exit immediately, yielding a standard war of at- trition (analogous to the two-player single-prize variety considered here) in which the N + 1 remaining bidders compete for N prizes. To circumvent this, Bulow and Klemperer (1999) perturbed the game in an ingenious manner: A conceding player continues to pay a (per- haps small) fraction of her fighting costs until the game ends. The perturbed game has a symmetric equilibrium involving rapid exit until the N + 1 highest-valued players remain.39 Thus, their perturbation would appear to “select” a unique equilibrium. Unfortunately, their approach does not solve the equilibrium-selection problem, as symmetry is used as an ad hoc selection criterion.40 In an (even slightly) asymmetric war of attrition the selected equilibrium may be very far from symmetric.41 The revenue rankings of dif- ferent auction formats may then be overturned. For instance, Krishna and Morgan (1997) considered the symmetric equilibrium of a model with affiliated values, and found that (from a revenue perspective) the war of attrition outperforms other standard auction formats. In contrast, the present paper suggests that the war of attrition may raise no revenue at all. 7.3. Macroeconomic Stabilization. Alesina and Drazen (1991) used the war of attrition to model a process of macroeconomic stabilization. They considered an economy with a rising debt-to-GNP ratio and hence a need for fiscal stabilization (Drazen and Helpman 1987, 1990). Such a stabilization requires the consent of two socioeconomic groups, and a concession by one side consists of an agreement “to bear a disproportionate share of the tax increase necessary to effect a stabilization.” Crucially, however, the Alesina-Drazen (1991) model is symmetric: The utility losses suffered due to the tax increases are drawn from the same distribution for both socioeconomic groups. They (and indeed subsequent authors, such as Casella and Eichengreen 1996) focused, therefore, on a symmetric equilibrium. In some situations, players might be expected to be “approximately” symmetric. In the Alesina- Drazen (1991) application, however, the different socioeconomic interest groups are likely to be asymmetric ex ante, and hence stabilization is likely to occur far more rapidly.42 39They refer to this as “instant sorting” (exit of K −1) until “one too many” (N +1) remain. In contrast, the present paper suggests “complete ex ante sorting” (exit of the K ex ante weakest) until N players remain. Bulow and Klemperer (1999, p. 178, note 15) acknowledge this possibility, noting that “[a]symmetric perfect- Bayesian equilibria include those in which K (pre-identified) firms quit in zero time . . . [e]quilibria of this kind seem particularly natural if (in contrast to our model) there any asymmetries between players.” Here I suggest that stochastic strength is exactly the kind of asymmetry to generate this result. 40 For instance, when N = K = 1 their model reduces to a classic war of attrition. This classic war of attrition has multiple equilibria. Their perturbation has no bite in the last stage of their game. 41 In other papers, Klemperer (1998) and Bulow, Huang, and Klemperer (1999) paid great heed to the potential influence of small asymmetries between bidders in common-value settings. 42 In their discussion of European deficits following World War I, Alesina and Drazen (1991) noted the “dominant position of the Conservatives” in Britain, which they claimed “led to a rapid stabilization by means that favored the Conservatives’ traditional constituencies.” The interpretation offered here is that the Conservatives were stochastically stronger in that situation. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 25 7.4. Technological Standards. The use of a symmetric specification, and a focus on sym- metric equilibria, is also present in studies of standards adoption. Farrell and Saloner (1988) viewed the adoption of a technological standard as a “battle of the sexes” game: Firms wish to coordinate on a some common standard, but prefer their own standard. They then modelled a standards committee as a war of attrition, in which each participant “holds out,” hoping that the opponent will agree to coordinate on the preferred standard. Crucially, they focus on the symmetric mixed-strategy equilibrium to this game. In an extension, Farrell (1996) considered an incomplete-information version of the same standards-adoption game. Once again, however, he chose the symmetric equilibrium as the vehicle for his analysis. With ex ante asymmetries, the situation may change: There may be immediate concession in favor of the stochastically stronger standard, even if it is suboptimal ex post. 7.5. Private Provision of a Public Good. An early economic application of the war of attrition was provided by Bliss and Nalebuff (1984). They considered N impatient individu- als, each of whom is capable of providing a public good. The first person to “concede” pays a private cost to supply the good, yielding a prize (the opportunity to free ride) for each of the N − 1 remaining players. The authors’ specification is, once again, symmetric, and they examined a symmetric equilibrium. The player with the lowest private cost of provision, relative to her benefit from the good, will provide the good following some delay. The results of the present paper suggest that, in an asymmetric version of the same game, the public good would be provided relatively quickly (hence increasing efficiency) by the ex ante most efficient player (who might be less efficient ex post, and hence this may reduce efficiency.)43 7.6. Concluding Remarks. I have suggested that the war of attrition may well end with the instant exit of one of the participants. A player is stochastically stronger when she is much more likely to experience very large valuations than her opponent. When this condition holds, and the classic war of attrition is perturbed using a “credibility” device, the selected equilibrium involves the instant exit of the stochastically weaker player. Important lessons emerge: First, ex ante perceptions of “strength” may be more important than a player’s ex post valuation for a prize. Second, a focus on the symmetric equilibrium of a symmetric game may be misleading, since small asymmetries may dramatically affect the outcome. I have already mentioned two qualifications to the results. First, players might not be ranked via the stochastic-strength measure of Definition 3. Second, the sensitivity analysis of Section 6 suggests that instant exit only occurs when the perturbation to the classic war of attrition (a critical feature, due to the equilibrium selection problem) is very small indeed. 43In a complete-information version of a similar model, Bilodeau and Slivinski (1996) obtained a unique equilibrium by imposing a time limit. They concluded that the public good is provided immediately by the most efficient player. Their assumption of complete information, however, meant that they did not distinguish between ex ante and ex post notions of strength. 26 DAVID P. MYATT I raise a third problem here: The results suggest that wars of attrition should end relatively quickly, and thus that fighting will not be seen. In contrast, the classic war of attrition is used to model situations in which delays are observed. If this paper is to be believed, then why might delay occur in a concessionary environment? One possibility is, of course, that the assumptions of the model fail. Another response, however, is to embed the war of attrition into a larger game. The key determinant of the outcome here is the stochastic strength of a player. Each participant, therefore, has a strong incentive to appear stronger ex ante. Thus a stochastically weaker player may wish to delay concession in the hope that a change in the players’ perceived relative strengths may leave her in the stochastically stronger position. To flesh out this idea, suppose that the war of attrition takes place in two stages. In the first stage, if both players fight, then a signal of the relative strengths emerge. This signal might be generated from conflict between them,44 and the players may then update to obtain an interim assessment of each others’ strength. If the signal is sufficiently precise to allow the overturning of an initial stochastic-strength ranking, then a stochastically weaker player has an incentive to participate in this first stage. Notice that this story involves direct learning: An additional signal changes the players’ beliefs. In contrast, the classic war of attrition involves only learning by revelation: A player updates her belief based upon her opponent’s continued presence in the game. Of course, a complete story would combine both direct learning and revelation in a unified model. Nevertheless, the results offered here suggest that such a unified model may be needed to explain some instances of costly delay. Appendix A. Proofs Propositions 1–3 are somewhat standard, and hence their proofs are omitted from the paper. For completeness, I have collected them together in the not-for-publication Appendix B. Propositions 4–6 are proved separately for the case T < ∞ and ξ = β = 0 (Section A.1) and the case T = ∞ and max{ξ, β} > 0 (Section A.2). Finally, Section A.3 contains the proof of Proposition 7 together with further details of the sensitivity analysis presented in Section 6. A.1. Proofs of Propositions 4–6 for T < ∞ and ξ = β = 0. Recall that, when T < ∞, it is a maintained assumption that u1 = u2 = ∞. Furthermore, following from an argument given in the main text, u∗1 = u∗2 = u∗ = 2(T − t) < ∞. 44Using the terminology of Maynard Smith (1974), this initial stage is a “tournament” rather than a “display.” In a recent working paper, Lee (2002) models direct learning by firms competing to obtain a monopoly position, using continuous time filtering techniques (Bolton and Harris 1999, Keller and Rady 1999, Moscarini and Smith 2001). As time goes on, participants observe a Wiener process, with a drift determined by the relative profitability of the two firms, and thus learn directly, rather than by revelation. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 27 Proof of Proposition 4 for T < ∞. Take Equation 6 and integrate by parts: Z u∗ Z u∗ fi (x) 1 Λi (u) = dx = − d log[1 − Fi (x)] u x(1 − Fi (x)) u x u∗ Z u∗ log[1 − Fi (x)] =− + log[1 − Fi (x)] d(1/x) x u u Z u∗ log[1 − Fi (u)] log[1 − Fi (u∗ )] log[1 − Fi (x)] = − ∗ − dx. u u u x2 Suppose, contrary Proposition 4, that u∗2 = u2 , so that F2 (u∗2 ) = 0, and hence log[1 − F2 (u∗2 )] = 0. Since F1 FSD F2 it must be that u1 ≥ u2 , and hence u∗1 ≥ u1 ≥ u2 = u∗2 . Λi (u∗i ) is well defined for i ∈ {1, 2}, and evaluation of Λ2 (u∗2 ) yields Z u∗1 Z u∗ ∗ log[1 − F2 (u∗ )] log[1 − F2 (x)] log[1 − F2 (x)] Λ2 (u2 ) = − ∗ − 2 dx − dx, u u∗2 x u∗1 x2 while the evaluation of Λ1 (u∗1 ) yields u∗ log[1 − F1 (u∗1 )] log[1 − F1 (u∗ )] log[1 − F1 (x)] Z Λ1 (u∗1 ) = − − dx. u∗1 u∗ u∗1 x2 Λ1 (v(t)) = Λ2 (w(t)) holds in the limit as t → 0. Equate Λ1 (u∗1 ) and Λ2 (u∗2 ) to obtain  Z u∗ 1 − F1 (u∗ )  1 1 log[1 − F (x)] 2 ∗ log ∗ − 2 dx = u 1 − F2 (u ) u∗2 x Z u∗ log[1 − F1 (u∗1 )]   1 1 − F1 (x) − log dx. u∗1 u∗1 x 2 1 − F2 (x) Since F1 (u) < F2 (u) for all u > u1 , the left-hand side of the equation is strictly positive and the right-hand side is strictly negative. This is a contradiction, and hence u∗2 > u2 .  Proof of Proposition 5 for T < ∞. I claim that u∗2 ≥ u∗1 ≥ 0. To prove this, note that ti (u) = 0 for all u ≤ 0, and hence u∗i ≥ 0 for i ∈ {1, 2}. Suppose, contrary to the claim, that u∗1 > u∗2 ≥ 0. Since u∗1 > 0, both Λ1 (u∗1 ) and Λ2 (u∗1 ) are well defined. Proposition 2 then implies that Λ2 (u∗1 ) − Λ1 (u∗1 ) < 0. This is inconsistent with F1 HRD F2 , since Z u∗   ∗ ∗ 1 f2 (x) f1 (x) Λ2 (u1 ) − Λ1 (u1 ) = − dx > 0. u∗1 x 1 − F2 (x) 1 − F1 (x) Hence I have a contradiction. Thus the claim is true. With the claim in hand, I turn to the proposition itself. Suppose that u > u∗2 ≥ u∗1 ≥ 0. Both Λ1 (u) and Λ2 (u) are well defined. Since F1 HRD F2 , and following the logic employed above, Λ2 (u) − Λ1 (u) > 0. Set t = t1 (u), so that v(t) = u. Proposition 2 insists that Λ2 (w(t)) − Λ1 (v(t)) = 0. This means that w(t) > u, and that t2 (u) < t = t1 (u), which is the desired result. Next, suppose that u∗2 ≥ u > u∗1 ≥ 0. This directly implies that t2 (u) = 0, t1 (u) > 0, and hence t1 (u) > t2 (u). 28 DAVID P. MYATT Finally, suppose that u∗2 ≥ u∗1 ≥ u > 0, so that t1 (u) = t2 (u) = 0. Since u > u1 ≥ u2 , this means that both players exit at time zero with positive probability, and with positive valuations. This cannot be optimal for either player, and hence cannot be the case.  Proof of Proposition 6 for T < ∞. First, I show that u → ∞ as T → ∞. Second, I show that, when Player 1 is stochastically stronger than Player 2, Λ2 (u) − Λ1 (u) → ∞ as T → ∞. Third, I use this fact to prove the statements given in the proposition. For the first step, I must show that, as the time limit grows arbitrarily large, the valuation at which the equilibrium stopping rules jump up to the time limit grows without bound. Integrating first order conditions in the usual way, Z u 1 − F1 (u∗1 ) 1 − F1 (u∗1 )     f1 (x)w(t1 (x)) ∗ ∗ t1 (u) = dx ≤ u2 log = u log , u∗1 1 − F1 (x) 1 − F1 (u) 1 − F1 (u) where the second equality follows from u∗2 = u∗1 = u∗ . Letting u → u∗ , t1 (u) → t and hence 1 − F1 (u∗1 )   ∗ t ≤ u log ≤ −u∗ log[1 − F1 (u∗ )] 1 − F1 (u∗ ) Suppose that the claim (u∗ → ∞) is false. Then I may construct a sequence of time limits {T } satisfying T → ∞ such that, in the equilibria, u∗ remains bounded throughout the sequence. It follows from the inequality above that t remains bounded throughout the sequence. But if t remains bounded, then u∗ = 2(T − t) → ∞ as T → ∞: A contradiction. For the second step in the proof, I recall that Player 1 is stochastically stronger than Player 2, by assumption. This means that the hazard rates of F1 and F2 are bounded apart for large valuations. Hence, for some λ > 0 and some U > max{u1 , u2 , 0}, f2 (u) f1 (u) − > λ for all u ≥ U. 1 − F2 (u) 1 − F1 (u) Following from the first step in the proof, u∗ = u∗1 = u∗2 > U for all T sufficiently large. Hence restrict attention to such T . For any u ∈ [max{u1 , u2 }, u∗ ) satisfying u > 0, Z u∗   1 f2 (x) f1 (x) Λ2 (u) − Λ1 (u) = − dx u x 1 − F2 (x) 1 − F1 (x) Z max{u,U }   Z u∗   1 f2 (x) f1 (x) 1 f2 (x) f1 (x) = − dx + − dx. min{u,U } x 1 − F2 (x) 1 − F1 (x) max{u,U } x 1 − F2 (x) 1 − F1 (x) Fix u and U , and allow T to grow large. The first term remains constant. As T → ∞, u∗ → ∞, and hence the second term satisfies Z u∗ u∗     1 f2 (x) f1 (x) − dx ≥ λ log → ∞. max{u,U } x 1 − F2 (x) 1 − F1 (x) max{u, U } This completes the second step in the proof. INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 29 For the third step, begin with the case where min{u1 , u2 } > 0. If u∗1 > u∗2 > 0, then Λ2 (u∗1 ) − Λ1 (u∗1 ) = Λ2 (u∗1 ) − Λ2 (u∗2 ). If u∗1 remains bounded as T → ∞, then the right hand side remains bounded, whereas the left hand side diverges to ∞ from the second step of the proof. This is a contradiction, and hence u∗1 → ∞. Thus, for T sufficiently large, I can ensure that Λ2 (u∗1 ) − Λ1 (u∗1 ) is positive. But the left hand side is negative, hence this is a contradiction. Thus u∗2 ≥ u∗1 > 0, and Λ2 (u∗2 ) − Λ1 (u∗2 ) = Λ1 (u∗1 ) − Λ1 (u∗2 ). Once again, if u∗2 remains bounded as T → ∞, then I obtain a contradiction: The right hand side remains bounded, and the left hand side diverges. I conclude that u∗2 → ∞, and hence for T sufficiently large u∗1 = u1 . This establishes the first claim in Proposition 6. For the second claim, note first that u∗1 = u∗2 = 0. Fix u > 0, and restrict to large enough T so that u∗ > u. Proposition 2 implies that Λ1 (u) = Λ2 (w(t1 (u))) and hence Z w(t1 (u)) f2 (x) Λ2 (u) − Λ1 (u) = Λ2 (u) − Λ2 (w(t1 (u))) = dx. u x(1 − F2 (x)) The left hand side diverges to ∞ as T → ∞, and hence w(t1 (u)) → ∞. Thus, conditional on u1 = u > 0, Pr[t2 ≥ t1 ] = Pr[u2 ≥ w(t1 (u))] → 0 as T → ∞. This establishes the second claim in the proposition. Finally, I turn to the third claim. If u ≤ 0, then the claim is true. Hence restrict to u > 0. If the claim is false, then I may construct a sequence T → ∞ such that t2 (u) ≥ ε > 0 throughout the sequence, and hence u > u∗2 . The stopping rule for Player 2 satisfies f2 (x)Λ−1 Z u Z u 1 (Λ2 (x)) f2 (x) t2 (u) = dx ≤ v(t2 (u)) dx ≤ −v(t2 (u)) log[1 − F2 (u)]. u∗2 1 − F2 (x) u∗2 1 − F2 (x) Employing Proposition 2 once more, Z u f1 (x) Λ2 (u) − Λ1 (u) = Λ1 (v(t2 (u))) − Λ1 (u) = dx. v(t2 (u)) x(1 − F1 (x)) Once again, the left-hand side diverges as T → ∞. For the equality to be maintained, it must be the case that v(t2 (u)) → 0. But this implies that, for sufficiently large T , t2 (u) < ε. This is a contradiction, and hence t2 (u) → 0 as T → ∞. This proves the third claim of the proposition.  A.2. Proofs of Propositions 4–6 for T = ∞ and max{ξ, β} > 0. Without loss of generality, I set ξ = 0 and β > 0 for all of these proofs. Recall that, since T = ∞, u∗i = ui . 30 DAVID P. MYATT Proof of Proposition 4 for T = ∞. When ui < ∞, integrate by parts to obtain Z ui log[1 − (1 − β)Fi (u)] log β log[1 − (1 − β)Fi (x)] Λi (u) = − − dx. u ui u x2 When ui = ∞, the same equality holds so long as the term log β/ui is set to zero. Suppose, contrary to the proposition, that u∗2 = u2 . This means that F2 (u∗2 ) = 0 and hence log[1 − (1 − β)F2 (u∗2 )] = 0. Since F1 FSD F2 it must be the case that u1 ≥ u2 , and hence u∗1 ≥ u1 ≥ u2 = u∗2 , and also u1 ≥ u2 . With these observations in hand, and for u∗1 < u2 , Z u∗1 Z u2 ∗ log β log[1 − (1 − β)F2 (x)] log[1 − (1 − β)F2 (x)] Λ2 (u2 ) = − − 2 dx − dx, u2 u∗2 x u∗1 x2 while the evaluation of Λ1 (u∗1 ) yields log[1 − (1 − β)F1 (u∗1 )] log β Λ1 (u∗1 ) = − u∗1 u1 Z u2 Z u1 log[1 − (1 − β)F1 (x)] log[1 − (1 − β)F1 (x)] − 2 dx − dx. u∗1 x u2 x2 Λ1 (u∗1 ) and Λ2 (u∗2 ) are well defined and equal. Equating them, obtain Z u2   Z u∗1 1 1 − (1 − β)F1 (x) log[1 − (1 − β)F2 (x)] 2 log dx = dx u∗1 x 1 − (1 − β)F2 (x) u∗2 x2  Z u1 log[1 − (1 − β)F1 (u∗1 )]  1 1 log[1 − (1 − β)F1 (x)] + ∗ + log β − − dx. (8) u1 u2 u 1 u2 x2 Since F1 FSD F2 the left hand side of Equation 8 is strictly positive:   1 − (1 − β)F1 (x) F1 (x) < F2 (x) ⇒ log > 0 ∀x ∈ (u∗1 , u2 ). 1 − (1 − β)F2 (x) Both of the first two terms on the right of Equation 8 are (at least weakly) negative. Hence, if the equation is to hold, the remaining terms must be strictly positive. Hence,   Z u1 Z u1   1 1 log[1 − (1 − β)F1 (x)] 1 1 1 log β − > dx ≥ log β dx = log β − . u 2 u1 u2 x2 u2 x 2 u2 u 1 Clearly, this is a contradiction, and Equation 8 cannot hold. The derivations above were for the case u2 > u∗1 . Suppose instead that u∗1 ≥ u2 . I may write Λ2 (u∗2 ) and Λ1 (u∗1 ) as Z u2 ∗ log β log[1 − (1 − β)F2 (x)] Λ2 (u2 ) = − − dx u2 u∗2 x2 Z u1 ∗ log[1 − (1 − β)F1 (u∗1 )] log β log[1 − (1 − β)F1 (x)] and Λ1 (u1 ) = ∗ − − dx. u1 u1 u∗1 x2 INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 31 Equating these two, as before, obtain Z u2 log[1 − (1 − β)F2 (x)] log[1 − (1 − β)F1 (u∗1 )] dx + u∗2 x2 u∗1 Z u1   log[1 − (1 − β)F1 (x)] 1 1 = dx − log β − . u∗1 x2 u2 u 1 The left hand side is strictly negative. Hence,   Z u1   1 1 log[1 − (1 − β)F1 (x)] 1 1 log β − > dx ≥ log β ∗ − ⇒ u∗1 < u2 , u2 u1 u∗1 x2 u1 u 1 which is, of course, a contradiction. I have a reached a contradiction in all cases, and hence the original supposition that u∗2 = u2 is false. This completes the proof.  Proposition 5 gives conditions under which t1 (u) > t2 (u), so that, when the players share the same prize valuation u, Player 1 fights for longer and wins the prize. For u ∈ (u∗1 , u∗1 )∩(u∗2 , u∗2 ) is is clear that this inequality will hold if and only if Λ2 (u)−Λ1 (u) > 0. Thus, in the following lemmas, I establish the properties of Λ2 (u) − Λ1 (u) when T = ∞. Lemma 1. If u1 > u2 > u > 0 and u ≥ max{u1 , u2 }, then limβ→0 [Λ2 (u) − Λ1 (u)] = ∞. Proof. Since u1 > u2 I may choose ũ satisfying u2 < ũ < u1 , and write Λ1 (u) Λ2 (u) − Λ1 (u) = Λ2 (u) − Λ1 (ũ) × . Λ1 (ũ) I now construct a lower bound for Λ2 (u), Z u2 (1 − β)f2 (x) Λ2 (u) = dx u x(1 − (1 − β)F2 (x)) 1 u2 (1 − β)f2 (x)   1 − (1 − β)F2 (u) Z 1 > dx = log , u2 u 1 − (1 − β)F2 (x) u2 β and an upper bound for Λ1 (ũ), Z u1 (1 − β)f1 (x) Λ1 (ũ) = dx ũ x(1 − (1 − β)F1 (x)) 1 u1 (1 − β)f1 (x)   1 − (1 − β)F1 (ũ) Z 1 < dx = log . ũ ũ 1 − (1 − β)F1 (x) ũ β Employing both of these inequalities, I obtain     1 1 − (1 − β)F2 (u) 1 1 − (1 − β)F1 (ũ) Λ1 (u) Λ2 (u) − Λ1 (u) > log − log × u2 β ũ β Λ1 (ũ)   log[1 − (1 − β)F2 (u)] Λ1 (u) log[1 − (1 − β)F1 (ũ)] 1 Λ1 (u) 1 = − − log β − . u2 Λ1 (ũ) ũ u2 Λ1 (ũ) ũ 32 DAVID P. MYATT I turn now to the ratio Λ1 (u)/Λ1 (ũ). Since u < u2 < ũ this satisfies Λ1 (u) Λ1 (u) − Λ1 (ũ) =1+ → 1 as β → 0. Λ1 (ũ) Λ1 (ũ) To verify this claim, notice that Λ1 (ũ) → ∞ as β → 0 and Z ũ Z ũ (1 − β)f1 (x) f1 (x) Λ1 (u) − Λ1 (ũ) = dx → dx < ∞. u x(1 − (1 − β)F1 (x)) u x(1 − F1 (x)) This means that    log[1 − F2 (u)] log[1 − F1 (ũ)] 1 1 1 lim inf [Λ2 (u) − Λ1 (u)] ≥ − + lim log − = ∞. β→0 u2 ũ β→0 β u2 ũ This completes the proof.  Lemma 2. Suppose that u1 = u2 = ∞. Suppose that, for some pair UH and UL > 0 satisfying UH ≥ UL ≥ max{u1 , u2 }, F1 (x) < F2 (x) for all x ≥ UH and f1 (x)/(1 − F1 (x)) < f2 (x)/(1 − F2 (x)) for all x ≥ UL . Then, for all u ≥ UL , lim inf β→0 [Λ2 (u) − Λ1 (u)] > 0. Thus, for this lemma, F1 FSD F2 and F1 HRD F2 in the upper tails of the distributions. Proof. Fix u ≥ UL and U > max{u, UH }. Integration by parts yields   Z ∞   1 1 − (1 − β)F1 (u) 1 1 − (1 − β)F1 (x) Λ2 (u) − Λ1 (u) = − log + log dx u 1 − (1 − β)F2 (u) u x2 1 − (1 − β)F2 (x)   Z U   1 1 − (1 − β)F1 (u) 1 1 − (1 − β)F1 (x) > − log + 2 log dx u 1 − (1 − β)F2 (u) u x 1 − (1 − β)F2 (x) where the inequality follows from the fact that the integrand of the second term on the right hand side of the equality is strictly positive for all x ≥ UH and hence all x ≥ U . For the terms on the right hand side of the inequality, I take limits as β → 0. First,     1 1 − (1 − β)F1 (u) 1 1 − F1 (u) log → log , u 1 − (1 − β)F2 (u) u 1 − F2 (u) and for the second term, Z U   Z U   1 1 − (1 − β)F1 (x) 1 1 − F1 (x) 2 log dx −→ 2 log dx u x 1 − (1 − β)F2 (x) u x 1 − F2 (x)  Z U    1 − F1 (u) 1 1 − F1 (u) 1 1 > log dx = log − . 1 − F2 (u) u x2 1 − F2 (u) u U To justify the limit as β → 0, notice that the integrand is continuous in x and β and converges to a continuous limit. The range of integration is the compact set [u, U ], and hence I may interchange order of limit and integration. The inequality follows from the fact that (1−F1 (x))/(1−F2 (x)) is strictly increasing for all x ≥ u, following from the assumption that the hazard of F1 is strictly lower than that of F2 for such x. I may conclude that, for INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 33 fixed u and any U chosen sufficiently large,   1 1 − F1 (u) lim inf [Λ2 (u) − Λ1 (u)] > − log . β→0 U 1 − F2 (u) Of course, this holds for all larger U as well. Hence, allowing U → ∞, I conclude that lim inf [Λ2 (u) − Λ1 (u)] ≥ 0. β→0 Notice that this weak inequality continues to hold for ũ > u. Of course, Z ũ   1 (1 − β)f2 (x) (1 − β)f1 (x) Λ2 (u) − Λ1 (u) = [Λ2 (ũ) − Λ1 (ũ)] + − dx, u x 1 − (1 − β)F2 (x) 1 − (1 − β)F2 (x) and the second term is strictly positive in the limit as β → 0. Hence the weak inequality derived above may be replaced with a strict inequality. This concludes the proof.  Lemma 3. Suppose that Player 1 is stochastically stronger than Player 2, and fix u > 0 satisfying max{u1 , u2 } > u ≥ min{u1 , u2 }. Then limβ→0 [Λ2 (u) − Λ1 (u)] = ∞. Proof. If u1 > u2 , then the claim follows directly from Lemma 1. Suppose instead that u1 = u2 = ∞. Player 1 is stochastically stronger than Player 2 by assumption (F1 AHRD F2 ) and hence there exists U > 0 such that, for all x ≥ U and some λ > 0, f2 (x) f1 (x) − > λ. 1 − F2 (x) 1 − F1 (x) Set UL = max{u, U }. Since the hazard rates are bounded apart in the tails, there exists some UH ≥ UL such that F1 (x) < F2 (x) for all x ≥ UH . This continues to be true for all larger UH . Now, Z UL   1 (1 − β)f2 (x) (1 − β)f1 (x) Λ2 (u) − Λ1 (u) = − dx u x 1 − (1 − β)F2 (x) 1 − (1 − β)F1 (x) Z UH   1 (1 − β)f2 (x) (1 − β)f1 (x) + − dx + Λ2 (UH ) − Λ1 (UH ) UL x 1 − (1 − β)F2 (x) 1 − (1 − β)F1 (x) Taking limits as β → ∞, the first term satisfies, Z UL   Z UL   1 (1 − β)f2 (x) (1 − β)f1 (x) 1 f2 (x) f1 (x) − dx → − dx u x 1 − (1 − β)F2 (x) 1 − (1 − β)F1 (x) u x 1 − F2 (x) 1 − F1 (x) which is a finite limit, and independent of UH . This follows, since the integrand is continuous in x and β, converges to a well-defined and continuous limit as β → 0, and the range of integration is a compact set. Thus the interchange of limit and integral is valid. Similarly, Z UH   Z UH   1 (1 − β)f2 (x) (1 − β)f1 (x) 1 f2 (x) f1 (x) − dx → − dx UL x 1 − (1 − β)F2 (x) 1 − (1 − β)F1 (x) UL x 1 − F2 (x) 1 − F1 (x) Z UH   1 UH ≥λ dx = λ log . UL x UL 34 DAVID P. MYATT Hence, Z UL   1 f2 (x) f1 (x) lim inf [Λ2 (u) − Λ1 (u)] ≥ − dx β→0 u x 1 − F2 (x) 1 − F1 (x)   UH + λ log + lim inf [Λ2 (UH ) − Λ1 (UH )] UL β→0 The first term is finite and independent of UH . The third term is positive, following an application of Lemma 2. The third term may be made arbitrarily large via a sufficiently large choice of UH . Hence Λ2 (u) − Λ1 (u) → ∞ as β → 0.  With these lemmas in hand, I may prove Propositions 5 and 6. Proof of Proposition 5 for T = ∞. Set UH = UL = u. Since F1 HRD F2 , the conditions of Lemma 2 are met. Hence Λ2 (u)−Λ1 (u) > 0 for β sufficiently small. Following the arguments given in the proof for T < ∞, it must be the case that t1 (u) > t2 (u).  Proof of Proposition 6 for T = ∞. The proof mirrors the third step in the proof for the case T < ∞. For instance, consider the case where min{u1 , u2 } > 0. If u∗1 > u∗2 , then Λ2 (u∗1 ) − Λ1 (u∗1 ) = Λ2 (u∗1 ) − Λ2 (u∗2 ). If u∗1 remains bounded away from u1 as β → 0, then the right hand side remains bounded, whereas the left hand side diverges to ∞ from Lemma 3. This is a contradiction, and hence u∗1 → ∞. Thus, for β sufficiently small, I can ensure that Λ2 (u∗1 ) − Λ1 (u∗1 ) is positive. But the left hand side is negative, hence this is a contradiction. Thus u∗2 ≥ u∗1 > 0. Continuing in this manner, in an identical manner to the case T < ∞, I may establish that u∗2 → u2 and hence, for β sufficiently small, that u∗1 = u1 . This establishes the first claim in Proposition 6. Other claims follow in a similar fashion.  A.3. Sensitivity Analysis. The proof of Proposition 7 and other numerical calculations for the uniform case are based on the following lemma. Lemma 4. Setting ξ = 0 without loss of generality, for Example 1 Λi (u) satisfies   1−β ui × [ui − βui − (1 − β)u] Λi (u) = log (9) ui − βui uβ(ui − ui ) Proof. Fi (x) = (x − ui )/(ui − ui ) and fi (x) = 1/(ui − ui ) so that ui − βui − (1 − β)x 1 − (1 − β)Fi (x) = , u i − ui which implies that (1 − β)fi (x) 1−β = x(1 − (1 − β)Fi (x)) x[ui − βui − (1 − β)x] INSTANT EXIT FROM THE ASYMMETRIC WAR OF ATTRITION 35 Write Equation 9 as     1−β ui Λi (u) = log + log [ui − βui − (1 − β)u] − log u , ui − βui β(ui − ui ) and differentiate to obtain   ∂Λi (u) 1−β (1 − β) 1 1−β =− + = . ∂u ui − βui ui − βui − (1 − β)u u u[ui − βui − (1 − β)u] Furthermore, evaluate Λi (u) at u = ui to obtain Λi (ui ) = 0. Hence Equation 9 is correct.  Proof of Proposition 7. Suppose that u1 > u2 . Then Player 1 is stochastically stronger than Player 2, and Proposition 6 applies directly. Next consider the case where u1 = u2 = u but u2 < u1 . Hence u1 and u2 share the same distribution above u1 , and F1 is really a truncation of F2 . Write Λi (u) as     1−β u(u − βui − (1 − β)u) Λi (u) = log − log β u − βui u(u − ui )     1 u(u − u) 1−β → log − log β u u(u − ui ) u − βui so that Λi (u) tends to a finite term (the first) plus a term that diverges to ∞ (the second). Hence, take the difference Λ2 (u) − Λ1 (u) to obtain   1 (u − w)v u − u1 (1 − β)(u2 − u1 )β log β Λ2 (w) − Λ1 (v) → log × − . u (u − v)w u − u2 (u − βu1 )(u − βu2 ) The last term tends to zero, since β log β → 0 as β → 0. In equilibrium, Λ1 (v(t)) = Λ2 (w(t)). Hence w(t) v(t) u − u1 v(t) = × < ⇒ w(t) < v(t) u − w(t) u − v(t) u − u2 u − v(t) Notice that in this example F1 FSD F2 , and hence Player 2 must “instantly exit” with positive probability. Thus v(0) = u1 . Hence w(0) u1 u − u1 u1 u1 = × = ⇔ w(0) = u − w(0) u − u 1 u − u2 u − u2 1 + (u1 − u2 )/u which completes the proof.  Next, keeping Lemma 4 in hand, with u1 = u2 = 0, Λi (u) reduces to   1−β ui − (1 − β)u Λi (u) = log . ui βu Via Proposition 2, I may solve for v(t) in terms of w(t). Simple algebra reveals that "  u /u #−1 v(t) u2 − (1 − β)w(t) 1 2 = (1 − β) + β . u1 βw(t) 36 DAVID P. MYATT This is the expression used for the generation of the first pane of Figure 1. Similarly, to construct the second pane, notice that I will be fixing the value of u1 = v(t), hence "  u /u #−1 w(t) u1 − (1 − β)v(t) 2 1 = (1 − β) + β . u2 βv(t) w(t)/u2 is the probability that Player 2 does not fight until time t in equilibrium, and hence is the probability that Player 1 actually wins. For the exponential distributions (Example 2), the following lemma pins down the solutions—the proof follows from simple algebra. Lemma 5. Setting ξ = β = 0 but T < ∞, for Example 2 Λi (u) = λi log[u∗ /u] and p λ 2 − λ1 ∗ λ1 ∗ 1 + 16T λ1 λ2 /(λ1 + λ2 ) − 1 log w(t) = log u + log v(t) where u = . (10) λ2 λ2 4λ1 λ2 /(λ1 + λ2 ) Proof. The first claim follows from simple integration. From Λ1 (v(t)) = Λ2 (v(t)) I obtain λ1 log[u∗ /v(t)] = λ2 log[u∗ /w(t)], and obtain the first part of Equation 10. 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