Asymmetric all-pay auctions with spillovers

Betto, Maria; Thomas, Matthew W. Article Asymmetric all-pay auctions with spillovers Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Betto, Maria; Thomas, Matthew W. (2024) : Asymmetric all-pay auctions with spillovers, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 1, pp. 169-206, https://doi.org/10.3982/TE5108 This Version is available at: https://hdl.handle.net/10419/296457 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 169–206 1555-7561/20240169 Asymmetric all-pay auctions with spillovers Maria Betto Department of Economics, Northwestern University Matthew W. Thomas Department of Economics, Northwestern University When opposing parties compete for a prize, the sunk effort players exert during theconflictcanaffectthevalueofthewinner’sreward.Thesespilloverscanhave substantial influence on the equilibrium behavior of participants in applications such as lobbying, warfare, labor tournaments, marketing, and R&D races. To understand this influence, we study a general class of asymmetric, two-player all-pay auctions where we allow for spillovers in each player’s reward. The link between participants’ efforts and rewards yields novel effects; in particular, players with higher costs and lower values than their opponents sometimes extract larger payoffs. Keywords. All-pay, contests, auctions, spillovers, war of attrition. JEL classification. C65, C72, D44, D62, D74. 1. Introduction All-pay auctions, or contests, model strategic interactions among players who must expend some non-refundable effort in order to win a prize. They have been applied in diverse settings such as labor (Rosen (1986)), R&D races (Che and Gale (1998), Dasgupta (1986)), and litigation (Baye, Kovenock, and de Vries (2005)). For tractability, the recent literature mostly assumes that players’ actions affect their opponent’s probability of winning, but not the value of the prize. Yet in many settings, such spillover effects on the prizes themselves arise naturally. For example, consider the setting in Che and Gale (1998), where two lobbyists compete in an all-pay auction to win an incumbent politician’s favor through campaign contributions. If the politician were instead a candidate running for office, then she would only be able to provide the reward if successfully elected. In this case, it is natural to assume that total campaign contributions increase the candidate’s chances of prevailing. Therefore, each lobbyist’s contributions increase her opponent’s value for winning the politician’s political favor. This raises new questions: is it better to curb one’s own contributions to make their opponent lose interest or is it preferable to ramp up the competition? These questions have been largely left unanswered. Maria Betto: [email protected] Matthew W. Thomas: [email protected] We thank Wojciech Olszewski, Alessandro Pavan, Marciano Siniscalchi, Bruno Strulovici, and Asher Wolinsky for invaluable comments throughout the writing process. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5108 170 Betto and Thomas Theoretical Economics 19 (2024) In other settings, spillovers may be designed. Consider an all-pay version of a standard labor tournament in which division managers apply effort toward some production technology in order to win a promotion awarded to the most productive division. To maximize aggregate effort, a principal might choose to make the value of this promotion depend on everyone’s performance in the contest. For example, if the promotion is for a partnership or involves stock options, the prize will be increasing in the efforts of all players. The effect that such compensation schemes have on the equilibrium has not yet been studied. This paper fully identifies the equilibrium strategies and payoffs in general twoplayer auctions with spillovers and establishes their uniqueness.1We consider games with (i) complete information, (ii) deterministic prizes, (iii) at least partially sunk investment costs, and (iv) a general dependence of each participant’s value for the prize on both players’ actions. The key contribution of this paper lies in incorporating (iv). Indeed, all-pay contests without spillovers were extensively studied by Siegel (2009,2010). These papers fully characterize equilibrium strategies and payoffs in games where contestants incur some (partially) unrecoverable cost, such as effort, in order to compete for prizes. We generalize the two-player single-prize version of their model to allow for general spillovers to affect the winner’s payoff. Our paper also has some overlap with the symmetric linear contests with spillovers studied in Baye, Kovenock, and de Vries (2012). Unlike their work, however, we restrict attention to the all-pay case, but allow for asymmetric equilibria and nonlinear payoffs. Even in the symmetric, linear all-pay auction with spillovers, we note that no previous paper that we are aware of has established equilibrium uniqueness. The addition of spillovers can have a significant impact on equilibrium behavior. First, players with strictly higher costs can have higher payoffs than those with lower costs, even if their value functions for the prize are identical. In fact, in some settings, players could increase their payoffs if they were allowed to commit to a schedule of costly handicaps (see Section 4). Thus, trying to favor an “underdog” participant in a contest by means of reducing their costs may very well have the opposite effect, and in fact decrease their welfare in equilibrium. This is also important in settings in which players can commit to increasing their costs (e.g., by selecting an inefficient technology), as they may choose to do so. Another contribution of this paper is the procedure to construct equilibrium strategy profiles. The equilibrium strategy distributions of asymmetric all-pay contests have two distinct parts: the densities and a mass-point at 0. In the literature on all-pay contests without spillovers, starting with Baye, Kovenock, and de Vries (1996), expected payoffs are obtained independently of the equilibrium distribution. This independence is exploited to derive the probability mass at 0 for the weaker player from the payoffs, which is then used to compute the densities. In the presence of spillovers, however, a player’s payoffs cannot be derived without the equilibrium strategy of their opponent. Because of this, the same process cannot be followed. To overcome this difficulty, we introduce 1This paper also establishes the existence of equilibrium, though this result has already been proven; see Olszewski and Siegel (2023), for example. Our method, however, differs substantially from the previous literature. Theoretical Economics 19 (2024) Asymmetric all-pay auctions 171 an algorithm that works in exactly the opposite order: first, it solves for the density independently of the mass-point, and then uses this density to find the probability mass at 0. Our method capitalizes on the theory of Volterra integral equations (VIEs), which are integral equations with a unique fixed-point that can be obtained via iteration. To the best of our knowledge, these techniques have not previously been applied to the determination of equilibrium mixed-strategy profiles.2 The game we study is general enough to encompass many different applications in which spillovers matter. In particular, investment wars, contests with winner’s regret, and militaristic conflicts all fit our framework, since spillovers are key in each of these settings. Our model also subsumes a natural extension to the war of attrition. which, unlike the classical model, yields a unique equilibrium on a bounded support. We are also able to use the same framework to describe wars of attrition where rational agents face uncompromising (never-yielding) types with positive probability, as in Abreu and Gul (2000)andKambe (2019). Our approach identifies why these games admit unique equilibria when the regular war of attrition does not: the addition of an uncompromising type introduces an unavoidable cost that depends on a player’s own score, and we show that this single characteristic is sufficient in ensuring a unique equilibrium. Finally, we extend the analysis to more than two players. The uniqueness result does not hold when the number of bidders exceeds two. We are nonetheless able to characterize a class of asymmetric equilibria when (appropriately normalized) costs are ranked. In this case, we show that only two players participate in equilibrium. In addition, we are able to fully characterize the unique symmetric equilibrium for all-pay auctions with (i) more than two identical players, (ii) multiple homogeneous prizes, and (iii) spillovers generated by the first runner-up. This setting accommodates a broad class of games including wars of attrition and auctions with winner’s regret with any number of players and prizes. This extends the usefulness of our novel methodology. The paper is organized as follows. We introduce the model, the equilibrium concept, and the assumptions in Section 2. We construct the equilibrium and prove its uniqueness in Section 3.Section4presents sufficient conditions under which a player has a positive expected payoff. This includes an example where a player with higher costs and lower values receives a positive expected payoff, while her opponent receives 0. Section 5contains useful results on closed forms that allows for simplified equilibrium computations in certain special cases. We illustrate their usage in the following Section 6, which is dedicated to applications. Sections 6.2,6.4,and6.3 in particular showcase closed-form solutions. Section 6.1 introduces a general perturbation of the classic war of attrition that ensures the equilibrium is unique. This perturbation admits the war of attrition with the possibility of an uncompromising type as a special case. In Section 7, we extend the analysis to contests with more than two players. Uniqueness no longer holds generally, though we are still able to find the unique symmetric equilibrium of a n-player, m-prize all-pay auction with spillovers. Finally, in Section 8,wereviewthe related literature and discuss the results. 2Few other works in economics use VIE methods in general. We note McAfee, McMillan, and Reny (1989) and McAfee and Reny (1992) as some early examples. More recently, Gomes and Sweeney (2014)alsoused VIEs, to compute the unique efficient equilibrium bidding functions in generalized second-price auctions. 172 Betto and Thomas Theoretical Economics 19 (2024) 2. Model We focus, for now, on auctions with two participants. Extensions with more players are considered in Section 7, where we show that the symmetric equilibrium of an auction with any number of identical players and prizes is just a transformation of the equilibrium of the two-player case. An asymmetric auction with spillovers is a family {I,{˜ Si}i∈I,{ui}i∈I},wherethefollowing statements hold: (a) The index set of players is I:={1, 2}. (b) For each i∈I,˜ Si:=[0, ∞)is player i’s action space, i.e., her set of available scores (or bids). We use a tilde because a later assumption will allow us to replace the action set with a bounded interval. We let s−iand ˜ S−idenote the action and action space, respectively, of player j=i. (c) For each i∈I,ui:˜ S→Ris player i’s payoff, where ˜ S:=i∈I˜ Si.Lets:=(si;s−i) denote an arbitrary element of ˜ S. Then, for each (si;s−i), we further define ui(si;s−i):=pi(si;s−i)vi(si;s−i)−ci(si), where (i) pi(si;s−i)denotes the probability that iwins the prize given the score profile (si;s−i),withpi(si;s−i)=1−p−i(s−i;si)and pi(si;s−i)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1ifsi>s −i αi∈[0, 1]if si=s−i 0ifsi<s −i; (ii) vi:˜ S→R+maps each score profile (si;s−i)to player i’s value vi(si;s−i)from winning the prize; (iii) ci:˜ Si→R+outputs player i’s private cost ci(si)given her submitted score si. Definition 1 (Two-Player All-Pay Auction With Spillovers). A two-player all-pay auction is said to have spillovers if, for some i∈Iand si∈˜ Si,thereexistss−i,ˆ s−i∈˜ S−isuch that vi(si,s−i)=vi(si,ˆ s−i), i.e., the prize’s value for at least one player and an action of that player are not constant in their opponent’s action. Accommodating spillovers is the distinguishing feature of our analysis. As is standard, we are interested in characterizing the Nash equilibrium of these general contests. Definition 2 (Best Responses). Consider a two-player all-pay auction {I,{˜ Si}i∈I, {ui}i∈I}. For each i∈I,let˜ Sidenote the set of probability distributions on ˜ Siand let ˜ S:=i∈I˜ Si.Playeri’s best response set bi(G−i)to G−i∈˜ S−iis given by bi(G−i):=argmax s∈˜ Si˜ S−i ui(s;s−i)dG−i(s−i). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 173 Definition 3 (Nash Equilibrium). Consider the two-player all-pay auction {I,{˜ Si}i∈I, {ui}i∈I}. A Nash equilibrium of this game is a profile G:=(G i)i∈I∈i∈I(˜ Si),where, for each i∈I,G i’s induced probability measure assigns measure 1 to bi(G −i). 2.1 Assumptions The following assumptions are imposed throughout whenever a two-player all-pay auction is invoked. Appendix III shows that none of these assumptions is superfluous to our results. Assumption 1 (A1, Smoothness). The function vi(si;y)is continuously differentiable in siand continuous in yfor all i∈I,si∈˜ Si,andy∈˜ S−iwith si≥y. The function ci(si)is continuously differentiable in sifor all i∈I,si∈˜ Si. Assumption 2 (A2, Monotonicity). For all i∈Iand si>0,c i(si)>0and v i(si;y)<c  i(si) for almost all y,wherev i(s;y):=∂vi(si;y)/∂si. Assumption 3 (A3, Interiority). For all i∈I, vi(0, 0)>c i(0)=0and lim si→∞ sup y∈˜ S−i vi(si;y)<lim si→∞ci(si). Versions of Assumptions A1,A2,andA3are adopted by most papers in the all-pay auction literature. Assumption A2formalizes the sense in which these contests are allpay, since bids are costly for both the winner and the loser.3Assumption A3ensures that bids are positive and bounded. Note that, for each i∈I,thereexistTi∈˜ Sisuch that player iwill never choose a score s≥Ti. Thus, we can restrict the action space to Si:=[0, Ti]. Assumption 4 (A4, Discontinuity at Ties). For all i∈Iand s∈Si∩S−i, vi(s;s)>0. Assumption A4is a novel, yet natural assumption. It states that agents would prefer to win a tie than to lose one. It is satisfied if the prize is always valuable (i.e., winning is better than losing) or if there are no spillovers. To see that it is never violated in the absence of spillovers, note that Tiis less than or equal to any xsatisfying vi(x;y)≤ci(x) for all y≤x. If there are no spillovers and vi(s)≤0forsomes≤Ti,thenci(s)≤0. Therefore, s=0, which violates Assumption A3. Note that this assumption is equivalent to assuming a discontinuity in payoffs at ties because Assumptions A1and A3guarantee that vi(s;s)=0 implies Assumption A4. 3We note that Assumption A2does exclude situations where a higher score is not necessarily more costly. Siegel (2014) discusses contests with nonmonotonic costs, allowing for competitors with head starts and the provision of performance-based subsidies. These contingencies are excluded from our analysis. 174 Betto and Thomas Theoretical Economics 19 (2024) 3. Characterization of equilibrium By standard arguments contained in the Appendix, any pair of equilibrium strategies will be mixed with support on some interval [0, ¯ s], and at most one player will have a mass-point at 0. Players must, therefore, be indifferent between all points of their interval support: ¯ ui(G−i):=s 0 vi(s;y)dG−i(y)−ci(s)for all s∈[0, ¯ s].(1) Any pair of distributions (G1,G2)thatsatisfy(1) is an equilibrium. This paper’s main contribution to the literature is in characterizing the solution to this system of equations and in showing that it is unique. Theorem 1. Every two-player all-pay auction has a unique Nash equilibrium (G i)i∈I∈ i∈I(Si)in mixed strategies. Furthermore, G i(s)=s 0˜ gi(y)dy +¯ si ¯ s˜ gi(y)dy,(2) where ˜ gi(s)solves ˜ gi(s)=c −i(s) v−i(s;s)−s 0 v −i(s;y) v−i(s;s)˜ gi(y)dy,(3) ¯ sisolves ¯ si 0˜ gi(y)dy =1,and¯ s=mini∈I¯ si. The solution admits the representation ˜ gi(s)=c −i(s) v−i(s;s)+s 0 r−i(s;y)c −i(y) v−i(y;y)dy, where r−i(s;y):=−k0 −i(s;y)+k1 −i(s,y)−k2 −i(s;y)+··· for k0 −i(s;y):=v −i(s;y) v−i(s;s)and kn −i(s;y),n=1, 2, , defined recursively by kn −i(s;y):=s y v −i(s;z) v−i(s;s)kn−1 −i(z;y)dz. We outline the proof here with an emphasis on the general methodology. We show in the Appendix that in any equilibrium, players choose strictly increasing, continuous mixed strategies with common support on some interval [0, ¯ s],asin(1), and that at most one participant can have a mass-point at 0. Moreover, differentiating (1) yields (3), which must be satisfied on [0, ¯ s]in equilibrium for some ¯ s(Lemma 0in the Appendix). The key step is recognizing that we can apply results about Volterra integral equations (VIEs) to show that (3) has a unique solution. The relevant result is summarized in the following lemma. For a proof, see, e.g., Brunner (2017). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 175 Lemma 1(Volterra (1896)). Let K(s;y)and f(s)be continuous functions. Then the integral equation g(s)=f(s)+s 0 K(s;y)g(y)dy for all s∈[0, ¯ s](4) has a solution gthat is unique almost everywhere. Moreover, (4) defines a contraction mapping, implying the solution can be found by iteration. This iteration reduces to g(s)=f(s)+s 0 R(s;y)f(y)dy, where R(s;y)is the unique resolvent kernel defined by R(s;y)=∞  m=0 Km(s;y), where K0≡Kand Kmis defined recursively for m=1, 2,  as Km(s;y)=s y Km−1(s;z)K(z;y)dz. Note that (4)isthesameas(3)forf(s):=c −i(s)/v−i(s;s)and K(s;y):=−v −i(s;y)/ v−i(s;s). So Lemma 1implies that only one pair of functions (˜ g1,˜ g2)solves (3). Next we show that the unique solutions are densities, i.e., for each ithere is an interval [0, ¯ si] where ˜ giis nonnegative and integrates to 1. Lemma 2. Assume a two-player all-pay auction where (˜ gi)i∈Isatisfies the indifference condition in (3). Then, for each i∈I,thereexists¯ si∈Sisuch that ¯ si 0˜ gi(y)dy =˜ Gi(¯ si)=1(5) and ˜ gi(s)is positive for s≤¯ si. Lemma 2is proven in the Appendix. We must now ensure that the two densities have the same support. The next key insight is that there is exactly one way to do this. Recall that at most one player can have an mass-point and that this mass-point must be at 0 (Lemma 0). If ¯ s1=¯ s2, then there is a unique equilibrium without any masspoint. Otherwise, order the players such that ¯ s1<¯ s2. Then give player 2 a mass-point of size 1 −˜ G2(¯ s1). By construction, both players’ densities integrate to 1 on the common support [0, ¯ s1]. The above can be performed via the following steps: Step 1. Find each ˜ gi(s).4 Step 2. Integrate each ˜ gi(s)to find ¯ sigiven by (5). 4Analytically, it can be expressed as a series or in closed form when possible (see Section 5)ornumerically (see Appendix IV). 176 Betto and Thomas Theoretical Economics 19 (2024) Step 3. Take ¯ s=mini¯ siand give each player a mass-point at 0 of size 1−˜ Gi(¯ s), which is positive for at most one player. The three steps are illustrated by Figure 1. Since the cumulative distribution functions are useful, we sometimes use the alternate expression presented in Corollary 1. Corollary 1. Consider a two-player all-pay auction where vi(s;y)is continuously differentiable in both arguments for all i∈I(Assumption A1guarantees differentiability in the first argument). Then we can alternatively express the unique equilibrium as Gi(s)=˜ Gi(si)−˜ Gi(s)+˜ Gi(s), where ˜ Gi(s)=c−i(s) v−i(s;s)+s 0 ∂v−i(s;y) ∂y ˜ Gi(y) v−i(s;s)dy.(6) The solution admits the series representation ˜ Gi(s)=c−i(s) v−i(s;s)+s 0 c−i(y) v−i(y;y) R−i(s;y) v−i(s;s)dy, where R−i(s;y):=K0 −i(s;y)+K1 −i(s,y)+K2 −i(s;y)+··· for K0 −i(s;y):=∂v−i(s;y)/∂y and Kn −i(s;y),n=1, 2, , defined recursively by Kn −i(s;y):=s y ∂v −i(s;z) ∂z Kn−1 −i(z;y) v−i(z;z)dz. We end this section with a note on parallels between our methodology and that used in the incomplete-information all-pay auction literature. The similarities are formal in nature, and arise because both problems involve solving a pair of differential (in the incomplete-information case) or integral (in our case) equations. In the incomplete-information setting of, e.g., Amann and Leininger (1996), the unique equilibrium—in pure strategies—is obtained as the solution to a pair of differential equations, which arise from taking first-order conditions of each players’ expected payoffs. To back out the mass of players types’ that bid 0, the authors then make use of the boundary condition where each players’ top type must, in equilibrium, choose identical top bids. In our setting with complete information, the unique equilibrium is instead in mixed strategies. It is obtained as the solution to a pair of integral equations, which arise from indifference, rather than first-order conditions: players’ payoffs must be invariant to any choice of bids within their mixed-strategy supports (1). Wearethenabletopindown Theoretical Economics 19 (2024) Asymmetric all-pay auctions 183 Proposition 4 (Multiplicative Margin of Victory Spillovers). Assume a two-player contest such (i) for some i,(vi(s;s))−1(∂vi(s;y)/∂y)=:ψi(s−y)depends only on the score differential s−y, and (ii) ∂vi(s;y)/∂y and ψiand ci(s)/vi(s;s)areofexponentialorder. Then, for all s∈(0, s], ˜ g−i(s)=L−1⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ xLci(s) vi(s;s) 1−Lψi(s)⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ and ˜ G−i(s)=L−1⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Lci(s) vi(s;s) 1−Lψi(s)⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , where Land L−1denote the Laplace and inverse Laplace transforms, respectively. Proposition 4can be used whenever the prize value is of the form vi(s;y)= v1 i(s)v2 i(s−y). For an application where we solve an all-pay auction with winner’s regret, see Section 6.4. 6. Applications 6.1 War of attrition with costly preparation The canonical war of attrition (WoA) is a game between two players i=1, 2. Each picks a score, which represents an exit time, in [0, ∞)and the player ito select the largest score siwins an amount that is decreasing in the loser’s choice s−iand constant in her own. Aplayer’spayofffunctionisthusgivenby ui(si;s−i)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fi(s−i)if si>s −i i(si)if si<s −i αifi(s−i)+(1−αi)i(si)if si=s−i, where fiand iare strictly decreasing, continuously differentiable functions such that fi(s)> i(s),lims→∞i(s)=−∞,i(0)=0, and αi=1−α−i∈(0, 1). The typical WoA admits multiple equilibria and, therefore, does not satisfy the assumptions in Section 2.1. In particular, it violates monotonicity (Assumption A2)and interiority (Assumption A3) because the payoff of the winner is constant (and, therefore, nondecreasing) in the player’s own score. We propose a general perturbation that selects a unique equilibrium of the WoA, and show that such a perturbation is solvable under our framework.8 8The problem of equilibrium selection in WoAs has been widely studied in the literature (Georgiadis, Kim, and Kwon (2022), Myatt (2005)). One way to select a unique equilibrium is to truncate the game, as in Ghemawat and Nalebuff (1985), so that at some point in finite time both players prefer to exit. A different 184 Betto and Thomas Theoretical Economics 19 (2024) Suppose the winner’s outcome is decreasing in her own score, even if this dependence is minimal, ui(si;s−i)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fi(s−i)−εi(si)if si>s −i i(si)−εi(si)if si<s −i αifi(s−i)+(1−αi)i(si)−εi(si)if si=s−i for any strictly increasing continuously differentiable function εiwith εi(0)=0and lims→∞εi(s)>f i(0)for all i. We denominate this variant a WoA with costly preparation, as there is some small preparation cost ε(s)incurred to set score s, i.e., the maximum amount of time sfor which one wishes to participat. For example, a company engaged in a price war might have to build up inventory in advance or secure a costly line of credit. A WoA with costly preparation fits the two-player all-pay auction with spillovers where vi(si;s−i):=fi(s−i)−i(si) ci(si):=εi(si)−i(si), which satisfy Assumptions A1–A4. Therefore, this game has a unique equilibrium, and there exists some ¯ ssuch that no player bids above ¯ s.Theorem1further allows us to characterize the equilibrium and Proposition 2gives a closed-form expression for the equilibrium strategies. As the preparation costs become small (with ε i(s)→0 uniformly for all s), the unique equilibrium of a WoA with costly preparation approaches the mixed-strategy equilibrium of the classic WoA. This is proven in the Appendix. The WoA with costly preparation generalizes other perturbations that have a unique equilibrium. For example, Abreu and Gul (2000)andKambe (2019)extendtheWoAto let a rational player’s opponent be of an uncompromising type with positive probability, where “uncompromising” describes someone who bids (or exits at) infinity. Let zidenote the (known) probability that player iis of an uncompromising type. Against such an opponent, a rational or compromising player loses with certainty. This is a special case of the WoA with costly preparation where εi(s):=−(z−i/(1−z−i))i(s). This relationship sheds light on the uniqueness of equilibrium found in the WoA with an uncompromising type. Indeed, by adding the possibility of a never-yielding opponent, we effectively introduce an unavoidable cost that depends on the player’s own score. As was shown in the WoA with costly preparation, this characteristic is actually sufficient for a unique equilibrium. way to select for an equilibrium, which we discuss in more detail, is to introduce a small probability that a player never exits. See, for example, Abreu and Gul (2000), Kambe (2019), Kornhauser, Rubinstein, and Wilson (1989). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 185 6.2 Offensive/defensive balance Military strategists generally agree that warfare is naturally asymmetric: the defending party can usually prevail with less expenditure of resources than the attacker (Clausewitz (1982)). More generally, scholars have tried to identify which factors influence the socalled offensive/defensive balance, that is, the many elements of military technology that generate either offensive or defensive advantages, and thus affect the probability of war (Levy (1984)). Our model is able to capture both the defensive advantage and the role of the prize-depleting nature of war in the offensive/defensive balance debate. An attacker (a) invades a defender’s (d) territory, which is worth V. Both combatants purchase costly scores in [0, ∞), and the combatant with the higher score wins. A score of sicosts cisi,whereci>0 is a positive constant, for player i∈I:={a,d}. Furthermore, a’s score inflicts δasadamage to the territory. Assuming the defender also inflicts a cost of δdsdonto the attacker does not change the analysis. If the attacker wins, it internalizes all costs faced by the defender, as these costs effectively depleted the resources available from the territory. Consider the payoff functions ua:[0, ∞)→Rfor the attacker, ua(sa,sd)=pa(sa,sd)(V−δasa)−casa, and the payoff function ud:[0, ∞)→Rfor the defender, ud(sa,sd)=1−pa(sa,sd)(V−δasa)−cdsd, where pa(·):[0, ∞)2→[0, 1]denotes the probability that the attacker is victorious. Accordingly, we let pa(sa,sd)=1wheneversa>s d,pa(sa,sd)=0whensa<s d,and pa(sa,sd)=λ∈[0, 1]whenever sa=sd. When we transform this model into our framework, we get ci(si):=cisiand va(sa;sd)=vd(sd;sa):=V−δasa. Assume it costs weakly more to attack than to defend (i.e., ca≥cd). The attacker does not have any spillovers, while the defender is harmed by her opponent. We are able to leverage the linearity of payoffs in this case to obtain a closed-form solution to the problem using Proposition 2. The defender receives positive payoffs if and only if ¯ sd=V ca+δa <V δa1−exp−δa cd=¯ sa, which holds whenever δa>0andca≥cd.Inthiscase, Ga(s)=1+cd δa logcaV (ca+δa)(V−δas)and Gd(s)=cas V−δas. The probability P(sa>s d|δa,ca,cd)that the attacker succeeds, in equilibrium, is given by P(sa>s d|δa,ca,cd)=cd δ2 aδa+calogca ca+δa<cd 2ca≤1 2, 186 Betto and Thomas Theoretical Economics 19 (2024) where the supremum is reached as δa→0. If the war damages the territory at least as much as it costs the attacker to inflict such damage (δa≥ca), a tighter bound is obtained: P(sa>s d|δa,ca,cd)<1−log(2)<1 3. Even if ca=cd, the defender is more than twice as likely to win than the attacker is. In our model, the stronger position of the defensive party comes as a byproduct of the inverse relationship between the attacker’s strength and the erosion of the prize’s value. This provides an alternate explanation for why it is typically easier to defend than to attack, something usually attributed to the high costs of maintaining long supply lines and of keeping seized territories (Glaser and Kaufmann (1998)). The defender’s stronger position also suggests that any positive participation cost in a war contest imposed on the aggressor would be effective in discouraging aggression.9 6.3 War of investment Investment has long been considered as a method of committing to entry deterrence (Dixit (1980)), while the war of attrition is a popular model of exit (Fudenberg and Tirole (1986)). Our model can combine the two attributes into a single model of competition in continuous time, where players invest to stay in the game, but are able to recoup part of that investment if their opponent invests less. Wars of investment can also be used to model Cold-War-style defense spending and competition between technology companies and R&D races. Assume two competitors, 1 and 2, invest in capital siat cost ci(si).Thecapitalisnecessary to engage in competition and depreciates at a constant rate. Competition results in zero profits. However, the winner is able to extract monopoly profits and benefits from the remaining capital according to an increasing function vi(si−s−i). More concretely, assume payoffs are u(si;s−i)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ vi(si−s−i)−ci(si)if si>s −i −ci(si)if si<s −i αivi(0)−ci(si)if si=s−i for any αi∈[0, 1). If Assumptions A1–A4are met, there is a unique equilibrium of capital investments in mixed strategies on finite support. Moreover, the equilibrium admits a closed-form solution by Proposition 3. Example 2. Let vi(s;y):=eρi(s−y)ωiand ci(s)=eρis−1, where ωi,ρi∈(0, 1)for each i∈I:={1, 2}.Then ˜ gi(s)=ρ−i ω−i , 9In the more general nonlinear model, where the value of the territory after invasion is given by vδ(sa) and the cost of choosing score sito player iis given by a continuously differentiable function ci:[0, ∞)→ R+satisfying the required Assumptions A1–A4,cd(s)≤ca(s)is sufficient to ensure that ˜ Ga(s)<˜ Gd(s)for all s>0. This guarantees the defender’s payoff remains positive, with Gd(s)=ca(s)/vδ(s). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 187 so the equilibrium strategies, excluding the possible mass-point at 0, will be uniform with ˜ Gi(s)=(ρ−i/ω−i)s. The pair of ratios ωi/ρiis, therefore, a sufficient statistic for the equilibrium of this game. Assume, without loss of generality that this ratio is weakly larger for player 1. Then the maximum duration of the game is player 2’s ratio ¯ s=ω2/ρ2. The equilibrium is fully characterized by the overall strength of the players ¯ sand the competitive balance δ:=(ω2/ρ2)/(ω1/ρ1)∈(0, 1]. Because the strategies are uniform, player 1’s average commitment duration is half of the strength. Player 2 on the other hand has a mass-point of size G2(0)=1−δ, which decreases as the competition becomes more balanced. Overall, the conflict is expected to last for Emin(s1,s2)=¯ s 01−G1(y)1−G2(y)dy =δ¯ s 3 total periods. The relationship between overall power and war duration is one to one. The duration is also increasing in the competitive balance. So a large strength differential implies the conflict will typically be short-lived, whereas close contests can have delayed resolutions. ♦ 6.4 All-pay auction with winner’s regret Winner’s regret is the remorse that the winner has from spending more than is necessary to win a contest or auction. This phenomenon has mostly been studied in the context of winner-pay first-price, auctions (Engelbrecht-Wiggans (1989), Filiz-Ozbay and Ozbay (2007)). We instead apply our framework to model winner’s regret in an all-pay auction. Let each player i∈I:={1, 2}choose a score in [0, ∞). Suppose ivalues the prize at μi(si)[1−hi(si−s−i)],whereμi(si)is the player’s objective value of the prize and hi(si−s−i)is the share of the winnings that is unappreciated due to regret. Each player pays the cost ci(si)whether he/she wins or loses. So payoffs are u(si;s−i)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ μi(si)1−hi(si−s−i)−ci(si)if si>s −i −ci(si)if si<s −i αiμi(si)−ci(si)if si=s−i for any αi∈[0, 1). We assume all functions are continuously differentiable with c(s)>0 and h i(s)≥0. Moreover, μ(0)>h (0)=c(0)=0andc(s)>μ (s)for each s,sothat lower bids are preferable even with no regret. Intuitively, the regret function hshould not exceed 1, though this is not a technical requirement. We can solve for the equilibrium with two players using Proposition 4and we can extend this equilibrium to the game with any number of identical players or prizes using Theorem 4. 188 Betto and Thomas Theoretical Economics 19 (2024) Example 3. Let μi(s):=ωi∈(0, 1 2),hi(s)=s2/2, and ci(s):=s−s2/2fors∈[0, 1].Then ˜ gi(s)=e−s ω−i . Without loss of generality, let ω1≥ω2, implying player 1 receives a nonnegative payoff. Player 1 will thus play a truncated exponential distribution with parameter 1 and support [0, −log(1−ω2)]. Her expected score will be E[s1|ω1,ω2]=1+1−ω2 ω2log(1−ω2), which depends negatively on her opponent’s payoff scaling factor ω2.Thisislowerthan in the same game without regret. The player with zero expected payoffs will place a mass-point at 0 of size G2(0)=1−ω2 ω1 , which is exactly the same size as if there were no regret. Player 2 will have an expected score E[s2|ω1,ω2]=ω2 ω11+1−ω2 ω2log(1−ω2), which is also less than in the same game without regret. The expected sum of the two scores score is E[s1+s2|ω1,ω2]=1+ω2 ω11+1−ω2 ω2log(1−ω2), which is decreasing in ω1and increasing in ω2. In contests such as labor tournaments, a large productivity differential between participants in the form of a high ω1and low ω2depresses aggregate effort. This is true in a contest with no spillovers, but the partial derivative of ω1is larger in absolute value when there is regret. That is, the effect is exacerbated by the fact that the stronger player is penalized for winning by a large margin. ♦ 7. More players In contests with spillovers and more than two players, many of the results considered here are violated. Existence still holds (see Olszewski and Siegel (2023)), but uniqueness does not. Moreover, expected payoffs will now depend on which equilibrium is played.10 When the normalized costs are ranked, Theorem 2 in Siegel (2010) and Theorem 2 in Siegel (2009) show that only two players ever participate in the equilibrium of a contest for a single prize. This effectively collapses the problem into a two-player contest. A version of this condition holds in our setting. We still require normalized costs to be ranked in some sense, but in a way that takes the spillovers into account. 10This is also true of contests with no spillovers if monotonicity does not hold (Siegel (2009, Example 2)). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 189 Theorem 3. Assume iand j,i= j,aretwoofthen>2players in a contest satisfying Assumptions A1–A4. Suppose that player ihas a positive payoff in the two-player contest where iand jare the participants, and that the following “ranked costs” condition holds for all k/∈{i,j},s∈˜ Sk,si∈˜ Si,andsj∈˜ Sj ck(s) vk(s;s{i,j})≥cj(s) vj(s;s{i}), (11) where sHis a vector of opponent scores that is 0for all players not in set H. Then there exists an equilibrium where only players iand jparticipate. To understand condition (11), consider the candidate equilibrium where players i and jcompete using their two-player strategies, and player kdoes not participate. By not participating, player kearns a payoff of 0—the same payoff as player j.Condition (11) says that if she enters, player k’s normalized cost will be higher at every point than player j’s already is. Therefore, her payoff from participating is at most 0 (player j’s payoff). So there is no profitable deviation for any player. Note that it is possible for kjand jkin the sense of (11) when spillovers decrease the value of the prize. In this case, there are multiple equilibria where different pairs of players participate. In the absence of spillovers, multiple equilibria also arise with three or more players. However, if payoffs are asymmetric, there can be at most one equilibrium where the support of each player’s strategy is a union of intervals. Additionally, the payoffs of each player are consistent across all equilibria. Neither of these properties holds in contests with spillovers. Payoffs generally vary across equilibria in which different players participate. 7.1 Symmetric equilibria The same method used to find the equilibrium of two-player auctions with spillovers can be applied more generally to find symmetric equilibria of all-pay auctions with n>2 identical players and m<nprizes, where the value of the prize for any given participant depends on his/her own score and on the score of the first runner-up (the player with the m+1th highest bid). More specifically, each prize has value v(s;y),wheresis the player’s own score and yis the score of the first runner-up. When there is only one prize, this amounts to saying that its value depends only on the two highest bids. Spillovers depend only on the score of the runner-up in many games such as the all-pay auction with winner’s regret and any game with a structure that resembles a war of attrition. For example, bargaining games and free riding games frequently have this structure where the last holdout to comply delays the prize for the winners. In the case where there is one prize, spillovers that depend on the first runner-up capture the margin of victory, which is relevant in many applications including elections and R&D races. Formally, we define a symmetric auction with runner-up spillovers as a family {I,P,{˜ Si}i∈I,{ui}i∈I}, where the following conditions hold: (i) The index set of players is I:={1, 2, ,n}with n≥2. 190 Betto and Thomas Theoretical Economics 19 (2024) (ii) The index set of prizes is P:={1, 2, ,m}with m<n. (iii) For each i∈I,˜ Si:=[0, ∞)is player i’s action space. We let s−idenote an arbitrary element of ˜ S−i:=j=i˜ Sj.Wefurtherlets(j)denote the jth highest score. (iv) For each i∈I,ui:˜ S→R,where ˜ S=i∈I˜ Si. For each s:=(si;s−i)∈˜ S,wefurther define ui(s):=pi(s)v(si,s(m+1))−c(si), where (a) pi(s)denotes the probability that iwins a prize given the score profile s,withi∈Ipi(s)=mand pi(s)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1ifsi≥s(m)>s (m+1) 1 {k∈I:sk=s(m)}if si=s(m)=s(m+1) 0ifsi≤s(m+1)<s (m); (b) v:[0, ∞)2→R+maps each pair of scores (si,s(m+1))to player i’s value v(si;s(m+1))from winning the prize; (c) c:[0, ∞)→R+outputs player i’s private cost c(si)given her submitted score si. In contrast with the two-player case introduced in Section 2, here we assume all players are symmetric in the sense that they have identical value (v)andcost(c) functions. Moreover, all prizes are equally valuable to each player iconditional on (si,s(m+1)). In this context, we are able to use the two-player, one prize equilibrium characterized in Theorem 1to construct the symmetric equilibria of a symmetric n-player, mprize all-pay auction with spillovers Theorem 4 (Equilibrium of a Symmetric n-Player, m-Prize All-Pay Auction With Runner-up Spillovers). Consider a symmetric n-player, m-prize all-pay auction with runnerup spillovers. Assume vand csatisfy Assumptions A1–A4.Letˆ Gbe defined as in Corollary 1. That is, let ˆ Gbe the equilibrium cumulative distribution function of a two-player all-pay auction with spillovers, ˆ G(s)=c(s) v(s,s)+s 0 c(y) v(y,y) R(s,y) v(s,s)dy, with R(s,y)=K0(s,y)+K1(s,y)+K2(s,y)+··· for K0(s,y)=∂v(s,y)/∂y and Kt(s,y),t=1, 2, , defined recursively by Kt(s,y):= s y(∂v(s,z)/∂z)(Kt(z,y)/v(z,z))dz. Then the symmetric equilibrium of the n-player, m-prize all-pay auction with runnerup spillovers is given by the unique Gthat solves ˆ G(s)= n−1  j=n−mn−1 jG(s)j1−G(s)n−j−1. (12) Theoretical Economics 19 (2024) Asymmetric all-pay auctions 191 To see why Theorem 4holds, consider the expected payoff of a player iwho bids s, s 0 v(s;y)dˆ G(y)−c(s), where ˆ Gis the probability measure of the mthlargestscoreoutofthen−1playersin I\{i}. In a symmetric equilibrium, ˆ Gis the n−mth order statistic of a sample of n−1 draws from the equilibrium distribution G, giving us (12). The right-hand side of (12)is increasing in G(s)and, thus, may be inverted to obtain Ggiven ˆ G. Atthesametime, ˆ Gis the equilibrium of a symmetric two-player auction, since each player’s indifference condition is identical to (3). This allows us to use Theorem 1to find ˆ G. Theorem 4shows that the equilibrium in the two-player case is also the symmetric equilibrium of the game with any number of players and prizes, subject to a particular monotone transformation. Intuitively, increasing the number of players (or decreasing the number of prizes) reduces the scores of each player. Asymmetric equilibria are more difficult to characterize when either spillovers or incomplete information is present. In a world without spillovers but complete information, the equilibrium of an n-player, m-prize contest is unique under mild conditions, and Siegel (2010) was able to provide an algorithm for its construction. Under incomplete-information, asymmetric equilibria with more than two players are notoriously difficult to analyze. A general characterization is still an open question, as far as we are aware.11 8. Related literature and conclusions Throughout this paper, we characterized and established uniqueness for the equilibrium of two-player contests using techniques from the theory of integral equations. We then extended these results to symmetric contests with more players and prizes to characterize the symmetric equilibrium. This allowed us to derive insights on equilibrium payoffs, winners and losers, and on the importance of spillovers for applications. This model does not require or imply that the results of a contest are known in advance. In fact, players are always uncertain of their own victory. However, this uncertainty stems from not knowing the resources that your opponent dedicated to the contest. The fact that ranked normalized costs are not enough to establish dominance demonstrates how spillovers can favor high-cost, low-value players who nevertheless have a marginal cost advantage over their opponent when bids are high. In particular, the results in this paper suggest several potential consequences of legal structures, conflicts and competition. This paper is most closely related to two others. Baye, Kovenock, and de Vries (2012) also considers spillovers in two-player contests, but focuses on symmetric equilibria and linear symmetric costs and valuations. We show that there are no asymmetric equilibria 11See Kirkegaard (2013) and Parreiras and Rubinchik (2010) for analyses on particular equilibria of N≥3 players all-pay incomplete-information auctions. 192 Betto and Thomas Theoretical Economics 19 (2024) in this two-player case and extend the analysis to include asymmetric players and general functional forms for the prize values. This allows us to establish equilibrium uniqueness, express novel results about payoffs, and characterize the equilibrium in different applications (Section 6). The second paper that approaches a similar question to our own is Xiao (2018). The author, however, focuses on constant prize values and separable spillovers in the cost functions, which are independent of winning or losing. This independence significantly restricts the equilibrium effects of the spillovers. Linearly separable spillovers on the cost have no effect on the equilibrium, while multiplicatively separable spillovers scale the cost of bids by an endogenous constant. This is not true when spillovers are in the prize value. Fu and Lu (2013) also analyze two-player all-pay auctions with linear spillovers in the costs. They assume that each contestant is a firm with a minority stake in their opponent’s profits. As such, even when a firm loses the auction, they still get to keep a share of the prize. On the other hand, regardless of winning or losing, they must also share in the cost of effort incurred by their opponent; hence, the existence of cost spillovers. Because these spillovers are linear and do not affect the prize of the winner, they have no effect on the equilibrium distributions, as was also noted in Xiao (2018). Our paper is also connected more broadly to the literature of spillovers in other auction and auction-like frameworks. Hodler and Yekta¸s(2012), for example, use a linear first-price auction with spillovers to model war. The authors refer to this as an all-pay contest, but only the winner actually pays because of the way funds are handled. Notably, spillovers have been given comparatively more attention in the Tullock contest framework. In these contests, each participants’ probability of winning is given by pi(si;s−i)=sr i/(sr i+sr −i)if (s1,s2)=(0, 0)and by pi(si;s−i)=1/2if(s1,s2)=(0, 0).Here r∈(0, ∞)is a parameter that controls how much one’s probability of winning responds to an increase in scores. The all-pay auction is a Tullock contest where r=∞;when r=1, we have a Tullock lottery instead. Chowdhury and Sheremeta (2011a) study a generalized Tullock lottery in which payoffs linearly incorporate one’s own effort and the effort of the rival. Their paper studies symmetric payoff and cost structures, and, as is usual in Tullock-type contests, both players are able to extract positive payoffs. The authors obtain asymmetric, pure-strategy equilibria, even when players are identical (Chowdhury and Sheremeta (2011b)). In contrast, our all-pay framework yields a unique symmetric mixed-strategy equilibrium when players are identical, and expected payoffs are zero. Damianov, Sanders, and Yildizparlak (2018) allows for players’ efforts to produce either positive (productive) or negative (destructive) externalities in a two-player Tullock lottery. The author finds that spillovers can either accentuate or reduce the competitive balance between participants when contrasted to a comparable fixed-prize contest. However, unlike our results, no reversal can ever occur: the “favored” player is always more likely to win and has a higher expected payoff.12 12There are many other examples of spillovers in Tullock-type contests; see, e.g., Chung (1996) and Hirai and Szidarovszky (2013). Theoretical Economics 19 (2024) Asymmetric all-pay auctions 199 All that remains is to show that Gcan be recovered from ˆ G(s). To see that this is the case, we rewrite (12)as ˆ G(s)=E[G(s)],where E[p]= n−1  j=n−mn−1 jpj(1−p)n−j−1. That is, E[p]is equal to the survival function of the binomial distribution evaluated at n−m. This function is known to be strictly increasing in pfor p∈[0, 1]. Therefore, Eis invertible. Appendix II: Optimal contest design In this section, we consider how a designer should bias a contest to increase the scores. Several papers have analyzed this problem of assigning prizes to maximize total scores, or the average score of the winner. For example, Mealem and Nitzan (2014)consider prize redistribution in a two-player all-pay auction with fixed values and symmetric costs. They show that equalizing the prize values maximizes the total scores and that the contest yields weakly more total score than any similar Tullock-type lottery contest. Che and Gale (2003) investigate the optimal design of contests for innovation procurement and find that the procurer might want to limit the maximum prize available to the most efficient firms—effectively eliminating any positive rents—so as to increase their own expected maximum surplus. The problem of optimal contest design in all-pay auctions with spillovers has not been previously analyzed. This is relevant because principals are constrained in the prizes that they can offer. Many of the tools that principals use to make prizes have spillovers. For example, if an employer chooses to construct a compensation package using a cash bonus and stock options, then the inclusion of the stock options will generate spillovers. This section analyzes the optimal prize choice when prizes can be constructed from multiple instruments. Let i⊂R˜ Sidenote the set of prize functions available to the designer for player i, and let V:i∈I˜ Si×i∈Ii→Rdenote the designer’s payoff function; i.e., given the pair of scores s:=(s1,s2)and the pair of value functions v=(v1(·;·),v2(·;·)),V(s,v)denotes the designer’s derived net benefit from the contest. We make the following (mild) assumptions. Assumption 1 (Completeness, D1). For each i∈I, the set of prizes iis convex and its closure contains an element with vi(·;·)≡0. Assumption 2 (Productive Scores, D2). For each i∈Iand v∈i∈Ii,thedesigner’s objective function V(s,v)is strictly increasing in si. Assumption 3 (Costly Prizes, D3). For each i∈I,s∈i∈I˜ Siand v−i∈−i,V(s,v)is decreasing in vi.14 14That is, if vi,ˆ vi∈iare such that vi(s;y)≤ˆ vi(s;y)for all (s,y)∈˜ Siט S−i,thenV(s,(vi,v−i)) ≥ V(s,(ˆ vi,v−i)). 200 Betto and Thomas Theoretical Economics 19 (2024) The primary complication with the construction in this paper is that the mass-point is difficult to compute. Fortunately, if the mechanism designer can discriminate between the two players, an optimal mechanism will have no atoms in many specifications. This is formalized in the following proposition. Proposition 5. Assume a two-player contest where a fully informed principal with payoff function Vchooses the prize vi∈ifor each i∈I. Assume that iand Vsatisfy Assumptions D1–D3, and that for all iand all vi∈i, Assumptions A1–A4hold. Then no contestant in equilibrium can have a positive payoff. Equivalently, no player will have a point-mass as part of his/her strategy. Proposition 5implies that there will be no strictly dominant player in any discriminating contest design problem where the principal benefits from the efforts of participants and pays for prizes. This proposition comes from the fact that the equilibrium strategy of the dominant player is locally invariant to changes in her prize value. Intuitively, for any contest with a strictly dominant player, there exists a more competitive contest where his/her prize is reduced and scores are larger. Proof of Proposition 5. Take an optimal choice of v:=(vi)i∈I∈i∈Ii. Suppose, by contradiction, that player ihas a strictly positive payoff. Her strategy is defined by ˜ gi(s)=c −i(s) v−i(s;s)−s 0 v −i(s;y) v−i(s;s)˜ gi(y)dy, which does not depend on vi. Because player −ihas an atom, we know that ˜ Gi(¯ s)− ˜ G−i(¯ s)>0. Therefore, there exists a γ∈(0, 1)such that ˜ Gi(¯ s)=˜ G−i(¯ s)/γ. Then the principal could offer (γvi,v−i)without changing the equilibrium strategy of player i. By costly prizes, Assumption D3, this is weakly preferable given a fixed distribution of s−i. By construction, player 2’s new equilibrium strategy is ˜ G−i(¯ s)/γ. This first-order stochastically dominates player −i’s original strategy. In fact, it is the same distribution, but with the mass-point removed. The productive scores assumption implies that this mechanism is strictly preferred. Proposition 5demonstrates that the expected welfare of all agents is 0 in a large class of contest design problems.15 It also suggests the optimality, from a design perspective, of handicapping the most efficient players (as in the players with lower costs and lower marginal costs) The idea is very much analogous to the conclusion in Che and Gale (2003); for example, handicapping the player who has the technological upper hand causes the less efficient player to become more aggressive and to choose higher scores than they would otherwise. 15Which is not to say that there are no settings where it would not apply. For example, the designer could wish to maximize the agents’ expected welfare. In this case, the principal’s objective function would violate costly prizes. It would usually also violate productive scores. Theoretical Economics 19 (2024) Asymmetric all-pay auctions 201 Appendix III: Removing assumptions An auction that satisfies all but one of Assumptions A1–A4may have equilibria that fail to meet our characterization. An auction that fails to satisfy any of Assumptions A2–A4 can have multiple equilibria. Eliminating Assumption 1(Smoothness). We assume continuous differentiability of viand ci. Continuity is not sufficient to ensure that the equilibrium has interval support. For example, consider the case where the prize is fixed at vi=1 and the costs are given by the density function of some distribution that uniformly assigns probability 1 to a dense subset of [0, 1]with Lebesgue measure 0.16 This cost function is continuous because the distribution assigns uniform weight to infinitely many points. It is also strictly increasing because the support is dense. However, it is not absolutely continuous. Then the aforementioned distribution is an equilibrium, which has support only on a set of measure 0. Eliminating Assumption 2(Monotonicity).Thecasewherev i(si;y)>c  i(si)for some siis considered in Siegel (2014) without spillovers. In this case, the equilibrium distribution has gaps and is thus not an interval. In the presence of spillovers, nonmonotonicity may generate pure-strategy equilibria or result in non-uniqueness. For example, consider the symmetric game where v1(s;y)=v2(s;y)=v(s;y)=%1+s−yif s≤1 (3−s)s−yif s>1 and c1(s)=c2(s)=c(s)=s. Note that this prize value satisfies all assumptions except for monotonicity, which is violated on [0, 1]. There are two asymmetric pure-strategy equilibria where one player bids 0 and the other player bids 1.17 Eliminating Assumption 3(Interiority). Consider the symmetric all-pay auction with spillovers, where v1(s;y)=v2(s;y)=v(s;y)=2√yand c1(s)=c2(s)=c(s)=s. Then there is a pure-strategy equilibrium where both players play 0. Moreover, there is also a mixed-strategy equilibrium at G 1(x)=G 2(x)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ √xif x∈[0, 1] 0ifx<0 1ifx>1, i.e., equilibria are no longer unique. In the other case where costs are no higher than the prize value in the limit (that is, limsi→∞supy∈˜ S−ivi(si;y)≥limsi→∞ci(si)), players might find it profitable to submit unbounded bids. This can result in nonexistence. 16For example, a uniform distribution over the countable union of cantor sets shifted by each of the rationals modulo 1. 17This game also has a symmetric mixed-strategy equilibrium. 202 Betto and Thomas Theoretical Economics 19 (2024) Eliminating Assumption 4(Discontinuity at Ties). Consider the symmetric allpay auction with spillovers, where v1(s;y)=v2(s;y)=v(s;y)=1y≤14&1−yand c(s)=s. Then there is a symmetric equilibrium where G 1(x)=G 2(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1−√1−x 2if x∈[0, 1) 0ifx<0 1ifx≥1 such that both players have an atom at 1, the point where the prize is worth 0 in the event of a tie. The usual argument that the two players cannot have atoms at the same point fails here because a small increase in either player’s bid does not increase the probability of winning a prize of positive value. There are also two asymmetric equilibria in this game of the form G i(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1−√1−x 2if x∈[0, 1) 0ifx<0 1ifx≥1 G −i(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1−√1−x 2if x∈[0, 1] 0ifx<0 1ifx>1. That is, one player has an atom at 0 while the other has an atom at 1. Any convex combination of the symmetric equilibrium and the above is also an equilibrium. Appendix IV: Numerical approximation Iteration method It is possible to approximate the solution by iterating numerically on the sequence ˜ gn+1(s)=1 v(s;s)c(s)−s 0 v(s;y)˜ gn(y)dy starting from ˜ g0=0 to find the true ˜ g. There is a much simpler and faster way. Matrix method 1 Consider our original equation s 0 v−i(s;y)˜ gi(y)dy =c(s) and consider this 3 ×3 discrete approximation of this problem for s∈[0, 1]: 1 3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v−i1 3,1 300 v−i2 3,1 3v−i2 3,2 30 v−i1, 1 3v−i1, 2 3v−i(1, 1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  ! " V ·⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ˜ gi1 3 ˜ gi2 3 ˜ gi(1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦  ! " g ≈⎡ ⎢ ⎢ ⎢ ⎢ ⎣ c−i1 3 c−i2 3 c−i(1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦  ! " c . So we can approximate ˜ gi(s)with g=3V−1c. Theoretical Economics 19 (2024) Asymmetric all-pay auctions 203 Matrix method 2 To get a good estimate, we do the same thing with an N×Ngrid for Nlargeonsomeinterval[0, T]:18 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ˜ gi1 N ˜ gi2 N . . . ˜ gi(T) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≈N ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v−i1 N,1 N0··· 0 v−i2 N,1 Nv−i2 N,2 N··· 0 ... v−iT,1 Nv−iT,2 N··· v−i(T;T) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −1 · ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ c−i1 N c−i2 N . . . c−i(T) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Getting the actual strategies Once you get (˜ g1,˜ g2), you just have to perform the following steps:19 Step 1. 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