A note on contests with a constrained choice set of effort
11 A note on contests with a constrained choice set of effort Doron Klunover a , John Morgan b December 2019
Abstract
We apply a fundamental property of the payoff function in a contest with two identical risk - neutral players, in which the choice set of effort is constrained to show that, under the usual concavity assumption, a Nash equilibrium in pure strategies exists and, except for a special case, is unique. It is shown that all equilibria are near the unconstrained equilibrium. Perhaps surprisingly, this is not the case when players have different prize evaluations. Keywords:
Contests; Constrained choice set of effort JEL classification numbers: C72, F51 . A previous version of this paper was circulated under the name: "Contest with a non-convex strategy space". a Corresponding author: Department of Economics, Ariel University, 40700 Ariel, Israel. E-mail address: [email protected]. b Haas School of Business and Department of Economics, University of California, Berkeley, California 94720. Introduction
The standard assumption in contest theory that players have a convex choice set may be reasonable when units of effort are sufficiently small, but there are many important contexts in which this does not hold. For instance, it may be possible for a country to contain an overseas crisis using a small number of aircraft, but getting them there and maintaining them may require the use of an aircraft carrier. Thus, the actual choice is between a large operation and no operation at all. Likewise, there may be a large fixed cost in supplementing air power with "boots on the ground". More generally, the choice set may be further constrained by rules, laws and regulations, technological constraints, political constraints and the like. Our model extends the contest literature, which usually assumes away problems of “granularity” of effort, or alternatively, imposes a particular constraint, such as entry costs, caps on expenditure, budget constraints, a discrete strategy space, etc. This is further discussed in the review of the literature. In particular, we consider a contest between two identical risk-neutral players and a logit contest success function (CSF), in which the player's impact function is concave and the choice set of effort is constrained. Specifically, the choice set can be any subset of the real line, as long as it satisfies a (perhaps) weak technical assumption. We apply a fundamental and quite interesting property of payoff functions in contests (which, as far as we know, has not been discussed previously) in order to show that a Nash equilibrium in pure strategies exists and, except in a special case, is unique. Specifically, equilibrium effort is one of the two feasible elements that are closest to the unconstrained equilibrium. We note that in some circumstances small changes in the strategy space produce large changes in effort. Although we focus on equilibrium in pure strategies and do not fully characterize the set of equilibria in mixed strategies, we show that such equilibria, if they exist, are also limited to the same two effort sets and therefore are also near the unconstrained equilibrium. Regarding players with different prize evaluations, we present a counterexample, in which the choice set is first fixed such that the equilibrium effort, perhaps surprisingly, is not close to the unconstrained equilibrium, and then fixed such that a Nash equilibrium in pure strategies does not exist. Therefore, in a sense, our results can be viewed as a robustness check for the well-known equilibrium in symmetric unconstrained contests; however, in the case of asymmetric contests, it may imply that the unconstrained equilibrium can be quite fragile. The rest of the paper is structured as follows: In the remainder of this section, we present a survey of the literature. In section 2, we describe the general model. Sections 3 characterizes equilibria in pure strategies and discusses the case of mixed strategies. An example in which players have different prize evaluations is presented in section 4. Section 5 concludes.
Review of the literature
Our model is related to the extensive and still growing literature on contests, which has been surveyed by Konrad (2009) and more recently by Corchón and Serena (2018). The assumption commonly made in the literature is that players have a convex strategy space. There are exceptions, however. In settings where the concavity assumption (as specified above) is relaxed, standard solution methods may fail. In this case, researchers approximate the strategy space by considering the limit of a sequence of discretized versions of the game. Baye et al. (1994) were the first to apply this method to characterizing the mixed strategy equilibria in the Tullock contest (Tullock, 1980), in which return on effort is greater than two. Discretization of the strategy space is also shown to be important in congestion models (see Otsubo and Rapoport, 2008) and suitable for experiments (see Rapoport and Amaldoss, 2004; Dechenaux et al., 2015). Che and Gale (1998) studied the first-price all-pay auction with the particular constraint of caps on players' expenditure. Moldovanu and Sela (2001) design the optimal allocation of prizes in all-pay auctions with entry fees. Other studies have introduced different budget constraints into the Colonel Blotto game (Kvasov, 2007; Roberson and Kvasov, 2012; Klumpp et al., 2019). We Technically, the equilibrium characterization offered in these works represents the limit of a discrete strategy space rather than a convex one. It is an open question as to whether the limit equilibrium coincides with a real equilibrium. See Dasgupta and Maskin (1986) for a careful analysis of the issue. For the complete characterization of these equilibria, see Alcade and Dahm (2010) and Ewerhart (2015). Other studies that have applied the same method include Amaldoss and Jain (2002) and Dechenaux et al. (2006). A closely related model is Dubey (2013), who analyzed an all-pay auction with asymmetric players and incomplete information, in which players have a binary choice set. examine a more general type of constrained choice set within the canonical model of contests presented by Dixit (1987), Tullock (1980) and Skaperdas (1996). The model
Two risk-neutral players compete for a prize with a common value v . Each player 𝑖 ∈ {1,2} chooses an irreversible non-negative effort 𝑒 𝑖 from the choice set 𝑆 ⊆ [0,∞) . The probability of player 𝑖 ∈ {1,2} winning the prize is determined by the following logit CSF: (1) 𝑝(𝑒 𝑖 , 𝑒 𝑗 ) = { 𝑓(𝑒 𝑖 ) 𝑓(𝑒 𝑖 )+𝑓(𝑒 𝑗 ) , if (𝑒 𝑖 , 𝑒 𝑗 ) ≠ (0,0) , if (𝑒 𝑖 , 𝑒 𝑗 ) = (0,0) } , where 𝑓 ′ (⋅) > 0, 𝑓′′(⋅) ≤ 0 and 𝑓(0) = 0 . We therefore assume that f ( e ) (usually referred to as the player's impact function) is twice differentiable and concave . The CSF in (1) has been axiomatized and is commonly used in the literature (see Skaperdas, 1996). Furthermore,
Baye and Hoppe (2003) show that a contest with a CSF which is a special case of (1) and a discrete strategy space is strategically equivalent to innovation tournaments and patent-race games. Player i's problem is therefore: Note that Baye and Hoppe (2003) focus on showing strategic equivalence rather than analyzing equilibrium behavior; therefore, for the case of non-increasing return on effort they refer the reader to the well-known unconstrained equilibrium. (2) max 𝑒 𝑖 𝐸𝜋(𝑒 𝑖 ; 𝑒 𝑗 ) such that 𝑒 𝑖 ∈ 𝑆 , where 𝐸𝜋(𝑒 𝑖 , 𝑒 𝑗 ) = 𝑓(𝑒 𝑖 ) 𝑓(𝑒 𝑖 ) + 𝑓(𝑒 𝑗 ) 𝑣 − 𝑒 𝑖 ∀𝑖 ∈ {1,2}. For the case in which
𝑆 = [0,∞) , let
Ri(e j ) be player i ’s best response function. Yildirim (2005) then notes the following important facts . Fact 1
Given e j , 𝐸𝜋(𝑒 𝑖 ; 𝑒 𝑗 ) is strictly concave in e i . (Yildirim 2005, Fact 1) Fact 2
Ri(e j ) is strictly increasing if the pair (e i ,e j ) is such that e i >e j ; it reaches its maximum if e i =e j ; and it is strictly decreasing if e i 𝑆 = [0,∞) , then there is a unique symmetric interior solution which we denote by 𝑒 𝑐∗ . Define 𝑒 and 𝑒 ̅ as follows. If 𝑒 𝑐∗ is not in S , then 𝑒 and 𝑒̅ are such that ∄𝑒 ∈ 𝑆| 𝑒 ∈ ( 𝑒, 𝑒 ̅) , where ≤ 𝑒 < 𝑒 𝑐∗ < 𝑒̅ . If 𝑒 𝑐∗ is in S , then 𝑒 ≡ 𝑒 𝑐∗ ≡ 𝑒 ̅ . In the remainder of the paper, we impose the following restriction on S . Assumption 1 The elements 𝑒 and 𝑒 ̅ exist. Assumption 1 requires that the bracketing of the strategy space be a closed set. Otherwise, equilibrium in pure strategies may not exist. Nash Equilibria In the following, we focus on pure strategies and later in this section we briefly discuss the case in which mixed strategies are allowed. First, pursuant to (2), we present the following useful and perhaps important property of the player's payoff function: Notice that the facts in Yildirim (2005) are more general since they refer also to the case in which 𝑓 𝑖 (𝑒) ≠ 𝑓 𝑗 (𝑒) . (3) 𝐸𝜋(𝑥, 𝑦) − 𝐸𝜋(𝑦, 𝑦) = 𝑝(𝑥, 𝑦)𝑣 + 𝑦 − (𝑣2 + 𝑥) = 𝑣2 − 𝑥 − (𝑣(1 − 𝑝(𝑥, 𝑦)) − 𝑦) = 𝐸𝜋(𝑥, 𝑥) − 𝐸𝜋(𝑦, 𝑥). As far as we know, this result is presented here for the first time and may have wider applications beyond the scope of this paper. For instance, it implies that in a contest where choice is limited to a set of two effort levels, there always exists a dominant (possibly weak) strategy. Building on (3) and Facts 1 and 2, we introduce Lemmas 1 and 2 which present important equilibrium features and some additional technical results needed for the rest of the analysis. Following that, Proposition 1 characterizes the complete equilibria in pure strategies. Lemma 1 Under Assumption 1, each player‘s equilibrium effort is an element from the set { 𝑒, 𝑒 ̅} . All proofs appear in the Appendix. Note that Lemma 1 implies that if 𝑒 𝑐∗ is in S , then it is the unique equilibrium. More generally, while Lemma 1 significantly reduces the candidates for equilibrium strategies, it remains to characterize the resulting set of equilibria. It proves useful to define a threshold effort level 𝑒 ̂ with the following properties: (1) When her rival chooses 𝑒 , a player is indifferent between choosing the threshold and 𝑒 . (2) Likewise, when her rival chooses 𝑒 ̂, a player is again indifferent between choosing 𝑒 ̂ and 𝑒 . Formally, 𝑒̂ solves 𝐸𝜋(𝑒̂, 𝑒′) = 𝐸𝜋(𝑒, 𝑒′) for 𝑒′ ∈ {𝑒̂, 𝑒}. The The proof for Lemma 1 considers only pure strategies. However, in a separate proof in the Appendix, we show that Lemma 1 holds also when mixed strategies are allowed. expression 𝑒̂ is well-defined as long as 𝑒 is not too large. Otherwise, no interior solution exists for solving the indifference condition. Lemma 2 formally establishes the properties of the threshold. Lemma 2 Under Assumption 1, if 𝑒 ≤ 𝑣/2, then there exists a unique 𝑒 ̂ ∈ [0, 𝑒 𝑐∗ ] which is monotonically decreasing in 𝑒 ; otherwise ∄𝑒̂. Note that Lemma 2 implies that the upper bound of 𝑒 for which there exists 𝑒̂ , is independent of the particular structure of CSF, as long as it satisfies (1). In Proposition 1, we characterize the equilibria structure. Proposition 1 In a contest with a CSF as shown in (1) and a choice set S that satisfies Assumption 1, the Nash equilibria in pure strategies are as follows: a. Effort is e when 𝑒 >v/2 or 𝑒 ̂ Let S ={0} ∪ [ v/2+ε,v ], where ε >0 . Then, there exists a unique effortless equilibrium. However, if we let ε <0 (such that v/2+ε ≥ e c* ), then the player exerts v/2+ ε and there is almost full rent dissipation. Note that Lemma 1 implies that any equilibrium in mixed strategies, if it exists, must also be close to e c* . Specifically, such an equilibrium must be a convex combination of e and 𝑒̅ for which the player’s best responses are e and 𝑒̅ . In this note, we focus on equilibria in pure strategies and therefore leave the analysis of mixed strategies equilibria for future research. Players with different prize evaluations In this section, we present an example in which the CSF is a special case of (1) and the choice set is constrained, but players have different prize evaluations. First, we fix S such that the equilibrium effort of one of the players is not close to her unconstrained equilibrium effort, and then we consider a different S for which an equilibrium in pure strategies does not exist. Example 2 Assume that f(e)=e and 𝑆 = [0,∞) , and let the valuations of the prize by players 1 and 2 be 1 and 2, respectively. The equilibrium effort is then (2/9, 4/9). Now, assume instead that S= {0.18, 0.2, 5/9}. Then the matrix of the players' payoffs is: player 2/ player 1 Notice that 5/9 is a dominant strategy for player 2 and player 1's best response to 5/9 is 0.18. Therefore, player 1's equilibrium effort violates Lemma 1. Now assume instead that S= {1/9, 0.2, 2/3}. Then the matrix of the players' payoffs becomes: player 2/ player 1 1/9 0.2 2/3 1/9 0.388, 0.888 0.246, 1.085 0.031, 1.047 0.2 0.442, 0.603 0.3, 0.8 0.03, 0.871 2/3 0.19, 0.174 0.102, 0.261 -0.16, 0.33 Notice that for player 2, 1/9 is dominated by 2/3, and for player 1, 0.2 dominates 2/3. However, a Nash equilibrium in pure strategies does not exist. Conclusion We characterized Nash equilibria in pure strategies in a class of symmetric two-player contests in which the assumption of a convex strategy space has been relaxed. A (perhaps) desirable property of the model is that the results can be tested directly in a lab experiment, in which the choices in a participant's choice set of expenditure do not have to be in discrete units. A direction for future research is to extend the analysis to equilibria in mixed strategies and asymmetric contests. Acknowledgements This work has benefitted from useful discussions with Nirit Agay, Christian Ewerhart, Arthur Fishman, Aart Gerritsen, Kai Konrad, Dan Kovenock, Ella Segev, Marco Serena, and Huseyin Yildirim. We also thank the audiences at the Public Choice Society meeting, the Contests: Theory and Evidence Conference , the AEA annual meeting, and a seminar held at the Max Planck Institute of Tax Law and Public Finance. All remaining errors are those of the authors. Doron Klunover thanks the Max Planck Institute of Tax Law and Public Finance for its support and hospitality (part of this paper was written during his visit there). Appendix: Proof of Lemma 1: Fact 2 implies that: (A. 1) 𝑒 𝑐∗ = 𝑚𝑎𝑥𝑅 𝑖 (𝑒 𝑗 ) = 𝑅 𝑖 ( 𝑒 𝑐∗ ) . By Fact 1 and (A.1), in any equilibrium, effort e satisfies: (A. 2) 𝑒 ≤ 𝑒̅ . In what follows, we show that equilibrium effort cannot be smaller than e . Fact 1 and Fact 2 imply that: (A. 3) 𝐸𝜋(𝑥, 𝑥) > 𝐸𝜋(𝑦, 𝑥) ∀ 𝑒 𝑐∗ ≥ 𝑥 > 𝑦 ≥ 0. Applying (3) to (A.3) results in: (A. 4) 𝐸𝜋(𝑥, 𝑦) > 𝐸𝜋(𝑦, 𝑦) ∀ 𝑒 𝑐∗ ≥ 𝑥 > 𝑦 ≥ 0 . Let 𝑒 ≥ 𝑥 ≥ 𝑦 ≥ 0 . By (A.3) and (A.4), ( x,y ) can be an equilibrium only when y=x=e . Moreover, let 𝑒 ̅ > 𝑒 𝑐∗ . If y 𝐸𝜋(𝑒, 𝑦) >𝐸𝜋(𝑒̅, 𝑦) , which contradicts the initial assumption that ( ē ,y ) is an equilibrium. Thus, any e in S such that e According to the proof of Lemma 2, if 𝑒 ̅ > 𝑣/2 , then 𝐸𝜋(𝑒, 𝑒) > 𝐸𝜋(𝑒̅, 𝑒) for all 𝑒; otherwise 𝑒 >< 𝑒 ̂ ↔ 𝐸𝜋 ( 𝑒, 𝑒 ) >< 𝐸𝜋 ( 𝑒 ̅ , 𝑒 ) . Lemma 1 implies that the set of equilibria in the original contest, in which the choice set is S , is a subset of the set of equilibria in the 2x2 contest. In what follows, it is shown that these two sets coincide. If ( e,e ) is an equilibrium in the 2x2 contest, then 𝐸𝜋(𝑒, 𝑒̅) ≥ 𝐸𝜋(𝑒̅, 𝑒̅) which by Fact 1 and (A.1) implies that 𝐸𝜋(𝑒, 𝑒̅) ≥ 𝐸𝜋(𝑒̅, 𝑒̅) > 𝐸𝜋( 𝑦 , 𝑒̅) for all y> 𝑒 ̅ . Therefore, by (A.3), if ( e,e ) is an equilibrium in the 2x2 contest, then it is an equilibrium in the original contest. In addition, if ( 𝑒 ̅ , 𝑒 ̅) is an equilibrium in the 2x2 contest, then 𝐸𝜋(𝑒̅, 𝑒̅) ≥ 𝐸𝜋(𝑒, 𝑒̅) which by Fact 1 implies that 𝐸𝜋(𝑒̅, 𝑒̅) ≥ 𝐸𝜋(𝑒, 𝑒̅) > 𝐸𝜋( 𝑦 , 𝑒̅) for all y References Alcade, J., Dahm, M., 2010. Rent seeking and rent dissipation: a neutrality result. Journal of Public Economics, 94, 1–7. Amaldoss, W., Jain, S., 2002. David vs. Goliath: An analysis of asymmetric mixed- strategy games and experimental evidence. Management Science, 48, 972-991. Baye, M.R., Hoppe, H.C., 2003. 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