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Contests with a Non-Convex Strategy Space

We characterize the Nash equilibria of a class of two-player contests with"lumpy"effort. Our main result shows that under quite reasonable conditions, constraints on the players' choice sets heighten competition. This stands in sharp contrast to the conventional wisdom that regulation decreases competition, but appears to be consistent with anecdotal evidence. Examples of this phenomenon include arms control, rules and regulations in sports, and drug wars. We also show that a constraint on available strategies can benefit a player.


Introduction
The standard assumption in contest theory that players have a convex choice set may be a reasonable assumption when units of effort are sufficiently small, but there are many important contexts in which this assumption 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 main result suggests that under reasonable conditions such strategic constraints heighten competition.
We use a stylized model that may need significant modification to completely describe any given situation. Nevertheless, numerous real-world examples in various contexts can be explained by the model. Moreover, the model improves on those in the existent literature, which usually assume away problems of "granularity" of effort in contests. This is further discussed in the review of the literature.
In particular, we consider two types of simultaneous two-player contests with non-increasing return on effort and a prize with a common value (or similar values).
The first is a symmetric strategy contest in which both players have the same non-convex choice set of efforts (as perhaps in the case of the Cold War). The structure of the choice set that we consider is quite general. For instance, it can contain two separate convex subsets or it may be entirely discrete. The second type is an asymmetric strategy contest in which one player has a convex choice set of efforts while the other player's is non-convex, such as the case of the Taliban forces that have more flexible options than the Western coalition it faces. We present a complete characterization of the Nash equilibria of these contests.
Our main result shows that under quite reasonable conditions deleting strategies from the players' strategy space heightens competition. In general, this result offers insight into the effect of strategic constraints on various situations that can be viewed as contests. One example is the offside rule in soccer that limits the set of possible positions for scoring a goal, but is considered to improve the competitiveness of the game. There are similar rules in other (team and individual) sports such as swimming, basketball, etc. An additional example in a different context is the Strategic Arms Limitation Talks (SALT) between the Cold War superpowers that accelerated development of new nuclear delivery technologies.
Other attempts at arms control apparently had a similar outcome. 1 Furthermore, in the asymmetric strategy contest, the strategic constraints allow the constrained player to make some sort of commitment. However, given that commitment usually has no value in the symmetric convex contest, she does not benefit from such constraints. Nevertheless, we show that under some circumstances it is preferable to be the constrained player.
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. Section 3 characterizes equilibria in the symmetric strategy contest and derives conditions under which constraints on available strategies heighten competition. Section 4 analyzes the asymmetric strategy contest and section 5 further generalizes our results. Section 6 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 assumptions commonly made in the literature are that players have a convex strategy space and that effort exhibits non-increasing returns to scale.
These assumptions are usually made for the sake of tractability and analytic convenience.
In settings where effort exhibits increasing returns, standard solution methods fail. In this case, researchers approximate the strategy space by considering the limit of a sequence of discretized versions of the game. 2 Baye et al. (1994) were the first to apply this method to characterizing the mixed strategy equilibria in the Tullock 2 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. contest (Tullock, 1980), in which return on effort is greater than two. 3 Later studies that applied the same method include Amaldoss and Jain (2002), Rapoport and Amaldoss (2004) and Dechenaux et al. (2006). In their models, the non-convexity of strategies is merely a tool used to examine the equilibrium rather than a feature of interest on its own. In contrast, non-convex strategy spaces are the present model's main focus, rather than a convenience. To be precise, we focus on contests with nonincreasing return on effort and a linear cost function for effort, which implies that the effective cost of effort is increasing and convex, 4 as in the canonical models presented by Dixit (1987), Tullock (1980) and Skaperdas (1996). To the best of our knowledge, the convexity assumption regarding the strategy space has not been relaxed previously. A closely related model is Dubey (2013). He analyzed an all-pay auction with asymmetric players and incomplete information, in which players have a binary choice set. The restriction on the strategy space is intended mainly to permit equilibrium characterization with asymmetries and private information rather than being of interest in itself.
We examine more general types of non-convexities and contest success functions.
The tradeoff for adopting greater generality is that we must exclude private 3 See Alcade and Dahm (2010) and Ewerhart (2015) for the complete characterization of these equilibria. 4 Without loss of generality, we also assume that marginal cost equals 1. information as a possibility. Thus, one can view Dubey (2013) as the private information analog to our model. 5

The model
Two risk-neutral players compete for a prize (the objective of success) with a common value v. Each player ∈ {1,2} chooses an irreversible non-negative effort from the choice set . There are two types of choice sets: 1) a convex interval = [0,∞) and 2) a non-convex subset of Sc, (⊂ ). Other than non-convexity, is completely arbitrary with no additional restrictions. For example, it may contain two separate convex subsets or, alternatively, may be completely discrete. The usual form of analysis assumes that both players have the same convex choice set. We instead consider two alternatives: 1) Both players have the same non-convex choice set; and 2) one player's choice set is convex while the other's is non-convex. We shall refer to the first situation as a symmetric strategy contest and the second as an asymmetric strategy contest.
Let ( , ) be the probability that player ∈ {1,2} will win. We shall make use of the term contest success function (or CSF) to describe p. It is well known that some restrictions are needed to ensure that there is a unique equilibrium when identical competitors with convex strategy spaces compete. 6 We retain these restrictions on the CSF throughout the analysis. This is done for both analytic 5 Notice that discretization of the strategy space is shown to be important in congestion models (see Otsubo and Rapoport, 2008) and suitable for experiments (see Rapoport and Amaldoss, 2004). 6 See Dixit (1987) and Baye and Shin (1999).
tractability and, more importantly, to allow for a ready comparison of our results to those in the literature, thus ensuring that any difference depends purely on dropping the convexity assumption. To complete the model, we assume that is twice continuously differentiable when + > 0, and that it also satisfies the usual properties: (1) ( , ) = 1 − ( , ) and ≡ > 0 and ≡ 2 2 < 0 in the interior of its domain.
The remainder of the paper characterizes the equilibria when we relax the assumption of continuous available strategies. In the next section, we compute the analog of the main results in contest theory, in a setting where the two competitors are identical.

Symmetric strategy contests
As a benchmark, Figure 1 illustrates the case in which strategy spaces are convex. It is well known that in this case there is a unique symmetric interior solution which we denote by * . We now compare this to a situation in which the set of available strategies is identical, but non-convex. We reproduce the standard equilibrium analysis with only this modification as follows: First, we present Lemmas 0-2 which present important equilibria features and some additional technical results needed for the rest of the analysis. We then present Proposition 1 which shows that under quite reasonable conditions a constraint on the player's choice set increases competition.
Finally, at the end of this section, Proposition 2 characterizes the complete equilibria structure more generally (without requiring the conditions in Proposition 1) and is followed by Corollary 1 which generalizes the result shown in Proposition 1.
Our analysis therefore begins with Lemma 0. Suppose that the players' choice sets are S1=S2=Snc. When ec * is contained in Snc, it is trivial to show that the choice of ec * by both players remains an equilibrium. However, it is unclear whether the BR 2 BR 1 * * e 1 e2 deletion of strategies might also produce new equilibria. In Lemma 0, we claim that, regardless of the structure of Snc, no new equilibria appear, but to formally demonstrate this will require Lemma 1, which applies regardless of whether or not ec * is in Snc. 7 For now, we postpone the discussion and merely note the following: Lemma 0 In the symmetric strategy contest, if ec * is contained in Snc, then it is the unique equilibrium.
Herein we assume that ec * is not contained in Snc and do not impose any other restrictions on the structure of Snc.
To this end, define and ̅ to be the efforts that bracket the equilibrium point under convex effort. That is, ∄ ∈ | ∈ ( , ̅), where 0 ≤ ≤ * ≤ ̅ . We restrict what follows to the non-trivial case, in which and ̅ exist.
We are now in a position to present Lemma 1: Lemma 1 In a symmetric strategy contest with non-convex effort, each player's equilibrium effort is an element from the set { , ̅}.
All proofs appear in the Appendix. Lemma 1 implies that, regardless of the full set of elements of Snc, it suffices to consider only the symmetric 2x2 contest where choice is limited to { , ̅}. 8 This greatly simplifies the analysis. Also, notice that Lemma 0 follows immediately since, when ec * is contained in Snc, the set { , ̅} becomes a singleton. 9 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 ̂ with the following properties: (1) When her rival chooses , a player is indifferent between choosing the threshold and .
(2) Likewise, when her rival chooses ̂, the player is again indifferent between choosing ̂ and . Lemma 2 There exists a unique which is not greater than v/2. If ≤ , then there exists a unique ̂∈ [0, * ) which is monotonically decreasing in ; otherwise ∄.
To obtain the next result in Proposition 1, we need additional commonly used assumptions (taken from Skaperdas, 1996, who defined them more generally for the n player case), that impose technical restrictions on the CSF. The following 9 Note that when ec * is contained in Snc, by the definition of e and , the set { , ̅ } is not uniquely defined; it can be either { * , ̅ } when * < ̅ or { , * } when < * or { * }. Lemma 1 holds in all of these cases.
Assumptions 1 and 2 together require that adding a third player to the two player contest does not affect any two players' winning probabilities in a head to head contest among themselves. Loosely speaking, this can be viewed as an extended version of the "independence of irrelevant alternative" property (for a detailed explanation, see Skaperdas, 1996). In addition, Assumption 3 requires homogeneity in effort.
Formally, given that a third player enter the contest, let ( , − ) be the winning probability of player i where − is all effort except and p i,j her wining probability in the head to head contest against player j≠i for all i∈ {1,2,3}. 10 Then: Skaperdas, 1996; A4).
Note that although technically these assumptions assume the existence of a third player, we analyze the two player contest. Specifically, when referring to these assumptions, we formally consider a three player contest in which one of the players exert zero effort.
We formally define "a symmetric bracketing of the strategy space" as − * = * − which allows to state the following proposition:

Proposition 1 Under Assumptions 1-3, in any symmetric strategy contest with non-convex
choices, a symmetric bracketing of the strategy space heightens competition. Specifically, the unique equilibrium effort in pure strategies is then . 11 Notice that Skaperdas (1996, Theorem 2) shows that Tullok's CSF (Tullok, 1980) is the only one that satisfies both (1) and Assumptions 1-3. 12 This class of CSF is perhaps the most commonly used CSF in the literature and is formally defined as: Note that it is well known that * = 4 in this case.
Proposition 1 can perhaps explain the outcome of the Washington Naval Treaty, in which the victors in WWI agreed to limit the arms race by imposing specific constraints on naval construction. However, in many ways, the development of aircraft carriers by the US and Japan was accelerated by these constraints, since budgets and hulls were now diverted from banned capital ship development to aircraft carriers and submarines. In particular, several American and Japanese carriers were constructed from battlecruiser and battleship hulls. As is well known, 11 Notice that pi=xi/(xi+ xj) is an exception, since in that special case a symmetric bracketing of the strategy space results in a continuum of equilibria consisting of arbitrary weights given to { , ̅ }. 12 Skaperdas (1996) axiomatize the Tullock CSF for the n-player contests. For this axiomatization, in addition to Assumptions 1-3, he imposes three more assumptions on p which are satisfied in (1) for the two player case. aircraft carriers turned out to be one of the deadliest weapons of the 20 th century. A further example is perhaps the Anabolic Steroid Control Act (ASCA) passed in 1990 which was intended to end "cheating" in sports. However, far from eliminating the "cheating", it is commonly argued that the use of stronger and riskier anabolic steroids and performance-enhancing drugs (PEDs) that clearly (and unethically) heighten competitive ability has proliferated since the ASCA became law. 14 Furthermore, Baye and Hoppe (2003) show that a contest with the CSF in (3) and a discrete strategy space is strategically equivalent to innovation tournaments and patent-race games. 15 Therefore, the result in Proposition 1 can also shed light on strategic constraints in these important and commonly analyzed games.
More generally, in Proposition 2 we characterize the equilibrium structure for any symmetric strategy contest and in Corollary 1 we note the sufficient and necessary condition under which constraints on the players' strategy space heighten competition.
Proposition 2 A symmetric strategy contest with non-convex choices has a unique equilibrium in pure strategies: a. Effort is e when >δ or ̂ <e b. Effort is when ̂ >e. 16 It therefore follows that: 14 See Ryvkin (2013) for a model of contests with doping. 15 Note that Baye and Hoppe (2003) focus on showing strategic equivalence rather than analyzing equilibrium behavior, therefore for the CSF in (3) and r≤2 they refer the reader to the well-known equilibrium of the standard convex contest. 16 Notice that =e is an exception since in that special case there is a continuum of equilibria consisting of arbitrary weights given to { , ̅ }. Also notice that when ≠e, if by mixing e and ̅ the player obtains an expected effort for which e and ̅ are the best responses, then it is an equilibrium, in addition to the one obtained by pure strategies. However, for a given e and ̅ , such effort may not exist. This is demonstrated in the proof.

Corollary 1 In any symmetric strategy contest with non-convex choices, constraints on the players' strategy space heighten competition when e<̂.
Proposition 2 and Corollary 1 imply that a gradual deletion of effort from the choice set can boost effort. To see this, let ̂ <e in which case the equilibrium effort is . After deleting the equilibrium effort and its neighborhood, equilibrium usually remains at e. However, a sequence of such actions will eventually result in a jump to . The basic intuition behind this result is related to the non-monotonic relation between the player's effort and her expected net payoff. In particular, Eπ(ei;ej) has an approximately inverted U-shape, which means that each level of Eπ(ei;ej) (apart from its maximum) can be obtained by responding with either of two different (nontangent) values of ei to ej. Furthermore, these two values of ei are negatively correlated, such that the smaller is the smaller of the two efforts, the larger will be the larger one. The result is illustrated in the following numerically simple (and somewhat extreme) example: Example 1 Assume that p(e,0)=1 for all e>0, and let Snc={0}∪ [v/2+ε,v], where ε>0.
Then, there exists an effortless peaceful equilibrium. However, if we let ε<0 (such that v/2+ε≥ec * ), then the player exerts v/2+ε and the conflict becomes far more intense since there is almost full rent dissipation.
It is important to note that our model-unlike a model with a single decision maker-involves strategic interaction. In the case of a single decision maker, Peltzman (1975) shows that mandatory wearing of seat belts can result in riskier driving. This may be the result of less effort being exerted while driving which is the opposite result to that of our model. In addition, although we derive conditions under which constraints on the players' strategy space heighten competition, we acknowledge that this may not always be the case. For instance, a technological improvement that makes it possible to manufacture a cheaper and stronger version of a drug such as heroin can perhaps be viewed as "a smoothing of the strategy space". Nevertheless, the war between the drug cartels may escalate as a result of that technological improvement.

Asymmetric strategy contests
In this section, we analyze an asymmetric strategy contest in which player 1 has a convex choice set, i.e. S1=Sc, while player 2 has a non-convex choice set, i.e. S2=Snc.
Given the structure of player 1's best response function, BR1(e2) (see Figure 1), and using parts of the proof of Lemma 1, Corollary 2 is immediate.

Corollary 2
In any asymmetric strategy contest with non-convex choices: (i) Player 2's equilibrium effort is an element from the set { , ̅}.
Corollary 2 implies that in an asymmetric strategy contest, and regardless of player 2's full set of elements, it is sufficient to identify the signs of ( , 1 ( ̅ )) − ( ̅ , 1 ( ̅ )) and ( , 1 ( )) − ( ̅ , 1 ( )). We restrict what follows to the non-trivial case, in which ̅ < . 17 In order to analyze the asymmetric strategy contest, we impose Assumptions 1-3 on the CSF in the rest of this section.
( , 1 ( ̅ )) − ( ̅ , 1 ( ̅ )), ( , 1 ( )) − ( ̅ , 1 ( )) 0 W Proposition 3 implies that, a gradual deletion of effort from the choice set can boost effort, as in the case of the symmetric strategy contest. However, unlike the case of the symmetric strategy contest, in which equilibrium in pure strategies is usually unique, there are two different Nash equilibria in pure strategies when is contained in the interval [ 0 , 00 ], where this interval is a singleton only when the exponent r in (3) is equal to 1.
Notice that player 2 will always prefer to have a convex choice set of efforts, or more precisely, to be able to choose ec * . This is because in the symmetric convex contest, commitment has no value, and therefore a player does not benefit from any type of bracketing in the strategy space (Dixit, 1987;Baik and Shogren, 1992;Possajennikov, 2009). 20 Nevertheless, in an asymmetric strategy contest, a player will prefer to be constrained in some circumstances. To see this, consider the following: When the equilibrium effort is ( 1 ( ), ), player 1's expected net payoff is larger than her rival's (and even larger than the payoff in the standard symmetric convex contest). 21 Indeed, in this case, player 1 benefits from the "handicap" of her rival.
However, this may not be the case when equilibrium effort is ( 1 ( ̅ ), ̅ ). We demonstrate this in example 2.

Non-convex strategy contests with approximately identical players
One might attribute the "jumps" in effort to the assumption of identical players (apart from the asymmetry in the choice sets presented in section 4). Indeed, such jumps might not occur when the difference between the players is sufficiently large.
For instance, a significant difference in prize valuation probably results in a "sticky" equilibrium, in which small changes in available strategies are irrelevant to the players' behavior. However, this is not the case with smaller asymmetries. In particular, while focusing on pure strategies, Proposition 4 generalizes the results of Proposition 2 to the case in which there is a small difference in the players' prize valuation. We subsequently discuss the intuition behind this result and show that it applies to other parameters of the model as well. . Furthermore, it is also possible that ( 1 * , 2 * ) = ( 1 ( ̅ ), ̅ ) and player 1's expected net payoff will be larger than player 2's. In particular, this can happen when r, the exponent in the CSF in (3), is sufficiently small (such that p1≈p2 and e1<e2).

then it is the equilibrium effort in the corresponding contest in which both players' prize valuation is in V. 23
The logic behind the result in Proposition 4 is as follows: the value of the player's "continuous best response function" at any given equilibrium shown in Proposition 2 (i.e., BR(e) for eϵ{e, ̅ }) is always between e and ̅ . Therefore, given the bracketing of the strategy space and that BRi(ej) is continuous in player i's prize valuation, a small change in this valuation will not change her best available response. The same logic applies to Proposition 3 and for small changes in the player's cost and effectiveness of effort.
As noted at the beginning of this section, a sufficient degree of asymmetry may lead to completely different equilibria. For instance, consider the following example: Example 3 Assume the CSF in (3)  strategy space be 2/3 instead of 5/9. In this case, there is no Nash equilibrium in pure strategies. Thus, unlike in the common valuation case, Nash equilibrium in pure strategies may not exist when prize valuations are sufficiently different.
We therefore note that there are qualifications that apply to the results when the players are significantly different.

Conclusion
We characterized Nash equilibria in the case of contests in which the convexity assumption of the strategy space has been relaxed. Although the model is quite stylized, it extends models in the literature which generally assume a convex strategy space. The main implication of the results-and one that seems counterintuitive-is that constraints on the players' strategies can heighten competition. Nonetheless, the result is consistent with anecdotal evidence.
Furthermore, in the first-price all-pay auction, it has been shown that excluding the set of players that mostly evaluates the prize (Baye et al., 1993), or putting caps on players' expenditure, (Che and Gale, 1993) actually increases aggregate expenditure.
Therefore, although we analyzed a different class of contests, in a sense, our results complement the existing literature on contests that shows that different types of constraints on players may heighten competition. It is also shown that a constraint on a strategy space can benefit a player.

Acknowledgements
This work has benefitted from useful discussions between the first author and Nirit Finance for its support and hospitality (part of this paper was written during his visit there).

Appendix:
Proof of Lemma 1: It is well known that in the standard symmetric strategy convex contest, player i's best response function, BRi(ej) (pictured in Figure 1), has the following property: 24 which implies that: Thus, it is immediate that in a symmetric strategy non-convex contest, any equilibrium effort e satisfies: Given that ( ; ) is concave in in the interior of its domain and therefore, for a given > 0, ( ( ); ) is the unique maximum of ( ; ), it is immediate from (A.1) that: We now present the following important identity: 25 The rest of the proof is carried out in two steps: In the first, we restrict the analysis to pure strategies, and, in the second, we extend the analysis to mixed strategies.
We thus begin by assuming that the player restricts her actions to pure strategies. In what follows, we show that the equilibrium effort cannot be smaller We now proceed to the second step of the proof, in which the player can mix pure strategies. We show that such an equilibrium can only exist when the player mixes e and ̅. Assume that the expected equilibrium is (qa+ (1-q) 0<q,q'<1, and without loss of generality, a>b and c>d. Given that for ej>0, Eπ(ei;ej) is concave in ei with a unique maximum at Eπ(BRi(ej);ej) (henceforth referred to as "the structure of Eπi"), it is immediate that:
Now let a≠c (and therefore b≠d). Then, either b>c or d>a; otherwise, given the structure of Eπi, it is impossible that both a and b are the best responses to q'c+(1-q')d, and c and d are the best responses to qa+(1-q)b. 26 Without loss of generality, let b>c. Then: Thus, by the structure of Eπi, Eπ(a, where by (A.5), However, the structure of Eπi when b>c also implies that: Notice that ( ̅ , 0) ≤ 1 and thus the RHS of the second equality in (A.9) is bounded from above by 2 , where at ̅ = 2 the LHS equals to 2 . Also, by (A.5), (0, * ) < ( * , * ) ↔ (0,0) < ( * , 0), which implies that at ̅ = * the LHS of the second equality in (A.9) is smaller than the RHS (i.e., * < ( ( * , 0) − 1 2 )). Therefore, since the LHS in the second equality of (A.9) is increasing and linear in ̅ and the RHS is non-decreasing and weakly concave in , ̅ 28 by the Intermediate Value Theorem, there exists a unique ̅ ≡ ∈ ( * , 2 ] that solves (A.9) such that 27 Notice that this argument holds also when q or q' equals zero or one. 28 Since ( ̅ , ) is increasing and concave in ̅ for > 0 and continuous in , ( ̅ , 0) must be nondecreasing in ̅ . follows that only at this specific β is the RHS of (A.12) identical to that of (A.11).
Therefore, given that ̂ is defined as: ̂= * − 2(1 − )∆, for a given ̅ , ̂ is obtained by a unique β.  (3), =1/2 and ec * =rv/4. Therefore, when r≤1, ≤ for all such that − * = * − . 33 Notice that in addition to the equilibrium in the case of pure strategies determined by the inequality in (A.14), an equilibrium may exist in mixed strategies. More precisely, if there exists a convex combination qe+(1-q) ̅ , 0<q<1 for which the player's best responses are e and ̅ , then it is an equilibrium in the case of mixed strategies. Such an equilibrium can be constructed as follows: Pick a tuple (a,c,b) such that a<ec * , c<b and Eπ(a,c)=Eπ (b,c), and let a=e and b= ̅ . Then, c=qa+(1-q)b is the player's expected equilibrium effort. However, for a given e and ̅ , such an equilibrium may not exist. For instance, by the continuity of Eπi in the interior of its domain, the player always prefers responding with e rather than ̅ to any convex combination of e= ec * -ε and a sufficiently large ̅ .