Group contests with private information and the “Weakest Link”

Journal Pre-proof Group contests with private information and the “Weakest Link”

Stefano Barbieri, Dan Kovenock, David A. Malueg, Iryna Topolyan

PII:

S0899-8256(19)30139-3

DOI:

https://doi.org/10.1016/j.geb.2019.09.008

Reference:

YGAME 3052

To appear in:

Games and Economic Behavior

Received date:

16 May 2018

Please cite this article as: Barbieri, S., et al. Group contests with private information and the “Weakest Link”. Games Econ. Behav. (2019), doi: https://doi.org/10.1016/j.geb.2019.09.008.

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Group Contests with Private Information and the “Weakest Link” Stefano Barbieri†

Dan Kovenock‡ Iryna Topolyan¶

David A. Malueg§

September 26, 2019

Abstract We study weakest-link group contests with private information. We characterize all pure-strategy Bayes-Nash equilibria: various degrees of coordination are possible, from every cost type choosing a distinct effort level to all cost types coordinating on a single effort level. Such coordination may not enhance welfare. If groups are symmetric except for group size, players in the smaller group bid more aggressively than those in the larger group, but when asymmetries regard multiple dimensions, no clear-cut conclusions are evident. As an additional avenue for cooperation, we investigate cheaptalk sharing of private information among teammates, who then coordinate on the effort level most preferred by the player with the largest announced cost. A single group sharing information does better. But, with respect to the equilibrium in which all types choose a distinct effort, when players of both groups cooperate in this fashion all within-group gains are lost to increased competition between groups. JEL Codes: D61, D82, H41 Keywords: all-pay auction, groups, Bayes-Nash equilibrium, weakest link, incomplete information, coordination

Declaration of interest:none

∗ We would like to thank the Editor and three anonymous referees for their detailed and insightful comments that substantially improved the paper. † Department of Economics, 206 Tilton Hall, Tulane University, New Orleans, LA 70118; email: [email protected]. ‡ Economics Science Institute, One University Drive, Chapman University, Orange, CA 92866; email: [email protected]. § Department of Economics, 3136 Sproul Hall, University of California, Riverside, CA 92521; email: [email protected]. ¶ Department of Economics, 2906 Woodside Dr., University of Cincinnati, Cincinnati, OH 45221; email: [email protected].

1

Introduction

Contests between groups are ubiquitous. Examples spanning a wide range of environments include lobbying, R & D races, political competition, warfare, and sports, among others. There are many dimensions to modeling a contest (Konrad, 2009). The winning group may be awarded a single prize or multiple prizes, which could be a public good consumed nonexclusively by all members, a private good, or a mixture of the two. Depending on how monolithic the competing entities are, one can distinguish between individualistic and group contests. In modeling group contests, a central issue is how individual efforts of group members determine that group’s performance. To capture different degrees of complementarity of efforts within a group, several types of the effort aggregation function have been introduced, such as the “best shot” (a group’s effort equals the largest effort within the group), the “weakest link” (a group’s effort equals the smallest effort within the group), and the additive aggregation function (a group’s effort equals the sum of efforts within the group). Furthermore, based on the rule that determines the winner (called the contest success function, or csf), one can model a contest as perfectly or imperfectly discriminating. The former is based upon the auction-type csf, while the latter is epitomized by a Tullock contest. We model group competition as a weakest-link perfectly discriminating contest with a public good prize. Such strong complementarity arises naturally in sports contests such as synchronized swimming or synchronized diving. Competitions between chamber orchestras or barbershop quartets also provide examples in which an entire performance can be ruined by the lackluster effort of a single out-of-tune member. Similarly, an interdisciplinary research proposal may require effort along several dimensions, each addressed by a subject-area specialist in a group, and a serious flaw in even a single dimension may suffice to doom the entire proposal, thus creating strong complementarity among individual efforts. Our study focuses on the interplay between the weakest-link effort technology and players being privately informed about their own marginal costs of effort while having only a vague idea of their teammates’ and rivals’ costs. With the weakest-link technology some players’ 1

efforts within a group may be wasted due to a lower contribution by a teammate. This suggests potential gains from coordination among teammates. One expects a single coordinating group to benefit, so we focus (not exclusively, though) on symmetric within-group coordination. Most of the paper focuses on the case of two groups only; Section 6.2 provides extensions. We concentrate on pure strategy Bayes-Nash equilibria, show that teammates always employ symmetric strategies, and demonstrate that a wide range of coordination is possible, from all cost types coordinating on a single effort level to every cost type choosing a distinct effort level. This multiplicity sharply contrasts with the standard uniqueness result for individualistic, two-player perfectly discriminating contests.1 Furthermore, increasing the degree of coordination may either improve or worsen welfare, and we provide a sufficient condition for each outcome to occur. One may also expect coordination to be more difficult in larger groups. To analyze this issue, we explore the effects of symmetrically increasing group sizes. Again, this may increase or decrease players’ welfare, and we are able to provide a sufficient condition for each outcome to occur. Interestingly, this condition is identical to the one we found relating the degree of equilibrium coordination within each group with welfare in the contest. We extend the analysis to asymmetric contests and find that, if groups differ only in size, then players in the smaller group are more aggressive and their group is more likely to win. But if asymmetries arise along multiple dimensions (e.g., group sizes and cost distributions), then no clear-cut conclusions are evident. Finally, as an additional way to explore cooperation within groups, we look at teammates’ incentives to exchange private information via cheap talk. We find that, although welfare is always higher in the cooperating group provided that its competitors are not engaged in (within-group) information sharing, if cooperation happens in both groups, then all gains are lost to increased competition (note that we compare cooperation via cheap-talk with the equilibrium without communication in which every cost type chooses a distinct effort level.) 1

See, e.g., Amann and Leininger (1996).

2

An interesting connection arises with the revenue equivalence theorem from auction theory (see e.g., Myerson, 1981): its assumptions do not hold when comparing equilibria with different coordination levels, allowing our finding that increasing the degree of coordination may improve or worsen welfare. But the assumptions of the revenue equivalence theorem apply when comparing cooperation via cheap-talk with the equilibrium without communication in which every cost type chooses a distinct effort level, yielding our finding that welfare is unchanged. Although group contests with complete information,2 as well as individualistic contests with private information,3 have been studied extensively, issues surrounding player-level private information4 and especially information exchange in group contests remain underresearched. While Brookins and Ryvkin (2015) and Einy et al. (2015) derive some general existence results for a broad class of group contests with private information, we focus on the properties of equilibria and look into whether private information hampers or facilitates players’ ability to coordinate in the presence of the weakest-link technology. The papers most closely related to our research are Chowdhury et al. (2016), and Barbieri and Malueg (2016) and (2018). The first analyzes weakest-link competition with full information. Our interest in private information is compelling, given how much coordination is important in a weakest-link setup, and given that even a little private information might make coordination much harder. Focus is the major difference between this study and Barbieri and Malueg (2016), which analyzes a setup similar to ours using similar techniques, but with the best-shot aggregator. There, a great deal of attention is devoted to the “group size” effect. Indeed, with the best-shot technology the intuition is not a priori clear, and it turns out that the smaller or larger group might be more likely to win. Here, given our weakest-link setup, this issue is less interesting: as discussed, the smaller group is more likely 2 See e.g., Baik et al. (2001), Baik (2008), Chowdhury et al. (2013), Barbieri et al. (2014), Topolyan (2014), Chowdhury and Topolyan (2016), and Chowdhury et al. (2016). 3 See e.g., Morath and M¨ unster (2008), Parreiras and Rubinchik (2010), Ryvkin (2010), Kirkegaard (2013a) and (2013b), and Wasser (2013a) and (2013b). 4 Eliaz and Wu (2018) study imperfectly discriminating group contests with group-level private information, where the prize value, albeit stochastic, is the same for all group members.

3

to win. More interesting is the interplay of within-group cooperation/coordination vs. competition. That is why we completely characterize equilibria, including those with (partial) coordination on an effort level, and why we consider information transmission via cheap-talk messages within groups. Neither of these avenues is explored (and cheap-talk alone would not help) in the best-shot setup of Barbieri and Malueg (2016). To the best of our knowledge, our study is the first to look at players’ incentives to exchange private information via cheap talk in a group contest setting.5 While Barbieri and Malueg (2018) considers incentives for information sharing in a weakest-link public good setting, we show that the competition across groups we introduce here has important effects. In Barbieri and Malueg (2018), cheap-talk always improves welfare and increases the level of public good provided. In this paper, if cheap-talk happens only in one group, it is possible for the effort of the group that communicates not to change, while that of the group that does not communicate goes down. So, realized group performance can go down after cheap-talk coordination. And if cheap-talk coordination occurs in both groups, then, because of increased competition, utility may not go up at all. The rest of the paper is organized as follows. In the next section, we describe our model, describe the conditions that equilibrium effort distributions must satisfy, and characterize equilibria with a finite effort support. Equilibria with infinite supports are characterized in Section 3 for symmetric groups and in Section 4 for the general setup in which groups can be asymmetric. In Section 5 we explore the possibility of information sharing within a group via cheap talk. Section 6 analyzes extensions to our main setup, and Section 7 concludes.

2

The model

Suppose two groups compete for a single prize in an all-pay auction. The prize is a public good that is consumed nonrivalrously and nonexclusively by all members of the winning 5 Information sharing in individualistic contests is examined inter alia by Kovenock et al. (2015), who find that the exchange of private information among competing entities unambiguously decreases welfare.

4

group. Group l has nl members who share a common valuation of the prize, vl , l = 1, 2. Denote by Il the index set of players in group l. Effort cost is a private information characteristic, independently distributed across players. Within each group l we assume the marginal cost of effort for each player is drawn according to the same atomless cumulative distribution (cdf) Fl over the interval [c, c¯], where c ≥ 0; furthermore, we assume that Fl possesses density fl , which is strictly positive on (c, c¯), l = 1, 2. The assumption of common support is not at all critical, but it is convenient as it shortens the statement of several theorems.6 Distributions, valuations, and group sizes are common knowledge. Each player is informed of the realization of her own cost, but has no information about the realizations of others’ costs, even of fellow group members. Within each group, players’ individual efforts are transformed into the group effort via the weakest-link effort technology; that is, the group effort equals the minimum effort exerted by members in the group. The group with the larger weakest-link effort wins the contest; ties are broken with equal probability in favor of each group. Regardless of victory or defeat, each player sustains the full cost of her effort. Therefore, the realized payoff to a member of group l with cost c exerting effort x is vl − cx if her group wins the contest and −cx if it loses. In what follows we focus on Bayes-Nash equilibria in pure strategies.7 Definition 1. A pure strategy of player i in group l is an Fl -measurable function gli (cli ) that prescribes to every realized cost of effort cli the corresponding effort level gli (cli ). Note that a pure strategy gli generates the bidding distribution Hgli on R+ as follows: Hgli (x) = μFl [cli : gli (cli ) ≤ x] , 6

Intuitively, what matters is the ratio of costs and benefits, and we allow v1 and v2 to vary between groups without constraints, even to the point where c/v1 > c¯/v2 , so that cost-benefit ratios do not overlap in the two groups. 7 We use the standard notion of Bayes-Nash equilibrium (see for instance ch. 8 in Mas-Colell et al., 1995). Standard arguments show that equilibrium strategies must be non-increasing in cost.

5

where μFl is the probability measure on [c, c¯] generated by Fl . Since gli is Fl -measurable, Hgli is well-defined. In what follows, if the strategy gli is part of an equilibrium and confusion does not arise, we simplify notation and denote Hgli with Hli . Similarly, we denote the equilibrium cdf of the −{i}

overall lowest effort in group l with Hl . Furthermore, we indicate with Hl

the equilibrium −{i,j}

cdf of the lowest effort among all members of group l excluding player i. Similarly, Hl

excludes players i and j. Denote the top end of the support of Hl with x¯l and the bottom end with xl , l = 1, 2; that is, xl ≡ min{x | x ∈ supp Hl } and x¯l ≡ max{x | x ∈ supp Hl }. For concreteness, consider a member i of group 1; calculations for group 2 follow similarly. With a contribution of γ < x¯, player i with cost c obtains equilibrium interim utility 

U1i (γ; c) ≡ v1 1 −

−{i} H1 (γ)



 H2 (γ) + v1

γ 0

−{i}

H2 (s) dH1

(s) − cγ.

(1)

Given the behavior of the other players, player i with cost c would now choose γ to maximize U1i (γ; c). Definition 2. A Bayes-Nash equilibrium is called degenerate if the supports of H1 and H2 are singletons, i.e., all types in a group bid the same amount. It is called semi-degenerate if one group’s support is a singleton while the other group’s support is not. Otherwise, it is called non-degenerate. We begin by characterizing degenerate and semi-degenerate equilibria.8 Thereafter, we study our topic of primary interest, the non-degenerate equilibria. The following result is an extension of Theorem 2.8 of Chowdhury et al. (2016); the proof is immediate and here omitted. The idea is simply that at degenerate equilibria all players coordinate on an effort that the highest-cost player in the contest would be willing to exert. While other types would be willing to exert greater effort, the weakest-link aggregation rule prevents such deviations 8

Both classes of equilibria have “mass points.” In general, a mass point at γ in the bidding distribution of a group occurs when a player bids γ with strictly positive probability, and all other players in that group bid at least as high with strictly positive probability.

6

from being worthwhile. Theorem 1 (Degenerate equilibria). Degenerate equilibria exist if and only if both n1 and n2 are at least 2. Now suppose n1 ≥ 2 and n2 ≥ 2. All degenerate symmetric Bayes-Nash v v  1 2 equilibria are as follows: g(c) ≡ λ ∀ c ∈ [c, c¯], where 0 ≤ λ ≤ min , . 2¯ c 2¯ c The following theorem characterizes all semi-degenerate equilibria (this and most subsequent proofs are in the Appendix). Note that members of the team not using a degenerate strategy all use the same strategy. Theorem 2 (Semi-degenerate equilibria). Semi-degenerate equilibria exist if and only if both n1 and n2 are at least 2 and v1 = v2 . Now suppose n1 , n2 ≥ 2, let v1 > v2 without loss of generality, and let a be the unique solution to a F2−1 (a ) =

v2 c¯. v1

The set of all

semi-degenerate equilibria is the following. Fix a ∈ [a , 1). Then, • In group 1, each player contributes x¯ =

v2 an2 −1 2F2−1 (a)

regardless of her cost type.

• In group 2, cost types c < F2−1 (a) contribute x¯ while cost types c > F2−1 (a) contribute 0; cost type c = F2−1 (a) is indifferent between contributing x¯ and 0. While degenerate and semi-degenerate equilibria are easy to describe, we now establish general properties of non-degenerate equilibrium. We begin by establishing properties of the cdfs H1 and H2 . Lemma 1 (Necessary conditions for non-degenerate equilibrium cdfs). In any non-degenerate Bayes-Nash equilibrium, the pdfs of groups’ weakest-link efforts, H1 and H2 , display these properties: 1. x1 = x2 = 0 and x¯1 = x¯2 = x¯ > 0, for some x¯ > 0. 2. There is no mass point for H1 or H2 in (0, x¯). Moreover, not more than one group can have a mass point at zero. 3. H1 and H2 have the same support S. 7

4. S contains no “holes” except possibly at the top, i.e., [0, x¯] \S = (a, x¯) for some a ≤ x¯. Furthermore, one distribution admits a mass point at x¯ if and only if the same is true for the other distribution and a < x¯. Lemma 1 implies H1 and H2 are continuous and strictly increasing on [0, x¯), which in turn implies that individual strategies are strictly decreasing when taking values in (0, x¯). It turns out this is sufficient to establish that teammates use essentially the same strategy.9 Lemma 2 (Equilibrium strategies are symmetric within a team). Fix a group l, l = 1, 2. In any non-degenerate Bayes-Nash equilibrium, for any two agents i and j in group l, we have gli = glj = gl , except on a set of measure zero.

3

Symmetric teams

We now focus on the case in which n1 = n2 = n, v1 = v2 = v, and F1 = F2 = F and characterize all nondegenerate equilibria. By Lemmas 1 and 2 this requires identifying two strategies, g1 and g2 , respectively, for members of groups 1 and 2. We begin by showing, in fact, that players use common strategies, that is, g1 = g2 . Lemma 3 (Nondegenerate equilibria are symmetric). In any non-degenerate Bayes-Nash equilibrium, we have gli = g, for some common function g, except possibly on a set of measure zero, i = 1, . . . , n and l = 1, 2. The following corollary of Lemmas 1 and 2 establishes additional properties of g in a non-degenerate equilibrium; the proof is immediate and has been omitted. Corollary 1. If g is a symmetric non-degenerate Bayes-Nash equilibrium strategy, we have the following: 1. g(¯ c) = 0. 9

The conclusion of Lemma 2 also applies to degenerate and semi-degenerate equilibria.

8

2. g admits no mass at z ∈ [0, x¯), where x¯ = g(c).10 3. g is continuous on (c0 , c¯], where c0 = sup {c : g(c) = x¯}.11 Corollary 1 implies that a symmetric equilibrium bidding strategy is continuous, with the possible exception of a jump to a flat spot in the vicinity of c, which would place a mass at x¯. We consider in turn these equilibria without and with a mass point.

3.1

The symmetric equilibrium strategy without mass

Theorem 3. There exists a unique non-degenerate Bayes-Nash equilibrium without mass at the upper bound. It is symmetric and has strategy given by 

c¯

g(c) = vn c

F (τ )2(n−1) f (τ ) dτ for all c ≤ c ≤ c¯. τ

(2)

Proof. First, note that Lemma 3 implies that any non-degenerate equilibrium is symmetric. The existence part of the proof is by construction. Let g be the symmetric equilibrium strategy. Let cM 1 = max{c12 , ..., c1n } denote a random variable which is the maximum of the random variables c12 , ..., c1n . Note that the relevant distributions for cM 1 are the cdf     M n−1 M n−2 and the density q(cM f (cM Q(cM 1 ) = F (c1 ) 1 ) = (n − 1) F (c1 ) 1 ). The payoff of player 1 with marginal cost c11 that acts like type ca11 can then be expressed as V11 (ca11 , c11 )



ca 11

= c

=

v (1 −

M (F (ca11 ))n ) q(cM 1 ) dc1



c¯

+ ca 11

n 1 − [F (ca11 )]2n−1 v − c11 g(ca11 ). 2n − 1



n M a q(c1 ) dcM v 1 − F (cM 1 ) 1 − c11 g(c11 ) (3)

The first expression in (3) is easily interpreted as the interim probability that group 1 wins when player 1 in group 1 acts as if her cost is ca11 . If player 1 in group 1 exerts least effort, 10

We say g admits mass at z whenever there exist c1 , c2 such that c1 < c2 and g(c) = z for all z ∈ (c1 , c2 ). It should be clear that when g admits no mass at x ¯, it is continuous everywhere, in which case the support of H does not have “holes.” When g does admit mass at x ¯, there is a hole in the support of H equal to (a, x ¯) for some a < x ¯. 11

9

her group will not win. The probability that one of the 2n − 1 players other than player 1 in group 1 has marginal cost exceeding ca11 is 1 − [F (ca11 )]2n−1 . Given this, group 1 wins if the highest-cost player is in group 2, which happens with probability

n , 2n−1

since players act

symmetrically. Now, taking the derivative of (3) with respect to the type ca11 , we have ∂V11 (ca11 , c11 ) = −nv [F (ca11 )]2(n−1) f (ca11 ) − c11 g  (ca11 ). ∂ca11 Thus, the FOC, evaluated at ca11 = c11 , yields g  (c11 ) = −

nv [F (c11 )]2(n−1) f (c11 ). c11

Now g(·) is derived by integrating this equation from c to c¯ and using the boundary condition g(¯ c) = 0. To see that the FOC is sufficient, note that deviations to contributions larger than g(c) are never profitable because they increase cost and do not increase the probability of winning. Moreover, using the FOC, we have ∂V11 (ca11 , c11 ) = −nF (ca11 )2(n−1) f (ca11 )v − c11 g  (ca11 ) ∂ca11  c11 2(n−1) a a f (c11 ) a − 1 , = nv [F (c11 )] c11 which is positive for ca11 < c11 and negative for ca11 > c11 , implying that the solution of the FOC identifies a best response. Necessity of the FOC for the maximum implies that the constructed equilibrium is unique within the class of symmetric non-degenerate equilibria without mass at the upper bound, which, combined with Lemma 3, implies uniqueness within the class of non-degenerate equilibria without mass at the upper bound. Example 1. Suppose F (c) = ca on [0, 1], where a > 0. We then have, for a = 1/(2n − 1),

g (c) =

nav 1 − c(2n−1)a−1 , (2n − 1)a − 1 10

which we obtain using (2). Similarly, if a = 1/(2n − 1), then g(c) = −nv log(c)/(2n − 1). Depending on the parameters a and n, the equilibrium strategy may be concave, linear, or convex in c. It is natural to ask, “How does expected utility change as the number of agents per group increases?” In any symmetric equilibrium, each group’s ex ante payoff gross of costs is v/2. Therefore, an equivalent question is: “How does expected cost change with n?” On the one hand, if we adopt a public good point of view, free-riding within a group becomes more intense as the group gets larger. This pushes efforts down. On the other hand, if we adopt an auction point of view, as free-riding plagues both groups symmetrically, perhaps now some cost types find it in their interest to overtake their weakened competitors and therefore increase their efforts. Indeed, this is what happens in Example 1 if we set a = 1 √ and compare n = 2 and n = 3. One can calculate that g(c)|n=3 ≶ g(c)|n=2 ⇐⇒ c ≶ 1/ 3. It turns out that in Example 1 these two effects exactly cancel out, in expectation. Indeed, we have 

c¯ c

nav cg(c)f (c) dc = (2n − 1)a − 1



1



1 − c(2n−1)a−1 aca dc =

0

av , 2(a + 1)

which is independent of the number of agents per group n. The following proposition shows sufficient conditions such that either effect dominates. Proposition 1 (Effects of increasing group size on utility). Consider the symmetric equilibrium described in Theorem 3. Increasing the number of team members n per team decreases (increases) each agent’s ex ante expected utility if

τ c

W (τ ) ≡

cf (c)dc

τ F (τ )

(4)

is increasing (decreasing) in τ .12 12

W can be interpreted as the ratio of the expected cost to τ , conditional on realized cost being below τ . It is related to the mean-advantage-over-inferiors function δ introduced by Bagnoli and Bergstrom (2005),

11

As discussed before Proposition 1, because of competition across groups, increasing n may have the counterintuitive result that some agents increase their contribution. This

c¯ ambiguity may extend to aggregate effort within a group, Eg agg ≡ n c g(c)f (c)dc, as we show next with two examples. Consider F (c) = ca on [0, 1]—the setup in Example 1. If a = 1/3, Eg agg |n=2 = 4v > 3v = Eg agg |n=3 , but the comparison reverses for a = 1/2, for which we have Eg agg |n=2 = 2v < 9v/4 = Eg agg |n=3 .13 Despite the ambiguity for Eg agg ,

c¯ the next result shows that group performance Eg min ≡ c g(c)dF n (c)—that is, the expected minimum contribution of players within a group—decreases with n. Proposition 2 (Effects of increasing group size on group performance). Consider the symmetric equilibrium described in Theorem 3. Increasing the number of team members n per team decreases ex ante expected group performance. We close this section with another natural question: “How do common changes in the distribution of types affect individual contributions?” The easiest comparison occurs when we consider a first-order stochastic dominance (FOSD) relation. One might conjecture that as higher-cost types become more likely in the sense of FOSD, the probability that a player’s effort will be wasted increases, so contributions should decrease at all cost levels. However, this is not always the case. For a counterexample, fix n = 2 and consider F (c) = ca on [0, 1]—the setup in Example 1—and compare a = 1 with a = 2, which characterize two cdfs related by FOSD. We have g(c)|a=1 = v(1 − c2 ) and g(c)|a=2 = 4v(1 − c5 )/5. These two contribution functions cross, with g(c)|a=1 > g(c)|a=2 for small c, and g(c)|a=1 < g(c)|a=2 for large c. This counterintuitive result occurs because, as the increased probability of wasted efforts plagues both groups symmetrically, sufficiently large cost types may find it in their interest to try to overtake their weakened competitors and therefore increase their efforts. as W (x) = 1 − δ(x)/x. 13 The calculations follow easily because, using (2) and changing the order of integration, we obtain   c¯  c¯ τ nf (c)dc  c¯  c¯ 2 F (τ )2(n−1) f (τ ) n c agg 2 dτ f (c) dc = v F (τ )2(n−1) f (τ ) dτ = v F (τ )2n−1 f (τ )dτ. = vn n Eg τ τ c c c c τ

12

3.2

The symmetric equilibrium strategy with mass at the maximum effort

We now explore the possibility of equilibrium effort distributions with a mass point at the top. It turns out that, varying the size of this mass point, one links the degenerate equilibria in Theorem 1, in which all agents coordinate on the same effort, with the equilibrium in Theorem 3, in which agents with a different marginal cost realization choose a different effort level. Therefore, the size of the mass point can be thought as a convenient parameter that describes the degree of coordination within teams. Let g be the symmetric equilibrium strategy such that: for some c0 > c all types in [c, c0 ) exert the maximum effort x¯ (to be determined), g exhibits a jump discontinuity at c0 ∈ (c, c¯), and g is continuous on (c0 , c¯]. Define x = limc→c+0 g(c), so that x < x¯. As before, cM 1 =   M M n−1 max{c12 , ..., c1n }, so that the relevant distributions for cM 1 are the cdf Q(c1 ) = F (c1 )   M n−2 and the density q(cM f (cM 1 ) = (n − 1) F (c1 ) 1 ). On (c0 , c¯] the equilibrium strategy coincides with that derived in the previous subsection because the (greater) efforts of lower-cost players are not determinative. If all types in [c, c0 ) exert effort x¯, then for n ≥ 2 these types have no incentive to increase effort above x¯. Moreover, they are willing to exert exactly effort x¯ if the type c0 player is just indifferent between x¯ and x . The relevant interim payoffs for the type-c0 player are U1 (x ; c0 ) =

nv 1 − (F (c0 ))2n−1 − c0 x 2n − 1

(by (3))

and

U1 (¯ x; c0 ) = v [F (c0 )]

n−1



1 1 − [F (c0 )]n 2





c¯

+ c0



n M v 1 − F (cM ¯, q(c1 ) dcM 1 ) 1 − c0 x

where the latter equation uses the tie-breaking rule that if all players choose effort x¯, then

13

each group has a 1/2 chance of winning. Indifference for a type-c0 player implies v x; c0 ) = c0 (¯ x − x ) − (F (c0 ))2n−1 . 0 = U1 (x ; c0 ) − U1 (¯ 2

(5)

Note that in equilibrium g could be either left- or right-continuous, with g(c0 ) equalling either x or x¯. For any c ∈ (c0 , c¯] the equilibrium strategy is given by (2), and 



c¯

x = lim g(c) = vn c↓c0

c0

F (τ )2(n−1) f (τ ) dτ . τ

Finally, x¯ is determined from (5): 

c¯

x¯ = vn c0

v F (τ )2(n−1) f (τ ) dτ + [F (c0 )]2n−1 . τ 2c0

(6)

We thus have the following result. Theorem 4. Fix n ≥ 2. There is a continuum of non-degenerate Bayes-Nash equilibria, indexed by c0 ∈ (c, c¯), with mass at the upper bound. Given any c0 ∈ (c, c¯), any cost type c0 < c ≤ c¯ contributes

 g(c) = vn c

c¯

F (τ )2(n−1) f (τ ) dτ . τ

(7)

Types in [c, c0 ) contribute 

c¯

x¯ = vn c0

F (τ )2(n−1) f (τ ) v [F (c0 )]2n−1 , dτ + τ 2c0

while type c0 is indifferent between contributing x¯ and x = vn



c¯ c0

F (τ )2(n−1) f (τ ) dτ . τ

Thus, equilibrium mass on x¯(c0 ) increases from 0 to 1 as c0 increases from c to c¯. Notice that as c0 approaches c, the equilibrium converges to the one of Theorem 3, while as c0 14

approaches c¯, the equilibrium converges to the degenerate equilibrium where all cost types contribute

v . 2¯ c

Figure 1 illustrates the nature of equilibria described by Theorems 3 and 4—

linearity is inessential. g(·) x¯(c0 ) x

c

cˆ

c

c0

c¯

Figure 1: Two equilibrium strategies in the symmetric contest, without (black) and with (red) mass at the upper bound We first ask whether players benefit from playing a “jump equilibrium” rather than the strictly decreasing one. For c > c0 there is no change in interim utilities since (2) and (7) are the same and the equilibrium behavior of lower-cost teammates is irrelevant. However, for c < c0 , interim utility in the jump equilibrium is  V (c; c0 ) =



c0 c

x¯(c0 ) dc +

c¯

g(c) dc,

(8)

c0

using the Envelope Theorem and the fact that the interim utility of type c¯ is zero. In Figure 1, we can then compare the interim utility of type c in the two equilibria by comparing the areas under the two equilibrium strategies, from c to c¯. Therefore, from Figure 1 it is clear that all types c ∈ [ˆ c, c0 ) prefer the jump equilibrum to the strictly decreasing one. An so do types c < cˆ that are sufficiently close to cˆ. However, types near c may or may not prefer the jump equilibrium. But if type c prefers the jump equilibrium, then so do all types c < c0 . More generally, one may ask, “How does ex ante expected utility change as c0 changes?” In a symmetric equilibrium, this is equivalent to asking: “how does expected cost change

15

with c0 ?” Defining the expected effort cost as EC(c0 ) and the largest effort as x¯(c0 ) we have  EC(c0 ) =



c0 c

c¯ x(c0 )f (c) dc +

c¯

cg(c)f (c) dc; c0

in the Appendix, we verify that     c0 f (c0 ) F (c0 ) v 2n−2 EC (c0 ) = [F (c0 )] + cf (c) dc . f (c0 )F (c0 ) − 2 c0 (c0 )2 c 

(9)

It turns out that for the power distributions in Example 1 we have EC  (c0 ) = 0. Indeed,

c 2a−1 a a+1 f (c0 ) c0 , c0 + F(c(c0 )02) = (a + 1)ca−2 , if F (c) = ca , then c 0 cf (c)dc = a+1 0 , and f (c0 )F (c0 ) = ac0 so in (9) the term in square brackets equals zero. Proposition 3 (Effects of increasing mass at the upper bound on utility). Consider the class of equilibria described in Theorem 4. Increasing the size of the mass point by increasing c0 decreases (increases) each agent’s ex ante expected utility if W (τ ), defined in (4), is increasing (decreasing) in τ . The reason for the different outcomes in Proposition 3 is that increasing c0 brings about two countervailing changes in equilibrium strategies. Because x ¯(c0 ) is decreasing in c0 ,14 if the jump point is increased from c0 to c0 , then the contribution of types immediately to the left of c0 is increased, while that of types closer to c is decreased. One may trace the effects of these two changes in interim utilities. From (8) we have ∂V (c; c0 ) = x¯(c0 ) − g(c0 ) + x¯ (c0 )(c0 − c) ∂c0   v v v 2n−1 2n−2 2n−1 = [F (c0 )] − (c0 − c) [F (c0 )] f (c0 ) + [F (c0 )] 2c0 2c0 2(c0 )2   c c0 − c v f (c0 ) ; [F (c0 )]2n−1 − = 2c0 c0 F (c0 ) as expected, types close to c0 benefit from the increase in c0 , but types close to c may be 14

This is easily deduced from (6).

16

damaged. Another way to see that utility may be different for two equilibria with different c0 is to note that, while the revenue equivalence theorem is valid in our setup, its assumptions may not be satisfied. For example, the assumption that agents perceive the same probability of victory at the interim stage in two equilibria with different c0 is not satisfied, because the probability of ties is different.15 We now analyze how coordination affects group performance and aggregate efforts. Proposition 4 (Effects of increasing mass at the upper bound on efforts). Consider the class of equilibria described in Theorem 4. Increasing the size of the mass point by increasing c0 decreases (increases) each group’s ex ante expected performance if and only if

F (c0 ) c0 f (c0 )

is greater than (less than) n−1. And increasing c0 always decreases

ex ante expected aggregate group efforts. Note that the fact that expected aggregate (and individual) effort always decreases with c0 is compatible with expected cost increasing or remaining constant. This is because, as illustrated in Figure 1, it is types with the lowest marginal cost of effort that decrease their efforts as c0 increases.

4

Asymmetric teams: different values, sizes, or distributions

We now consider the general case in which n1 and n2 , v1 and v2 , and F1 and F2 are not necessarily the same and characterize all nondegenerate equilibria. By Lemmas 1 and 2 this requires identifying two strategies, g1 and g2 ; in equilibrium, all members of group i use gi , i = 1, 2. In contrast with the symmetric case, now one group can have a mass point at 0 in its distribution of minimum effort. And, as in the symmetric case, mass points at 15 Likewise, revenue is not constant across the degenerate equilibria in Theorem 1 because the agent with the worst type receives different interim utility.

17

the largest effort remain possible. In our derivations, we use a construction due to Amann and Leininger (1996). For the range of efforts where both g1 and g2 are strictly decreasing, consider the function ξ(c11 ) = g2−1 (g1 (c11 )), which maps any cost type c11 of player 1 in group 1 who puts effort g1 (c11 ) into that cost type of player 1 in group 2 who exerts the same effort, i.e., c21 = ξ(c11 ) = g2−1 (g1 (c11 )). Note that since both g1 and g2 are strictly decreasing, ξ is strictly increasing and, by the chain rule, ξ  (c) = (g2−1 ) (g1 (c))g1 (c).

(10)

Mass points at zero or at the largest contribution affect the range of cost types where ξ is well-defined. We consider first equilibrium strategies that do not put mass at the top, then those that do.

4.1

Equilibrium strategies without mass at the maximum effort

The following result, with proof in the Appendix, characterizes equilibrium. Theorem 5. Assume limx→c f2 (x) = 0. There exists a unique non-degenerate Bayes-Nash equilibrium without mass at the upper bound. Let ξ be the unique strictly increasing solution to the differential equation ξ  (c) =

n1 v2 cf1 (c) , n2 v1 ξ(c)f2 (ξ(c))

(11)

together with the initial condition ξ(c) = c. Each player in group 1 that exerts a positive effort uses strategy

 g1 (c) = −

c¯ c

g1 (τ ) dτ,

(12)

where g1 (τ ) = 0 if g1 (τ ) = 0, and otherwise g1 (τ ) = −

n1 v2 F2 (ξ(τ ))n2 −1 F1 (τ )n1 −1 f1 (τ ) . ξ(τ ) 18

(13)

Each player in group 2 that exerts positive effort uses strategy

g2 (c) = g1 ξ −1 (c) .

(14)

Theorem 5 does not describe which team, if either, puts positive probability on zero or how large the atom of probability is. These considerations are resolved by the solution to the differential equation (11) with initial condition ξ(c) = c. For example, if in tracing this solution one finds a c < c¯ such that ξ(c ) = c¯, this means that group-1-player types in [c , c¯] never win; therefore, these types exert zero effort. And if instead ξ(¯ c) = c for some c < c¯, then it is group-two-player types [c , c¯] that exert zero effort. The following proposition presents a sufficient condition for each case, in addition to determining which group contains the more “aggressive” players. Proposition 5. The equilibrium described in Theorem 5 has the following properties: 1. If ∀c ∈ [c, c¯] we have

v1 n1

<

v2 n2

·

f1 (c) , f2 (c)

then g1 (c) < g2 (c) ∀c ∈ (c, c¯). Every player in

group 1 puts mass at zero, i.e., H1i (0) > 0 ∀i ∈ I1 , while H2j is atomless for any j ∈ I2 . 2. If ∀c ∈ [c, c¯] we have

v1 n1

=

v2 n2

·

f1 (c) , f2 (c)

then g1 (c) = g2 (c) and neither group puts mass at

zero. 3. If ∀c ∈ [c, c¯] we have

v1 n1

>

v2 n2

·

f1 (c) , f2 (c)

then g2 (c) < g1 (c) ∀c ∈ (c, c¯). Further, H1i is

atomless for any i ∈ I1 , while players in group 2 put mass at zero. While we state Proposition 5 for general distributions, two important applied considerations arise when F1 = F2 . First, Proposition 5 shows that, if v1 = v2 , then players in the smaller group bid more aggressively than those in the larger group.16 Second, notice that if per-member values across the two groups are identical, i.e., if v1 /n1 = v2 /n2 , then ξ becomes the identity map, and every player uses the same strategy in equilibrium as in 16

This result is analogous to the one in Barbieri et al. (2014) that shows, in the semi-symmetric equilibria of the group best-shot all-pay auction with complete information and symmetric valuations, that players in the larger group put mass at zero.

19

Theorem 3. The fact that per-member values are key in our pure public good setup is very surprising. Indeed, this turns out to be a coincidence. What truly matters for g1 = g2 is that v1 × n2 /(n1 + n2 − 1) = v2 × n1 /(n1 + n2 − 1), i.e., the values for victory, multiplied by the probability of victory for an agent who is not the overall weakest link, must be the same in the two groups. To see the technical intuition for why in this case g1 = g2 is consistent with equilibrium, note that, if strategies are the same, then the utility of an agent in group 1 that has marginal cost c11 and acts like type ca11 can be written in an analogous manner to (3): V11 (ca11 , c11 ) = v1



n2 1 − [F (ca11 )]n1 +n2 −1 − c11 g(ca11 ). n 1 + n2 − 1

The interpretation is similar to that given after (3): if player 1 in group 1 is the one with the overall largest cost, her group will not win. Given that this does not happen, group 1 wins if the highest-cost player is in group 2, which happens with probability n2 /(n1 + n2 − 1) since players act symmetrically. Proceeding similarly for group 2, we see that symmetry of strategies is consistent with equilibrium only if v1 × n2 /(n1 + n2 − 1) = v2 × n1 /(n1 + n2 − 1). We now illustrate the results in this section with two examples. The first one applies Theorem 5 and Proposition 5, and further determines which group is more likely to win. Example 2. F1 and F2 are uniform on [0, 1], n1 = 1, and n2 = 2. We use Theorem 5 to find the equilibrium without mass at the maximum effort. Here, (11) reads as ξ(c) ξ  (c) =

which can be solved as

 ξ(c) =

v2 c, 2v1

v2 c. 2v1

Consistent with Proposition 5, we see that equilibrium depends on the relation between v2 and 2v1 . We only consider the case v2 ≤ 2v1 , since the other is dealt with in a similar fashion. If v2 < 2v1 , then ξ(1) < 1, so group-2 players drop out for costs sufficiently close to 1, while

20

the group-1 effort distribution is atomless. Therefore, g1 (c) < 0 ∀c ∈ [0, 1), and using (12), (13), and (14), we obtain g1 (c) = v2 (1 − c) and

g2 (c) =

⎧   ⎪ √ ⎪ v2 ⎨ 2v1 v2 − c 2v1

if c ≤

⎪ ⎪ ⎩0

if c >

 

v2 2v1 v2 . 2v1

Note that if v2 = 2v1 , then members of groups 1 and 2 end up using the same strategy. If v2 < 2v1 , then g2 (c) < g1 (c) for all c ∈ (0, 1), in accordance with Proposition 5. In either case, since g2 (c) ≤ g1 (c) and n2 > n1 , it follows that group 1 is more likely to win. We conclude this section with a second example, which illustrates how to proceed when Proposition 5 does not apply. In particular, we show that it is not true that members of the smaller group are always more aggressive, even for identical values, when cost distributions differ. Example 3. F1 (c) = ca and F2 (c) = cb on [0, 1] with a > 0 and b > 0, n1 = 1, and n2 = 2. While now Proposition 5 does not apply, one can still work with Theorem 5 to find the equilibrium. Here, (11) reads as ξ  (c) =

which can be solved as

 ξ(c) =

v2 caca−1 , 2v1 ξ(c)b(ξ(c))b−1

v2 a b + 1 2v1 a + 1 b

1 b+1

a+1

c b+1 .

√ For the sake of concreteness, set v1 = v2 = 1, a = 1/2, and b = 1. Then ξ(c) = c3/4 / 3, and since ξ(1) < 1, g1 has no flat spot and g2 does. Now, using (12), (13), and (14), we obtain g1 (c) = (1 − c1/2 ) and

g2 (c) =

⎧ ⎪ ⎪ ⎪ ⎨1 − (3c2 )1/3

√ if c ≤ 1/ 3

⎪ ⎪ ⎪ ⎩0

otherwise. 21

In contrast with equilibrium when Proposition 5 applies, the strategies described above cross, with g1 (c) ≶ g2 (c) for c ≶ 1/9.

4.2

Equilibrium strategies with mass at the maximum effort

We next derive jump equilibria analogous to those in Section 3.2 when groups use different strategies, now assuming both groups have at least two members. For each l = 1, 2, let gl be an equilibrium strategy such that all group l types in [c, c0l ) exert the maximum effort x¯ (to be determined), gl exhibits a jump discontinuity at some c0l ∈ (c, c¯) and is continuous on (c0l , c¯]. By Lemma 1, the upper bound x of the continuous part of the support should be the same for both groups. Thus x = limc→c+01 g1 (c) = limc→c+02 g2 (c) and x < x¯. As before, let cM = max{cl2 , ..., clnl }, so that the relevant distributions for cM are the cdf l l     M nl −1 M nl −2 and the density ql (cM fl (cM Ql (cM l ) = Fl (cl ) l ) = (nl − 1) Fl (cl ) l ). On (c01 , c¯] the equilibrium strategy coincides with that derived in the previous subsection because the (greater) efforts of lower-cost players are not determinative. If all types in [c, c01 ) exert effort x¯, then for n1 ≥ 2 these types have no incentive to increase effort above x¯. Moreover, they are willing to exert exactly effort x¯ if the type c01 player is just indifferent between x¯ and x . Similar reasoning applies to group 2. The relevant interim payoffs for the type-c01 player are 



U1 (x ; c01 ) = v1

c¯ c01

M M Pr g2 (c2j ) < g1 (cM 1 ), for some j = 1, . . . , n2 q1 (c1 ) dc1 + v1 (1 − [F2 (c02 )]n2 ) [F1 (c01 )]n1 −1 − c01 x

and  x; c01 ) = v1 U1 (¯



M M Pr g2 (c2j ) < g1 (cM 1 ), for some j = 1, . . . , n2 q1 (c1 ) dc1 c0 1  1 n2 [F1 (c01 )]n1 −1 − c01 x¯, + v1 1 − [F2 (c02 )] 2 c¯

22

where the latter equation uses the tie-breaking rule that if all players choose effort x¯, then each group has a 1/2 chance of winning. Indifference for a type-c01 player in group 1 implies x; c01 ) = c01 (¯ x − x ) − 0 = U1 (x ; c01 ) − U1 (¯

v1 (F1 (c01 ))n1 −1 (F2 (c02 ))n2 . 2

(15)

Similarly, indifference for type c02 in group 2 yields x − x ) − 0 = c02 (¯

v2 (F1 (c01 ))n1 (F2 (c02 ))n2 −1 . 2

(16)

Equations (15) and (16) together imply

v2 c01 F1 (c01 ) = v1 c02 F2 (c02 ).

(17)

Because the left- (right-) hand side of (17) is strictly increasing in c01 (c02 ), for a given c01 ∈ [c, c¯] there exists at most one c02 satisfying (17). In particular, when c01 = c, we have c02 = c. Consider the following possibilities: 1. v1 < v2 . When c01 = c¯, there is no c02 that solves (17). Define the function ν1 (c) = cF1 (c), which is strictly increasing in c. Then there exists a unique ν1−1 ( vv12c¯ ). It can be verified that when c01 > ν1−1 ( vv12c¯ ), (17) never holds. When c ≤ c01 ≤ ν1−1 ( vv12c¯ ), there exists a unique c02 that solves (17). 2. v1 = v2 . Define ν2 (c) = cF2 (c), and using the fact that ν2 is strictly increasing, establish that for every c01 ∈ [c, c¯] there exists a unique c02 that solves (17). The corresponding solution ranges from c to c¯ as c01 ranges from c to c¯. Note that when c < c01 < c¯, it is generally not true that c02 = c01 unless F1 = F2 . 3. v1 > v2 . Similarly, for every c01 ∈ [c, c¯] there exists a unique c02 that solves (17). The corresponding solution ranges from c to ν2−1 ( vv21c¯ ) as c01 ranges from c to c¯. 23

Note that, quite interestingly, group size does not enter (17); what matters is the difference in values and cost distributions. Of course, in equilibrium g1 could be either left- or right-continuous, with g1 (c01 ) equaling either x or x¯. The same is true about g2 . For any c ∈ (c01 , c¯] the equilibrium strategy g1 is given by (12), where g1 (τ ) = 0 if g1 (τ ) = 0, otherwise g1 (τ ) is determined by (13), and ξ is the unique solution to the initial value problem given by (11) together with the initial condition ξ(c01 ) = c02 . Then x = lim g1 (c). c↓c01

Finally, x¯ is determined from (15). We thus have the following result. Theorem 6. Fix n1 ≥ 2 and n2 ≥ 2. For each i = 1, 2, let νi (c) = cFi (c). There is a continuum of non-degenerate Bayes-Nash equilibria, indexed by c01 ∈ C ⊆ (c, c¯), with mass at the maximum effort. The range of the mass at the maximum effort in group 1 is as follows. 1. If v1 < v2 , then C = (c, ν1−1 ( vv12c¯ )]. 2. If v1 ≥ v2 , then C = (c, c¯). Given any c01 ∈ C, any cost type c ∈ (c01 , c¯] in group 1 uses strategy g1 (c) = −

c¯ c

g1 (τ ) dτ,

with g1 (τ ) = 0 if g1 (τ ) = 0, and otherwise g1 (τ ) = −

n1 v2 F2 (ξ(τ ))n2 −1 F1 (τ )n1 −1 f1 (τ ) , ξ(τ )

where ξ is the unique solution to the initial value problem given by (11) together with the   initial condition ξ(c01 ) = c02 ≡ ν2−1 vv21 c01 F1 (c01 ) . Cost type c02 < c ≤ c¯ in group 2 that exerts positive effort uses strategy

g2 (c) = g1 ξ −1 (c) .

24

(18)

Group-1 types in [c, c01 ) and group-2 types in [c, c02 ) contribute

x¯ = lim g1 (c) + c↓c01

v1 [F1 (c01 )]n1 −1 [F2 (c02 )]n2 , 2c01

while group-1 type c01 and group-2 type c02 are indifferent between contributing x¯ and x = lim g1 (c). c↓c01

(19)

Theorem 6 implies that, when v1 < v2 , the equilibrium mass that group-1 members place on x¯(c01 ) varies between 0 and some intermediate level17 (less than 1), while equilibrium mass in group 2 varies between 0 and 1. Notice that as c01 approaches c, the equilibrium converges to the one without coordination of Theorem 5, while as c01 approaches the highest possible level ν1−1 ( vv12c¯ ), the equilibrium converges to the semi-degenerate equilibrium of Theorem 2 corresponding to k = 2, where all players in group 2 contribute c¯ while players in group 1 contribute 0 or c¯ with some positive probabilities. The case v1 > v2 provides analogous insights. When v1 = v2 , equilibrium masses on x¯(c01 ) by groups 1 and 2 increase from 0 to 1 as c01 increases from c to c¯. When c < c01 < c¯, those two masses are not equal, generally. As c01 approaches c¯, the equilibrium converges to the degenerate equilibrium where all cost types contribute

v . 2¯ c

As c01 approaches c, the equilibrium converges to the one of Theorem

5. Therefore, similarly to Section 3, we can interpret increases in c01 and c02 as increases in the degree of coordination on effort levels within each group. The following result is analogous to Proposition 5 and provides sufficient conditions for ordering groups’ strategies. Proposition 6. Consider an equilibrium of Theorem 6 with cost cutoffs c01 and c02 . This equilibrium has the following property. 17

  This level equals F1 ν1−1 ( vv12c¯ ) .

25

1. If

F1 (c01 ) F2 (c01 )

>

v1 v2

and

v1 n1

<

v2 n2

·

f1 (c) f2 (c)

for all c ≥ c01 , then g1 (c) < g2 (c) ∀c ∈ (c01 , c¯). Every

player in group 1 puts mass at zero, while the group-2 effort distribution is atomless at zero. 2. If

F1 (c01 ) F2 (c01 )

=

v1 v2

and

v1 n1

=

v2 n2

·

f1 (c) f2 (c)

for all c ≥ c01 , then g1 (c) = g2 (c) ∀c ∈ (c01 , c¯) and

neither group puts mass at zero. 3. If

F1 (c01 ) F2 (c01 )

<

v1 v2

and

v1 n1

>

v2 n2

·

f1 (c) f2 (c)

for all c ≥ c01 , then g1 (c) > g2 (c) ∀c ∈ (c01 , c¯). Every

player in group 1 uses a strategy that is atomless at zero, while players in group 2 put mass at zero. One might conjecture that a “stronger” group (in the sense of higher valuation or smaller size) puts greater mass at the upper bound of the support. However, this is not true in general. First, group size does not enter (17). Second, Table 1 below shows that, if F1 = F2 , then the mass at the top effort level can be largest for the group with the lowest valuation.18 Table 1: Examples of upper bound mass when F1 (c) = c, F2 (c) = c2 on [0, 1]; v1 = 1, and v2 = 2 c01 0.10 0.20 0.30

c02 0.27 0.43 0.56

F1 (c01 ) F2 (c02 ) 0.10 0.07 0.20 0.19 0.30 0.32

We conclude this section by remarking on two issues regarding equilibria with mass at the top effort. 1. Increases in coordination in group 1, i.e., increases in c01 , are only consistent in equilibrium with increases in coordination in group 2, i.e., with increases in c02 . This holds because the left- and right-hand side of (17) are strictly increasing in c01 and c02 , respectively. In other words, (increased) coordination in a group is only possible in equilibrium if there is (increased) coordination in the competitor group. 18

Simulations show that first-order stochastic dominance does not provide clear-cut conclusions, either.

26

2. Changing degrees of coordination changes the expected probability of victory of each group. It is important to note how in the symmetric equilibria of Section 3 these issues are trivial, while in the present asymmetric framework the size of the effects are highly dependent on distributions and parameter values.19

5

Cheap-talk communication within groups

In this section, we consider the effects of group members coordinating their efforts within each group via cheap-talk communication. The complementarity of efforts of the weakest-link technology offers strong rewards to coordination, since wasted contributions are eliminated and agents are willing to contribute more, because they do not fear that teammates will not match their effort. As demonstrated in Barbieri and Malueg (2018), it is possible to design simple and intuitive schemes that provide agents with the appropriate individual incentives to coordinate their behavior in a weakest-link public good model. Here, we focus on one such scheme. Before jointly deciding on a contribution level, teammates exchange information. In particular, within each team agents simultaneously exchange public messages about their cost realization: possible messages are the elements of [c, c¯]. Then teammates all contribute the effort level most preferred by the agent with the highest announced cost. In their public good model, Barbieri and Malueg (2018) show that truth-telling is a part of an equilibrium. The same result holds in our contest setup. Intuitively, one can draw a relationship to the optimality of truthful bidding in the secondprice auction: a lie can only change the allocation for the worse. Consider an agent with true cost c, and let the highest cost reported by his teammates be cM . Also, as described above, let all of this agent’s teammates coordinate on the effort most preferred by who announces the highest cost. If cM > c, then a lie can only increase the overall reported maximum cost 19

The working paper version of this document, Barbieri et al. (2018), provides a detailed example illustrating these issues.

27

above cM . This lowers teammates’ contributions, so it is never profitable. And if cM < c, then by telling the truth the overall reported maximum cost is c, so a truthful agent gets his teammates to contribute at his most preferred effort level, which the agent then matches. Overstating c damages this agent, as teammates would contribute less. Note that this agent gains nothing by understating c, because this agent then would not match his teammates’ increased effort level. Rather, he would stick to the optimal contribution for c, which is unaffected by the increased teammates’ efforts. We formalize this argument in Proposition 8 when both groups share information among their own members. But we first explore the consequences of information sharing when only one group does so. In their public good model (only one group exists), Barbieri and Malueg (2018) show that information sharing allows the agent with the realized highest cost to contribute more, as now he is sure to be pivotal with probability one. Therefore, the peformance of the group increases. Furthermore, all other team members do not have to waste effort above that of the weakest link and all agents benefit. In our symmetric setup with 2 groups of n agents each, it turns out that a similar result goes through if one group coordinates its efforts as described above. But one must also consider competitive effects. In the rest of this section, we focus on equilibria without mass points at the maximum effort.20 Proposition 7. Consider a symmetric environment with 2 groups of n agents each. Assume that within group 1 cost realizations are revealed among teammates and agents coordinate on the effort most preferred by the agent with the largest cost, while agents in group 2 behave as in Sections 3 and 4, i.e., they do not share information or coordinate efforts. With respect to the situation in which neither group coordinates described in Theorem 3, the probability of victory of group 1 increases and the interim expected utility of any agent in group 1 strictly increases, for any cost type strictly larger than c. Notice that Proposition 7 only states the the probability of victory of group 1 increases, 20

To avoid unnecessary technical complications, we further assume the density f is strictly positive on [c, c¯], not just (c, c¯) as in previous sections.

28

not that the performance of group 1 increases. Indeed, the following example shows that communication in group 1 may result only in the discouragement of agents in group 2. Example 4. F( c) = c on [0, 1], n = 2, and v = 1. After cheap-talk messages and coordination, it is as if the number of agents in group 1 is reduced to 1, and the cost distribution in this group becomes that of the maximum of two uniforms. Therefore, the situation is the  3 same described in Example 3 with a = 2 and b = 1. This results in ξ(c) = 23 c 2 . Substitution into (12)–(14) yields g1 (c) = 1 − c2 , and

g2 (c) =

⎧ 3 ⎪ ⎪ ⎨1 − c3 3 2

if c ≤

⎪ ⎪ ⎩0

if c >

2

 

2 3 2 . 3

Importantly, the equilibrium without cheap talk described in Example 1 for this setup also has g(c) = 1 − c2 . Therefore, cheap-talk communication leaves the strategy of the group that comunicates unchanged, while that of the group that does not communicate goes down. The common-sense result in Proposition 7 leaves open an important question in our contest setup: since the reduction of free-riding within teams is expected to induce harsher competition across teams, do agents actually benefit if they all coordinate their efforts in their respective groups via cheap talk? We again answer this question in our symmetric setup with 2 groups of n agents each. First, we determine the equilibrium effort level most preferred by the agent with the largest cost realization cM . Denote the effort level that all teammates provide after exchanging information in a symmetric equilibrium as gM (cM ). If the highest cost agent in group 1 contributes x, in equilibrium she receives utility −1 v(1 − F (gM (x))n ) − cM x;

29

so the FOC yields −1 −1 (x))n−1 f (gM (x)) −vnF (gM

1 −1  gM (gM (x))

= cM .

In a symmetric equilibrium the above can be evaluated at x = gM (cM ), so  (cM ) = − gM

vnF (cM )n−1 f (cM ) . cM

(20)

c) = 0, the above displayed equation then yields this analog of (2): Noting that gM (¯ M

gM (c ) = vn



c¯ cM

F (τ )n−1 f (τ ) dτ. τ

(21)

As expected, the comparison of (2) and (21) reveals the highest-cost agent is more aggressive in the presence of coordination via cheap talk, as gM (c) > g(c) ∀c ∈ [c, c¯), and so each team’s performance is larger for any realization of the costs of its members. Note as well that, while the weakest link works harder, communication means that the other team members do not have to waste effort above that of the weakest link. Second, we now show that truth-telling is (part of) an equilibrium when agents report their cost realization to their teammates, given the above contribution strategies, assuming all received messages are believed to be true, and given messages and behavior in the other team. Consider the utility of a group-1 agent with realized cost c who manages to induce a maximal realization of the cost z in her group: VM (z, c) = v(1 − F (z)n ) − cgM (z).

We then have vnF (z)n−1 f (z) ∂VM (z, c)  =− z − cgM (z) ∂z z  = gM (z)(z − c);   

by (20)

<0

30

therefore, VM (z, c) is strictly increasing in z for z < c and strictly decreasing for z > c. This implies that any lie when reporting one’s true cost to teammates can only damage this agent. Indeed, if this agent’s cost realization is maximal in the group, then z = c is optimal by the above reasoning. And if the maximum cost c > c belongs to another agent, then a lie can only move z further to the right of c . But this is not profitable since VM (z, c) decreases in z for z > c. We can now compare payoffs in the equilibrium described above, where agents contribute gM (cM ) described in (21), and in the equilibrium of Theorem 3. Since in any symmetric equilibrium, each group’s ex ante payoff gross of costs is v/2, we can focus on the comparison of expected costs. Using (21), an agent’s expected cost in the game with cheap talk equals 



c¯

c

c c

c

n−1





c¯

n−1

gM (c)dF (τ ) + gM (τ )dF (τ ) f (c)dc c ⎡ ⎤  c¯  c¯  c ⎣gM (c)F (c)n−1 + gM (¯ c) F (¯ c)n−1 − gM (c)F (c)n−1 − F (τ )n−1 gM (τ )dτ ⎦ f (c)dc =    c c =0

 =

c¯





c¯

c vn c

c

 F (τ )2(n−1) f (τ ) dτ f (c)dc, τ

where the first equality follows after integration by parts and the second by (21). But the above displayed equation is the expected cost in the equilibrium described in Theorem 3; indeed, the term in square brackets above is g(c) in (2). In other words, all within-team gains are lost to increased competition between teams. Proposition 8. In a symmetric environment, agents’ ex ante utility is the same in the symmetric equilibrium with cooperation with the effort strategy described in (21) and in the symmetric equilibrium without cooperation described in Theorem 3. Two interesting contrasts arise between our group contest and the public good model of Barbieri and Malueg (2018). First, cheap talk may not increase group performance, as Example 4 shows. Second, Proposition 8 shows that cheap talk may not increase agents’ 31

utility. It is well worth pointing out that in Proposition 8 we are comparing the equilibrium without communication and without mass at the top described in Section 3, with that with communication described in this section, that is the two equilibria with the least and the most cooperation. However, without communication, other non-degenerate symmetric equilibria with mass at the top exist, and these equilibria are not generally ranked in terms of payoffs, so the comparison with communication becomes more complex.21 One can also interpret Proposition 8 in light of the revenue equivalence theorem. In both equilibria, two quantities are the same across the sharing and non-sharing regimes: 1) the interim probabilities of victory for any type, and 2) the utility for the worst types. Then, ex ante expected utility and therefore ex ante expected efforts have to be the same across the two equilibria.

6 6.1

Extensions More than two groups

Our arguments in Theorem 3 are easily extended to the situation where there are more than two competing groups, as long as these groups are identical and all agents act symmetrically.22 Denote with N ≥ 2 the number of competing groups. The analogue of (3) is V11 (ca11 , c11 )



ca 11

= c

v (1 −

N −1 (F (ca11 ))n )

M q(cM 1 ) dc1 +



c¯ ca 11



n N −1 M a v 1 − F (cM q(c1 ) dcM 1 ) 1 −c11 g(c11 ).

The FOC then yields the following equivalent of (2):  g(c) = vn(N − 1)

c¯ c

(1 − F (τ )n )N −2 F (τ )2(n−1) f (τ ) dτ for all c ≤ c ≤ c¯. τ

21 A way to rank these equilibria, and therefore compare them to the equilibrium with communication, is to use the sufficient condition in Proposition 3, if it applies. 22 Without assuming symmetry this becomes a qualitatively harder problem and much of the tractability is lost, see, e.g., Parreiras and Rubinchik (2010).

32

Furthermore, one can show that Proposition 1 is unchanged. For a uniform distribution on [0, 1], n = 2, and N = 3, the above displayed equation solves as g(c) =

v 3

(1 − 3c4 + 2c6 ) . If

we compare this to Example 1, where g(c) = v(1 − c2 ), we see that increasing the number of competing groups from 2 to 3 reduces agents’ efforts.

6.2

Non-linear costs

If we assume symmetry, we can easily show that our assumption of linear costs is not essential. For example, we can let the cost of contribution x be c ρ(x), where ρ is a convex function with ρ(0) = 0. Focusing on the symmetric teams case and strictly decreasing equilibrium strategies, the utility of type c11 that acts like type ca11 can be expressed as V11 (ca11 , c11 ) =

n 1 − [F (ca11 )]2n−1 v − c11 ρ(g(ca11 )), 2n − 1

which is the analogue of (3). The FOC, along with the boundary condition g(c) = 0 then yields

 ρ(g(c)) = vn c

c¯

[F (τ )]2(n−1) f (τ ) dτ, τ

which is the analogue of (2). Of course, g can be retrieved by inverting ρ. Furthermore, one can show that both Propositions 1 and 8, which deal with agents’ utility in equilibrium, are unchanged.

6.3

Differential uncertainty about teammates and competitors

In Sections 3 and 4, we made the assumption that each player has the same information about members of his and of the other group. Section 5, through cheap talk, effectively considers the other extreme assumption: players know the costs of their teammates. It is important from the point of view of applications to consider an intermediate situation. In this section, we assume that with probability p teammates on a given team become informed of each other’s costs and play as described in Section 5, while with probability 1 − p those 33

agents are only informed of their own cost. These probabilities are common and independent across teams. A full treatment of this issue is deserving of a separate analysis. But here we provide a class of strategies for which our previous work can be used essentially unchanged, and that consitutes an equilibrium under a condition we determine. We restrict attention to symmetric groups and strategies without mass points at the top. We have in mind the case of two effort functions, gi and gu —one for the case in which agents are informed, and one when they are uninformed of their teammates’ costs—with non-overlapping effort ranges. In particular, the range of gu is [gu (¯ c) = 0, gu (c)] and the range of gi is “higher”: c), gi (c)]. [gu (c) = gi (¯ Given these strategies, consider first the case in which information happens to be revealed within a group, group 1, say. Denote with cM 1 the realized maximum cost in group 1. As in Section 5, agents in this group coordinate and they all contribute gi (cM 1 ). With probability 1−p, agents in group 2 are uninformed, so group 1 wins. With probability p, agents in group 2 are informed as well, so group 1 wins with probability (1 − F n (cM 1 )). Similar considerations hold for group 2. Therefore, it is as if groups for which information is revealed are competing as described in Section 5, but for an “effective” prize of size v × p, as with probability (1 − p) they will win regardless of effort choice, as long as it stays above gu (c) = gi (¯ c). Thus, the relevant FOC resembles (20): gi (cM ) = −

v(1 − p)nF (cM )n−1 f (cM ) , cM

c) = gu (c), rather than gM (¯ c) = 0. but now the boundary condition is gi (¯ Consider now the case in which information happens not to be revealed within a group. The same reasoning above implies that it is as if groups are competing in the same manner described in Section 3 for an “effective” prize of size v × (1 − p), as with probability p they are sure to lose regardless of their effort choice. As the final condition remains the same of

34

Section 3, i.e., gu (¯ c) = 0, the strategy gu can be calculated from (2) as  gu (c) = v(1 − p)n

c

c¯

F (τ )2(n−1) f (τ ) dτ for all c ≤ c ≤ c¯. τ

We now consider non-local deviations. The reasoning described in Section 3 implies that, if information is not revealed in group 1, type c has no profitable deviation to any effort in the range [0, gu (c)]. And deviations to larger efforts are not profitable, as no teammate contributes more than gu (c) in equilibrium. The reasoning described in Section 5 implies that, if information is revealed in group 1, type c has no profitable deviation to any effort in the range [gi (¯ c) = gu (c), gi (c)]. But deviations to lower efforts are potentially profitable. We characterize the payoff of a deviation to c) = 0, gu (c)] through the equivalent choice of a type c˜ to mimick (and exert effort γ ∈ [gu (¯ effort gu (˜ c) = γ). We then have that the deviation payoff of type c that acts as c˜ is c) + v(1 − p)(1 − F n (˜ c)), D(c, c˜) = −cgu (˜

as this type is sure to be the weakest link within his group. A sufficient condition for the deviation to be non-profitable is for D(c, c˜) to have a maximum at c˜ = c, for any c. This is the case for a uniform distribution of costs on [0, 1] and two agents per group, as we obtain ∂D(c, c˜) c) − v(1 − p)2(˜ c) = 2v(1 − p)˜ c (c − 1) < 0. = −cgu (˜ ∂˜ c Furthermore, we can calculate that gu (c) = v(1 − p)(1 − c2 ), and gi (c) = v(1 + p − 2cp). As p → 0, observed equilibrium behavior converges to that described in Example 1, with contribution function g(c) = v(1 − c2 ). As p → 1, observed equilibrium behavior converges

35

to a contribution function g(c) = 2v(1 − c), which is the same one obtains in Section 5 from equation (21).

7

Conclusion

This paper contributes to the emerging literature on group contests with incomplete information and is the first attempt to study players’ incentives to share information in a group contest. We analyzed the interplay between the weakest-link effort aggregation and private information about cost of effort. A remarkable finding is that players within a group always choose symmetric strategies. What is surprising is that agents in the same group that have the same realized cost contribute the same amount, despite the uncertainty about teammates’ contributions. This result is important because it allows a characterization of all equilibria without assuming symmetry, in contrast with the more restrictive analysis in Barbieri and Malueg (2016) for the best-shot aggregator, which has to focus on equilibria that are (essentially) symmetric.23 The literature on full-information weakest link contests (Chowdhury et al., 2016) establishes existence of pure strategy equilibria (in which the support of a player’s strategy is a singleton). In our setting, where each player is privately informed about her own cost of effort but is unsure about the other players’ costs, a player’s level of effort depends on the private realization of her cost. Interestingly, we show that, even when the cost of effort is a private information, it is still possible for the support of a player’s strategy to be a singleton. In such a situation, all players coordinate on the effort that the highest-cost player is willing to exert. Although other types would be willing to exert greater effort, the weakest-link technology makes such deviations unprofitable. Because dispersion in the cost of effort may yield dispersion of the effort levels, the sup23

This also contrasts with the full-information results in Baik et al. (2001) and Chowdhury et al. (2013) that assume alternative effort aggregation technologies, such as “best shot” or additive; they show existence of a champion within each group who exerts a positive level of effort, as well as free riders who contribute zero.

36

port of the equilibrium effort distribution may also be a compact interval, not just a singleton. For this kind of equilibrium in a symmetric setup, we show that the effect of symmetrically increasing group size on ex ante welfare is ambiguous. An increase in the group size exacerbates the free-riding problem; however, players may benefit since the contest becomes less competitive. For a large class of cost distributions (namely, power distributions), the two countervailing effects cancel out so that changes in the group size are welfare-neutral. We characterize the condition under which an increase in the group size increases (decreases) the ex ante expected payoff. It is also possible that the support of the effort distribution consists of an interval plus an isolated point. In this case, all players with sufficiently low realized cost contribute the same amount, which is discretely greater than the effort level chosen by any other cost type. The effort distribution then has a mass point at the top. One may hypothesize that such coordination of the more productive types is welfare-enhancing, as it partially eliminates the waste of effort that comes as a consequence of the weakest-link effort technology when efforts are dispersed. However, the effect of the size of the mass point at the top on ex ante welfare is, once again, ambiguous. The intuition is that, although coordination helps reduce the waste of effort, higher cost types among those that do coordinate experience a discrete jump in their contribution level, while lower cost types contribute less than what they would in the absence of coordination.24 As a limiting case, even all players coordinating on the same effort level may not improve ex ante welfare. Interestingly, the same condition that characterizes the effect of the group size on welfare also characterizes the welfare effect of the size of the mass point at the top. It is also interesting to investigate the implications of asymmetries between groups. If groups are symmetric except for group size, players in the smaller group bid more aggressively than those in the larger group, supporting the classical group-size paradox (see Olson, 1965). Surprisingly, this result is analogous to the finding of Barbieri et al. (2014), who find 24

This, of course, happens because all types that coordinate do so at the level of effort that the highest-cost type among them is willing to exert.

37

that, in the semi-symmetric equilibria of the group best-shot all-pay auction with complete information and symmetric valuations, players in the larger group put mass at zero. When asymmetry between the two groups is along multiple dimensions, such as group size, valuation and cost distribution, the interplay between those three factors determines which group bids more aggressively. Just as in the case of symmetric groups, it is possible that the effort distributions for both groups possess a mass point at the top. Remarkably, the relative size of the mass points is independent of the group sizes. One might suspect that a “stronger” group (in the sense of higher valuation or smaller size) puts greater mass at the top. This, however, is not true in general. Simulations show that first-order stochastic dominance does not provide clear-cut insights either. To investigate the role of information exchange, we consider the case in which players within a group may exchange information about their cost realization (using cheap talk), and all coordinate on the effort most preferred by the agent with the largest cost. We show that a player is always better off reporting truthfully. Given that players in the rival group do not coordinate, a group enjoys a greater probability of victory from coordination, compared to the benchmark case of no coordination, and almost every cost type has a higher interim expected utility. This is, of course, due to the mitigation of free-riding within the coordinating group. A striking result is the comparison of the equilibrium with the most cooperation—that in which players within each group communicate and coordinate on a single effort—and the equilibrium with the least cooperation—that in which players do not communicate and all realized cost levels choose a different effort. All within-group gains from cooperation are lost to increased competition between groups, so that the ex ante utility is the same in the two equilibria.

38

Appendix Proof of Theorem 2. First, we build a semi-degenerate equilibrium in which group-1 members use a degenerate strategy. Then, we will show that semi-degenerate equilibria in which group-2 members use a degenerate strategy do not exist. Let all group-1 members bid x¯1 > 0 with probability 1. Then, it must be that x ¯1 = x¯2 = x¯ in equilibrium. If instead x¯1 > x¯2 , then for all cost levels, each player in group 1 could improve her payoff by reducing her bid to (¯ x1 + x¯2 )/2, contradicting the assumption of equilibrium. Similarly, if x¯1 < x¯2 , then a positive mass of group 2 types bid in the interval ((¯ x1 + x¯2 )/2, x¯2 ], and all of these types could increase their interim payoffs by reducing their bids to (¯ x1 + x¯2 )/2. The assumption of semi-degenerate supports implies x1 = x¯ and x2 < x¯. For group-2 members, any bid in (0, x¯) is sure to lose and so yield a negative interim payoff. Therefore, any types of group-2 players not bidding x¯ must bid 0. Thus, x2 = 0 and, by the property of the weakest-link aggregator, each player i in group 2 must put mass ai at x¯, with 0 < ai ≤ 1, for otherwise group-1 agents would have a profitable downward deviation. Therefore, for any i, type F2−1 (ai ) must obtain a non-negative payoff:  (v2 /2) j∈I2 ,j=i aj − x¯F2−1 (ai ) ≥ 0, or, multiplying both sides by ai , v2  aj ≥ x¯ai F2−1 (ai ), 2 j∈I

∀i ∈ I2 .

2

Since we are looking for a semi-degenerate equilibrium, there exists at least one agent i such that ai < 1, so type F2−1 (ai ) ∈ (c, c¯) is indifferent between the bids of zero and x ¯ and we have

v2  aj = x¯ai F2−1 (ai ). 2 j∈I

(A.1)

2

But the above two displayed equations imply ai ≤ ai < 1 for all i ∈ I2 , because xF2−1 (x) is a strictly increasing function of x and the left-hand sides are the same. But then (A.1)

39

applies to i as well; hence, ai = ai = a for all i ∈ I2 . Equation (A.1) now implies that

x¯ =

v2 an2 −1 . 2F2−1 (a)

The utility of player i in group 1 who contributes x¯ when her cost is c¯ equals c¯ 1 1 v2 x, c¯) = v1 ( an2 + (1 − an2 )) − x¯c¯ = v1 (1 − an2 ) − an2 −1 −1 , U1,i (¯ 2 2 2 F2 (a) while a deviation to an arbitrarily small but positive effort generates utility that approaches U1,i (0+ , c) = v1 (1 − an2 ),

as the contribution approaches 0. Therefore, a cost type c¯ in group 1 will not deviate to a x, c¯) ≥ U1,i (0+ , c), or contribution lower than x¯ if and only if U1,i (¯ aF2−1 (a) ≥

v2 c¯. v1

(A.2)

Note that aF2−1 (a) is strictly increasing in a with a maximum at c¯ for a = 1. When v1 > v2 , there exists a unique a < c¯ where a F2−1 (a ) =

v2 c¯, v1

(A.3)

such that (A.2) is satisfied if and only if a ≥ a . The unique a determined by (A.3) is the lower bound on the mass that each player in group 2 places at x¯. If n1 and n2 are both greater than 1, then the above discussion exhausts all possible deviations, since upwards deviations from x¯ are not profitable by the properties of the weakest-link aggregator. This concludes the proof that the strategy described in the statement of the theorem is an equilibrium. We now turn to the necessary conditions in the statement of the theorem. First, note that if at least one group is composed of only one agent, then this agent has a strictly profitable 40

upwards marginal deviation from x¯, and no semi-degenerate equilibrium can exist. Second, when v2 ≥ v1 , (A.2) is never satisfied for the admissible range of a for a semi-degenerate equilibrium. This implies that semi-degenerate equilibria do not exist when v2 = v1 and, if they exist for v2 = v1 , then the degenerate strategy must be adopted by the members of the group with the larger valuation. Proof of Lemma 1. 1. Suppose x¯1 > x¯2 . Then, by the properties of the weakest-link aggregax2 , x¯1 ] with strictly positive probability, so each is better tor, all players in group 1 bid in ( x¯1 +¯ 2

off by reducing such bids to

x ¯1 +¯ x2 2

and we have a contradiction to equilibrium. Therefore

x¯1 = x¯2 = x¯. By definition of non-degenerate equilibrium, x¯ > xl , l = 1, 2. In the rest of the proof, many arguments rest on agents increasing their probability of success through an increase in their bid. Since x¯1 = x¯2 = x¯, the properties of the weakest-link aggregator imply that all agents contribute with strictly positive probability amounts arbitrarily close to x¯. Therefore, all agents contributing any amount strictly lower than x¯ can increase the probability of winning by increasing their efforts. Next suppose x1 > x2 . Then no agent in group 2 bids in (0, x1 ) because such bids are costly but yield zero chance of winning; therefore, 0 = x2 and H2 (0) > 0. Furthermore, there must be a positive mass for H2 in [x1 , x1 + ε], for any ε > 0, otherwise one of the group 1 agents contributing at or near x1 could profitably reduce her contribution. If there is a mass point for H2 at x1 , then there must be a mass point for H1 at x1 , for otherwise a group 2 agent contributing x1 with positive probability could profitably lower her contribution to zero. But if H1 and H2 have mass points at x1 , then a group 2 agent contributing x1 with positive probability could increase her payoff by increasing this bid to x1 + δ, for δ > 0 sufficiently small, and we reach a contradiction. And if, instead, H2 has no mass point at x1 , then, for δ > 0 sufficiently small, there exists a strictly positive mass of group 1 agents who are contributing between x1 and x1 + δ who, by reducing their contribution to ε, can obtain a discrete savings in cost while sustaining an infinitesimal drop in the probability of winning, thus raising their payoffs. This too is a contradiction, so it must be that x1 = x2 = x. 41

We now show x = 0, proceeding again by contradiction. If x > 0 and both groups have a mass point at x, then there is a profitable upwards deviation to x + ε, for ε > 0 sufficiently small. And if group 1, say, does not have a mass point at x, then agents in group 2 bidding sufficiently close to x can reduce their bids to zero and save a discrete amount on their bidding costs while suffering an infinitesimal decrease in their probability of winning. In either case, a contradiction arises so x must be zero. 2. By contradiction, suppose that H2 has a mass point at γ ∈ (0, x¯). Then no agent in group 1 should contribute with strictly positive probability in (γ − ε, γ], for ε > 0 sufficiently small. To see this, note that all agents in group 1 are bidding in a neighborhood of x¯ with strictly positive probability. So if an agent raises her bid from anything in (γ − ε, γ] to γ + δ, her probability of winning increases discontinuously while, for δ > 0 sufficiently small, cost increases only marginally. Having thus established that no contributions in group 1 fall in (γ − ε, γ] with strictly positive probability, we see agents in group 2 bidding γ have a strictly profitable deviation to γ − ε/2 : they save on the cost with no repercussions on the probability of winning. To see that at most one group puts mass at zero, note that if both groups had a mass point at 0, then there would be a profitable upwards deviation to ε, for ε > 0 sufficiently small. 3. Denote the support of Hi with Si , i = 1, 2. Let t ∈ S1 be such that t ∈ / S2 . Then there exists a non-empty interval (a, b) ⊂ [0, x¯] such that t ∈ (a, b), (a, b) ⊆ S1 by Part 2 above, , b) have a strictly profitable and (a, b) ∩ S2 = ∅. Therefore, group 1 agents bidding in ( a+b 2 deviation to

a+b , 2

since this does not change the probability of victory but reduces cost.

4. Suppose to the contrary that there exists an interval (a, b) with 0 < a < b < x¯ such that (a, b) ∩ S = ∅, but [b, b + ε) ⊂ S, for some ε > 0. Then agents bidding in [b, b + ε) have a strictly profitable deviation to

a+b 2

unless a mass point for H1 and H2 exists at b. But, by

Part 2, that is possible only if b = x¯. Suppose now that H1 admits a mass point at x¯. Then, for ε > 0 sufficiently small, no contribution in group 2 can fall in (¯ x − ε, x¯), for otherwise a profitable deviation to x¯ results. But since x¯ must remain in S2 , then H2 has a mass point

42

at x¯. And since x¯ − ε < x¯, then there exists a < x¯ such that neither H1 nor H2 puts positive probability on (a, x¯). Suppose now that H1 does not admit a mass point at x¯. Then there is a strictly positive probability of group 1’s contributions falling in (¯ x − ε, x¯), for any ε > 0, which implies there cannot be a mass point for H2 at x¯. Therefore, a = x¯, i.e., S = [0, x¯]. Proof of Lemma 2. We focus on members of group 1; considerations for group 2 follow similarly. With a contribution of γ < x¯, player i with cost c obtains equilibrium utility

Ui (γ; c) ≡ v(1 −

−{i} H1 (γ))H2 (γ)



γ

+v 0

−{i}

H2 (s) dH1

(s) − cγ,

(A.4)

a continuous function of both γ and c. Integration by parts yields  Ui (γ; c) = v

0

γ

−{i}

(1 − H1

−{i}

(s)) dH2 (s) − cγ + vH2 (0)(1 − H1

(0)),

(A.5)

so that ∂ −{i} Ui (γ; c) = v(1 − H1 (γ)) dH2 (γ) − c. ∂γ

(A.6)

Furthermore, note that for any two teammates i and j, i = j, we have −{i}

1 − H1

−{i,j}

(γ) = (1 − H1

(γ))(1 − H1j (γ)).

(A.7)

We complete the proof through a series of steps. The following step shows that, essentially, equilibrium strategies can have no jumps, except possibly at x ¯. Step 1. Consider 0 ≤ γl < γh < x¯. H1i cannot put zero probability over (γl , γh ) while at the same time putting strictly positive probability on (γl − ε, γl ] and [γh , γh + ε) for ε > 0 sufficiently small. Proof. We proceed by contradiction. If H1i puts zero probability over (γl , γh ), then, by continuity of Ui , a necessary condition for equilibrium is that for some cˆi we have Ui (γl ; cˆi ) =

43

Ui (γh ; cˆi ), and lim

γ↓γl

∂ ∂ Ui (γ; cˆi ) ≤ 0 ≤ lim Ui (γ; cˆi ). γ↑γh ∂γ ∂γ

(A.8)

By Lemma 1, the support of H1 includes (γl , γh ); therefore, there must exist one other agent in group one, j, say, that bids on a dense subset of (γl , γl + ε). In this interval, j’s FOC, using (A.6) and (A.7), reads as −{i,j}

−1 (γ) = v(1 − H1 g1j

−{i,j}

= v(1 − H1

(γ))(1 − H1i (γ)) dH2 (γ) (γ))F1 (ˆ ci ) dH2 (γ).

(A.9)

Similarly, some group one agent k bids in (γh − ε, γh ), with strategy g1k satisfying the FOC −{i,k}

−1 (γ) = v(1 − H1 g1k

(γ))F1 (ˆ ci ) dH2 (γ).

(A.10)

Now suppose player i with cost cˆi deviates to γ ∈ (γl , γl + ε). In this range, (A.6), (A.7), and (A.9) imply ∂ −{i,j} Ui (γ; cˆi ) = v(1 − H1 (γ))(1 − H1j (γ)) dH2 (γ) − cˆi ∂γ −1 g1j (γ) (1 − H1j (γ)) − cˆi = F1 (ˆ ci ) −1 (γ) g1j −1 F1 (g1j (γ)) − cˆi . = F1 (ˆ ci )

(by (A.6) and (A.7)) (by (A.9))

−1 Define cˆj ≡ limγ↓γl g1j (γ). Now the first limit in (A.8) yields

0 ≥ lim γ↓γl

−1 g1j (γ) ∂ 1 −1 Ui (γ; cˆi ) = lim F1 (g1j (ˆ cj F1 (ˆ (γ)) − cˆi = cj ) − cˆi F1 (ˆ ci )). γ↓γl F1 (ˆ ∂γ ci ) F1 (ˆ ci )

Consequently, because cF1 (c) is strictly increasing at cˆi , it follows that cˆj ≤ cˆi . An analogous

44

−1 argument for player k yields cˆk ≡ limγ↑γh g1k (γ) ≥ cˆi . Therefore,

cˆj ≤ cˆk .

(A.11)

If j = k then we have a contradiction, because g1j is strictly decreasing when taking values in (γl , γl + ε), weakly decreasing otherwise, and γl < γh . Therefore, j = k. But we know g1j is strictly decreasing when taking values in (γl , γl + ε) and g1k is strictly decreasing when taking values in (γh − ε, γh ). Therefore, (A.11) implies H1j (γ) > H1k (γ)

∀γ ∈ (γl , γh ).

(A.12)

To conclude the proof, note that equilibrium requires Uj (γl ; cˆj ) ≥ Uj (γh ; cˆj ), and Uk (γh ; cˆk ) ≥ Uk (γl ; cˆk ). Using (A.5), we have  0 ≤ Uj (γl ; cˆj ) − Uj (γh ; cˆj ) = −v

implying



γh

v γl

−{j}

(1 − H1

γh γl

−{j}

(1 − H1

(s)) dH2 (s) + cˆj (γh − γl ),

(s)) dH2 (s) ≤ cˆj (γh − γl ).

(A.13)

Similarly,  0 ≤ Uj (γh ; cˆk ) − Uj (γl ; cˆk ) = v

implying

γl

 cˆk (γh − γl ) ≤ v

γh

γh γl

−{k}

(1 − H1

−{k}

(1 − H1

(s)) dH2 (s) − cˆk (γh − γl ),

(s)) dH2 (s).

Because cˆj ≤ cˆk , (A.13) and (A.14) imply 

γh γl

(1 −

−{j} H1 (s)) dH2 (s)

 ≤

45

γh

γl

−{k}

(1 − H1

(s)) dH2 (s),

(A.14)

or, using (A.7),



γh γl

−{j,k}

(1 − H1

(s))(H1j (s) − H1k (s)) dH2 (s) ≤ 0, ♦

which is impossible by (A.12).

We now show that if one equilibrium strategy has a jump from γl to x¯, then all strategies have the same jump point. Step 2. Consider 0 ≤ γl < γh < x¯. H1i cannot put zero probability over (γl , x¯), while at the same time putting strictly positive probability on (γl − ε, γl ], for ε > 0 sufficiently small, if H1 puts strictly positive probability on (γl , γh ). Proof. First note that, for any ε > 0, all agents put strictly positive probability on (¯ x − ε, x¯] in equilibrium, by Lemma 1. Proceed again by contradiction and denote with cˆi the type at which the jump of g1i from x¯ to γl occurs. Given Step 1, there are two cases to consider. In the first, (γl , x¯] ⊂ supp H1t for all other agents t = i in group 1. In the second, there exists some other agent k in group 1 with H1k putting zero probability on (γh , x¯), but strictly positive probability on (γh − ε, γh ]. The first case is handled as in Lemma 1. Indeed, if one starts from condition (A.8), substituting x¯ for γh , then all other steps leading to a contradiction follow unchanged. In the second case, we have Ui (¯ x; cˆi ) ≥ Ui (γh ; cˆi ). Furthermore, letting cˆk denote the cost at which g1k jumps from x¯ to γh , we see that agent k’s indifference between γh and x¯ means x; cˆk ). These conditions, together with (A.5) and (A.7), yield Uk (γh ; cˆk ) = Uk (¯ x; cˆi ) − Ui (γh ; cˆi ) 0 ≤ Ui (¯  x¯ −{i} (1 − H1 (s)) dH2 (s) − cˆi (¯ x − γh ) =v γh  x¯ −{i,k} (1 − H1 (s))(1 − H1k (s)) dH2 (s) − cˆi (¯ x − γh ) =v γh  x¯ −{i,k} ck ) (1 − H1 (s)) dH2 (s) − cˆi (¯ x − γh ) = vF1 (ˆ γh

46

(A.15)

and

x; cˆk ) − Uk (γh ; cˆk ) 0 = Uk (¯  x¯ −{i,k} ci ) (1 − H1 (s)) dH2 (s) − cˆk (¯ x − γh ). = vF1 (ˆ

(A.16)

γh

Subtracting (A.16) from (A.15), we obtain  ck ) − F1 (ˆ ci ))v 0 ≤ (F1 (ˆ

x ¯ γh

−{i,k}

(1 − H1

(s)) dH2 (s) + (ˆ ck − cˆi )(¯ x − γh ).

This inequality implies cˆi ≤ cˆk . Moreover, note that i should not want to contribute in (γl , γl +ε), and that Lemma 1 shows that there must exist some agent j other than i such that (γl , γl +ε) is contained in the support of H1j . As in the proof of Step 1, limγ↓γl

∂ U (γ; cˆi ) ∂γ i

≤ 0,

−1 (γ). We then reach condition (A.11), which leads (A.6), and (A.9) imply cˆi ≥ cˆj ≡ limγ↓γl g1j

♦

to the same contradiction as in Step 1. We now show that the lowest contribution of all agents is zero. Step 3. For any agent i in group 1, we have limc↑¯c g1i (c) = 0.

Proof. Proceed by contradiction and suppose g1i (¯ c) = γl > 0. Since H1 puts strictly positive probability on (γl − ε, γl ), there must exist some agent j contributing in (a dense −1 (γ) and note that cˆj < c¯. As in the proof of subset of) this interval. Let cˆj ≡ limγ↑γl g1j

Step 1,

0 ≤ lim γ↑γl

∂ −{i,j} (γ))(1 − H1j (γ)) dH2 (γ) − c¯ Ui (γ; c¯) = lim v(1 − H1 γ↑γl ∂γ −{i,j}

= v(1 − H1

(γl ))F1 (ˆ cj ) dH2 (γl ) − c¯.

At the same time, agent j’s FOC yields

0 = lim γ↑γl

∂ −{i,j} (γ))(1 − H1i (γ)) dH2 (γ) − cˆj Uj (γ; cˆj ) = lim v(1 − H1 γ↑γl ∂γ 47

(A.17)

−{i,j}

= v(1 − H1 =

(γl )) dH2 (γl ) − cˆj

c¯ − cˆj F1 (ˆ cj )

(by (A.17))

> 0,

♦

which is impossible. Therefore, it must be that g1i (¯ c) = 0.

Step 4. If g1i and g1j take value on a common interval (γl , γh ), then g1i = g1j on this interval, except for a set of measure zero. Proof. By contradiction, consider any γ ∈ (γl , γh ) such that γ = g1i (ˆ ci ) = g1j (ˆ cj ), but cˆj > cˆi . Almost always, the first-order conditions

∂ U (γ; cˆi ) ∂γ i

=

∂ U (γ; cˆj ) ∂γ j

= 0 must hold;

equations (A.6) and (A.7) then imply, dividing one FOC by the other, that F1 (ˆ 1 − H1j (γ) cj ) cˆi = , = i cˆj 1 − H1 (γ) F1 (ˆ ci ) or cˆj F (ˆ cj ) = cˆi F (ˆ ci ), in contradiction of the assumption that cˆj > cˆi .

♦

Steps 1, 2, and 3 imply that for any two agents i and j in group 1, the supports of H1i and H1j are the same; by Lemma 1 they equal S, and S is either [0, x¯], or [0, a] ∪ {¯ x}, for some a ∈ (0, x¯). Step 4 then establishes that identical teammates use essentially the same strategy, which concludes the proof of Lemma 2. Proof of Lemma 3. By Lemma 2 we know players within a team use the same strategy. Let gl denote the common strategy of players on team l, l = 1, 2. Moreover, for a nondegenerate equilibrium the range of the strategies is common and the range where they are strictly decreasing is common. Let γ be such that in a neighborhood of gl−1 (γ) strategy gl is strictly decreasing, l = 1, 2. Consider player 11 in group 1. Define cM 1 = max{c12 , . . . , c1m }, which has the associated cdf Q(c) = [F (c)]n−1 and density q(c) ≡ Q (c). For player 11 in group 1 with cost c11 acting

48

like a type ca11 , the associated payoff is V11 (ca11 , c11 )

=

−c11 g1 (ca11 ) 

c¯

+ ca 11



ca 11

+ 0

M v · Pr(g2 (c2j ) < g1 (ca11 ), for some j = 1, . . . , n) q(cM 1 ) dc1



M M v · Pr g2 (c2j ) < g1 (cM 1 ), for some j = 1, . . . , n q1 (c1 ) dc1

= −c11 g1 (ca11 ) + vQ(ca11 ) [1 − Pr(g2 (c2j ) ≥ g1 (ca11 ), for all j = 1, . . . , n)]  c¯ 

 M v · 1 − Pr g2 (c2j ) ≥ g1 (cM q1 (cM + 1 ), for all j = 1, . . . , n 1 ) dc1 ca 11

 n ! = −c11 g1 (ca11 ) + vQ(ca11 ) 1 − F (g2−1 (g1 (ca11 )))  c¯ n !  M +v 1 − F (g2−1 (g1 (cM q(cM 1 ))) 1 ) dc1 .

(A.18)

ca 11

The relevant partial derivative is " # n−1 −1  ∂V11 n−1   a a −1 a −1 a a = −g (c ) c + nv [F (c )] (g (c ))) f (g (g (c ))) g (g (c )) . F (g 11 1 1 1 1 1 11 11 2 11 2 11 2 11 ∂ca11 (A.19) Where g1 is strictly decreasing, in equilibrium ca11 = c11 , so the FOC ∂V11 /∂ca11 |ca

11 =c11

=0

implies n−1   nv [F (c11 )]n−1 F (g2−1 (g1 (c11 ))) f (g2−1 (g1 (c11 ))) g2−1 (g1 (c11 )) + c11 = 0.

(A.20)

A similar analysis for player 21 in group 2 yields the condition n−1   [F (c21 )]n−1 f (g1−1 (g2 (c21 ))) g1−1 (g2 (c21 )) + c21 = 0. nv F (g1−1 (g2 (c21 )))

(A.21)

Following Amann and Leininger (1996), where g1 is strictly decreasing we define the function ξ(c11 ) = g2−1 (g1 (c11 )); thus, if a group-1 player with cost c exerts effort γ, then so, too, does a group-2 player with cost ξ(c ). Note that where g1 and g2 are strictly decreasing,

49

their inverse functions g1−1 and g2−1 are well-defined and decreasing. And since both g1 and g2 are decreasing, ξ is increasing. By the chain rule, ξ  (c) = (g2−1 ) (g1 (c))g1 (c), so (g2−1 ) (g1 (c11 )) = ξ  (c11 )/g1 (c11 ). Now the FOC (A.20) can be rewritten as nvF (c11 )n−1 F (ξ(c11 ))n−1 f (ξ(c11 )) ξ  (c11 ) = −c11 g1 (c11 ).

(A.22)

For c21 = ξ(c11 ) = g2−1 (g1 (c11 )), the second FOC (A.21) can be rewritten as nvF (c11 )n−1 F (ξ(c11 )n−1 f (c11 ) = −ξ(c11 )g1 (c11 ).

(A.23)

From (A.23), solve for g1 (c11 ) and substitute the solution into (A.33) to obtain c11 f (c11 ) = ξ(c11 )f (ξ(c11 )) ξ  (c11 ).

(A.24)

Let x¯ ≡ g1 (c) = g2 (c). For any c ∈ [c, c¯], define 

c

xf (x) dx.

M (c) = c

Observe that M (·) is nonnegative, finite, and strictly increasing on the support [c, c¯]. Now there are two cases to consider, either g1 and g2 place no mass at x¯ or they do place mass at x¯. First, suppose the equilibrium places no mass at x¯, so ξ(c) = c. Suppose g1 is strictly decreasing at c11 . Then, integrating both sides of (A.24), we have  M (c11 ) =



c11

c11

xf (x) dx = 

c

ξ(x)f (ξ(x)) ξ  (x) dx

c ξ(c11 )

=

yf (y) dy

(where y = ξ(x))

c

50

= M (ξ(c11 ));

because M is strictly increasing, it follows that ξ(c11 ) = c11 , in turn implying g1 (c) = g2 (c) for all c. If instead each strategy places mass at x¯, then there is some c˜l > c such that gl (c) = x¯ for all c < c˜l and limc↓˜cl gl (c) = x for some x < x¯, l = 1, 2 (see Lemma 1). A group-l player with type c˜l must be indifferent between efforts x¯ and x . From (A.18), a group-1 player with type c˜1 has payoff c1 x + vQ(˜ c1 ) {1 − [F (˜ c2 )]n } + v U1 (x ; c˜1 ) = −˜



n !  M 1 − F (ξ(cM q(cM 1 )) 1 ) dc1

c¯ c˜1

if he exerts effort x , and he has payoff $

1 U1 (¯ x; c˜1 ) = −˜ c1 x¯ + vQ(˜ c1 ) 1 − [F (˜ c2 )]n 2

%



c¯

+v c˜1

n !  M 1 − F (ξ(cM q(cM 1 )) 1 ) dc1

if he exerts effort x¯ (the “1/2” arises because, if all players exert effort x¯, each team wins with probability 1/2). Indifference between x and x¯ now implies v v c1 )F (˜ c1 )n−1 F (˜ x − x ) = Q(˜ c2 )n = F (˜ c2 ) n . c˜1 (¯ 2 2 For members of group 2, analogous reasoning yields v c˜2 (¯ x − x ) = F (˜ c2 )n−1 . c1 )n F (˜ 2 Therefore, v c1 )n F (˜ x − x )F (˜ c1 ) = F (˜ c2 )n = c˜2 (¯ x − x )F (˜ c2 ), c˜1 (¯ 2 which, because F is strictly increasing, in turn implies c˜2 = c˜1 . Let c˜ ≡ c˜1 . Using the condition ξ(˜ c) = c˜, we can proceed as above to integrate (A.24) from c˜ to c to conclude that ξ(c) = c for all c > c˜. It then follows that g1 (c) = g2 (c) for all c > c˜; and for all c < c˜ it is 51

the case that g1 (c) = g2 (c) = x¯. Proof of Proposition 1. Expected cost is 



c¯

cg(c)f (c) dc =



c¯



c vn

τ 

c

c

c¯

=v

n 

c

τ

c¯

=v

n 

c c¯

=v c

=

v 2



c

c¯

c

 F (τ )2(n−1) f (τ ) dτ f (c) dc τ

c¯ c

cf (c)dc τ cf (c)dc

τ F (τ )

F (τ )2(n−1) f (τ ) dτ

(by (2)) (reversing the order of integration)

F (τ )2n−1 f (τ ) dτ

n W (τ ) d[F (τ )2n ] 2n W (τ ) d[F (τ )2n ].

c

Since F (τ )2(n+1) first-order stochastically dominates F (τ )2n , if W (τ ) is increasing in τ, then expected cost is larger with n + 1 agents than with n. Therefore each agent’s expected utility is smaller with more agents per group. The rest of the proof follows along the same lines and is here omitted. Proof of Proposition 2. Using (2) and changing the order of integration, we obtain

Eg

min



c¯

=

vn

2

c



τ

c¯

n

=v c

v = 3



c¯ c



c

c¯ c

 F (τ )2n−2 f (τ ) dτ F (c)n−1 f (c)dc τ

nF (c)n−1 f (c)dc τ F (τ )

F (τ )2n−1 f (τ )dτ

d [F (τ )3n ] . τ F (τ )

So, the expected group performance is smaller with n + 1 agents than with n. We can see this in the above displayed equation by noting that F (τ )3(n+1) first-order stochastically dominates F (τ )3n and τ F (τ ) at the denominator is increasing in τ . Therefore, despite the fact that some cost types may increase their contribution, the expected aggregate effort in the symmetric equilibrium without mass is decreasing in the group size.

52

Verification of equation (9). Starting with  EC(c0 ) =



c0

c¯ x(c0 )f (c) dc +

c

c¯

cg(c)f (c) dc, c0

we obtain 





EC (c0 ) = c0 x¯(c0 )f (c0 ) + x¯ (c0 )

c0

c

cf (c) dc − c0 g(c0 )f (c0 ) 

x(c0 ) − g(c0 )) + x¯ (c0 ) = c0 f (c0 )(¯



c0

cf (c) dc c

  v v v 2n−1 2n−2 2n−1  = f (c0 ) [F (c0 )] + g (c0 ) + (2n − 1) [F (c0 )] f (c0 ) − [F (c0 )] 2 2c0 2(c0 )2  c0 × cf (c) dc c

   c0 v v v 2n−1 2n−2 2n−1 = f (c0 ) [F (c0 )] − [F (c0 )] f (c0 ) + [F (c0 )] cf (c) dc 2 2c0 2(c0 )2 c   c0   v f (c0 ) F (c0 ) 2n−2 f (c0 )F (c0 ) − = [F (c0 )] + cf (c) dc . 2 c0 (c0 )2 c Proof of Proposition 3. Using (4), we have 

W (τ ) =

τ f (τ )τ F (τ ) − [F (τ ) + τ f (τ )]

τ c

cf (c)dc

(τ F (τ ))2

;

therefore, if W  (c0 ) > 0, then the term in square brackets in (9) is positive, so expected cost increases in c0 . Since in a symmetric equilibrium expected utility and cost move in opposite directions, we have established that if W  (τ ) > 0, then expected utility decreases in c0 . The rest of the proof follows along similar lines. Proof of Proposition 4. Using Theorem 4, expected group performance in an equilibrium with mass characterized by c0 ∈ (c, c¯) is & Eg

min

=



c¯

vn c0

v (F (τ ))2n−2 f (τ ) dτ + F (c0 )2n−1 τ 2c0 53

' F (c0 )n



c¯

+

vn c0

2



c¯ c

 F (τ )2n−2 f (τ ) dτ F (c)n−1 f (c)dc. τ

Then we have ∂Eg min = ∂c0



F (c0 )2n−2 f (c0 ) v (2n − 1)F (c0 )2n−2 f (c0 )c0 − F (c0 )2n−1 −vn + · c0 2 c20   c¯ F (τ )2n−2 f (τ ) v + vn F (c0 )2n−1 n (F (c0 ))n−1 f (c0 ) dτ + τ 2c0 c0  c¯ F (τ )2n−2 f (τ ) 2 n−1 dτ − vn F (c0 ) f (c0 ) τ c0  F (c0 ) v 3n−2 F (c0 ) = (n − 1)f (c0 ) − . 2c0 c0

F (c0 )n

Therefore, increasing c0 and putting higher mass at the maximum effort increases (decreases) the expected group performance if and only if

F (c0 ) c0 f (c0 )

is greater than (less

than) n − 1. This last condition is equivalent to cdf of the maximum of n − 1 costs being inelastic (elastic) at c0 . As far as aggregate effort, using Theorem 4, we obtain

Eg

agg

2



= vn F (c0 )

c¯ c0

(F (τ ))2n−2 f (τ ) vn dτ + F (c0 )2n + τ 2c0



c¯

vn c0

2



c¯ c

 F (τ )2n−2 f (τ ) dτ f (c)dc. τ

Then we have 

(F (τ ))2n−2 f (τ ) F (c0 )2n−1 f (c0 ) dτ − vn2 τ c0 c0  c¯ vn 2nF (c0 )2n−1 f (c0 )c0 − F (c0 )2n (F (τ ))2n−2 f (τ ) 2 · dτ + − vn f (c ) 0 2 c20 τ c0 vnF (c0 )2n =− 2c20

∂Eg agg = vn2 f (c0 ) ∂c0

c¯

< 0.

Therefore, higher mass at the top effort level leads to a lower aggregate effort.

54

Proof of Theorem 5. Consider player 11 in group 1. Let cM 1 = max{c12 , . . . , c1n1 }, with cdf H1 . So H1 (c) = [F1 (t)]n1 −1 . For player 11 in group 1 with cost c11 announcing type ca11 the associated payoff is V11 (ca11 , c11 )

=

−c11 g1 (ca11 ) 

1

+ ca 11



ca 11

+ 0

M v1 · Pr(g2 (c2j ) < g1 (ca11 ), for some j = 1, . . . , n2 ) h1 (cM 1 ) dc1

M M v1 · Pr g2 (c2j ) < g1 (cM 1 ), for some j = 1, . . . , n2 h1 (c1 ) dc1

= −c11 g1 (ca11 ) + v1 H1 (ca11 ) [1 − Pr(g2 (c2j ) ≥ g1 (ca11 ), for all j = 1, . . . , n2 )]  1 

 M + v1 · 1 − Pr g2 (c2j ) ≥ g1 (cM h1 (cM 1 ), for all j = 1, . . . , n2 1 ) dc1 ca 11

 n ! = −c11 g1 (ca11 ) + v1 H1 (ca11 ) 1 − F2 (g2−1 (g1 (ca11 ))) 2  1  n2 !  M + v1 1 − F2 (g2−1 (g1 (cM h1 (cM 1 ))) 1 ) dc1 . ca 11

The relevant partial derivative is " # n2 −1 −1  ∂V11 n1 −1   a a −1 a −1 a a F = −g (c ) c + n v [F (c )] (g (g (c ))) f (g (g (c ))) g (g (c )) . 11 2 1 1 2 1 2 1 1 1 11 11 2 11 2 11 2 11 ∂ca11 (A.25) In equilibrium ca11 = c11 , so we rewrite the FOC ∂V11 /∂ca11 |ca

11 =c11

as

 n −1  n2 v1 [F1 (c11 )]n1 −1 F2 (g2−1 (g1 (c11 ))) 2 f2 (g2−1 (g1 (c11 ))) g2−1 (g1 (c11 )) + c11 = 0. (A.26) Similarly, write V21 (ca21 , c21 ) for player 1 in group 2 and take the FOC to get n −1   n1 v2 F1 (g1−1 (g2 (c21 ))) 1 [F2 (c21 )]n2 −1 f1 (g1−1 (g2 (c21 ))) g1−1 (g2 (c21 )) + c21 = 0. (A.27) Observe that (g2−1 ) (g1 (c11 )) = ξ  (c11 )/g1 (c11 ) and obtain from (A.26) the first optimality

55

condition: n2 v1 F1 (c11 )n1 −1 F2 (ξ(c11 ))n2 −1 f2 (ξ(c11 )) ξ  (c11 ) = −c11 g1 (c11 ).

(A.28)

For c21 = g2−1 (g1 (c11 )) = ξ(c11 ), (A.27) yields the second optimality condition: n1 v2 F2 (ξ(c11 ))n2 −1 F1 (c11 )n1 −1 f1 (c11 ) = −ξ(c11 )g1 (c11 ).

(A.29)

From (A.29), express g1 (c11 ) and substitute into (A.28), yielding ξ  (c11 ) =

n1 v2 c11 f1 (c11 ) . n2 v1 ξ(c11 )f2 (ξ(c11 ))

(A.30)

Let us introduce a change of variable χ(c11 ) ≡ ξ 2 (c11 ), then χ (c11 ) = 2ξ(c11 )ξ  (c11 ), and equation (A.30) can be written as χ (c11 ) =

2n1 v2 c11 f1 (c11 ) . ( n 2 v 1 f2 χ(c11 )

(A.31)

Note that since no group puts mass at x¯, by Part 3 of Lemma 1 we have ξ(c) = c and, therefore, χ(c) = c. Equation (A.31) together with the initial condition χ(c) = c constitute an initial value problem. The Lipschitz condition is satisfied at c, provided limx→c f2 (x) = 0. Therefore, under this condition there exists a unique solution for χ by the Picard-Lindel¨of theorem (Coddington and Levinson, 1955, Theorem 3.1, p.12). Since ξ is nonnegative, there ( exists a unique solution for ξ, defined as ξ(c11 ) = χ(c11 ). Next, we solve for g1 from (A.29) as

 c) = − g1 (c11 ) = g1 (c11 ) − g1 (¯

Finally, solve for g2 from

c¯

c11

  g2 (c21 ) = g1 ξ −1 (c21 ) .

56

g1 (τ )dτ

(A.32)

Proof of Proposition 5. Let Note that ξ  (c) =

n1 v2 f1 (c) n2 v1 f2 (c)

v1 n1

<

v2 n2

·

f1 (c) . f2 (c)

We begin by showing that ξ(c) > c ∀c ∈ (c, c¯).

> 1, therefore by continuity of ξ, there exists > 0 such that

ξ(c) > c for all c ∈ (c, c + ]. Suppose by contradiction ξ(c1 ) ≤ c1 for some c1 > c. By continuity of ξ, then there exists some smallest cost c such that ξ(c ) = c . Since ξ(c) > c ∀c ∈ (c, c ), a necessary condition for ξ(c ) = c is ξ  (c ) ≤ 1. But (11) implies ξ  (c ) =

n1 v2 f1 (c ) n2 v1 f2 (c )

> 1, a contradiction. Therefore ξ(c) > c for all c ∈ (c, c¯]. We derive two

implications. First, since ξ(c) = g2−1 (g1 (c)) and g2 is strictly decreasing, applying g2 to both sides of ξ(c) > c we obtain g1 (c) < g2 (c). Second, ξ(¯ c) > c¯. Therefore, by continuity there exists c < c¯ such that ξ(c ) = c¯, so group-1-player types in [c , c¯] never win. Therefore, these types exert zero effort. The results for the remaining two cases can be proven analogously. Proof of Proposition 6. Fix an equilibrium with mass at the highest effort, with the corresponding cutoffs c01 and c02 . Assume

F1 (c01 ) F2 (c01 )

>

v1 v2

and

n2 v1 n1 v2

<

cf1 (c) cf2 (c)

for all c ≥ c01 . We now

show ξ(c) > c for all c ∈ (c01 , c¯]. Using the initial condition of Theorem 6, observe that   c02 > c01 if and only ν2−1 vv21 c01 F1 (c01 ) > c01 . Since ν2−1 is increasing, this inequality is equivalent to v2 c01 F1 (c01 ) > ν2 (c01 ) = c01 F2 (c01 ), v1 which is equivalent to v1 F1 (c01 ) > . F2 (c01 ) v2 Therefore, our first assumption implies c02 > c01 , thus ξ(c01 ) > c01 . Note that ξ  (c01 ) = n1 v2 f1 (c01 ) n2 v1 f2 (c01 )

> 1, therefore by continuity of ξ, there exists > 0 such that ξ(c) > c for all

c ∈ (c01 , c01 + ]. The rest of the proof now follows just as that of Proposition 5. Proof of Proposition 7. We begin by deriving equilibrium, along the lines of Theorem 5, and we show that the probability of winning for group 1 increases with respect to the symmetric case in Theorem 3. Consider the utility of an agent in group 1 (the organized group) with

57

cost type c that pretends to be c . &



Vo (c , c) = −c

c

g1 (c )dF

c

&



c

+v c



1−F

n−1

 (z) +

c

n

'

c¯

g2−1 (g1 (c ))

g1 (z)dF



n−1

(z)

dF n−1 (z) +



c¯

1−F

c

n

g2−1 (g1 (z))



' dF n−1 (z) .

The only difference with the formulation in Theorem 5 is that agents only end up contributing the amount preferred by the highest-cost type. The derivative w.r.t. c gives * )



∂Vo (c , c) 1

. = g1 (c )F n−1 (c ) −c − vnF n−1 g2−1 (g1 (c )) f g2−1 (g1 (c ))  −1 ∂c g2 g2 (g1 (c )) When the strategy is strictly decreasing, the truth-telling FOC (which is necessary and sufficient for the same reason described in Theorem 5) is



c = −vnF n−1 g2−1 (g1 (c)) f g2−1 (g1 (c)) Since ξ (c) ≡ g2−1 (g1 (c)), we have ξ  (c) ≡

g2

(

g1 (c) −1 g2 (g1 (c))

)

g2



g2−1

1

. (g1 (c))

, so the above displayed equation can

be rewritten as c · g1 (c) = −ξ  (c) vnF n−1 (ξ (c)) f (ξ (c)) ;

(A.33)

this is the analogue to equation (A.28). Consider now the utility of an agent in group 2 with cost type c that pretends to be c : 



V2 (c , c) = −cg2 (c )+v

& c

c



1−F

n



g1−1



(g2 (c ))



dF

n−1

 (z) +

c¯ c

1−F

n



g1−1

(g2 (z))



' dF

This is the same expression we derived in the proof of Theorem 5. Proceeding in the same fashion as done there, the truth-telling FOC, evaluated at ξ (c), yields ξ (c) g1 (c) = −vnF n−1 (ξ (c)) F n−1 (c) f (c) ,

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n−1

(z) .

which is exactly (A.29). Dividing (A.33) by (A.29) and rearranging, we obtain ξ  (c) =

cf (c) · F n−1 (c) , ξ (c) f (ξ (c))

(A.34)

which should be compared to (A.30), which if the teams are symmetric yields ξ(c) = c. Following the same logic as in the proof of Proposition 3, we see that after group 1 gets organized, (A.34) leads to ξ (c) < c for any c ∈ (c, c¯). Intuitively, that is because at any putative point c0 ∈ (c, c¯) with ξ(c0 ) = c0 , we obtain ξ  (c0 ) < 1, which immediately pushes ξ(c) below the 45 degree line. However, we cannot follow the same logic of the Proof of Proposition 3 to conclude also that ξ (¯ c) < c¯ because this would require ξ  (¯ c) ≥ 1, but this is c) = 1. Nonetheless, we can now not contradicted by (A.34) under ξ(¯ c) = c¯, which yields ξ  (¯ use another method of proof to show ξ (¯ c) < c¯, if f (¯ c) > 0, which we have assumed in this section. Suppose to the contrary that ξ(¯ c) = c¯. (ξ(¯ c) > c¯ is impossible because ξ (c) < c for any c ∈ (c, c¯) and ξ is continuous.) Using (A.34) we then have ξ  (¯ c) = 1, and differentiating c) = (n−1)f (¯ c) > 0. Define Δ(c) ≡ ξ(c)−c. Note that Δ(¯ c) = Δ (¯ c) = 0 (A.34) we obtain ξ  (¯ under the hypothesis ξ(¯ c) = c¯. A second-order Taylor expansion now yields 1 1 Δ(c) − Δ(¯ c) ≈ Δ (¯ c)(c − c¯) + Δ (¯ c)(c − c¯)2 = (n − 1)f (¯ c)(c − c¯)2 > 0. 2 2 Therefore, the above-displayed equation gives ξ(c) − c > 0 for some c in a left neighborhood of c¯, a contradiction to ξ(c) < c. We conclude that for any type c > c, the probability of victory of group 1 is strictly larger after group 1 gets organized. We now compare the interim equilibrium utility of an agent in the organized group, Wo (c) ≡ Vo (c, c), with that of an agent in the symmetric equilibrium of Theorem 3, Wno (c) ≡ V11 (c, c), as defined in (3), and we establish that Wo (c) > Wno (c), for all c > c. We begin with limc↑¯c Wo (c) = v (1 − F n (ξ (¯ c))) > 0 = Wno (¯ c). Now suppose that by contradiction c) = Wno (˜ c), with Wo (c) > Wno (c) for all c > c˜. Then, there exists some c˜ such that Wo (˜

59

using the envelope theorem, we must have 

c˜ c

g1 (˜ c)dF

n−1



c¯

(z) + c˜

 g1 (z)dF n−1 (z) = |Wo (˜ c)| ≤ |Wno (˜ c)| = g(˜ c),

where g is the equilibrium contribution function in the symmetric equilibrium of Theorem 3. But this contradicts Wo (˜ c) = Wno (˜ c), since it implies that c˜’s contribution under organization (and hence its cost) is weakly lower than if neither group is organized, and we have previously c) > established that the probability of winning is strictly larger under organization, so Wo (˜ Wno (˜ c).

References Amann, Erwin, and Wolfgang Leininger (1996), Asymmetric all-pay auctions with incomplete information: the two-player case, Games and Economic Behavior 14, 1–18. Bagnoli, Mark, and Ted Bergstrom (2005), Log-concave probability and its applications, Economic Theory 26, 445–469. Baik, Kyung Hwan (2008), Contests with group-specific public-good prizes, Social Choice and Welfare 30, 103–117. Baik, Kyung Hwan, In-Gyu Kim, and Sunghyun Na (2001), Bidding for a group-specific public-good prize, Journal of Public Economics 82, 415–429. Barbieri, Stefano, and David A. Malueg (2016), Private information group contests: Bestshot competition, Games and Economic Behavior 98, 219 – 234. Barbieri, Stefano and David A. Malueg (2018), On the voluntary provision of “weakest-link” public goods: The case of private information, Journal of Public Economic Theory, forthcoming.

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Barbieri, Stefano, David A. Malueg, and Iryna Topolyan (2014), The best-shot all-pay (group) auction with complete information, Economic Theory 57, 603 – 640. Barbieri, Stefano, Dan Kovenock, David A. Malueg, and Iryna Topolyan (2018), Group Contests with Private Information and the ’Weakest Link.’ Available at SSRN: https://ssrn.com/abstract=3179725 Brookins, Philip, and Dmitry Ryvkin (2015), Equilibrium existence in group contests, Economic Theory Bulletin 4, 265 – 276. Chowdhury, Subhasish M., and Iryna Topolyan (2016) The attack-and-defense group contests, Economic Inquiry 54, 548–557. Chowdhury, Subhasish M., Dongryul Lee, and Roman M. Sheremeta (2013), Top guns may not fire: best-shot group contests with group-specific public good prizes, Journal of Economic Behavior and Organization 92, 94–103. Chowdhury, Subhasish M., Dongryul Lee, and Iryna Topolyan (2016), The max-min group contest: Weakest-link (group) all-pay auction, Southern Economic Journal 83, 105 – 125. Coddington, Earl A. and Norman Levinson (1955). Theory of Ordinary Differential Equations. McGraw Hill, New York. Einy, Ezra, Ori Haimanko, Diego Moreno, Aner Sela, and Benyamin Shitovitz (2015), Equilibrium existence in Tullock contests with incomplete information, Journal of Mathematical Economics 61, 241–245. Eliaz, Kfir, and Qinggong Wu (2018), A simple model of competition between teams, Journal of Economic Theory 176, 372–392. Kirkegaard, Ren´e (2013a), Incomplete information and rent dissipation in deterministic contests, International Journal of Industrial Organization 31, 261–266. 61

Kirkegaard, Ren´e (2013b), Handicaps in incomplete information all-pay auctions with a diverse set of bidders, European Economic Review 64, 98–110. Konrad, Kai (2009), Strategy and Dynamics in Contests. Oxford University Press Inc., New York. Kovenock, Dan, Florian Morath, and Johannes M¨ unster (2015), Information sharing in contests, Journal of Economics and Management Strategy 24(3), 570–596. Mas-Colell, Andreu, Michael Whinston, and Jerry Green (1995), Microeconomic Theory. New York: Oxford University Press. Morath, Florian, and Johannes M¨ unster (2008), Private versus complete information in auctions, Economics Letters 101(3), 214–216. Myerson, Roger (1981), Optimal auction design, Mathematics of Operations Research 6, 58–73. Parreiras, Sergio, and Anna Rubinchik (2010), Contests with three or more heterogeneous agents, Games and Ecnomic Behavior 68 (2), 703–715. Ryvkin, Dmitriy (2010), Contests with private costs: Beyond two players, European Journal of Political Economy 26, 558-567. Topolyan, Iryna (2014), Rent-seeking for a public good with additive contributions, Social Choice and Welfare 42, 465-476. Wasser, Cedric (2013a), Incomplete information in rent-seeking contests, Economic Theory 53, 239-268. Wasser, Cedric (2013b), A note on Bayesian Nash equilibria in imperfectly discriminating contests, Mathematical Social Sciences 66, 180-182.

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