# Rent-Seeking Group Contests with Private Information

## Comments

## Transcription

Rent-Seeking Group Contests with Private Information

Rent-Seeking Group Contests with Private Information Jean-François Mercier∗ April 7, 2015 Abstract A model of rent-seeking group contest is developed. The contested good is a local public good. Individuals have private information concerning their valuation for the contested good. I restrict effort levels to be dichotomous, allowing me in turn tractability of the equilibria. I show existence of an equilibrium. All contestants exert positive expected effort in equilibrium. From simulation results I find that the presence of large groups of contestants decreases the average expected effort in equilibrium. I also show that the Olson’s paradox, which asserts that groups of large size are less effective at winning a contest than small groups, may or may not hold. If individuals valuations are drawn from distribution where large valuations have a sufficiently high density, the Olson paradox holds. Keywords: Group contests, Private information, Coalitions, Local public good, Threshold equilibrium JEL Classification Numbers: C70, D20, D74, H41 ∗ [email protected], Department of Economics, McGill University, Canada. I wish to thank my supervisor Licun Xue. I am also grateful to Rohan Dutta for enlightening discussions. I also thank Ching-Jen Sun for commenting on an earlier draft as well as Luis Corchon, Marco Serena, Peter Eccles, John Galbraith, Hassan Benchekroun and Ngo Van Long. 1 1 Introduction In this paper, I study a type of contest in which groups of individuals compete against each other in order to obtain a group-specific public good. As opposed to individual contests where groups are of size one, a group contest allows groups of individuals to aggregate their effort in the acquisition of a good that would benefit to all members of the winning group. Many real-life situations can be modelled as such a group contest. Two groups of lawyers may compete in legal proceedings. Individuals with similar socio-economic status may, as a group, engage in trying to influence legislators or other public officials in favor of a specific cause. Pharmaceutical firms may collaborate in order to win the race to find a cure to some disease. And so on. In these examples, a contestant may have multiple partners who by their actions, which are to exert some level of effort, affect the contestant’s probability of winning, that is, the contestant’s probability of consuming the public good. An example of contest for which my paper is closely related to is when a number of different firms may form joint ventures in order to participate in a contest for the production of a public project such as the construction of a bridge, for example. It is common practice, in the construction industry for example, that competing firms collaborate to the design of a project and submit a tender as one group. These firms may in other circumstances be competitors. Consequently, the amount of information that firms share when forming such a group may be limited or even null. Naturally, firms are usually uninformed about the extent to which their competitor(s) value(s) the contested good. However, it is safe to assume that in such a context, competing firms do know who they are competing against and the way in which their competitors collaborate. That is, they know the structure of the opposing groups. In this example, firms must behave strategically not only against the other groups of firms who may end up producing the project instead of them, but also against their own collaborators, because firms also have an incentive to free-ride within a group. By now, it is well understood that the greater a group is, the greater are the incentives to free-ride within that group. There is nonetheless another source of free-riding and it is the lack of information regarding valuations. Holding constant the size of a group if one firm sees its collaborators being more likely to derive high valuations for the project, then this firm will expect its collaborators to exert more effort on average, which in turn induces the firm to exert less effort. 2 The way in which such a project is attributed is not necessarily straightforward. It is often the case that the submission that is the less expensive will be the chosen one. This is likely to be true for simple projects, or at least projects that must be conducted in a very specific way. Submissions in this case are likely to be very similar and choosing among them becomes easy: choosing the less expensive submission. In this context, a straightforward way of modeling this situation is using a deterministic contest, where the good is allocated with probability one to the group that exerts the highest aggregate effort, which is in some sense an all-pay auction. However, when it comes to complex projects that require great skills and innovation, for instance a bridge or a skyscraper, it is most likely the case that the submissions will be different in many respects, hardening greatly the task of choosing one among them. In this context, the modeling approach is the use of a non deterministic contest where the highest bidder does not receive the good with probability one, but has the highest probability of receiving it. Most papers on Contest Theory have used the approach of non deterministic contests, either with full information or with private information, with individualistic or with group contests. I also follow this approach. My model is line with other papers such as Wasser (2013) and Einy et al. (2013) who analyse incomplete information in the case of individual contests and provide conditions for existence – and uniqueness in some families of Tullock contests – of an equilibrium. However, the functions that describe effort levels in equilibrium are not tractable. Barbieri & Malueg (2014) also develop a group contest model with private information. However, the main difference with my paper and their’s is that in Barbieri & Malueg (2014), the authors consider a Best-Shot All-Pay auction, which is a deterministic contest where ”each group’s performance equals the best effort (”best shot”) of its members, and the group with the best performance wins the contest.” A best-shot contest allows tractability of a group’s effort supply. Though, a shortcoming of this approach is the inevitable necessity to use a deterministic contest and a second one is the fact that the effort of a group is determined by only one group member as opposed to being determined by the aggregate effort of all group members. In my paper, I consider group contests in a non deterministic framework and I assume that the effort supply of any group is the sum of efforts of all group members. The key restriction that enables tractability of the solution is to assume, as in Dubey (2013), that the effort level is a dichotomous variable. 3 The essence of my solution is to rule out the possibility that contestants choose their level of effort in a continuous matter. The game induced by such a contest can be thought of as taking place before the realization of the valuations. I show that in equilibrium, agents’ actions consist of choosing, prior to the realization of their valuation, a threshold where one would exert effort given that the realization of the valuation is above or equal to that threshold and would not exert effort otherwise. I solve the game by computing for any individual, the specific realization of valuation that would make him indifferent between exerting effort or not. These specific valuations are, in fact, the equilibrium thresholds. The effort levels are then in terms of expected effort, that is, the probability that an individual’s realization of his valuation be above the equilibrium threshold. A nice feature of this model is that, in equilibrium, all contestants select a positive threshold which means that they all exert positive expected effort. With complete information and when the cost of exerting effort is a linear function of effort, it was shown in Baik (1993) that agents with low valuation for the public good have an incentive to let agents of their own group with high valuation exert effort so that low valuation individuals can reap the benefits without paying the cost of exerting effort. Such an extreme prediction in terms of participation is more plausibly the exception rather than the rule in many circumstances and applications. It was recently pointed out by Topolyan (2014) that ”there are many examples when all group members contribute to the collective cause [but the] existing theory is not fully capable of handling such situations. Whether there is an equilibrium where all players contribute is an interesting question.” Topolyan (2014) suggests, as a solution to this paradox, a model with a continuum of equilibria where all players make contributions. Though, it must be pointed out that the results in Topolyan (2014) are related to deterministic contests. In my paper, the solution that I propose comes about as a novel answer to this paradox. I also consider the impact of partition structures on contestants’ behavior in equilibrium. In a group contest, the particular way in which contestants are grouped together is described as a partition of the contestants. I argue that not only the number of individuals within one’s group matters when it comes to deciding the thresholds, but also the specific structure of the subpartition of competitors matters. For instance, consider an architectural firm A that participates alone into a contest for a public project financed by a local government. If the set of A’s opponents consists of firm B and C then 4 A’s behavior must depend on whether B and C form a coalition or if they are mutual competitors. The reason is that if B and C are grouped together, they both have an incentive to free-ride each other. Firm A is aware of this fact and can internalize that within the group {B,C} there are incentives to free-ride. Consequently, A can take advantage of that by increasing his level of expected effort. The characterization of the equilibrium allows me to describe the best-response behaviors of the agents. I show that agents who belong to the same group do indeed have an incentive to free-ride each other. For instance, let B and C form a group, while A is alone. If B exerts more expected effort then C will exert less expected effort and vice versa. I also show that competing firms, say A and B, may have behaviors that are either positively or negatively correlated. Behaviors are negatively correlated if groups are small relative to the size of all contestants. And behaviors are positively correlated otherwise – perhaps only on a strict subset of the effort space. Apart from Barbieri & Malueg (2014), group structures have not been taken into consideration in the literature on group contests. My results indicate that the presence of larger groups reduces the average level of effort. Thus, if a contest designer wishes to maximize the level of average effort, groups should be broken down to smaller groups whenever it is possible. For instance, the existence of a group of contestants should not be justified by the fact that group members find it beneficial to have the possibility to free-ride on co-members. Lastly, I check the model against the so-called Olson’s Paradox (Olson (1965)) which states that ”the free rider problem inside large groups is so acute that, in equilibrium, large groups exert less aggregate effort than small groups, which explains the success of the latter 1 .” I show that even though free-riding exists within groups, groups of larger size may still exert a larger aggregate expected effort in equilibrium in which case leads to a greater probability of winning the contest. However, this result does not alway hold. For distribution functions where large values have a sufficiently large density, then the Olson’s paradox holds. This contradicts what Esteban & Ray (2001) refer to as the ”common wisdom” which is that the ”Olson thesis holds when the prize is private but may be reversed when the prize is purely public”. 1 Quote taken from Corchón (2007). 5 2 Further Review of Literature This paper belongs to the literature on rent-seeking contests (Tullock (1980)). For a complete review on the theory of contests in economics, the reader is referred to Nitzan (1994), Corchón (2007), Konrad (2009) or Long (2013). More specifically, I build upon a family of papers where authors investigate contests with competing individuals divided into groups of contestants. The literature on group contests mostly investigates the existence and the characterization of the free-riding problem in games with complete information. Papers like Katz et al. (1990), Ursprung (1990), Baik (1993), Riaz et al. (1995) and Baik & Shogren (1998), although they consider various contest success functions, which are functions that map effort to a probability of winning, all establish that free-riding is an important feature of any equilibrium. For instance, Baik (1993), Baik et al. (2001), Baik (2008) show that only the individuals with the unique highest valuation within each group are contributing in equilibrium. Chowdhury et al. (2013) modify the Baik’s specification by assuming that the probability of group winning the contest is determined by the maximal individual effort within the group. It leads to an equilibrium where the free-riding takes yet another extreme form; at most one player in each group exerts positive effort, however, it is not necessarily the player with the highest valuation. In deterministic contests, Barbieri et al. (2013) and Topolyan (2014), show that equilibria exist where all the players contribute positively to the collective cause. It has also been shown that allowing for complementarity in the players’ efforts (Kolmar & Rommeswinkel (2013)), allowing for non linear cost of effort (Epstein & Mealem (2009)) or using a success function that depends on the minimal effort level within each group (Lee (2012)), also alleviates the severity of the free-riding problem. One aspect of contests on which we know much less about concerns what happens when the players have private information. There is a growing interest in individual contests with incomplete information. The literature includes Hurley & Shogren (1998a), Malueg & Yates (2004), and Sui (2009) who examine models where the players valuations are private and distributed according to a simple discrete distribution. There are also a number of papers where only one player is affected by the information asymmetry (Harstad (1995); Hurley & Shogren (1998a),Hurley & Shogren (1998b); Schoonbeek & 6 Winkel (2006); Pogrebna (2008); Wärneryd (2003)). Among the most recent papers, Fey (2008) analyses a contest with two players where the cost of exerting effort is privately known and proves the existence of an equilibrium for both discrete and continuous distributions. Wasser (2013) and Einy et al. (2013) generalize the analysis to more than two players and analyse existence and uniqueness of a Bayes-Nash equilibrium in different informational settings. Finally, Ryvkin (2010) addresses the issue of player heterogeneity and how it impacts the aggregate effort level in individual contests. A common result is that the equilibrium effort levels tend to be lower when information is private than when it is public. However, Ryvkin (2010) shows that the difference in equilibrium effort levels does not generalize to contests of more than two players. The case of group contests with incomplete information has been studied by Barbieri & Malueg (2014) and by Brookins & Ryvkin (2014). The latter points out the non tractability of the equilibrium but uses numerical techniques to depict the equilibrium strategies. There is a vast literature concerned with the Olson’s Paradox. The relevant stream of literature, for the purpose of the present paper, is the one related to the ways in which the paradox can be reversed. Chamberlin (1974) and McGuire (1974) suggested, without a formal demonstration, that the Olson’s paradox holds when the collective good, which is the good that any group aims at providing, has a sufficient private component to it. If the collective group is purely public, they suggested that the Olson’s paradox shall be reversed. More recently, counter arguments to the paradox have been proposed. Katz et al. (1990), Nti (1998) argue that instead, success in contests can be predicted by large valuations, small costs or contest success functions that favor certain agents. Esteban & Ray (2001) shows that if the cost of exerting effort is sufficiently convex, the paradox can be reversed even if the collective good is purely private. Pecorino & Temimi (2008) extend the model of Esteban & Ray (2001) to a game of pure public good provision and show that with fixed participation cost, large groups may fail at providing the public good. In my paper, I consider only pure (local) public goods. Nonetheless, the integration of private information and the consideration of different partition structures form a novel approach to the paradox. 7 3 The model Let P denote some partition of N such N = {1, ..., n} is a set of individuals. S 0 0 that ∀I, I ∈ P , I ∩ I = ∅ and I∈P I = N . The groups of individuals in N are grouped according to P and are participating to a rent-seeking contest where the prize is a local public good. Only one group can win the contest. Only the members of the winning group can consume it, the others are excluded2 . The valuation that individual i ∈ N has for the local public good is denoted θi ∈ Θ = [0, θ̄], where 0 < θ̄ < ∞. The list of valuations is denoted by θ ∈ Θn . ∀i ∈ N , θi is a random variable that is independently distributed and follows a probability distribution fi with cumulative distribution function Fi . The realization of θi is known to i and only to i. No communication within or across elements of a partition is allowed. Upon the realization of θ, each contestant decides whether to exert effort (ei = 1) or not (ei = 0). Assuming that the effort space is dichotomous is a simplification that makes it possible to compute effort levels, in equilibrium, in a tractable way. Morever, this assumption is arguably not too far from reality as it is potentially rarely the case that individuals in real life can adjust their effort levels in a continuous manner. The cost of exerting effort is c > 0 and is the same for all individuals. Let P (i) denote the element in P that contains player i and let |P (i)| = pi . For any given partition P and any list of effort levels e ∈ {0, 1}n , I assume that the probability that i consumes the local public good is (P P ej Pj∈P (i) if j∈P (i) ej > 0 e j∈N j πi (e, e−i ) = P 0 if j∈P (i) ej = 0 I follow the convention that e−i denotes the list of effort levels for all individuals other than i. The functional form of πi is slightly different from the usual form, which is known in the literature Pas the Tullock contest success function. It is a common assumption that if j∈N ej = 0 then all contestants consume the good with equal probabilities. However, with this assumption, it is possible that a group wins the good without having a single individual in the group exerting effort. This seems rather odd when, after all, a contest should take 2 Such a contest could be generalized to incorporate spillovers induced by the consumption of a local public by some I ∈ P . However, in this paper, spillovers are ruled out. Bloch & Zenginobuz (2007) consider such a scenario, although it is in different context. 8 place among individuals who signal their interest towards the good, which comes at the cost of at least signaling their interest to participate. The group contest can be represented as a Bayesian game. This game consists of a finite set N of players and a partition P of N . For all player i ∈ N , the set of possible actions is {0, 1}. Individual i is differentiated by his type, which is the value θi that i has for the contested good. Individuals are of types that are randomly distributed over Θ with fi (θi ) > 0 ∀θi ∈ Θ. Individual i’s information is the realization of his type and the distribution functions of all other players. Anything else (N, P or c) is common knowledge. In this game, a strategy for any player i is a function σi : Θ −→ {0, 1} Denote σ−i = (σj )j∈N \i and let E be the expectation operator over θ−i . Note that for any given list of strategies σ−i and any ei ∈ {0, 1}, we have that 0 ≤ E [πi (ei , σ−i )] ≤ 1, ∀i ∈ N and in particular, θ̄E [πi (1, σ−i )] − c ≤ 0 if c ≥ θ̄. If c > θ̄, i does not have an incentive to exert positive effort, no matter the realization of θi . From then on, it is assumed that c ∈ (0, θ̄]. Given σ−i , i’s objective is to choose a function σi such that ∀θi ∈ Θ, σi ∈ arg max {θi E [πi (ei , σ−i )] − c · ei } ei ∈{0,1} Thus, an equilibrium of C is a list σ ∗ such that ∀i ∈ N and ∀θi ∈ Θ, ∗ σi∗ ∈ arg max θi E πi (ei , σ−i ) − c · ei ei ∈{0,1} 3.1 Optimal strategies I will show that the search for equilibrium strategies can be simplified by looking only at cutoff strategies. 9 Definition 1. σi is a cutoff strategy for player i if there exists a cutoff xi ∈ R such that ( 0 if θi < xi σi = 1 if θi ≥ xi Remark 1. If a non-cutoff strategy differs from some cutoff strategy only by the fact there is a subset Θ0 ⊂ Θ with a mass of 0 (which means the probability that a realization of θi belongs to that interval is zero) on which and only on which the behaviors suggested by the strategies are different, then I say that the two strategies are essentially identical. So if an equilibrium in cutoff strategies exists, the essentially identical non-cutoff strategies can also be part of an equilibrium. I will ignore such non-cutoff strategies for obvious reasons. From then on, a non-cutoff strategy will be defined as any strategy σ where there exists 0 ≤ a < b < c < d < e < f ≤ θ̄ such that ∀θ ∈ [a, b] ∪ [e, f ] and ∀θ0 ∈ [c, d], σ(θ) 6= σ(θ0 ). Proposition 1. For any σ−i and for any non-cutoff strategy used by i, there always exists some cutoff strategy σi that makes i strictly better-off. Proof. Fix a list of strategies σ−i . We have that ∀θ ∈ Θn , E [πi (1, σ−i )] ≥ E [πi (0, σ−i )] , ∀i ∈ N (1) which is true since for any σ−i , i cannot decrease P (i)’s probability of winning by exerting effort, but it may be the case that σ−i is such that P (i) wins whether or not i exerts effort. Case 1 : E [πi (1, σ−i )] = E [πi (0, σ−i )]. Then it must be that E [πi (0, σ−i )] = 1 because the probability that P (i) wins the contest strictly increases if i changes his level of effort from ei = 0 to ei = 1 as long as E [πi (0, σ−i )] < 1. In this case, a non-cutoff strategy suggests that for some interval of values, i should exert effort and pay c even though i is already certain of consuming the good. It is then clear that i should not exert effort for any realization of θi and this would in turn make i strictly better-off. This is the behavior suggested by a cutoff strategy with a cutoff of 0. Case 2 : E [πi (1, σ−i )] > E [πi (0, σ−i )]. 10 Then there exists x̂i ∈ R+ such that x̂i E [πi (1, σ−i )] − c = x̂i E [πi (0, σ−i )] Re-write x̂i as follows: x̂i = c E [πi (1, σ−i )] − E [πi (0, σ−i )] (2) ∀θi < x̂i , we have that θi E [πi (1, σ−i )] − c < θi E [πi (0, σ−i )] and it is optimal for i not to exert effort. Otherwise, ∀θi ≥ x̂i , θi E [πi (1, σ−i )] − c ≥ θi E [πi (0, σ−i )] and it is weakly optimal for i to exert effort. Any non-cutoff strategy would suggest to either exert effort on some interval below x̂i and/or not to exert effort on some interval above x̂i . In this case, a cutoff strategy with cutoff x̂i also makes i strictly better-off. Remark 2. Since any realization of θ outside of Θn has probability 0, any two cutoff strategies σ̃i , σ̂i with respective cutoffs x̃i , x̂0i ≥ θ̄i are identical (with respect to the amount of expected utility they bring to i) to the cutoff strategy σ̄i with cutoff θ̄. However, for technical reasons, I will restrict the set of cutoffs to be a closed interval X ⊂ R+ such that Θ ⊂ X. In the quest for an equilibrium of C, Proposition 1 implies that for any σ−i , one can simplify the search only to cutoff strategies, and that no non-cutoff strategy can be part of an equilibrium. Note that a cutoff stragegy is fully described by its cutoff. We can use this fact to transform C into a a normalform game where actions are cutoffs chosen from some closed and convex set X ⊇ Θ. N and P are held constant. In this normal-form version of C, E [πi (ei , σ−i )] takes a specific form. Let L, M ⊆ N with |L| = l and |M | = m and fix the cutoff strategies of all individuals different from i to x−i . For j 6= i, since θj follows a cumulative distribution function Fj over Θ, i expects j to exert effort with probability (1 − Fj (xj )) and to not exert effort with probability Fj (xj ). Therefore, the probability that l individuals in N \ P (i) 11 and m individuals in P (i) \ i exert positive effort is equal to X Y Y Πi (l, m; x−i ) := (1 − Fk (xk )) Fk (xk ) L⊂N \P (i) · k∈N \P (i)∪L k∈L Y X M ⊂P (i)\i (1 − Fk (xk ) k∈M Y Fk (xk ) k∈P (i)\M ∪i Thus, n−pi pi −1 E [πi (ei , x−i )] = XX Πi (l, m; x−i )i (ei , l, m) (3) l=0 m=0 ( i (ei , l, m) = ei +m ei +m+l 0 if ei + m > 0 if ei + m = 0 One could interpret this game as follows. Prior to the realization of θ, any individual i ∈ N decides upon a threshold xi and commits to exert effort if the realized value of θi is greater or equal to xi and to not exert effort otherwise. Individual i maximizes his expected utility which depends on the one hand on the probability with which himself will exert effort and on the probability that the individuals other than i will exert effort in the contest. A nice feature of this approach is that xi can then be interpreted as a choice of expected effort that corresponds to (1 − Fi (xi )). Proposition 2. If σ is a list of cutoff strategies such that ∃i ∈ N for which E [πi (1, σ−i )] = E [πi (0, σ−i )], then σ cannot be an equilibrium. Proof. We already know that in this case, E [πi (0, σ−i )] = 1. If this is so, then it must be the case that |P (i)| > 1 and among the members of P (i) different from i, there must be some j whom ∀θj ∈ Θ, σj (θj ) = 1. If not, then there exists some realizations of θ for which no member of P (i) different from i would exert effort and this would imply in that case that E [πi (0, σ−i )] < 1. Now since c > 0, σj brings to individual j a negative expected utility for any realization of θj below c. Then there exists a beneficial deviation for j. For example, the strategy σj0 such that σj0 = σj for all θj ≥ c and σj0 = 0 otherwise makes j strictly better off. Therefore, σ cannot be an equilibrium of C. We are then left with a unique possibility: if σ is an equilibrium, then for any i ∈ N , it must be the case that E [πi (1, σ−i )] > E [πi (0, σ−i )]. In this 12 case, Proposition 1 implies that ∀i ∈ N , i’s best response function is given by equation (2). Using (3) and equation (2), we have that a list of equilibrium thresholds x∗ solves the following system of equations: n−pi pi −1 x∗i XX Πi (l, m; x∗−i ) (i (1, l, m) − i (0, l, m)) = c, ∀i ∈ N (4) l=0 m=0 where ( i (1, l, m) − i (0, l, m) = l (1+m+l)(m+l) 1 1+l if m > 0 if m = 0 Using (4), we can derive the reaction of i with respect to a change in any other j ∈ N . Let’s fix all the cutoffs different than i and j. The simplest case is a change in xj when j ∈ P (i). If xj increases (decreases), then j exerts effort with a smaller (greater) probability. Consequently, the probability of m being small is increased (reduced). This change in xj has the effect of shifting the probability weights towards low (high) values of m and consequently towards l high (low) values of (1+m+l)(m+l) . Then, i’s best response is to decrease (increase) xi . We then have that within a group, contestants are strategic substitutes to each other. If, otherwise, j 6∈ P (i) then it is uncertain whether i and j are complements or substitutes. It is clear that if P (i) contains only i, it means that m can only take the value of 0, then an increase (decrease) l in xj shifts the probability weights towards high (low) values of (1+l) and it has inevitably a negative (positive) impact on xi . A contestant that has no teammate always reacts negatively to the variations of opponents’ effort. When |P (i)| > 1, a change in xj may have either a negative impact or a l positive impact on xi . To see this, note that (1+m+l)(m+l) does not necessarily increase with l. We have that l l+1 l−m − = (1 + m + l)(m + l) (2 + m + l)(m + l + 1) (1 + m + l)(m + l)(2 + m + l) l We have that if l − m > 0 then (1+m+l)(m+l) diminishes with l. However, whether xi diminishes or increases with xj depends on whether l − m is more or less likely to be positive, which is determined by the list of cutoffs. xi and xj are negatively correlated if the list of cutoffs is such that the probability weights on the positive values of l − m is high enough so that the expected 13 l value of (1+m+l)(m+l) is negatively correlated with l.This is expected to be the case when m can take sufficiently small values compared to l. If this is the case then when j slacks off, i takes advantage of this by exerting even more effort. And when xj decreases, xi would increase because of a discouragement effect. Or put it differently, the benefit of not exerting effort becomes relatively larger than the benefit of exerting effort. In this context, i would rather win the good with a lesser probability as it decreases the probability of paying the cost of effort without winning the good. However, if |P (i)| is relatively large, then l − m is more likely to be negative. Consequently, for |P (i)| sufficiently large, an increase in xj leads, at some point, to an increased likelihood of having negative values of l − m, shifting l in turn the probability weights towards low values of (1+m+l)(m+l) . In this case, xi will react positively to a change in xj . This can be explained by the fact that, on the one hand, the size of P (i) is large enough so that i feels confident to slack off effort when j slacks off. On the other hand, the size of P (i) is large enough so that i reacts competitively to the changes in xj . Just as a relatively big firm in some market would not let smaller firms increase their market power without retaliation. 3.2 Existence of an equilibrium It is convenient to re-write (4) in the following way. If an equilibrium x∗ exists, then this equilibrium is characterized by a list x∗ of cutoffs that solves x∗i = E πi (1, x∗−i ) c , ∀i ∈ N − E πi (0, x∗−i ) Proposition 3. Given any list of continuous c.d.f. functions F , a Nash equilibrium of the game always exists. Proof. Construct the function Φ : X n −→ X n such that Φ(x) = Φi (x) i∈N 14 where ( Φi (x) = c E[πi (1,x−i )]−E[πi (0,x−i )] if E [πi (1, x−i )] − E [πi (0, x−i )] ≥ c/θ̄ θ̄ otherwise If this function possesses a fixed point, then an equilibrium exists. In Remark 2, it was assumed that X is a closed interval in R+ and so X n is compact. This function is also continuous on X n . To show this, note that E [πi (1, x−i )] − E [πi (0, x−i )] is obviously continuous on X n . The only way in which Φi may not be continuous is if E [πi (1, x−i )] − E [πi (0, x−i )] = 0. But by construction, Φi is identically equal to θ̄ for any x such that E [πi (1, x−i )] − E [πi (0, x−i )] < c/θ̄. Take any x such that E [πi (1, x−i )] − E [πi (0, x−i )] = 0. There always exists a neighborhood around x such that Φi is identically equal to θ̄. Thus, for any sequence (xn )∞ n=0 that converges to x, it must be the case that the ∞ sequence (Φi (xn ))n=0 converges to Φi (x). Since ∀i ∈ N , Φi is continuous on X n , then so is Φ. From Brouwer’s fixed point theorem, Φ has a fixed point and thus an equilibrium exists. Note that this result holds for any list of continuous c.d.f functions F . 3.2.1 On the symmetry of equilibrium(a) Proposition 3 states that an equilibrium exists but is quiet about the specific structure of the equilibrium(a). A common equilibrium structure is the symmetry of actions for players that are faced with the same strategic situation. Intuitively, in a symmetric equilibrium, all contestants whose actions entail the very same opportunity costs, i.e. who face the same strategic situation, behave identically in equilibrium. Consider an example with the partition {1}, {2} and both c.d.f. are equal to F and let’s assume without loss of generality that θ̄ = 1. The two equations that needs to be solved are given by equation (4): 1 x1 F (x2 ) + (1 − F (x2 ) = c 2 1 x2 F (x1 ) + (1 − F (x1 ) = c 2 15 For the common uniform distribution function, this system implies that x1 = x2 and the unique solution is given by 1 √ x1 = x2 = ( 1 + 8c − 1) 2 If we had restricted our search to symmetric equilibria, x1 = x2 would have been assumed at the start and consequently, only one equation would have been needed to solve for the equilibrium. However, if F is an exponential function, say F (x) = x6 , then the system has three solutions. The solutions are plotted in figure ?. Figure 1: Best response functions for c = 0.5 Consider another example with the partition {1, 2}, {3} with all c.d.f. being the uniform distribution. From (4), we have three equations to solve simultaneously. 1 1 x1 x2 x3 + x2 (1 − x3 ) + (1 − x2 )(1 − x3 ) = c 2 6 1 1 x2 x1 x3 + x1 (1 − x3 ) + (1 − x1 )(1 − x3 ) = c 2 6 1 1 x3 x1 x2 + (x1 (1 − x2 ) + (1 − x1 )x2 ) + (1 − x1 )(1 − x2 ) = c 2 3 16 We can see that the two first equations imply that x1 = x2 . This system is thus reduced to 1 1 = c x1 x1 x3 + x1 (1 − x3 ) + (1 − x1 )(1 − x3 ) 2 6 2 21 x3 x1 + x1 (1 − x1 ) + (1 − x1 ) = c 3 The solution to this systeme is depicted in figure ? Figure 2: Best response functions for c = 0.5 Although an equilibrium always exist for continuous distribution functions, multiplicity of equilibria cannot be ruled out. However, it is easy to see that a symmetric equilibrium must always exist. The reason is the following. i and j can be in a symmetric strategic situation in two different ways. The first is if j ∈ P (i) and if Fi = Fj . In this case, i, j have the same partners and compete against the same opponents. In the system of equation that determines the equilibrium, the equation corresponding to i’s strategic situation is symmetric to j’s. That is, xi and xj are interchangeable among these two equations. Thus it must be the case that, at least, xi = xj is a solution. The second is when j 6∈ P (i) but when |P (i)| = |P (j)| and the set of distribution functions for the individuals in P (i) \ i is same as for the individuals in P (j) \ j. Again in this case, a symmetric equilibrium must exist. 17 A sharper result can be derived for identical players who belong to the same group. If in the first case, the distribution functions Fi and Fj are equal to the uniform distribution function, then it must be the case that i and j behave identically in equilibrium. Proposition 4. Let i, j ∈ N and i 6= j. And let Fi = Fj be the uniform c.d.f. If j ∈ P (i) then, in equilibrium, xi = xj . Proof. Define M 0 ⊂ N with |M 0 | = m − 1. Without loss of generality, assume that θ̄ = 1. We have that Πi (l, m; x−i ) can be re-written as follows X Y Y Πi (l, m; x−i ) := Fj (xj ) (1 − Fk (xk )) Fk (xk ) L⊂N \P (i) X · M ⊂P (i)\{i,j} k∈P (i)\M ∪{i,j} X L⊂N \P (i) Y X Y Y (1 − Fk (xk )) (1 − Fk (xk )) Fk (xk ) k∈N \P (i)∪L k∈L k∈M 0 M 0 ⊂P (i)\{i,j} Fk (xk ) Y (1 − Fk (xk )) k∈M + (1 − Fj (xj )) · k∈N \P (i)∪L k∈L Y Y Fk (xk ) k∈P (i)\M 0 ∪{i,j} = Fj (xj )S(i) + (1 − Fj (xj ))S 0 (i) With this formulation, with Fj (xj ) = xj and with Fi (xi ) = xi , the system of equations that represents the equilibrium, contains the two equations for i and j are n−p i −1 Xi pX xi xj S(i) (i (1, l, m) − i (0, l, m)) l=0 m=0 n−pi pi −1 + (1 − xj ) XX 0 S (i) (i (1, l, m) − i (0, l, m)) l=0 m=0 18 =c and xj xi n−pi pi −1 XX S(j) (i (1, l, m) − i (0, l, m)) l=0 m=0 n−pi pi −1 + (1 − xi ) XX 0 S (j) (i (1, l, m) − i (0, l, m)) =c l=0 m=0 but since the terms xi and xj do not appear in S(i), S(j), S(i), S 0 (i), we have that S(i) = S(j) = S(i) = S 0 (i). These two equations reduce to n−pi pi −1 xi (1 − xj ) XX S 0 (i) (i (1, l, m) − i (0, l, m)) = c l=0 m=0 and n−pi pi −1 xj (1 − xi ) XX S 0 (j) (i (1, l, m) − i (0, l, m)) = c l=0 m=0 and then to xi (1 − xj ) = xj (1 − xi ) so it must be the case that xi = xj . 4 The symmetric equilibrium For simplicity, I will assume that all agents share the same c.d.f. and I will only consider the symmetric equilibrium. This assumption concerning the distribution functions allows me to simplify the problem since in a symmetric equilibrium, all agents who belong to a group of the same size – whether they belong to the same group or not – choose the same cutoff in equilibrium. 19 4.1 Olson’s Paradox The Olson’s paradox (Olson (1965)) suggests that groups of greater size may be less effective than groups of smaller size at providing a local public good. The incapacity to be as efficient as small groups comes from the fact that the free-riding incentives are more important the greater is the size of the group. Since research on group contests, for the most part, do not consider the possibility of private information, it is worthwhile to verify how the possibility of private information affects the ”group-size paradox”. This possibility comes as another source of free-riding. For instance, if a contestant has partners that are likely to have high valuations for the good, this gives an incentive to exert less expected effort. In fact, when this extra source of free-riding is added to the model, then the group-size paradox may not be reversed and even if the good is public. This also serves as a counter-example to what Esteban & Ray (2001) refer to as the common wisdom which says that the Olson’s paradox can hold as long as the good as some element of privateness in consumption. If the good is fully public, within a group, then the common wisdom is to agree that groups of larger size will be more efficient at providing the public good. Contrary to this result, I show a counter-example with convex distribution functions that actually goes in line with the Olson’s paradox and against the common wisdom. Recall that the choice of a threshold xi can be thought of as a choice of expected effort as the probability of exerting effort is then (1 − Fi (xi )). Consequently, since the effort level is either 0 or 1, the level of expected effort associated to a threshold of xi is (1 − Fi (xi )). In the context of this model, the Olson’s paradox is studied from an ex-ante perspective. The problem is then to see whether a group of a large size has an expected probability of winning that is greater than for smaller groups. For group P (i), the expected probability of winning given x is P (1 − Fj (xj )) Pj∈P (i) j∈N (1 − Fj (xj )) In the symmetric equilibrium, this is reduced to p (1 − Fi (xi )) Pi j∈N (1 − Fj (xj )) 20 Consequently, if P (i) has a larger expected probability of winning than P (j), then pi (1 − Fi (xi )) > pj (1 − Fj (xj )) Consider the partition {{1, 2}, {3}}. If all individuals have the distribution function F (θ) = θ2 and the cost of effort is 0.5, then at the symmetric equilibrium, the threshold chosen by group {1, 2} is 0.8972 and the one chosen by individual 3 is 0.6698. This gives group {1, 2} and {3} respective expected probability of winning of 0.3257 and 0.6743. If, instead, the distribution functions are F (θ) = θ, then the thresholds in equilibrium are respectively 0.7853 and 0.5835 and the winning probabilities are respectively 0.5076 and 0.4924. For the remainder of the paper, I will extract results from the model by making another simplifying assumption. Assumption 1. ∀i ∈ N , θi is independently and uniformly distributed over the unit interval [0, 1] and c ∈ (0, 1). 4.2 Expected effort as a function of c In this section, I depict equilibrium thresholds for various values of c ∈ [0, 1]. For the case of N = {1, 2, 3}, it is still possible to compute the thresholds in equilibrium as a function of the effort cost, c. However, for |N | ≥ 4, the equilibrium thresholds do not have closed-form solutions. With the model that I have developed, I am unable to make a general statement concerning equilibrium thresholds for general N and P . However, depicting numerically the thresholds for small increments of c is somewhat easy. For |N | ≤ 5, any symmetric equilibrium has at most two different thresholds as it is impossible to form more than two groups of different sizes. In what follows, I will discuss the case of |N | = 3 and |N | = 4. The similar patterns suggest that for |N | ≥ 5, no qualitative differences should be observed. 4.2.1 N = {1, 2, 3} In Figure 3, we can see that the incentive to free ride is the strongest when P = {1, 2, 3}. Individuals push their threshold up as mush as possible expecting that at least one individual among the two others will exert effort. 21 Figure 3: Equilibrium thresholds for N = {1, 2, 3} The opposite case is when P = {{1}, {2}, {3}}. In this case, individuals have no teammates to rely on. Consequently, they exert more expected effort. If we compare P = {{1}, {2}, {3}} with P = {{1}, {2, 3}} we can see that individual 1 exerts more expected effort in the latter. Note that, 1 faces two opponents in both cases. However, from the threshold values in equilibrium, individual 1 is able to internalize the fact that when 2 and 3 form a group, they have an incentive to free ride each other, leading to a reduction in their level of expected effort. This in turn gives 1 an incentive to take advantage of this by increasing his level of expected effort. In Figure 4, we can see that the average expected effort in N is the greatest when the partition is broken down to an individual contest. We can see this since the smallest average threshold is always smaller in the individual contest. 4.2.2 N = {1, 2, 3, 4} In Figure 5, I provide the result of simulations for small increments of the cost parameter c. The simulation shows that for most values of c, the highest equilibrium threshold is when the partition is {1, 2, 3, 4}. The next highest equilibrium threshold is set by individual 2 (and 3 and 4) in P = {{1}, {2, 3, 4}}. 22 Figure 4: Average equilibrium thresholds for N = {1, 2, 3} Then after that, individual 3 (and 4) in P = {{1}, {2}, {3, 4}}, all individuals in P = {{1, 2}, {3, 4}}, all individuals in P = {{1}, {2}, {3}, {4}}, individual 1 (and 2) in P = {{1}, {2}, {3, 4}} and lastly, individual 1 in P = {{1}, {2, 3, 4}}. We can see a clear pattern: on the one hand, individual i’s incentive to free-ride is greater, the larger is the size of P (i). On the other hand, individuals for whom the group size stays the same, will exert more expected effort in response to the increased free-riding incentives outside of their group. In Figure 6, we also see that the highest average expected effort is when the contest is individual. Moreover, we can see that the presence of larger group diminishes the average effort. This comes as no surprise as it has been shown by several authors that the presence of larger groups entails a larger incentive to free-ride. 4.3 Adding an extra player to the contest In this section, I analyse the impact of adding an extra individual into the contest that contains originally three contestants. From the perspective of 1, the extra individual, namely individual 4, enters the contest either as a teammate or as an opponent of 1. If 4 enters as an opponent of 1, the impact 23 Figure 5: Equilibrium thresholds for N = {1, 2, 3, 4} Figure 6: Average equilibrium thresholds for N = {1, 2, 3, 4} 24 on 1’s effort depends on whether 4 joins an existing group or enters the contest alone. In what follows, I compare a specific partition of {1, 2, 3} with all of its possible extensions from the inclusion of individual 4. There are mainly two elements to consider when choosing the right threshold: the marginal benefit of free riding and the marginal benefit of exerting effort. When an extra player is added to one’s group, then the marginal benefit of free-riding and the marginal benefit of exerting effort both increase for that one player. Naturally, these have opposing impacts on the expected effort supply. But it may be the case that the incentive to free ride dominates the incentive to exert effort, in which case individuals would decrease their threshold, or vice versa. When a player is added as one’s opponent, intuitively, that one player may have a lower marginal benefit of exerting effort as well as a lower marginal benefit of not exerting effort. This is so because it is expected that the winning probability of one group, in equilibrium, should decrease when there are more opponents. 4.3.1 P = {1, 2, 3} Table 1: Equilibrium thresholds (c = 0.5) P x∗1 x∗2 x∗3 x∗4 {1, 2, 3} 0.7937 0.7937 0.7937 {1, 2, 3, 4} 0.8409 0.8409 0.8409 0.8409 {{1, 2, 3}, {4}} 0.8403 0.8403 0.8403 0.6470 When 4 joins the group of three, the incentive to free ride is greater than if 4 enters the contest alone. When 4 enters the contest alone, we can see that 4 is considered as a negligible threat to 1 (and 2 and 3). The difference in the threshold of 1 in P 0 = {{1, 2, 3}, {4}} compared to P 0 = {1, 2, 3, 4} is nearly infinitesimal. Whether 1 sees individual as a teammate or as an opponent does not change his behavior. The size of of P (1) is large enough so that the equilibrium strategy of its members is almost invariant to the inclusion of a single opponent into the contest. However, the decrease in expected effort when the partition goes from {1, 2, 3} to {{1, 2, 3}, {4}} is puzzling. This goes against the intuition as one would imagine that with the inclusion of individual 4, 1 must exert more effort in order to secure a high 25 probability of winning. In this case, it seems that, from the perspective of 1, the increased probability that effort will be wasted, makes 1 more prudent, hence the reduced expected effort in equilibrium. 4.3.2 P = {{1, 2}, {3}} Table 2: Equilibrium thresholds (c = 0.5) P {{1, 2}, {3}} {{1, 2, 4}, {3}} {{1, 2}, {3, 4}} {{1, 2}, {3}, {4}} x∗1 x∗2 x∗3 x∗4 0.7941 0.7941 0.6186 0.8403 0.8403 0.6370 0.8403 0.7826 0.7826 0.7826 0.7826 0.8162 0.8162 0.7018 0.7018 When 4 joints P (1), 1 and his teammates all increase their threshold, since they now have an extra player on whom to free ride. Individual 3 can then increase slightly his threshold in response to this increased free riding incentive in the opposing group. In the case of 4 joining P (3), 1 and 2 increase their expected effort by lowering their threshold. This is what one should intuitively expect as the opponent’s group is now ”stronger”. Also, 3 now has a teammate on whom to free ride on, and so reduces his level of expected effort. The last case is reminiscent of the puzzling case in Table 1. Individual 4 enters the contest as a single contestant. One could think that the level of competition that 1 faces is even higher because the new contestant does not have the possibility to free ride. Intuitively, it makes sense to think of the subpartition {{3}, {4}} being more competitive than {3, 4}. Then one should expect members of P (1) to be more aggressive and decrease their threshold more than when 4 joins P (3). However, it is actually the opposite that happens. This could be explained by the fact that when 4 enters the contest alone, 3 still has no marginal benefit of free riding and a decreased marginal benefit of exerting effort which can only make him reduce his level of expected effort. Compared to when 4 joins P (3), not only 3 sees his marginal benefit of exerting effort increase but 3 also sees his marginal benefit of free riding increase from null to a positive value. This may have a mixed effect on the behavior of individuals in P (1). Clearly, in this case, the incentive to exert effort dominates for them. Now, when 4 enters the contest alone, 26 the effect on the individuals in P (1) is clearer: their opponents only have a reduced incentive to exert effort. This, in turn, reduces the marginal benefit of exerting effort since, ceteris paribus, it is more probable that they win. 4.3.3 P = {{1}, {2}, {3}} Table 3: Equilibrium thresholds (c = 0.5) P {{1}, {2}, {3}} {{1}, {2}, {3, 4}} {{1}, {2}, {3}, {4}} x∗1 x∗2 x∗3 x∗4 0.6914 0.6914 0.6914 0.7018 0.7018 0.8162 0.8161 0.7413 0.7413 0.7413 0.7413 When the contest is originally an individual contest, we may expect that an extra player entering the contest alone would induce individuals to exert more effort due to the increased competition. However, all players see their marginal benefit of exerting effort decrease and their marginal benefit of free riding is still null. Thus they can only decrease their threshold when 4 enters the contest alone. When 4 joins P (3), the incentive to free ride is evident for 3 and 4. Although, 1 and 2 increase their threshold, this change in the behavior is small. 5 Discussion So far, little have been said concerning the goal or the desires of the contest designer. If the contest designer is assumed to be benevolent, it may be irrelevant to to discuss his interest in the contest. Moreover, if a contest designer has no value for the contested good, then the designer should not care whether the good is allocated or not and nor should he care about who gets the good. However, the assumption made on the functional form of the contest win probabilities, which says that if no effort is exerted then the good is not allocated, reveals that effort is implicitly valued by the designer. The reason is that not allocating the good is socially inefficient. Thus, by deciding not to allocate the good, it is implicitly assumed that the designer prefers baring the inefficiency cost of unrealized utility than to allocate a good to 27 individuals who have not exerted effort. With this in mind, it reasonable to assume that the designer may prefer contests where individuals exert high expected effort. It was shown that groups of greater size can expect, ex-ante, and as long as the curvature of the c.d.f.’s do not add too much extra incentive to free-ride, to win the good with a higher probability. Although in this paper, I do not consider the process of coalition formation, we can still argue that individuals may prefer to belong to groups of greater size if it offers a greater expected probability of winning. Belonging to a large group may let team members diminish their expected effort since they can rely on a greater number of teammates to exert effort instead of them. From the perspective of the contest designer, this may not be desired. What a contest designer would want to avoid is a situation in which contestants team up for the ”wrong” reasons. That is to form a group mainly because of the two properties stated above: high expected probability of winning and free-riding. Free-riding drives down the expected efforts, which is bad for the contest designer. A good reason to form a group would be the necessity of forming a group. For instance, a group of architectural firms may belong to the same bidding group for a governmental contract. If they were to win the contest, the firms may produce the project jointly in a way that all firms are complementary to each other. There can be a firm specializing in drawing the plans for a building and another firm responsible of structural engineering and both of them are necessary to the delivery of the project and so without collaboration, the two firms could not participate to the contest alone. But in the case of two firms that are essentially identical in their specialization, and are grouped together for the simple reason of lowering the probability of exerting the cost of effort without reaping the benefits, then it seems justifiable, from the designer’s point of view, that these two firms be competitors instead of partners. 6 Concluding Remarks In this paper, I have developed a model of rent-seeking group contest with imperfect information. I have introduced a novel way to tackle the problem io non tractability of the equilibria in group contests: using dichotomous effort levels. I show that an equilibrium exists under fairly general assumptions. 28 This model is such that in equilibrium, all contestants exert positive expected effort. I have found that the average expected effort is maximized in individual contests. This result suggests that competing groups should be broken down into smaller groups whenever it is possible. Lastly, the Olson’s Paradox may or may not be invalidated. An interesting extension to allow individuals within a coalition to share information with the other members of the coalition. One could then verify if the free-riding incentives are less important in such a scenario. 7 Bibliography References Baik, Kyung Hwan. 1993. Effort levels in contests: The public-good prize case. Economics Letters, 41(4), 363–367. Baik, Kyung Hwan. 2008. Contests with group-specific public-good prizes. Social Choice and Welfare, 30(1), 103–117. Baik, Kyung Hwan, & Shogren, Jason F. 1998. A behavioral basis for bestshot public-good contests. Advances in Applied Microeconomics: Contests, 7, 169–178. Baik, Kyung Hwan, Kim, In-Gyu, & Na, Sunghyun. 2001. Bidding for a group-specific public-good prize. Journal of Public Economics, 82(3), 415–429. Barbieri, Stefano, & Malueg, David A. 2014. Private-Information Group Contests. Available at SSRN 2501771. Barbieri, Stefano, Malueg, David A, & Topolyan, Iryna. 2013. The Best-Shot All-Pay (Group) Auction with Complete Information. Available at SSRN 2326211. Bloch, Francis, & Zenginobuz, Unal. 2007. The effect of spillovers on the provision of local public goods. Review of Economic Design, 11(3), 199–216. Brookins, Philip, & Ryvkin, Dmitry. 2014. Equilibrium existence in group contests of incomplete information. Tech. rept. 29 Chamberlin, John. 1974. Provision of collective goods as a function of group size. American Political Science Review, 68(02), 707–716. Chowdhury, Subhasish M, Lee, Dongryul, & Sheremeta, Roman M. 2013. Top guns may not fire: Best-shot group contests with group-specific public good prizes. Journal of Economic Behavior & Organization, 92, 94–103. Corchón, Luis C. 2007. The theory of contests: a survey. Review of Economic Design, 11(2), 69–100. Dubey, Pradeep. 2013. The role of information in contests. Economics Letters, 120(2), 160–163. Einy, Ezra, Haimanko, Ori, Moreno, Diego, Sela, Aner, & Shitovitz, Benyamin. 2013. Tullock contests with asymmetric information. Epstein, Gil S, & Mealem, Yosef. 2009. Group specific public goods, orchestration of interest groups with free riding. Public Choice, 139(3-4), 357–369. Esteban, Joan, & Ray, Debraj. 2001. Collective Action and the Group Size Paradox. American Political Science Review, 95(3). Fey, Mark. 2008. Rent-seeking contests with incomplete information. Public Choice, 135(3-4), 225–236. Harstad, Ronald M. 1995. Privately informed seekers of an uncertain rent. Public Choice, 83(1-2), 81–93. Hurley, Terrance M, & Shogren, Jason F. 1998a. Effort levels in a Cournot Nash contest with asymmetric information. Journal of Public Economics, 69(2), 195–210. Hurley, Terrance M, & Shogren, Jason F. 1998b. Asymmetric information contests. European Journal of Political Economy, 14(4), 645–665. Katz, Eliakim, Nitzan, Shmuel, & Rosenberg, Jacob. 1990. Rent-seeking for pure public goods. Public Choice, 65(1), 49–60. Kolmar, Martin, & Rommeswinkel, Hendrik. 2013. Contests with groupspecific public goods and complementarities in efforts. Journal of Economic Behavior & Organization, 89, 9–22. Konrad, Kai A. 2009. Strategy and dynamics in contests. OUP Catalogue. 30 Lee, Dongryul. 2012. Weakest-link contests with group-specific public good prizes. European Journal of Political Economy, 28(2), 238–248. Long, Ngo Van. 2013. The theory of contests: a unified model and review of the literature. European Journal of Political Economy, 32, 161–181. Malueg, David A, & Yates, Andrew J. 2004. Sent Seeking With Private Values. Public Choice, 119(1-2), 161–178. McGuire, Martin. 1974. Group size, group homo-geneity, and the aggregate provision of a pure public good under cournot behavior. Public Choice, 18(1), 107–126. Nitzan, Shmuel. 1994. Modelling rent-seeking contests. European Journal of Political Economy, 10(1), 41–60. Nti, Kofi O. 1998. Effort and performance in group contests. European Journal of Political Economy, 14(4), 769–781. Olson, Mancur. 1965. The logic of collective action. Harvard University Press. Pecorino, Paul, & Temimi, Akram. 2008. The Group Size Paradox Revisited. Journal of Public Economic Theory, 10(5), 785–799. Pogrebna, Ganna. 2008. Learning the type of the opponent in imperfectly discriminating contests with asymmetric information. Available at SSRN 1259329. Riaz, Khalid, Shogren, Jason F, & Johnson, Stanley R. 1995. A general model of rent seeking for public goods. Public Choice, 82(3-4), 243–259. Ryvkin, Dmitry. 2010. Contests with private costs: Beyond two players. European Journal of Political Economy, 26(4), 558–567. Schoonbeek, Lambert, & Winkel, Barbara M. 2006. Activity and inactivity in a rent-seeking contest with private information. Public Choice, 127(1-2), 123–132. Sui, Yong. 2009. Rent-seeking contests with private values and resale. Public Choice, 138(3-4), 409–422. Topolyan, Iryna. 2014. Rent-seeking for a public good with additive contributions. Social Choice and Welfare, 42(2), 465–476. 31 Tullock, G. 1980. Efficient rent seeking. In: J.M. Buchanan, R.D. Tollison, & Tullock, G. (eds), Toward a theory of the rent-seeking society. College Station: Texas A&M University Press. Ursprung, Heinrich W. 1990. Public Goods, Rent Dissipation, and Candidate Competition. Economics & Politics, 2(2), 115–132. Wärneryd, Karl. 2003. Information in conflicts. Journal of Economic Theory, 110(1), 121–136. Wasser, Cédric. 2013. Incomplete information in rent-seeking contests. Economic Theory, 53(1), 239–268. 32