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Strategic Choice of Sharing Rules in Collective Rent-Seeking and the Group Size Paradox Pau Balart∗ Sabine Flamand† Orestis Troumpounis‡ February 2, 2014 Abstract We provide new insights regarding the occurrence of the group size paradox —a situation in which a small group outperforms a large one— when two groups of different size compete for a mixed public-private good by considering that (i) the sharing rules determining how the private part of the prize is allocated among group members are public information and thus their choice is strategic, and (ii) one of the two groups may set its sharing rule faster than the other. We show that regardless of the timing of the interaction, the large group uses its sharing rule as a mean to address free-riding among its members, thereby making the small group inactive for most degrees of privateness of the prize. In turn, the group size paradox occurs only when groups choose their sharing rules simultaneously and the good is sufficiently private. If the choice of sharing rules is sequential the group size paradox always vanishes regardless of the leader’s size. This results from an increase (decrease) in between-group competition when the large (small) group is the leader. Keywords: collective rent-seeking, group size paradox, strategic choice of sharing rules, strategic complements, strategic substitutes ∗ Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe, Spain, e-mail: [email protected] † Nova School of Business and Economics, Campus de Campolide, P-1099-032, Lisbon, Portugal, e-mail: [email protected] ‡ Department of Economics, Lancaster University, Bailrigg, Lancaster, LA1 4YX, United Kingdom, e-mail: [email protected] 1 1 Introduction Collective contests where group members organize and compete for a prize are a common phenomenon: research and development races, procurement contests, funds to be allocated among different departments within a university, regions within a country or countries in the EU, prizes and projects allocated among different divisions of a firm, or party members that participate in pre-electoral campaigns. Furthermore, one can interpret several prizes sought by competing groups as a mixture between a public and a private good. Regional governments competing for a budget typically use the latter to provide both monetary transfers and local public goods. Similarly, prizes in research and development races involve both reputational and monetary benefits for the winning team. More generally, any contest between groups for a monetary reward can be considered as involving a public good component as long as the winners enjoy some benefits in terms of status, reputation, or satisfaction related to the victory. The sharing rules determining the way the private part of the prize is allocated among the group’s members, the composition of the prize and the size of the competing groups are known to affect individual incentives to contribute to such contests. In turn, those incentives ultimately determine the occurrence of the group size paradox (GSP henceforth), a situation where a small group outperforms a large one. The notion of the GSP dates back to the seminal work by Olson (1965), who stressed the severity of the free-rider problem within large groups. Esteban and Ray (2001) and Nitzan and Ueda (2011) have recently studied the occurrence of the GSP for mixed public-private goods. In this paper, we consider two additional factors that may further affect the occurrence of the GSP, namely (i) the strategic choice of sharing rules when these are public information and (ii) the non-simultaneity of the choice of sharing rules. Let us further motivate both.1 1 Our analysis relates to the collective rent-seeking literature, which focuses on the ways to allocate a prize among group members, starting after Nitzan (1991a). This literature considers both exogenously given sharing rules under different specifications as well as the endogenous choice of such rules. This literature is large and we do not aim to review it. For exogenous sharing rules see for example Nitzan (1991a,b); Davis and Reilly (1999); Ueda (2002), and for endogenous sharing rules see for instance Lee (1995); Baik and Lee (1997); Lee and Hyeong Kang (1998); Baik and Lee (2001); Noh (2002); Baik and Lee (2007, 2012a); Nitzan and Ueda (2011). 2 Our model aims at filling the gap of the strategic choice of sharing rules within competing groups. While Nitzan and Ueda (2011) extend the analysis of Esteban and Ray (2001) by considering the possibility of endogenous sharing rules, they do so in a setup of imperfect information where sharing rules are private information, hence their choice does not involve a strategic interaction between groups. While it is often reasonable to think of groups’ internal organization (sharing rules) as private information, many other cases are better described by a situation of perfect information where the decisions regarding sharing rules are potentially strategic.2 Think for instance of university departments competing for funds, with the departments allocating part of these funds among their members according to a publicly known incentive scheme. More generally, any competition taking place between groups belonging to a common organization is often likely to involve common knowledge of the individual incentive schemes within each group. Likewise, it is not unreasonable to consider that in some instances, firms have some knowledge regarding the incentives schemes of their competitors. At the same time, we consider that a sequential choice of sharing rules is of further interest as it is often likely that one of the competing groups has the opportunity of establishing its sharing rule faster than its competitor. This may result from the organizational skills of the groups, their relative size, their access to information regarding the contest itself, etc.3 Interestingly, and as we mention in the discussion, if the two groups were able to endogenously decide the timing of the strategic interaction, in most instances it is the sequential version of the game that would prevail in equilibrium. Thus, the very fact that the equilibrium timing of the game may be a sequential one provides 2 In fact, in the literature on collective rent-seeking and sharing rules the assumption of perfect information as in this paper is the usual one. To the best of our knowledge, the only papers analyzing the case of private information are Baik and Lee (2007, 2012a); Nitzan and Ueda (2011). 3 The sequential choice of sharing rules in collective rent-seeking for the case of a pure private good has been studied by Gürtler (2005). He shows that if the more numerous group determines its sharing rule prior to the smaller group, rent dissipation in the group contest is higher than in an individual contest. The introduction of a mixed public-private good allows us to stress the strategic use of sharing rules by the large group to keep the small group inactive for most intermediate degrees of privateness. See also Noh (1999), who analyzes the interactions between group sharing rules, the competition between two groups over the common output, and the allocation of resources between productive and appropriate activities. He shows that the sequential choice of sharing rules and resource allocations results in the adoption of fully egalitarian sharing rules in both groups. 3 further justification for departing from the assumption of simultaneity. In turn, and as our results point out, such a sequential choice of sharing rules provides a new explanation for the complete elimination of the GSP. These new features are introduced in a standard model of collective rent-seeking, in which two groups of different size compete for a mixed public-private good. All individuals face the same linear cost of effort, and they have the same valuation of the prize. Individuals belong to one of the two competing groups, which have to select the sharing rule according to which the private part of the good will be divided among the group members. The dynamics of collective action in group contests for pure private and pure public goods have been widely investigated and are well understood. Esteban and Ray (2001) and Nitzan and Ueda (2011) have considered the possibility of mixed public-private goods, and find that no group can be inactive in equilibrium. In stark contrast with those findings, we show that regardless of the particular timing of the interaction between the two groups, the small group is inactive for most degrees of privateness of the contested prize. If such prize is a pure public good, the choice of sharing rules is irrelevant and both groups face the same winning probability. However, as long as the good exhibits an arbitrarily small degree of privateness, the large group is able to use its sharing rule as a coordination device so as to address free-riding among its members, thereby leading to the inactivity of the small group. In turn, the large group is able to provoke such an outcome because the sharing rules allow for net transfers from low to high performers within the group (i.e., the individual reward from the private component of the prize may be more than proportional to individual effort). This characteristic of the group sharing rules constitutes the fundamental difference between our analysis and the one of Nitzan and Ueda (2011) regarding the possibility of one group being inactive in equilibrium, as they exclude the possibility of such transfers. As it turns out, this extreme outcome occurs for most intermediate degrees of privateness of the prize, except the ones very close to full privateness. Conversely, if the good is mostly private, both groups are active 4 in equilibrium. For a given level of privateness, the most direct effect of group size is that it yields better odds at winning the competition between groups, since there is a larger number of potential contributors. Clearly, this effect goes against the occurrence of the GSP. Then, the degree of privateness of the contested good determines the relative meritocracy of the selected sharing rules and thus the level of individual effort in each group. More specifically, the returns to effort vary depending on the private-public nature of the good, and are smaller for the large group the higher the degree of privateness. Indeed, if the good is (at least partially) private, a larger group size also implies a lower individual share of the prize in case of victory. This is the group-size deterrence effect (Nitzan, 1991a), which is the driving force of the GSP in Esteban and Ray (2001). Furthermore, in the presence of (at least partially) meritocratic group sharing rules (i.e., individual reward depends on relative performance to some extent), the degree of privateness of the prize has yet another dimension that influences the occurrence of the GSP. As with such sharing rules, individuals receive more of the private good the more they contribute, it follows that the higher the degree of privateness of the good, the larger the incentives to compete with the other group members. We show that the endogenous choice of sharing rules, regardless of its timing, is such that the higher the degree of privateness, the more likely that the small group adopts a more meritocratic rule than the large one. This, in turn, reinforces the effect of privateness on the GSP.4 When the prize is sufficiently private and the choice of sharing rules is simultaneous, the size deterrence effect prevails, which decreases the attractiveness of a potential victory from the perspective of the large group’s members. As a result, the large group implements a less meritocratic rule than the small group, inducing low levels of effort that 4 Kolmar and Wagener (2013) analyze the adoption of efficient incentive schemes within competing groups, separate from the group sharing rule, and offer an explanation as to why the GSP may not hold: larger groups may end up with a better organizational structure precisely because their larger free-rider problem builds up higher pressure to establish efficient incentive schemes. Nitzan and Ueda (2009) consider the case where the intra-group division of the prize is determined anarchically and show that it strengthens the possibility of the GSP. For more on the relationship between group size and group success, see Kolmar (2013) and Konrad (2009). 5 ultimately sustain the GSP. Conversely, for a sufficient degree of publicness of the contested good, the size deterrence effect is relatively unimportant, which in turn eliminates the GSP.5 Our results under a simultaneous strategic choice of sharing rules substantially differ from the two closest works in the literature. Esteban and Ray (2001) assume that the private part of the prize is shared among group members on an egalitarian basis. In that case, for a linear cost of effort, the GSP takes place for any strictly positive degree of privateness of the good. On the other hand, Nitzan and Ueda (2011) show that the simultaneous non-strategic choice of sharing rules eliminates the GSP unless the prize is purely private. Our results for the simultaneous case are less extreme than these two. We show that the GSP may arise as long as the contested good is sufficiently private, but not necessarily a pure private good. While the two previous works provide further results on the influence of a convex cost of effort on the elimination of the GSP, the introduction of a strategic choice of sharing rules obliges us to restrict ourselves to the linear cost case.6 Introducing a non-simultaneous choice of sharing rules alters the previous result by fully eliminating the GSP regardless of the leader’s size. With sequential sharing rules, in fact, the two groups face the same probability of winning the prize whenever the latter is either a purely public or a purely private good. For all intermediate values of privateness, the large group outperforms the small one. If the large group is the leader, the intuition is similar to the standard Stackelberg duopoly model. That is, the large group commits to a more meritocratic sharing rule than in the simultaneous game (hence effort increases), while the small group selects a less meritocratic rule than in the simultaneous game (hence effort decreases), which prevents the occurrence of the GSP. The small group 5 Again, in the case of most intermediary degrees of privateness of the prize, the aggregate effort effect intensively dominates the size deterrence one, leading to situations where the small group is inactive in equilibrium independently of the timing of the strategic interaction between groups. 6 A convex cost of effort penalizes higher levels of individual effort, hence it works against the occurrence of the GSP. Corchón (2007) provides an intuition for the latter:“Intuitively, it is clear that Olson’s conjecture cannot hold if costs rise very quickly with effort: for instance if costs are zero up to a point, say Ḡ where they jump to infinity, all agents will make effort Ḡ and smaller groups will exert less effort than large ones.” Therefore, the elimination of the GSP that we find in the sequential case should also hold with convex costs. Likewise, the occurrence of the GSP in the simultaneous case should be less likely under the assumption of convex costs. 6 acting as the leader follows a very different strategy. It commits to a less meritocratic rule than in the simultaneous game, thereby lowering the level of competition between groups. In turn, the large group also responds with a less meritocratic rule than in the simultaneous game, but the reduction is relatively lower than the one implemented by the small group. As a result, total effort decreases relatively less in the large group, which again eliminates the GSP. Even tough the GSP vanishes independently of the leading group’s size, it does so for two very different reasons: the sharing rules are strategic substitutes from the perspective of the small group, while they are strategic complements from the perspective of the large group. The paper is structured as follows: In Section 2, we present the model. In Section 3, we characterize the equilibrium of the effort stage. In Section 4 and Section 5, we analyze the simultaneous and sequential choice of sharing rules, respectively. Section 6 concludes with a discussion. All proofs can be found in the Appendix. 2 The Model There are two groups i = A, B of size ni ∈ N+ and without loss of generality A is the large group (i.e., nA > nB ). All individuals within a group are identical and the cost of exerting effort is linear.7 The valuation of the contested (divisible) good is the same for all individuals, and is denoted by V . Individuals’ choice of efforts within each group takes place simultaneously given the sharing rules. Individuals are risk neutral, and the expected utility of individual k = 1, ..., ni in group i is given by 7 We abstract from the possibility of intra-group heterogeneity regarding lobbying effectivity, which can be reflected in different valuations of the prize by the members. One can ask whether a group whose members have highly unequal interests in the collective action will be more or less active. The literature has provided contrasted answers to the latter question, due to the difference in the assumed form of the effort cost function (see the discussion in Nitzan and Ueda (2013) and the references they cite). Assuming very weak and plausible restrictions on the form of the effort cost functions, Nitzan and Ueda (2013) show that if a group competes for a prize with sufficiently many rivals or with a very superior rival, unequal stakes among the members can enhance its performance. 7 ( EUki = where Ei = Pni j=1 eji ) eki 1 Ei αi + (1 − αi ) p + (1 − p) V − eki Ei ni EA + EB denotes the aggregate effort of group i and p ∈ [0, 1] measures the degree of privateness of the contested good (p = 0 corresponds to the case of a pure public good while p = 1 corresponds to the case of a pure private good). The sharing rule αi ∈ [0, ∞) is the relative weight given to meritocracy as opposed to egalitarianism in the division of the prize within the corresponding group. If αi = 1, the private part of the prize is shared according to relative effort, while if αi = 0, it is shared equally among the group members regardless of their relative contribution. Therefore, as long as αi > 0, high performers in group i receive a larger share of the prize than low performers. A value of αi larger than one implies that the prize is allocated more than proportionally to relative i V effort within group i. More precisely, when αi > 1, group i collects −(1 − αi ) npi (EAE+E B) from all of its members and allocates it according to relative effort within the group (Baik and Lee, 1997; Lee and Hyeong Kang, 1998). Consequently, for values of αi larger than one, an individual exerting no effort pays a transfer to the group members who exert positive effort. Thus, if the efforts of the active group members are sufficiently high, expected utility may be negative for an individual exerting no effort. Our game consists in the strategic choice of the groups’ sharing rules, followed by the simultaneous choice of individuals’ effort levels.8 We analyze both the simultaneous and sequential choice of sharing rules by the two groups, hence the game has two or three stages respectively. The equilibrium concept is subgame perfection in pure strategies. In the effort stage we focus on within-team symmetric equilibria. 8 As individuals are identical within groups, we assume that in each group, the choice regarding the sharing rule is made by the representative individual by maximizing his expected utility function. 8 3 Effort Stage Before analyzing the choice of sharing rules, we first solve the last stage of the game, which essentially extends the results of Ueda (2002) and Davis and Reilly (1999) by allowing for a mixed public-private prize. Proposition 1. Let χi (αA , αB ) = (1−αi )p(ni nj −nj )−(1−αj )p(ni nj −ni )−ni nj (1−p)− pni with i = A, B and j 6= i. There exists a unique within-team symmetric equilibrium in pure strategies in the effort subgame characterized as follows: • When χi (αA , αB ) < 0 for all i = A, B: eˆi = Λi Φi V ni [nj p+ni (2nj (1−p)+p)]2 • When χi (αA , αB ) ≥ 0 for group i = A, B: ẽ∗j = ẽi = 0 αj p(nj −1) V n2j with Λi = [nj (1 − p) + p] [ni (1 − p) + p + (ni − 1)pαi ] + (nj − 1) [ni (1 − p) + p] pαj and Φi = nj p(1 − αi ) + ni [nj (1 − p(1 − αi + αj )) + pαj ]. The condition χi (αA , αB ) < 0 determines whether group i is active in equilibrium. As can be seen from the analytical expression of χi (αA , αB ), the more egalitarian is the sharing rule within a group, the more likely that its members are inactive. Indeed, as free-riding incentives increase, individual effort eventually drops to zero. Furthermore, the more meritocratic the rule of its opponent, the more likely that a group is inactive. When both groups are active (i.e., χi (αA , αB ) < 0 for all i = A, B), equilibrium effort in each group is increasing in the relative meritocracy of its sharing rule and in the value of the prize. When only one group is active, its members compete in a standard Tullock contest with equilibrium effort increasing in the relative meritocracy of the sharing rule and decreasing in the size of the group. Definition 1. The GSP arises if and only if EA < EB . 9 Given the contest success function used for the competition between groups, the previous definition is equivalent to the small group facing a higher probability of winning than the large group. Using the above equilibrium for exogenous sharing rules, we obtain our first result regarding the GSP: Proposition 2. With exogenous sharing rules, • For p = 0, the GSP never arises. • For p ∈ (0, 1], the GSP arises if and only if αA < nA − nB + 2αB nA (nB − 1) 2nB (nA − 1) Esteban and Ray (2001) assume that the prize is shared on an egalitarian basis among the group members regardless of their individual contribution, and find that with a linear cost of effort, the GSP arises for all values of p except for the extreme case of p = 0 (i.e., when the good is purely public).9 In fact, and as our results confirm, when the prize is a pure public good the two groups face the same probability of winning.10 Introducing the possibility of (exogenous) group-specific sharing rules leads to a more nuanced result. In particular, for any degree of privateness of the good, the less (more) meritocratic the sharing rule of the large (small) group, the more likely that the GSP arises. As with sharing rules, individual reward depends on relative performance within the group to some extent, the degree of privateness of the prize has another dimension. Here, individuals receive more of the private good the more they contribute. Since they do not coordinate, there is an additional expenditure of effort besides the one arising from the 9 Pecorino and Temimi (2008) modify the model of Esteban and Ray (2001) to a standard public good setting, and show that their results are robust to the presence of small fixed costs of participation in the case of a pure public good, but not in the case of a fully rival good. Furthermore, when the degree of rivalry is sufficiently high, the introduction of small fixed costs of participation implies that collective action must break down in large groups. 10 On pure public goods, see also Baik (2008); Riaz et al. (1995), Ursprung (1990) and Katz et al. (1990), among others. They have shown that with a pure public good, a group with larger membership attains a winning probability larger than or equal to that of a smaller group. Hence the GSP is not valid in such cases. 10 classical effect on the group winning probability: the lower the degree of privateness of the good, the smaller the incentives to compete with the other members of your group. In the limit, when the prize is purely public, individuals do not get any reward linked to personal effort. The effect of relative publicness on the GSP (i.e., the size deterrence effect) is driven by this factor, which is absent in Esteban and Ray (2001).11 4 Simultaneous Choice of Sharing Rules We now consider the full two-stage game: In stage one the two groups A and B choose S S their sharing rules (αA , αB ) simultaneously, and in stage two they choose their effort levels (eSA , eSB ), also simultaneously. As the sharing rules are irrelevant in the case of a pure public good, we assume that p > 0. While the equilibrium concept is subgame perfection, in our formal results we present the equilibrium choice of sharing rules, as effort levels chosen in stage two can be derived directly from Proposition 1. Proposition 3. Let c. If the two groups choose their sharing rules simultaneously: • If 0 < p ≤ p1 , only group A is active and the continuum of sharing rules in the subgame perfect equilibrium is given by S – αA = nA nA −1 h 1−p p + (nB −1)αS B +1 nB i S – αB ∈ [ nB (1−p)+p , nB [nA (1−p)+3p−2]−2p ] p (nB −1)p • If p > p1 , both groups are active and the sharing rule for i = A, B in the unique subgame perfect equilibrium is given by ni 2ni nj (ni − 1) + Ai p + Bi p2 αi = (ni − 1) p [2ni nj − p(ni (2nj − 1) − nj )] where Ai = ni nj (9 − 4ni ) − nj (nj + 2) − 2ni and Bi = ni nj (2ni − 7) + nj (nj + 3) + 3ni − 2. 11 This additional effect of privateness on the GSP is also present in Nitzan and Ueda (2011), as they also consider partially meritocratic group sharing rules. 11 For any positive degree of privateness smaller than the threshold p1 , the large group implements a sufficiently meritocratic sharing rule so as to keep the small group inactive. This is in stark contrast with the case of a pure public good where both groups are active and face the same probability of winning the prize (Baik, 2008). Remarkably, an arbitrarily small and negligible degree of privateness of the prize is sufficient to allow the large group to prevent any activity by the small group. Indeed, the large group uses its sharing rule as a coordination device among its members to prevent the small group from exerting any effort, thereby securing its own victory. Observe, furthermore, that such preemptive behaviour by the large group occurs for most degrees of privateness, as the threshold p1 is in fact very close to one.12 Such a coordination mechanism among the members of the large group is possible to the extent that we allow the sharing rule αi to exceed one, in which case an inactive group member has to pay a transfer to the other (active) group members. As a result, selecting a very meritocratic sharing rule allows the large group to address free-riding among its members even for low degrees of privateness, thereby leading to high levels of individual effort and ultimately to the inactivity of the small group. The fact that one group may be inactive in a subgame perfect equilibrium constitutes a fundamental difference with Ueda (2002), who assumes both a pure private good and sharing rules such that there may be no such transfers of resources between individuals within a group. If the good is sufficiently private (p > p1 ), we are at an interior solution and both groups are active in equilibrium. The small group always implements a sharing rule such that the expected share of the private prize is more than proportional to individual effort S (αB > 1 for all p > p1 ). The relative meritocracy of the large group’s sharing rule is strictly decreasing in the degree of privateness of the prize and larger than one whenever 12 Notice that the numerical value of p1 is large and very close to one as the size of the groups increases, and/or when there is a large difference between group sizes. For instance, if nA = 15 and nB = 7 then p1 = 0.98. However, this critical value of the degree of privateness can be rescaled by assuming that the public good is congestible. This could be captured by a parameter c ∈ [0, 1) such that the utility attributed to the public good by any member of group i is (1 − p)V /nci . In that case the critical value n [ncA (1+nB )−nA ncB ] p1 is given by pc1 = nc nBB(1+n , which for the previous example rescales p1 to pc1 = 0.76 c c B )−nB (nA +nA nB ) A for c = 0.6. In order to avoid unnecessary complications in the main text, we consider the case of c = 0. 12 the good is sufficiently public. Interior (eA>0,eB>0) (eA>0,eB=0) 0 1 P2 P1 αA< αB αA>αB Figure 1: Simultaneous choice of sharing rules Figure 1 summarizes our comparisons: when the degree of privateness is larger than p1 , both groups are active, although the large group implements a more meritocratic rule S S than the small one (αA > αB ) as long as p ≤ p2 .13 As the good gets relatively more private (p > p2 ), it is the small group that implements the most meritocratic sharing rule S S (αA < αB ), which may lead to the emergence of the GSP for a sufficiently high degree of privateness, as the following proposition establishes: Proposition 4. With a simultaneous choice of sharing rules, the GSP arises if and only if p > pGSP = 2nA nB 14 . 1+2nA nB !"#%'(!$)% %&!"&!$ !"#!$#% % ./ .1 *+(,-. .0 / .,-. ,-. Figure 2: Simultaneous choice of sharing rules and the GSP 13 To avoid introducing more notation, we do not define formally the thresholds p2 and p3 presented in the interpretation of our results. They can be easily calculated and one can show that p1 < p2 < p3 < pGSP . 14 The numerical value of pGSP is large and very close to one as the size of the groups increases, and/or when there is a large difference between group sizes. For instance, if nA = 15 and nB = 7 then pGSP = 0.99. Again, this value can be rescaled if we assume that the public good is congestible. nA nB [ncB +2nA ncB −ncA (1+2nB )] , which for the previous example yields In that case pcGSP = nc nc (n −n 1+c A B )+nA (1+2nA )nB ]−nA nB (1+2nB ) B[ A c pGSP = 0.95 for c = 0.6. 13 The GSP arises above a high degree of privateness of the contested good (p > pGSP ). When the good is purely private, therefore, the GSP always occurs. As can be seen in Figure 2, the members of the large group exert more effort than the ones of the small group (eSA > eSB ) as long as p 6 p3 . Clearly, the GSP can only arise for degrees of privateness of the prize that guarantee eSA < eSB , hence it must exceed p3 . In order for the GSP to hold, individual effort by the members of the small group must be large enough so as to compensate for the smaller number of members in the group. One can identify two opposite effects of group size on returns to effort driving the previous results. On the one hand, there is an aggregate effort effect. As groups’ performance is additive in the contributions of their members, a larger group size implies better odds at winning the competition between groups. On the other hand, there is a group-size deterrence effect (Nitzan, 1991a). If the good is (at least partially) private, a larger size also translates into a lower share of the prize for each group member in case of victory.15 Clearly, this latter effect disappears when the good is purely public, while it exacerbates when the degree of privateness increases. For degrees of privateness close to p1 the size deterrence effect is not sufficient to compensate for the aggregate effort effect. This means, in turn, that effort is more valuable for the members of the large group than it is for the members of the small group. As a result, the large group implements a relatively more meritocratic sharing rule, yielding higher levels of effort by its members. When the degree of privateness exceeds p2 , the size deterrence effect increases accordingly, and effort progressively becomes less valuable for the large group members. Therefore, from p2 on, the large group selects a sharing rule that is more egalitarian than the one of the small group. However, as the two effects are at play, returns to effort are still higher for the members of the large group, so that they exert higher effort until a yet larger degree of privateness is reached. As the prize becomes more private, the large group progressively decreases the degree of 15 As we explained before, this effect of relative privateness on the GSP is also driven by the existence of (partially) meritocratic group sharing rules. 14 meritocracy of its sharing rule such that from p3 on, returns to effort are smaller for its members than for the members of the small group (from p > p3 it holds that eSB > eSA ). Before analyzing the sequential choice of sharing rules by the two groups, the following lemma will be useful in order to understand its implications for the GSP. Lemma 1. If both groups are active, the sharing rules are strategic complements from the perspective of the large group, whereas they are strategic substitutes from the perspective of the small group. Given its size advantage, the large group reacts to an increase in αB by increasing αA . Conversely, the small group chooses to decrease αB following an increase in αA . Intuitively, an increase in αB would not be worthwhile in terms of its effect on the group’s winning probability, given the size disadvantage of the small group. 5 Sequential Choice of Sharing Rules We consider the following three-stage game: In stage one the leader (group i ∈ {A, B}) chooses its sharing rule αiL . In stage two the follower (group j 6= i) chooses its sharing rule αjF . In stage three both groups simultaneously choose their effort levels (eLi , eFj ). As for the simultaneous case, we assume that p > 0 and we present the equilibrium choices of sharing rules (αiL , αjF ) in our formal results, as effort levels (eLi , eFj ) chosen in stage three can be derived directly from Proposition 1. 5.1 Large group A is the leader Proposition 5. Let p01 = nA (nA −nB −1) . 1+nA (nA −nB −1) If group A is the leader: • If 0 < p ≤ p01 , only group A is active and the continuum of sharing rules in the subgame perfect equilibrium is given by L – αA = nA nB (1−p)+p (nA −1) p 15 F – αB ] ∈ [0, nB (1−p)+p p • If p > p01 , both groups are active and the sharing rules in the unique subgame perfect equilibrium are given by L – αA = 1 + nA 1−p p F – αB = (nA −1)(nA −nB )p[nA (1−p)+p]−nA [nB (1−p)+p][nA (2−2nB (1−p)−3p)−(nB −2)p] 2nA (nB −1)[nA (1−p)+p]p Observe that when the large group is the leader it commits to a more meritocratic sharing rule than in the simultaneous case, inducing the small group to be inactive for an even larger range of p (i.e., p01 > p1 ). In other words, the large group acting as the leader is capable of tying the small group’s hands and oblige the latter to react with a sharing rule leading to its own inactivity, and this for a larger range of p. The remaining intuitions regarding the relative levels of meritocracy and individual efforts within each group are similar to the ones of the simultaneous case. The following lemma will be useful in order to understand the implications of the sequential interaction when the large group chooses first. L S F S Lemma 2. If p > p01 then αA > αA and αB < αB . When the large group has the leading advantage, we are in a situation that is very similar to a standard Stackelberg duopoly model. Once the large group commits to a L S more meritocratic rule than in the simultaneous case (αA > αA ), the follower reacts by F S selecting a less meritocratic rule than in the simultaneous case (αB < αB ). Indeed, recall that the sharing rules are strategic substitutes from the perspective of the small group (Lemma 1). For low values of privateness (i.e., p < p02 ), the large group ties more the prize to relative effort than the small group and each member of the large group exerts higher effort than each member of the small group.16 Once the prize is private enough these Again, to avoid introducing more notation, we do not define formally the thresholds p02 and p03 presented in the interpretation of our results. They can be easily calculated and one can show that p01 < p02 < p03 . 16 16 relationships alter. First the small group selects a more competitive sharing rule than the large group (from p > p02 ) and from a point on each member of the small group exerts higher effort than each member of the large group (from p > p03 it holds that eFB > eLA ). Proposition 6. If the large group is the leader and for all values of p including full privateness, the GSP never arises. Contrary to the simultaneous case, the GSP never takes place regardless of the degree of privateness of the prize, which is a direct consequence of Lemma 2. Indeed, as the large group acting as the leader selects a more meritocratic sharing rule than in the simultaneous case, and as the small group reacts with a less meritocratic one, it follows that aggregate effort increases in the large group, while it decreases in the small group. As it turns out, these effort variations are large enough to eliminate the GSP.17 5.2 Small group B is the leader Proposition 7. If group B is the leader: • If 0 < p ≤ p1 , only group A is active and the continuum of sharing rules in the subgame perfect equilibrium is given by – F αA = nA nA −1 h 1−p p + (nB −1)αL B +1 nB i L ∈ [0, nB [nA (1−p)+3p−2]−2p – αB ] (nB −1)p • If p > p1 , both groups are active and the sharing rules in the unique subgame perfect equilibrium are given by F – αA = nB [nA (1−p)+p][nB (2−2nA (1−p)−3p)−(nA −2)p]−(nB −1)(nA −nB )p[nB (1−p)+p] 2(nA −1)nB [nB (p−1)−p]p L – αB = 1 + nB (1−p) p 17 In fact, for both the cases of full publicness (p = 0) and full privateness (p = 1) of the prize, the two groups face the same probability of winning. 17 For intermediate degrees of privateness covering most of the interval of p (0 < p 6 p1 ), the small group is not able to take advantage of its leadership. Regardless of the choice of the small group in stage one, in stage two the large group selects a very meritocratic sharing rule such that the small group remains inactive in stage three. Observe that the threshold level of privateness that determines whether the small groups is active or inactive is identical to the one obtained in the simultaneous case (Proposition 3), that is, the small group cannot take advantage of its leadership by becoming active for any p 6 p1 . F S L S Lemma 3. If p > p1 then αA < αA and αB < αB . For all degrees of privateness of the prize yielding an interior equilibrium, and in stark contrast with the case of the large group being the leader, it turns out that when the small group has the leadership advantage it strategically chooses to lower competition L S 18 by selecting a less meritocratic rule than in the simultaneous game (αB < αB ). In turn, the large group also reacts with a less meritocratic rule than the one it selects when S F ). Indeed, recall that the sharing rules are strategic < αA the game is simultaneous (αA complements from the perspective of the large group (Lemma 1). Therefore, both groups adopt relatively more egalitarian rules, and their members exert a lower level of effort than in the simultaneous game ((eFA , eLB ) < (eSA , eSB )). Proposition 8. If the small group is the leader and for all values of p including full privateness, the GSP never arises. Observe that here, the GSP does not arise for a very different reason than when the large group is the leader. The small group selects a sharing rule such that its members are inactive when the prize is public enough (but not purely public), which clearly prevents the occurrence of the GSP as in the previous cases. Then, if p > p1 the small group acting 18 This is broadly consistent with previous findings on the effects of precommitment in contests with asymmetric players. Dixit (1987) establishes that with two asymmetric players, the “favorite” commits effort at a higher level than in a Nash equilibrium without commitment, and the “underdog” at a lower level. 18 as the leader strategically chooses to lower competition by selecting a less meritocratic rule than in the simultaneous game, which in turn induces the large group to adopt a similar behaviour. However, the reduction in meritocracy implemented by the small group is relatively larger than the corresponding reduction in the large group, and is such that the GSP vanishes. The fact that the leading advantage of the small group always eliminates the GSP is somewhat surprising. Indeed, one could expect that the small group could exploit such a leadership advantage by adopting more meritocratic rules, thereby inducing high levels of effort by its members, potentially giving rise to the GSP for a larger set of p0 s. As it turns out, the small group chooses instead to lower competition between the two groups, which in turn eliminates the GSP for all values of p.19 Hence, despite being the leader —and thus obtaining a higher payoff than in the simultaneous game— the small group chooses to eliminate the GSP. In turn, this suggests that the GSP is not always valuable from the perspective of the small group. 6 Discussion We have shown that the strategic choice of sharing rules has important implications regarding the occurrence of the GSP. In particular, the timing of the strategic interaction is important. While the GSP may occur for different degrees of privateness of the prize in the case of a simultaneous interaction, when there is a first mover, the GSP vanishes independently of the leader’s size. Furthermore, regardless of the order of moves, the large group selects a sharing rule leading to the inactivity of the small group for most values of privateness of the prize, even arbitrarily small ones. We have analyzed the relationship between the size of a group and its performance in terms of its probability of winning the prize. However, another relevant notion of group effectiveness relates its size to per-capita payoffs. Even though the large group always 19 As in the case where the large group is the leader, for both the extreme cases of full publicness (p = 0) and full privateness (p = 1) of the prize, the two groups face the same probability of winning. 19 outperforms (or ties) the small one in terms of aggregate effort (and thus in terms of winning probability) whenever the order of moves is sequential, the members of the small group achieve strictly higher expected utility than the members of the large group when the prize is sufficiently private. This is also the case when the choice of sharing rules is simultaneous, from a degree of privateness strictly smaller than the one required for the GSP to arise.20 In fact, irrespective of the order of moves, when the prize is sufficiently private so that both groups are active, individual utility is decreasing in privateness in the large group, while it is increasing in the small group. With this alternative notion of group effectiveness, therefore, the members of the small group always outperform the members of the large group in a competition over a pure private good regardless of the timing of the interaction. As a larger group size decreases the per-capita value of the private component of the prize, the large group has some intrinsic disadvantage in terms of payoffs. Although the size of the leader does not matter in terms of the GSP, it does have some implications regarding the expected utility of groups’ members and rent dissipation. Consider the case where both groups are active for any timing of the interaction. When the large group is the leader, all its members clearly have higher expected utility than in the simultaneous version of the game, while all the members of the small group (the follower) are worse off. Therefore, a transition from simultaneous to sequential competition where the large group is the leader never consists in a Pareto improvement, and increases rent dissipation. Conversely, when the small group is the leader, the members of both groups exert less effort than in the simultaneous game, and they have higher expected utility. Thus, going from simultaneous to sequential competition where the small group is the leader consists in a Pareto improvement and decreases rent dissipation. Given these implications, a natural extension of our model consists in endogenizing √ −1 √ The exact value of the threshold is given by p = 1 − 1 + ni ( nA + nB )2 − 1 when group i is the leader (i = A, B). Although the corresponding threshold for the simultaneous case cannot be derived analytically, one can show that it is unique given the strict monotonicity of expected utility with respect to privateness in both groups. 20 20 the timing of the strategic interaction between the two groups. More specifically, let us assume that prior to the choice of sharing rules, groups are able to declare their intention to be the leader or the follower of the game.21 If both groups choose the same order the simultaneous game is played, while if they agree on the order the sequential game is played. It is easy to verify that when both groups are active, the unique equilibrium of the game with such an endogenous choice of the timing structure gives the leadership to the small group.22 Therefore, with an endogenous timing selection previous to the rent-seeking activity, the GSP should never take place.23 21 In the context of contests, Baik and Shogren (1992) analyze a standard Tullock contest with two individuals, and find that the unique equilibrium with endogenous timing is a sequential game where the weak player is the leader (see also Leininger (1993) and Morgan (2003)). Gürtler (2005) analyzes endogenous timing in the context of collective rent-seeking, and our results do not differ qualitatively for the case of p = 1. Baik and Lee (2012b) study the endogenous timing of contests with delegation, and show that the delegate of the higher-valuation player chooses his effort level after observing his counterpart’s. The industrial organization literature has extensively considered endogenous timing extensions. See for instance Amir and Grilo (1999), Hamilton and Slutsky (1990) and Van Damme and Hurkens (1999) in the context of a duopoly model. 22 Since this equilibrium is unique it is worth mentioning that it yields lower aggregate effort than in the simultaneous game. Thus, if from the designer’s point of view effort is valuable, he should possibly oblige groups to choose their sharing rules simultaneously, or even force the large group to move first. If effort is instead considered as wasteful, the designer should opt for the design where the small group is the leader and thus should not intervene in the game. 23 If the small group is inactive, the simultaneous game may arise as an equilibrium in some particular cases, which clearly also prevents the occurrence of the GSP. 21 7 Appendix Proof of Proposition 1. This proof relies on an extension of Ueda (2002). In order to facilitate comparison, let us transform αi = 1 − α̂i . The expected utility of individual k = 1, ..., ni in group i = A, B is i [(1 − α̂i ) Pneiki e p + α̂i n1i p + (1 − p)] (EAE+E V − eki B) j=1 ji By the Kuhn-Tucker Theorem, the necessary and sufficient conditions to characterize the unique symmetric pure-strategy Nash equilibrium are the following: (i) eki = Ei /ni for each member of group i = A, B and (ii) V {(1 E − α̂i )p + α̂i p ni + (1 − p) − Ei [1 E − p(1 − 1 )]} ni − 1 ≤ 0 for all i Similarly to Ueda (2002) we can define: γ̂i = (1 − α̂i )p + We also define n̂i = 1 , 1−p(1− n1 ) α̂i p + (1 − p) ni which is always positive and larger than one. Then the i above necessary and sufficient conditions for the pure strategy Nash equilibrium can be written as V {γ̂i E − Ei n̂i } E − 1 ≤ 0 for i = A, B. This expression is analogous to expression (2) in Ueda (2002, p. 616), which guarantees that Proposition 1 and Corollary 1 in Ueda (2002) hold in our case. Then, in our two groups case, from Corollary 1(b), members of group i are inactive if and only if γ̂i ≤ γ̂j − 1 , ∀i 6= j n̂i Rewriting this condition in terms of αA , αB yields χi (αA , αB ) ≥ 0 in Proposition 1. Solving the system of equations arising from the first-order conditions in the interior and corner solutions, we find the equilibrium effort levels. 22 Proof of Proposition 2. By definition the GSP takes place if and only if EA < EB . Consider the equilibrium effort levels when both groups are active. Then ẼA = nA êA < ẼB = nB êB can be written as pA(αA , αB )B(αA , αB ) V >0 C where: A(αA , αB ) = − [nB (1 − p) + p] [nA (1 − p) + p + (nA − 1)pαA ] − (nB − 1) [nA (1 − p) + p] pαB < 0, B(αA , αB ) = nB (1 − 2αA ) + nA [2nB (αA − αB ) + 2αB − 1] 2 and C = [nA (2nB (p − 1) − p) − nB p] > 0 The above expression is equal to zero for p = 0, hence the GSP does not arise in that case. For other values of p we need to consider the sign of each expression. Since C is always positive and A(αA , αB ) is always negative, the GSP arises if and only if B(αA , αB ) < 0, which can be rearranged as αA < nA − nB + 2αB nA (nB − 1) 2nB (nA − 1) Proof of Proposition 3. From Proposition 1 and the equilibrium effort levels if both groups are active, the expected utility for group i = A, B is given by ˆ i (αA , αB ) = EU [ni nj +((αi −1)(ni −1)nj +αj ni (1−nj ))p]V ni [nj p+ni (2nj (1−p)+p)]2 [ni (2ni − 1)nj − Ap + B(ni − 1)p2 ] where A = ni (1 − αj − ni ) + nj (1 − αi ) + ni (4ni + αi + αj − 5)nj B = αi (nj − 1) + αj (nj − 1) + (ni − 1)(2nj − 1) From Proposition 1 and the equilibrium effort levels if only group i is active, the expected utility for group i is given by 23 ˜ i (αi ) = V − (ni − 1)(ni + αi ) pV EU n2i If group i is inactive then from Proposition 1 it must be that χi (αi , αj ) > 0. Notice that χi (αi , αj ) + χj (αi , αj ) = −nA [2nB (1 − p) + p] − nB p < 0. Hence if χi (αi , αj ) > 0 then χj (αi , αj ) < 0, meaning that if i is inactive j must be active. Therefore if i is inactive it holds that EUi (αi ) = 0. Solving χj (αi2 (αj ), αj ) = 0 we define αi2 (αj ) = ni ni −1 h 1−p p + (nj −1)αj +1 nj i . Given that χj (αi , αj ) is strictly increasing with respect to αi it is true that χj (αi , αj ) > 0 for all values αi > αi2 (αj ) and hence αi2 (αj ) is the minimum value of αi that guarantees that members of group j are inactive in equilibrium given any αj . Solving χi (αi3 (αj ), αj ) = 0 we define αi3 (αj ) = nj p+ni [nj +αj p−(αj +1)nj p] . (1−ni )nj p Given that χi (αi , αj ) is strictly decreasing with respect to αi then for all α̇i ∈ [0, αi3 (αj )] it is true that χi (α̇i , αj ) > 0 and hence group i is inactive. Comparing the expressions of αi2 (αj ) and αi3 (αj ) notice that αi2 (αj ) > αi3 (αj ) and therefore we can write the expected utility as follows: 0 if 0 ≤ αi ≤ αi3 (αj ) ˆ i (αA , αB ) if αi3 (αj ) < αi < αi2 (αj ) EUi (αi ) = EU EU ˜ i (αi ) if αi ≥ αi2 (αj ) Notice that EUi (αi ) is a continuous function since ˆ i (αA , αB ) = 0 and limα →α (α ) EU ˆ i (αA , αB ) = EU ˜ i (αi ) limαi →αi3 (αj ) EU i i2 j ˆ i (αA , αB ) is Moreover, the second derivative of EU 2nB (nA −1)2 [nB (p−1)−p]p2 V nA [nB p+nA (2nB (1−p)+p)]2 < 0. Thus, ˆ i (αA , αB ) is a strictly concave function in the unrestricted domain αi ∈ (−∞, +∞) EU and obtains a global maximum at: αi1 (αj ) = nj [ni (1−p)+p][nj (2−2ni (1−p)−3p)−(ni −2)p]−(nj −1)(ni −nj )p2 αj 2(ni −1)nj [nj (p−1)−p]p ˜ i (αi ) is strictly decreasing with respect to αi . Finally, notice that EU 24 If αi1 (αj ) ≤ αi3 (αj ) then for all αi > αi3 (αj ) the expected utility EUi (αi ) is strictly decreasing and hence negative. Therefore, EUi (αi ) takes its maximal value of zero for all values α̇i ∈ [0, αi3 (αj )]. Comparing the analytical expressions of αi1 (αj ) and αi3 (αj ) it is true that αi1 (αj ) ≤ αi3 (αj ) if and only if αj ≥ nj ni (1 − p) + p = α̂j nj − 1 p ˜ i (αi ) is strictly decreasing with respect to If αi1 (αj ) ≥ αi2 (αj ) and given that EU αi then EUi (αi ) has a unique global maximum at αi2 (αj ). Comparing the analytical expressions of αi1 (αj ) and αi2 (αj ) it is true that αi1 (αj ) ≥ αi2 (αj ) if and only if αj ≤ nj [ni (1 − p) + 3p − 2] − 2p = α̃j (nj − 1)p If αi3 (αj ) < αi1 (αj ) < αi2 (αj ) then EUi (αi ) has a unique global maximum at αi1 (αj ). Summing up, the best response of group i can be written as: αi2 (αj ) for αj ≤ α̃j αi (αj ) = αi1 (αj ) for α̃j < αj < α̂j α̇i ∈ [0, αi3 (αj )] for αj ≥ α̂j We now consider the nine possible combinations of the two groups’ best responses: Case 1: Both groups play the interior If i = A, B plays αi1 (αj ), it yields the following sharing rule: ni 2ni nj (ni − 1) + Ai p + Bi p2 αi = (ni − 1) p [2ni nj − p(ni (2nj − 1) − nj )] where Ai = ni nj (9 − 4ni ) − nj (nj + 2) − 2ni and Bi = ni nj (2ni − 7) + nj (nj + 3) + 3ni − 2. From the best responses, this is an equilibrium if and only if α̃i < αi < α̂i for i = A, B, which is true as long as 25 p> nB (nA − nB − 1) = p1 1 + nB (nA − nB − 1) Hence, both groups being active is an equilibrium if and only if p > p1 . Case 2: Both groups play the corner If i = A, B plays αi2 (αj ), there is no solution, as the best responses are two parallel lines with positive slope [(nj − 1)ni ] / [nj (ni − 1)]. Case 3: A plays the interior and B plays the corner If A plays αA1 (αB ) and B plays αB2 (αA ), it yields the following sharing rules: α A = 1 + nA αB = 1 p −1 nB [nA (1−p)+p] (nB −1)p From the best responses, this is an equilibrium if and only if αA 6 α̃A and α̃B < αB < α̂B . As the sharing rules above are such that αB > α̂B , the latter condition is never satisfied. Hence we do not reach an equilibrium. Case 4: A plays the corner and B plays the interior If A plays αA2 (αB ) and B plays αB1 (αA ), it yields the following sharing rules: αA = nA [nB (1−p)+p] (nA −1)p αB = 1 + nB 1 p −1 From the best responses, this is an equilibrium if and only if αB 6 α̃B and α̃A < αA < α̂A . As the sharing rules above are such that αA > α̂A , the latter condition is never satisfied. Hence we do not reach an equilibrium. Case 5: A plays the interior and B plays inactivity If A plays αA1 (αB ) and B plays α̇B ∈ [0, αB3 (αA )], we are at an equilibrium if and only if αA > α̂A and α̃B < αB < α̂B , which is never satisfied. Let αA = αA1 (αB ). Then, the condition αB < αB3 (αA ) reduces to αB < α̃B , hence we reach a contradiction. 26 Case 6: A plays inactivity and B plays the interior If A plays α̇A ∈ [0, αA3 (αB )] and B plays αB1 (αA ), we are at an equilibrium if and only if α̃A < αA < α̂A and αB > α̂B , which is never satisfied. Let αB = αB1 (αA ). Then, the condition αA < αA3 (αB ) reduces to αA < α̃A , hence we reach a contradiction. Case 7: Both groups play inactivity If i = A, B plays α̇i ∈ [0, αi3 (αj )], we immediately reach a contradiction, as αA 6 αA3 (αB ) and αB 6 αB3 (αA ) cannot hold simultaneously. The condition αA 6 αA3 (αB ) is equivalent to αB > nB [nA − (1 − αA )(nA − 1)p] nA (nB − 1)p Then, we have that αB3 (αA ) > nB [nA −(1−αA )(nA −1)p] nA (nB −1)p if and only if p > 2nA nB 2nA nB −nA −nB >1 Hence we reach a contradiction. Case 8: A plays the corner and B plays inactivity If A plays αA2 (αB ) and B plays α̇B ∈ [0, αB3 (αA )], we are at an equilibrium if and only if αA > α̂A and αB 6 α̃B , which holds if and only if αB ∈ [ nB (1−p)+p , α̃B ] and p 6 p1 . p Case 9: A plays inactivity and B plays the corner If A plays α̇A ∈ [0, αA3 (αB )] and B plays αB2 (αA ), we are at an equilibrium if and only if αA 6 α̃A and αB > α̂B , which is never satisfied. Let αB = αB2 (αA ). Then, αB > α̂B reduces to αA > [nA (1 − p) + p] /p. As α̃A < [nA (1 − p) + p] /p, the conditions for an equilibrium are never satisfied. Proof of Proposition 4. Clearly, if the small group is inactive, the GSP cannot occur. If p > p1 , from Proposition 2, the GSP arises if and only if 27 S αA < S nA − nB + 2αB nA (nB − 1) 2nB (nA − 1) Substituting for the equilibrium value of αiS (i = A, B) from Proposition 3, the above condition reduces to (nA − nB ) [2nA nB (1 − p) − p] <0 2(nA − 1)nB p which holds if and only if p > 2nA nB 1+2nA nB = pGSP . Proof of Lemma 1. Taking derivatives of αA1 (αB ) and αB1 (αB ) we directly obtain the result: (nB −1)(nA −nB )p 2(nA −1)nB [nB (1−p)+p] ∂αA1 (αB ) ∂αB = ∂αB1 (αA ) ∂αA A −nB )p = − 2nA(n(nAB−1)(n < 0, hence αA is a strategic substitute for group B. −1)[nA (1−p)+p] > 0, hence αB is a strategic complement for group A. Proof of Proposition 5. The best response of group B can be obtained from the simultaneous case: αB2 (αA ) for αA ≤ α̃A αB (αA ) = αB1 (αA ) for α̃A < αA < α̂A α̇B ∈ [0, αB3 (αA )] for αA ≥ α̂A The expected utility of group A being the leader is given by: 0 if αA ≤ α̃A ˆ A (αA , αB1 (αA )) if α̃A < αA < α̂A EUA (αA ) = EU EU ˜ A (αA ) if αA ≥ α̂A Notice that EUA (αA ) is a continuous function since ˆ A (αA , αB1 (αA )) = 0 and limα →α̂ EU ˆ A (αA , αB1 (αA )) = EU ˜ A (α̂A ) limαA →α̃A EU A A 28 ˆ A (αA , αB1 (αA )) is Moreover, the second derivative of EU [p(nA −1)]2 V 2n2A [nA (p−1)−p] < 0. Thus, ˆ A (αA , αB1 (αA )) is a strictly concave function in the unrestricted domain αA ∈ (−∞, +∞), EU L L and obtains a global maximum at αA , where αA = 1 + nA 1−p is the solution of the firstp order condition ˆ A (αA ,αB1 (αA )) ∂ EU ∂αA ˜ A (αA ) is strictly decreasing with = 0. Given that EU L respect to αA , α̂A strictly dominates any αA > α̂A . As αA > α̃A , it follows that EUA (αA ) L is maximized at min{αA , α̂A } and thus the corresponding expected utility is strictly posL itive. Observe then that αA < α̂A if and only if p > nA (nA −nB −1) 1+nA (nA −nB −1) = p01 . Therefore, • If p 6 p01 , group A selects α̂A and group B selects αB ∈ [0, nB (1−p)+p ], with the p upper bound being the solution of αB = αB3 (α̂A ). L L • If p > p01 , group A selects αA and group B selects αB1 (αA ), given by L αB1 (αA )= (nA −1)(nA −nB )p[nA (1−p)+p]−nA [nB (1−p)+p][nA (2−2nB (1−p)−3p)−(nB −2)p] 2nA (nB −1)[nA (1−p)+p]p F = αB L S Proof of Lemma 2. Using the equilibrium levels of (αA , αA ) we can see that L S αA ≥ αA ⇐⇒ (nA −nB )[nA (1+nA −nB )(p−1)−p] (nA −1)[nA (2nB (p−1)−p)−nB p] ≥0 The numerator of this expression is strictly positive if and only if p > 1+ nA (1+nA1−nB )−1 > 1, whereas the denominator is positive if and only if p > 2nA nB 2nA nB −nA −nB > 1. Therefore, the L S previous expression is strictly positive for any p ∈ [p1 , 1], which guarantees that αA ≥ αA when both groups are active. F S Using the equilibrium levels of (αB , αB ) we can see that F S αB ≤ αB ⇐⇒ (nA −nB )2 [nA (1+nA −nB )(p−1)−p]p 2nA (nB −1)[nA (p−1)−p][nA (2nB (p−1)−p)−nB p] The numerator is strictly negative for any p ∈ [0, 1 + ≤0 1 ], nA (1+nA −nB )−1 which includes the set [0, 1]. The demominator is only negative for values of p larger than one. Therefore, F S the previous expression is negative for any p ∈ [0, 1], which guarantees that αB ≤ αB . 29 Proof of Proposition 6. Clearly, if the small group is inactive, the GSP cannot occur. If p > p01 , from Proposition 2, the GSP arises if and only if L < αA F nA − nB + 2αB nA (nB − 1) 2nB (nA − 1) L F Substituting for the equilibrium value of αA and αB from Proposition 5, the above condition reduces to nA (nA − nB )(p − 1) [nA (2nB (p − 1) − p) − nB p] <0 2(nA − 1)nB p [nA (1 − p) + p] As the denominator is positive, the condition holds if and only if the numerator is negative, which requires p > 1. Hence we reach a contradiction. Proof of Proposition 7. The best response of group A can be obtained from the simultaneous case: αA2 (αB ) for αB ≤ α̃B αA (αB ) = αA1 (αB ) for α̃B < αB < α̂B α̇A ∈ [0, αA3 (αB )] for αB ≥ α̂B The expected utility of group B being the leader is given by: 0 if αB ≤ α̃B ˆ B (αA1 (αB )), αB ) if α̃B < αB < α̂B EUB (αB ) = EU EU ˜ B (αB ) if αB ≥ α̂B Notice that EUB (αB ) is a continuous function since ˆ B (αA1 (αB )), αB ) = 0 and limα →α̂ EU ˆ B (αA1 (αB )), αB ) = EU ˜ B (α̂B ) limαB →α̃B EU B B ˆ B (αA1 (αB )), αB ) is Moreover, the second derivative of EU [p(nB −1)]2 V 2n2B [nB (p−1)−p] < 0. Thus, ˆ B (αA1 (αB )), αB ) is a strictly concave function in the unrestricted domain αB ∈ EU 30 L L (−∞, +∞), and obtains a global maximum at αB , where αB = 1 + nB 1−p is the solution p of the first-order condition ˆ B (αA1 (αB )),αB ) ∂ EU ∂αB ˜ B (αB ) is strictly de= 0. Given that EU L creasing with respect to αB , α̂B strictly dominates any αB > α̂B . Observe that αB < α̂B L always holds, while αB 6 α̃B if and only if p 6 p1 . Therefore, • If p 6 p1 , group B selects αB ∈ [0, α̃B ] and group A selects αA2 (αB ). L L • If p > p1 , group B selects αB and group A selects αA1 (αB ), given by L αA1 (αB )= nB [nA (1−p)+p][nB (2−2nA (1−p)−3p)−(nA −2)p]−(nB −1)(nA −nB )p[nB (1−p)+p] 2(nA −1)nB [nB (p−1)−p]p F = αA S L ) we can see that , αB Proof of Lemma 3. Using the equilibrium levels of (αB L S αB ≤ αB ⇐⇒ (nA −nB )[(nA −nB −1)nB (p−1)+p] (nB −1)[nA (2nB (p−1)−p)−nB p] ≤0 The numerator of the above expression is positive if and only if p ≥ p1 . The denominator is negative if and only if p < 2nA nB , 2nA nB −nA −nB hence it is negative for any p ∈ (0, 1]. Therefore, the above expression is negative for any p ∈ [p1 , 1], which guarantees that L S αB < αB when both groups are active. L S Using the equilibrium levels of (αA , αA ) we can see that F S αA ≤ αA ⇐⇒ (nA −nB )2 p[(nA −nB −1)nB (p−1)+p] 2(nA −1)nB [nB (p−1)−p][nA (2nB (p−1)−p)−nB p] ≥0 The numerator of the above expression is non-negative if and only if p ≥ p1 . The F S denominator is positive for any p ≤ 1, which guarantees that αA < αA when both groups are active (i.e., p ∈ [p1 , 1]). Proof of Proposition 8. Clearly, if the small group is inactive, the GSP cannot occur. If p > p1 , from Proposition 2, the GSP arises if and only if F αA < L nA − nB + 2αB nA (nB − 1) 2nB (nA − 1) 31 F L Substituting for the equilibrium value of αA and αB from Proposition 7, the above condition reduces to (nA − nB )(p − 1) [nA (2nB (p − 1) − p) − nB p] <0 2(nA − 1)p [nB (1 − p) + p] As the denominator is positive, the condition holds if and only if the numerator is negative, which requires p > 1. Hence we reach a contradiction. 32 References Amir, R. and I. 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