Sequential Choice of Sharing Rules in Collective Contests

Transcription

1 Sequential Choice of Sharing Rules in Collective Contests Pau Balart Sabine Flamand Oliver Gürtler Orestis Troumounis February 26, 2017 Abstract Grous cometing for a rize need to determine how to distribute it among their members in case of victory. Considering cometition between two grous of different size, we show that grou sharing rules are strategic comlements for the large grou, while they are strategic substitutes for the small one, hence the timing of choice is crucial. For sufficiently rivate rizes, a switch from a simultaneous choice to the small grou being the leader consists in a Pareto imrovement and reduces aggregate effort. On the contrary, when the large grou is the leader aggregate effort increases. As a result, the equilibrium timing is such that the small grou chooses its sharing rule first. If the rize is not rivate enough, the small grou retires from the cometition and switching from a simultaneous to a sequential timing may reverse the results in terms of aggregate effort. The sequential timing also guarantees that the small grou never outerforms the large one. Keywords: collective rent seeking, sequential, grou size aradox, sharing rules, strategic comlements, strategic substitutes JEL classification: C72, D23, D72, D74 An earlier version of this aer was entitled Rent seeking in sequential grou contests. Deartment of Business Economics, Universitat de les Illes Balears, Carretera de Valldemossa, km 7.5 (Edifici Jovellanos), Palma de Mallorca, Sain, au.balart@uib.cat Deartment of Economics and CREIP, Universitat Rovira i Virgili, Av. de la Universitat 1, Reus, Sain, sabine.flamand@urv.cat University of Cologne, Albertus-Magnus Platz, D Cologne, Germany, hone: , fax: , oliver.guertler@uni-koeln.de Deartment of Economics, Lancaster University, Bailrigg, Lancaster, LA1 4YX, United Kingdom, o.troumounis@lancaster.ac.uk 1

2 1 Introduction Contests among organizations and grous of individuals are widesread. Examles include research and develoment races, re-electoral camaigns, rocurement contests, or even sort and art contests. In all the above, grous erformance deends on individual contributions of their members, which imlies that grous need to coordinate and establish some rules regarding their internal organization. As the literature on collective contests has ointed out, a key element of grous organization is related to the allocation of the rize among the winning grou members. 1 While contests among grous are generally thought as simultaneous, the timing in which contenders organize and imlement their internal rules need not necessarily be so. Indeed, there is no a riori reason to believe that before the actual cometition takes lace, the timing in which the involved organizations decide uon their governing rules coincides. Differences in the size or in the informational advantage of organizations, as well as the existence of some established organizations challenged by an entrant, may result in the sequential determination of their governing rules. Consequently, grous involved in a simultaneous cometition may well be deciding uon their internal rules in a sequential fashion. Similar to revious literature, we find that if the order of moves is determined endogenously, the equilibrium timing of the game is a sequential one, and this very fact constitutes a strong justification for dearting from the simultaneity assumtion (Baik and Shogren, 1992; Leininger, 1993; Morgan, 2003). In this aer, we contribute to the literature on collective contests by showing how different timings with resect to grous choice of sharing rules alter individual incentives and thus grou erformance. In our two grou model, most of our results arise from the fact that sharing rules are strategic substitutes for the small grou while they are strategic comlements for the large one. When both grous are active, the large grou acting as the leader behaves in a similar manner as in a standard Stackelberg duooly model. That is, the large grou commits to a more meritocratic sharing rule than in the simultaneous game (hence effort increases), while the small grou (whose sharing rule is a strategic substitute) selects a less meritocratic rule than in the simultaneous game (hence effort decreases). On the contrary, the small grou acting as the leader follows a very different strategy. It commits to a less meritocratic rule than in the simultaneous game, thereby weakening cometition between grous. In turn, the large grou (whose sharing rule is a 1 Starting with Nitzan (1991), the literature has considered both exogenous and endogenous sharing rules. For a recent survey on rize-sharing rules in collective rent seeking, see Flamand and Troumounis (2015). For surveys of the literature on individual contests in general, see among others Corchón (2007), Konrad (2009), Long (2013), Corchón and Serena (2016) and also Dechenaux et al. (2015) for the related exerimental evidence. On collective contests, see the recent survey by Kolmar (2013), and also Sheremeta (2015) for a recent review of exerimental evidence. 2

3 strategic comlement) also resonds with a less meritocratic rule than in the simultaneous game. Interestingly, and in contrast to the simultaneous timing (Balart et al., 2016), the small grou never outerforms the large one in terms of winning robabilities. In other words, Olson (1965) s celebrated grou size aradox (GSP) vanishes regardless of the leader s size. The exact nature of the rize is also key to grous erformance. Indeed, we show that when the rize is not rivate enough and the sharing rules allow for transfers, the sequential choice of sharing rules leads to monoolization, a situation in which one grou retires from the cometition (Ueda, 2002). Interestingly, the large grou takes advantage of its leadershi by reventing the small grou from being active for a greater range of rivateness of the rize comared to the simultaneous case (Balart et al., 2016), thereby making monoolization more likely. In contrast, the small grou is unable to take advantage of the timing structure when it is the leader, so that monoolization occurs in the same instances as in the simultaneous game. Our results clearly have imlications regarding aggregate effort exenditures. When the large grou is the leader, and rovided that the rize is rivate enough so that both grous are active, total rent-seeking exenditures increase with resect to the simultaneous case. Conversely, when the small grou is the leader, and again rovided that both grous are active, aggregate effort is lower comared to the simultaneous case. Situations are conceivable in which a designer has an imact on certain arameters of the contest like the order of moves. Thus, if from the designer s oint of view effort is valuable, he should ossibly oblige the two grous to choose their sharing rules simultaneously, or even force the large grou to move first. If effort is instead considered as wasteful, the designer should ot for the design where the small grou is the leader. In this latter case, in fact, the designer should not intervene in the game since the unique equilibrium of the game with an endogenous choice of the timing structure gives the leadershi to the small grou whenever both grous are active. Notice that in contrast to what haens when both grous are active, when the small grou is inactive switching from a simultaneous to a sequential timing may reverse the results in terms of aggregate effort. Thus, the degree of rivateness of the rize is a critical feature that should be taken into account to understand the imlications of different timing arrangements. Litigation is one examle that our model can be alied to. Since arties involved in a legal battle send irretrievable resources to revail in court, litigation has often been modeled as a rent-seeking contest (Baye et al., 2005; Farmer and Pecorino, 1999; Gürtler and Kräkel, 2010). Many law firms make use of incentive ay, conditioning lawyers comensation on their individual erformance (for examle, through bonus ayments), which is tyically measured by the number of billable hours. The structure of incentive 3

4 ay differs among firms, some firms require lawyers to bill at least 1,600 hours a year, others demand much higher numbers. The size of bonuses also differs among firms. In the US, many law firms rovide ublicly accessible information about their comensation ractices in the NALP Directory of Legal Emloyers. 2 Among other things, these firms ublicly state whether they ay bonuses to eligible lawyers, what factors a ossible bonus ayment is based on, and whether a minimum exists for billable hours. Hence, law firms, by simly checking the NALP Directory of Legal Emloyers, have a very good idea about comensation ractices at their cometitors, meaning that incentive schemes are ublicly known. Finally, not all law firms rovide the NALP Directory of Legal Emloyers with information about their comensation ractices at the same time. Accordingly, sequential determination of comensation rules is conceivable if a firm gathers information about comensation ractices at cometing firms, and then chooses and discloses its own comensation ractice. To have a concrete examle at hand, suose that there is a legal battle, and the involved arties are reresented by Baker & McKenzie and Shearman & Sterling, resectively, two law firms that are located in New York City. Baker & McKenzie reorts in the NALP Directory of Legal Emloyers that both base salaries and bonuses deend on the number of billable hours and that lawyers are exected to bill at least 2,000 hours a year. In contrast, Shearman & Sterling reorts that base salaries deend only on seniority and emhasizes that a minimum billable hour requirement does not exist. This means that at Baker & McKenzie firm rofits (that tyically deend on how successful the firm is doing in court) are more likely to be shared according to individual erformance, whereas at Shearman & Sterling individual erformance is relatively less imortant. In other words, the choice of different comensation contracts can be understood as a choice of different sharing rules. 2 The Model There are two grous i = A, B with n i N members and without loss of generality A is the large grou (i.e., n A > n B > 1). 3 Each member k = 1, 2,..., n i of grou i chooses his individual level of effort e ki 0 whose cost is linear. The valuation of the rize is the same for all individuals, and is denoted by V. 4 The robability that grou i wins the 2 Comensation ractices for this examle can be found at htt:// In a similar sirit, disclosure is often a feature of executive ay of CEOs and board of directors. In the UK, for instance, regulations that involve disclosure of remuneration were imlemented in 2013 (Guta et al., 2016). 3 If n A = n B the timing of choices of grou sharing rules is irrelevant (see footnote 14). 4 The assumtion of symmetric valuations is the standard one in the literature on endogenous sharing rules, mainly for tractability reasons. In order to analyze the choice of selective incentives in the resence 4

5 between-grou cometition is given by P i = { 1 for E 2 A = E B = 0 otherwise E i E A +E B where E i = n i k=1 e ki is the total effort of grou i. 5 Individuals are risk neutral, and the exected utility of member k = 1,..., n i of grou i is given by } Ei { [ ] e α ki i E i + (1 α i ) 1 n i + (1 ) E A +E B V e ki for E i > 0, E j 0 ( ) EU ki = 1 1 n i V for E i = E j = 0 0 for E i = 0, E j > 0 for i = A, B and i j. 6 The arameter (0, 1] denotes the degree of rivateness of the rize ( = 0 would corresond to the case of a ure ublic good while = 1 corresonds to the case of a ure rivate good). 7 Indeed, an imortant feature of collective contests is that the rize sought by cometing grous may often be interreted as a mixture between a ublic and a rivate good. Local governments cometing for funds tyically devote them to the rovision of both monetary transfers and local ublic goods. Similarly, rizes in research and develoment races involve both reutational and monetary benefits for the winning team. More generally, any rize sought by cometing grous may be interreted as a mixture between a ublic and a rivate good insofar as the winners derive some benefits in terms of status, reutation, or satisfaction following a victory. The sharing rule α i 0 reresents how tied the allocation of the rivate art of the rize is on individual contributions (i.e., meritocracy) as oosed to egalitarianism in grou i. Previous literature on sharing rules in collective rent seeking has considered both the cases of α i [0, 1] (Baik, 1994; Lee, 1995; Noh, 1999; Ueda, 2002) and α i [0, ) (Baik and Shogren, 1995; Baik and Lee, 1997, 2001; Lee and Kang, 1998; Balart et al., 2016). If α i > 1, the sharing rule of grou i allows for transfers among its members, as in of heterogeneity within grous, Nitzan and Ueda (2016) assume that the contested rize is a ublic good, and focus on cost sharing rules rather than rize sharing rules, which allows them to tackle the roblem analytically. They show that intra-grou heterogeneity revents the effective use of selective incentive mechanisms. 5 This tye of contest success function is widely used in the literature and has been axiomatized by Skaerdas (1996). Alternatively, many authors have used all-ay auctions to model cometition, which have been analyzed by Baye et al. (1996). 6 We assume that the cost of effort is linear for tractability. An imortant insight due to Esteban and Ray (2001) is that if the costs of effort are sufficiently convex, members of larger grous benefit from a cost advantage that reverse the GSP. In the resent analysis we wish to focus on the combined effects of the degree of rivateness of the rize and of the timing of choices. As we find that a sequential choice of sharing rules eliminates the GSP for any degree of rivateness of the rize, we conjecture that assuming convex effort costs would reinforce the result. 7 Notice that we exclude the ossibility of a ure ublic good, in which case sharing rules are irrelevant. For the analysis of these cases refer to the multile equilibria results in Katz et al. (1990). (1) 5

6 Hillman and Riley (1989). 8 In such case, grou i collects (1 α i ) E i n i (E A +E B V from each ) E of its members and allocates α i i V according to their relative contributions.9 (E A +E B ) We consider a multi-stage game where sharing rules are chosen rior to the choice of individual efforts. Both grous choose their sharing rule by maximizing their aggregate welfare (i.e., max k i EU ki). 10 strategies. The equilibrium concet is subgame erfection in ure 2.1 Effort Stage and Simultaneous Sharing Rules The simultaneous game consists in two stages: In stage one each grou i = A, B chooses its sharing rule α S i, while in stage two individuals choose their level of effort simultaneously and indeendently. At the effort stage, and given the chosen sharing rules, grous members choose their level of effort by maximizing (1) subject to efforts being non-negative. The equilibrium of this stage is solved in Balart et al. (2016) who generalize the results of Ueda (2002) and Davis and Reilly (1999) by allowing for a mixed ublic-rivate rize, thereby leading to the ossibility of monoolization. whenever More recisely, grou i retires from the contest α i n i [α j (n j 1) n j (1 )] n j (n i 1)n j with i = A, B and j i. If (2) holds, grou i is inactive (i.e., e ki = ẽ i = 0 for all k i) while the members of grou j comete in a standard n j -layers Tullock contest for a rize of valuation α j V (i.e., the rivate art of the rize that is allocated according to relative effort) and thus exert effort e kj = ẽ j = α j(n j 1) V k. (3) n 2 j 8 Cost-sharing in collective contests for urely ublic rizes can also be interreted in terms of withingrou transfers (Nitzan and Ueda, 2016; Vazquez, 2014). 9 When the sharing rules do not allow for transfers among individuals (i.e., α i [0, 1] for i = A, B), the equilibrium is identical regardless of the articular timing of the game. This occurs because the constraints on the sharing rules are binding irresective of the timing of the interaction, and thus allowing one grou to choose its sharing rule first does not alter the strength of cometition comared to the simultaneous case (formal results are available uon request). 10 As the aggregate welfare of grou i only deends on aggregate effort, and as in equilibrium aggregate effort is unique for any α i, it follows that the sharing rule α i that maximizes the exected utility of the reresentative individual in a within-grou symmetric equilibrium also maximizes k i EU ki. The sharing rules may thus be chosen by a benevolent leader maximizing the welfare of the grou, which can be justified by the fact that such leader cannot do well without the suort of the grou members, that is, without taking care of their welfare at least artially (Nitzan and Ueda, 2011). (2) 6

7 Conversely, if (2) is violated for i = A, B, then both grous are active and individual effort in the symmetric equilibrium is given by e ki = ê i = {ρ j[ρ i + α i (1 ρ i )] + (1 ρ j )α j ρ i }{n j (1 α i ) + n i [n j (1 (1 α i + α j )) + α j ]} n j [n i (ρ i + ρ j )] 2 (4) where ρ i = n i + (1 ). 11 As can be seen from (2), the less (more) meritocratic the sharing rule of grou i (j), the more likely that monoolization occurs, so that grou i is inactive. Let us define α i3 (α j ) as the value of α i that satisfies (2) with equality, reresenting the maximum value of α i that guarantees that grou i s members are inactive [ given any α j ]. By switching the subscrits in (2), we can also define α i2 (α j ) = n i 1 n i + (n j 1)α j +1 1 n j as the minimum value of α i guaranteeing that grou j s members are inactive given any α j. As α i2 (α j ) > α i3 (α j ) we can write grou i s exected utility as follows: 0 if 0 α i α i3 (α j ) EU i (α i ) = EU ˆ i (α A, α B ) if α i3 (α j ) < α i < α i2 (α j ) EU i (α i ) if α i α i2 (α j ) where EU ˆ i (α A, α B ) denotes individual ayoff in grou i when both grous are active, and EU i (α A, α B ) denotes individual ayoff in grou i when only this grou is active. 12 One can now obtain the best resonse corresondence for the simultaneous choice of sharing rules rior to the effort stage: α i2 (α j ) for α j α j α i (α j ) = α i1 (α j ) for α j < α j < ˆα j (5) α i [0, α i3 (α j )] for α j ˆα j where ˆα j = n j n i (1 )+ n j, α 1 j = n j[n i (1 )+3 2] 2 (n j, and α 1) i1 (α j ) corresonds to the first order condition from maximizing individual ayoff when both grous are active (i.e., EU ˆ i (α A, α B )). 13 Intuitively, the best resonse suggests that if grou j selects a very meritocratic rule (i.e., α j ˆα j ) then it is not worth being active for grou i, so that it selects any α i [0, α i3 (α j )] guaranteeing its inactivity. Conversely, if grou j selects a very egalitarian rule (i.e., α j α j ), grou i maximizes its ayoff by reventing grou 11 The symmetric equilibrium resented is unique if α i > 0 for both i = A, B. In case of urely egalitarian sharing rules, our setu is equivalent to the case of a ure ublic rize and multile equilibria may arise (see Balart et al for a detailed characterization). 12 The exact exressions for exected utilities are EU i (α i ) = V (ni 1)(ni+αi) V and ˆ EU i (α A, α B ) = 1+(αi 1)(1 ρi) αj(1 ρj) n j[n i(ρ i+ρ j)] 2 V [ n i (2n i 1)n j C + D(n i 1) 2], where ρ i = n i + (1 ), C = n i (1 α j n i )+n j (1 α i )+n i (4n i +α i +α j 5)n j and D = α i (n j 1)+α j (n j 1)+(n i 1)(2n j 1). 13 The analytical exression of α i1 (α j ) aears in the aendix. n 2 i 7

8 j from being active in the cheaest way (i.e., by choosing α i2 (α j )). For intermediate values of α j, grou i selects α i1 (α j ) and thus both grous are active. " &$ %" # ' 45$0/0("0$6&& * 0,+)"82/&7" # + " * & ' " #$ %" & ' " #$ %" & ' 9 " &, %" # ' " #( %" & ' ) 9 " &, %" # ' " &$ %" # ' " #( %" & ' " & ) "& 45$0/0("0$6&%#0,+& * 7" " * # + & '!"#$%&'&()*+&")*%#,*) &!"#$%&-&()*+&")*%#,*).#,#%#/012+0#,&")30#, Figure 1: Simultaneous choice of sharing rules for = 0.64 (left) and = 0.72 (right) and n A = 4, n B = 2. Given the above best resonses, Balart et al. (2016) show that when grous choose their sharing rules simultaneously, there exists a threshold 1 = n B(n A n B 1) 1+n B (n A n B 1) determining the occurrence of monoolization: If > 1, both grous are active and the sharing rule for i = A, B in the unique!! subgame erfect equilibrium is given by α S i = n i 2n i n j (n i 1) + C i + D i 2 (n i 1) [2n i n j (n i (2n j 1) n j )] where C i = n i n j (9 4n i ) n j (n j + 2) 2n i and D i = n i n j (2n i 7) + n j (n j + 3) + 3n i 2. If 0 < 1, only grou A is active and the continuum of sharing rules in the subgame erfect equilibrium is given by [ ] αa S = n A 1 n A + (n B 1)α B +1 1 n B αb S [ n B(1 )+, n B[n A (1 )+3 2] 2 (n B ] 1) Figure 1 summarizes the best resonses, the equilibrium sharing rules and the otential occurrence of monoolization. Observe that in both anels, the sloe of a grou s best resonse changes deending on the value of the other grou s sharing rule (as summarized in (5)). Notice also that the occurrence of monoolization deends on. In 8

9 the right anel where > 1, the interior best resonses α i1 (α j ) intersect outside of the monoolization region, so that in the deicted equilibrium both grous are active. In the left anel where 1, the interior best resonses α i1 (α j ) intersect within the monoolization region, yielding multile equilibria where only the large grou is active. 3 Results Our first result is essential to understand the imlications of a sequential choice of sharing rules: Proosition 1. If both grous are active, the sharing rules are strategic comlements from the ersective of the large grou, whereas they are strategic substitutes from the ersective of the small grou. The interior best resonse α i1 (α j ) is linear and has a ositive (negative) sloe for the large (small) grou. In other words, the sharing rules are strategic comlements (substitutes) for the large (small) grou irresective of the secific level at which we evaluate the best resonse. This, in turn, will determine how each grou chooses its sharing rule when given the oortunity to move first. More secifically, each grou being the leader recommits to a sharing rule such that the other grou (i.e., the follower) is induced to decrease its level of meritocracy comared to the simultaneous game. That is, the large (small) grou recommits to higher (lower) levels of meritocracy, thereby increasing (reducing) the strength of cometition. This result has some similitudes as well as some interesting differences with resect to the case of recommitment in effort levels by two layers studied by Dixit (1987). He finds that cross derivatives of equilibrium efforts are zero whenever the layers are symmetric and thus the timing of the game does not alter the recommitted effort levels. This is also the case regarding the choice of sharing rules rovided the two grous have the same size (i.e., n A = n B = n). 14 Further, Dixit (1987) shows that in a two-layer asymmetric contest efforts are strategic comlements from the ersective of the advantaged layer, while they are strategic substitutes from the ersective of the underdog. Similarly, in our context the sharing rules are strategic substitutes from the ersective of the small grou and strategic comlements from the ersective of the large grou. The large grou has an advantage in the sense that it always wins with higher robability conditional on the members of the two grous exerting the same level of individual effort. However, there is an imortant difference with resect to the strategic imlications found by Dixit 14 If the two grous have the same size (i.e., n A = n B = n), the equilibrium sharing rules are given by α A = α B = 1 + n 1 regardless of the order of moves, as in this case best resonses are indeendent of the other grou s sharing rule. 9

10 (1987). In his case, marginal returns to effort are increasing (decreasing) in the other layer s effort if it is small (large) enough. This is not the case with sharing rules, for which strategic behaviors only deend on grou size. 3.1 Sequential Timing In the following subsections where the game is sequential, we consider the following three stages: In stage one the leader (grou i {A, B}) chooses its sharing rule αi L. In stage two the follower (grou j i) chooses its sharing rule αj F. In stage three the members of the two grous simultaneously and individually choose their effort levels (e L i, e F j ). In what follows, we resent the equilibrium choices of sharing rules (αi L, αj F ) in our formal results. Effort levels (e L i, e F j ) chosen in stage three can be derived directly from exressions (3) and (4) Large grou A is the leader Proosition 2. Let 1 = n A(n A n B 1) 1+n A (n A n B 1) and grou A be the leader: If > 1, both grous are active and the sharing rules in the unique subgame erfect equilibrium are given by α L A = ρ An A α F B = n Bρ B [n A (2n B ρ B 2+)+(n B 2)] n A (1 ρ A )(n A n B )ρ A 2n A n B (1 ρ B )ρ A If 0 < 1, only grou A is active and the continuum of sharing rules in the subgame erfect equilibrium is given by α L A = ρ Bn B 1 ρ A α F B [0, ρ Bn B ] When the large grou is the leader and both grous are active, we are in a situation that is very similar to a standard Stackelberg duooly model. In the left anel of Figure 3, we dislay the equilibrium sharing rules when the large grou is the leader for = 1, and comare them with the ones arising under a simultaneous timing: Once the large grou commits to a more meritocratic rule than in the simultaneous case (αa L > αs A ), the follower reacts by selecting a less meritocratic rule than in the simultaneous case (αb F < αs B ). Indeed, recall that the sharing rules are strategic substitutes from the ersective of the small grou (Proosition 1). Furthermore, having the leadershi advantage allows the large grou to exclude the small grou from the contest for a larger range of than in the 10

11 simultaneous case (i.e., 1 > 1 ). In other words, the large grou acting as the leader is caable of tying the small grou s hands and oblige the latter to react with a sharing rule leading to its own inactivity, and this for a larger range of. In turn, the small grou is indifferent between choosing any of the sharing rules belonging to the interval such that it retires from the contest. Observe that being the first mover enables the large grou to select its referred equilibrium (i.e., the lowest α A ) out of the continuum arising in the simultaneous case. In other words, monoolization is achieved in the cheaest way for the large grou. The following result will be useful in order to understand the imlications of the sequential interaction: Proosition 3. When the large grou A is the leader: If > 1 then aggregate effort is strictly greater than in the simultaneous case. If 1 then aggregate effort is smaller than in the simultaneous case. If is such that both grous are active (i.e., > 1), we saw that the large grou increases the level of meritocracy of its sharing rule comared to the simultaneous case, which induces the small grou to do the oosite. Given that the large grou chooses to strengthen cometition between grous, these equilibrium sharing rules are such that aggregate rent-seeking exenditures are strictly greater than when the grous select their sharing rules simultaneously. If is such that monoolization occurs in both the simultaneous and sequential cases (i.e., 1 ), however, the above results are reversed. That is, the large grou reduces the level of meritocracy of its sharing rule with resect to the simultaneous case, so that aggregate effort is smaller. Finally, recall that when 1 < < 1 both grous are active in the simultaneous case while only the large one is active when being the leader. Yet, the total effort that the large grou s members exert in order to exclude the small grou from the cometition exceeds the sum of both grous efforts in the simultaneous case. Proosition 4. If the large grou is the leader and for all values of including full rivateness, the GSP never arises. The GSP refers to a situation where the small grou outerforms the large grou in terms of winning robabilities (Olson, 1965). When the large grou chooses its sharing rule first, and contrary to the simultaneous case, the GSP never takes lace regardless of the degree of rivateness of the rize. As the large grou acting as the leader selects a more meritocratic sharing rule than in the simultaneous case, and as the small grou reacts with a less meritocratic one, it follows that aggregate effort increases in the large 11

12 grou, while it decreases in the small grou. As it turns out, these effort variations are large enough to eliminate the GSP Small grou B is the leader Our results are significantly altered by giving the leadershi to the small grou, which again is a direct consequence of Proosition 1: Proosition 5. Let grou B be the leader: If > 1, both grous are active and the sharing rules in the unique subgame erfect equilibrium are given by α F A = n Aρ A [n B (2n A ρ A 2+)+(n A 2)] n B (1 ρ B )(n A n B )ρ B 2n A n B (1 ρ A )ρ B α L B = ρ Bn B If 0 < 1, only grou A is active and the continuum of sharing rules in the subgame erfect equilibrium is given by αa F = ρ B+(1 ρ B )α L B 1 ρ A ] αb [0, L n Aρ A 2ρ B 1 ρ B For all degrees of rivateness of the rize such that both grous are active, and in stark contrast to the case in which the large grou is the leader, it turns out that when the small grou has the leadershi advantage it strategically chooses to weaken cometition by selecting a less meritocratic rule than in the simultaneous game (αb L < αs B ). In turn, the large grou also reacts with a less meritocratic rule than the one it selects when the game is simultaneous (αa F < αs A ). Indeed, recall that the sharing rules are strategic comlements from the ersective of the large grou (Proosition 1). In the right anel of Figure 3, we dislay the equilibrium sharing rules when the small grou is the leader for = 1, and comare them with the ones arising under a simultaneous timing. For sufficiently low degrees of rivateness ( 1 ), the small grou is not able to take advantage of its leadershi. Regardless of the choice of the small grou in stage one, in stage two the large grou selects a highly meritocratic sharing rule so that the small grou retires from the contest in stage three. Observe that the threshold level of rivateness that determines whether the small grou is active or inactive is identical to the one obtained in the simultaneous case, that is, the small grou cannot take advantage of its leadershi for any 1. Proosition 6. When the small grou B is the leader: 12

13 If > 1 then aggregate effort is strictly smaller than in the simultaneous case. If 1 then aggregate effort is strictly smaller than in the simultaneous case for αb L < n B(1 )+, while it can be either greater or smaller than in the simultaneous case for αb L n B(1 )+. When is such that both grous are active, both grous adot relatively more egalitarian rules than in the simultaneous game, their members exert a lower level of effort, hence aggregate effort is smaller. As when the large grou is the leader, when is such that the small grou is inactive in both the simultaneous and sequential cases (i.e., 1 ), the above results may be reversed. More secifically, deending on which articular sharing rule is chosen by the small grou, aggregate effort may be greater in the simultaneous than in the sequential case. Proosition 7. If the small grou is the leader and for all values of including full rivateness, the GSP never arises. Observe that here, the GSP does not arise for a very different reason than when the large grou is the leader. The small grou selects a sharing rule such that its members are inactive when the rize is ublic enough (but not urely ublic), which clearly revents the occurrence of the GSP as in the revious cases. Then, if > 1 the small grou acting as the leader strategically chooses to weaken cometition by selecting a less meritocratic rule than in the simultaneous game, which in turn induces the large grou to adot a similar behavior. However, the reduction in meritocracy adoted by the small grou is relatively larger than the corresonding reduction in the large grou, and is such that the GSP vanishes. Figure 2 summarizes our findings regarding the sequential choice of sharing rules and the occurrence of monoolization and the GSP, while in Figure 3 we resent and comare the equilibrium choice of sharing rules under the three timing arrangements (for n A = 4, n B = 2 and = 1). Again, observe that the sloe of the best resonse is ositive for the large grou and negative for the small one. The simultaneous case is reresented by the intersection of the two best resonses. The sequential equilibria are determined by the tangency oint between the isorofit of the first mover and the best resonse of the follower. When the large grou is the leader, its equilibrium sharing rule is higher than in the simultaneous case, and the one of the small grou is lower. That is, the large grou chooses to strengthen cometition, thereby inducing the small grou to reduce the meritocracy of its sharing rule. As a result, the large grou moves to a higher level isorofit curve, while the oosite is true for the small grou. When the small grou is the leader, equilibrium takes lace at lower levels of meritocracy. When allowed to move 13

14 ;!:% -% %.% " %&/#0,1/'9,(% $#+;)%&#(%$)#*)+% &/#0,1/'0,(% %,% -%!"# :-<% %.% " %&/#0,1/'9,(% &/#0,1/'0,(% %!"#$$%&'(%$)#*)+%,% -%.-% % :% " %&/#0,1/'9,(% &/#0,1/'0,(% % Figure 2: Simultaneous versus sequential choice of sharing rules Figure 3: Simultaneous versus sequential choice of sharing rules with = 1 (for n A = 4, n B = 2). first, the small grou chooses to weaken cometition, which induces the large grou to adot a similar behaviour. As a result, both grous enjoy a greater ayoff than in the simultaneous case. 14

15 3.2 Further Comarisons and Endogenous Timing We have analyzed the relationshi between the size of a grou and its erformance in terms of its robability of winning the rize. However, another relevant notion of grou effectiveness relates its size to er-caita ayoffs. Even though the large grou always outerforms (or ties) the small one in terms of winning robability when the order of moves is sequential, the members of the small grou achieve strictly higher exected utility than the members of the large grou when the rize is sufficiently rivate. This is also true when the choice of sharing rules is simultaneous, from a degree of rivateness strictly smaller than the one required for the GSP to arise. 15 In fact, irresective of the order of moves, when the rize is sufficiently rivate so that both grous are active, individual utility is decreasing in rivateness in the large grou, while it is increasing in the small grou. With this alternative notion of grou effectiveness, therefore, the small grou s members always outerform the members of the large grou in a cometition over a ure rivate good regardless of the timing of the interaction. As a larger grou size reduces the er-caita value of the rivate comonent of the rize, the large grou has some intrinsic disadvantage in terms of ayoffs. Although the size of the leader does not matter in terms of the GSP, it does have some imlications regarding the exected utility of grous members and aggregate effort. Consider the case where both grous are active for any articular timing. When the large grou is the leader, all its members clearly have higher exected utility than in the simultaneous version of the game, while all the members of the small grou (the follower) are worse off. Therefore, a transition from simultaneous to sequential cometition where the large grou is the leader never consists in a Pareto imrovement, and increases aggregate effort. Conversely, when the small grou is the leader, the members of both grous exert less effort than in the simultaneous game, and they achieve higher exected utility. Thus, switching from simultaneous to sequential cometition where the small grou is the leader consists in a Pareto imrovement and reduces aggregate effort. Given these imlications, a natural question on which our model can shed light is the following: If grous were to also choose the timing of the game, which sequence would revail? To answer this question, let us assume that rior to the choice of sharing rules, grous are able to declare their intention to be the leader or the follower of the game. Suose the two grous have the ossibility of choosing between two dates to declare their sharing rules. If they choose the same date, sharing rules are chosen simultaneously. 15 The exact value of the threshold is given by = 1 [ 1 + n i [ ( na + n B ) 2 1 ]] 1 when grou i is the leader (i = A, B). Although the corresonding threshold for the simultaneous case cannot be derived analytically, one can show that it is unique given the strict monotonicity of exected utility with resect to rivateness in both grous. 15

16 If they choose different dates, sharing rules are chosen sequentially. summarized in the following roosition: The results are Proosition 8. If > 1 the unique equilibrium of the timing game is such that the small grou is the leader. If 1 the ayoff-dominant equilibrium of the timing game is such that the small grou is the leader. Suose that the rize is rivate enough so that both grous are active in the simultaneous setu. If the small grou selects its sharing rule at date one, the large grou refers to wait, whereas if the small grou selects its sharing rule at date two, the large grou takes the lead. Hence the large grou always refers the sequential version of the game. On the contrary, if the large grou selects its sharing rule at date one, so does the small grou, whereas if the large grou moves at date two, the small grou takes the lead. Therefore, the unique equilibrium of the game with an endogenous choice of the timing structure gives the leadershi to the small grou. This is also the case under monoolization, rovided that we select the ayoff-dominant equilibrium among all the ossible equilibria. 16 Further, with an endogenous timing selection revious to the rent-seeking activity, the GSP should never take lace. Again, the fact that the sequential game arises endogenously constitutes a strong justification for dearting from the simultaneity assumtion. This result is in line with the one in contests among individuals where the sequential timing also arises endogenously (Baik and Shogren, 1992; Leininger, 1993; Morgan, 2003). 17 In the resence of asymmetric layers, the higher valuation layer whose effort is a strategic comlement gives the leadershi to the underdog whose effort is a strategic substitute (Baik and Shogren, 1992; Leininger, 1993). We rovide an analogous result in the context of grou sharing rules: The large grou whose sharing rule is a strategic comlement gives the leadershi to the small grou whose sharing rule is a strategic substitute. 16 Given the multilicity of equilibria when 1, any timing can arise endogenously as an equilibrium deending on the articular sharing rule that grou B selects out of the continuum. A reasonable selection criterion which allows a comarison across the three versions of the game is ayoff dominance. Given that the small grou B is inactive regardless of the articular timing (and thus obtains zero rofits), ayoff dominance simly requires that monoolization occurs with the small grou selecting the smallest α B out of the continuum. Further, we also use ayoff dominance to select among the multile Nash equilibria of the timing game. 17 The industrial organization literature has also extensively considered endogenous timing. See for instance Amir and Grilo (1999), Deneckere and Kovenock (1992), Hamilton and Slutsky (1990) and Van Damme and Hurkens (1999) in the context of a duooly model. 16

17 4 Discussion Collective contests have drawn a lot of attention given their wide range of alications. In this aer, we contribute to this literature by studying the case in which grous are allowed to determine their sharing rules sequentially. We believe such an analysis is relevant for several reasons. First, we show that the sequential choice of sharing rules emerges endogenously when the grous are allowed to decide uon the timing of the game. Second, we rovide real life examles in which the comensations of organizations members are ublic information and are chosen in a sequential fashion. Thus, our framework fits well with comensation schemes in litigation firms or with the debate on the regulation of executive comensation disclosure. Third, we show that the timing according to which organizations of different sizes choose their sharing rules has several consequences of interest from the ersective of a contest designer. Desite being highly stylized, our model rovides interesting insights regarding the interaction between grous size, the nature of the rize and the timing of choices. The relative size of the first mover is key to our results. The large grou is more aggressive when allowed to move first than in the simultaneous contest, while the small grou acting as the leader follows the oosite strategy. This is a direct consequence of the fact that grou sharing rules are strategic comlements from the ersective of the large grou, while they are strategic substitutes from the ersective of the small grou. The ublic-rivate nature of the rize also lays an imortant role. When the large grou moves first and organizations comete for a rize that is close to the definition of a ure rivate good, aggregate effort is greater than in the simultaneous case. However, the oosite is true when the ublic comonent of the contested-rize is sufficiently high, or when the small grou moves first and the contested rize is sufficiently rivate. We also rovide interesting insights regarding the GSP and monoolization, two situations that can arise when the choice of sharing rules is simultaneous. First, the GSP never arises under a sequential timing regardless of the leader s size. When allowed to move first, the large grou increases the degree of meritocracy of its sharing rule, thereby increasing the effort of its members and reventing the small grou to outerform. On the contrary, the small grou acting as the leader does exactly the oosite, which reduces its members effort thus making it easier for the large grou to outerform. Regarding the occurrence of monoolization, the large grou takes advantage of its leadershi by excluding the small grou from the cometition for a larger set of arameters than in the simultaneous case. Yet, this is not the case for the small grou, as having a first mover advantage does not allow its members to reduce the set of arameters for which monoolization occurs. 17

18 5 Aendix Proof of Proosition 1. The best resonse for grou i s sharing rule when both grous are active arises from maximizing EU ˆ A (α A, α B ) and is equal to α i1 (α j ) = n j[n i (1 )+][n j (2 2n i (1 ) 3) (n i 2)] (n j 1)(n i n j ) 2 α j 2(n i 1)n j [n j ( 1) ] Taking derivatives of α A1 (α B ) and α B1 (α B ) we directly obtain the result: α A1 (α B ) α B = (n B 1)(n A n B ) > 0, hence α 2(n A 1)n B [n B (1 )+] B is a strategic comlement for grou A. α B1 (α A ) α A = (n A 1)(n A n B ) < 0, hence α 2n A (n B 1)[n A (1 )+] A is a strategic substitute for grou B. Proof of Proosition 2. Maximizing the exected utility of the reresentative individual of grou i in the within-grou symmetric equilibrium also maximizes the aggregate welfare of grou i. Therefore, in this roof and the subsequent ones, we focus on the sharing rule α i that maximizes the exected utility of the reresentative individual in grou i, which we denote by EU i (α A, α B ). The best resonse of grou B as resented in (5) is: α B2 (α A ) α B (α A ) = α B1 (α A ) α B [0, α B3 (α A )] for α A α A for α A < α A < ˆα A for α A ˆα A The exected utility of grou A being the leader is given by: 0 if α A α A EU A (α A ) = EU ˆ A (α A, α B1 (α A )) if α A < α A < ˆα A EU A (α A ) if α A ˆα A Notice that EU A (α A ) is a continuous function since lim αa α A ˆ EU A (α A, α B1 (α A )) = 0 and lim αa ˆα A ˆ EU A (α A, α B1 (α A )) = EU A (ˆα A ) Moreover, the second derivative of EU ˆ [(n A (α A, α B1 (α A )) is A 1)] 2 V < 0. Thus, 2n 2 A [n A( 1) ] EU ˆ A (α A, α B1 (α A )) is a strictly concave function in the unrestricted domain α A (, + ), and obtains a global maximum at αa L = 1 + n A 1, where αl A is the solution of the firstorder condition EU ˆ A (α A,α B1 (α A )) α A = 0. Given that EU A (α A ) is strictly decreasing with resect to α A, ˆα A strictly dominates any α A > ˆα A. As αa L > α A, it follows that EU A (α A ) is maximized at min{α L A, ˆα A} and thus the corresonding exected utility is strictly ositive. Observe then that α L A < ˆα A if and only if > n A(n A n B 1) 1+n A (n A n B 1) = 1. Therefore, 18

19 If 1, grou A selects ˆα A = n A n B (1 )+ (n A 1) leading to the inactivity of grou B, and grou B selects any α B [0, n B(1 )+ solution of α B = α B3 (ˆα A ) guaranteeing the inactivity of grou B. ], with the uer bound being the If > 1 there exists a unique equilibrium such that both grous are active. Grou A selects αa L = 1 + n A 1 and grou B selects α B1 (αa L ), given by α B1 (α L A ) = n A[n B (1 )+][n A (2n B (1 ) 2+3)+(n B 2)] (n A 1)(n A n B )[n A (1 )+] 2n A (n B 1)[n A (1 )+] = α F B Denoting ρ i = n i the main text. + (1 ) we can rewrite the equilibrium sharing rules as resented in Proof of Proosition 3. If 1 monoolization arises both in the simultaneous and in the sequential cases and there is a continuum of equilibria. By comaring the equilibrium suorts of αb S and αb F we can see that the lower bound of the former is equal to the uer bound of the latter which guarantees that αb F αs B. Similarly, αl A is equal to the lowest ossible value in the suort of αa S, which guarantees that αl A αs A. Given that α only grou A is active, equilibrium aggregate effort is equal to n A (n A 1) A V, which n 2 A is strictly increasing in α A, hence aggregate effort is greater in the simultaneous case. If 1 < 1 monoolization only arises in the sequential case. Equilibrium aggregate effort in the simultaneous case where both grous are active is given by EA S + ES B = n An B (1 n A n B )+[n A +n B +2(n A 3)n A n B +2n A n 2 B] [2n B 1+n A (2+n B (n A +n B 5))] 2 n A [2n B ( 1) ] n B V while equilibrium aggregate effort in the sequential case where grou B is inactive is given by EA L = (n B + n B )V. Therefore, equilibrium aggregate effort is greater in the simultaneous case if and only if [n A( 1) ][(n A n B 1)n B (1 ) ]V n A [2n B (1 )+]+n B 0. Notice that the denominator is always ositive and the first term of the nominator is always negative. The second term of the numerator is ositive if and only if 1 which is not in the analyzed interval. Hence, equilibrium aggregate effort is greater in the sequential case. If > 1 both grous are active in both the simultaneous and sequential cases. Thus we now have that equilibrium aggregate effort in the sequential case where the large grou is the leader is given by E L A + EF B = n A(n A +n B 1) n A (n A +n B 3) 2n A V. Therefore, equilibrium aggregate effort is greater in the sequential case if and only if (n A n B )[n A (1+n A n B )(1 )+]V 2n A [n A (2n B (1 )+)+n B ] > 0 which is always true for any > 0. 19

20 Proof of Proosition 4. Clearly, if the small grou is inactive, the GSP cannot occur. If > 1, the GSP arises if and only if α L A < n A n B + 2α F B n A(n B 1) 2n B (n A 1) Substituting for the equilibrium value of αa L and αf B condition reduces to from Proosition 1, the above n A (n A n B )( 1) [n A (2n B ( 1) ) n B ] 2(n A 1)n B [n A (1 ) + ] As the denominator is ositive, the condition holds if and only if the numerator is negative, which requires > 1. Hence we reach a contradiction. < 0 Proof of Proosition 5. The best resonse of grou B as resented in (5) is: α A2 (α B ) α A (α B ) = α A1 (α B ) α A [0, α A3 (α B )] for α B α B for α B < α B < ˆα B for α B ˆα B The exected utility of grou B being the leader is given by: 0 if α B α B EU B (α B ) = EU ˆ B (α A1 (α B ), α B ) if α B < α B < ˆα B EU B (α B ) if α B ˆα B Notice that EU B (α B ) is a continuous function since lim αb α B ˆ EU B (α A1 (α B ), α B ) = 0 and lim αb ˆα B ˆ EU B (α A1 (α B ), α B ) = EU B (ˆα B ) Moreover, the second derivative of EU ˆ [(n B (α A1 (α B ), α B ) is equal to B 1)] 2 V < 0. 2n 2 B [n B( 1) ] Thus, EU ˆ B (α A1 (α B ), α B ) is a strictly concave function in the unrestricted domain α B (, + ), and obtains a global maximum at αb L = 1+n B 1, where αl B is the solution of the first-order condition EU ˆ B (α A1 (α B ),α B ) α B = 0. Given that EU B (α B ) is strictly decreasing with resect to α B, ˆα B strictly dominates any α B > ˆα B. Observe that αb L < ˆα B always holds, while α L B α B if and only if 1. Therefore, If 1, grou B selects any α B inactive, and grou A selects α A2 (α B ) = [ 0, n B[n A (1 )+3 2] 2 (n B 1) n A n A 1 ] [ 1 + (n B 1)α B +1 n B ]. so that it remains If > 1 there exists a unique equilibrium such that both grous are active. Grou B selects αb L = 1 + n B (1 ) and grou A selects α A1 (αb L ), given by 20

21 α A1 (α L B ) = n B[n A (1 )+][n B (2n A (1 ) 2+3)+(n A 2)]+(n B 1)(n A n B )[n B (1 )+] 2(n A 1)n B [n B (1 )+] = α F A Denoting ρ i = n i the main text. + (1 ) we can rewrite the equilibrium sharing rules as resented in Proof of Proosition 6. If 1 monoolization arises both in the simultaneous and in the sequential cases and there is a continuum of equilibria. By comaring the equilibrium suorts of αb L [0, n B[n A (1 )+3 2] 2 (n B ] and α S 1) B [ n B(1 )+, n B[n A (1 )+3 2] 2 (n B ] we can see 1) that their uer bounds coincide. Therefore, αb L < αs B for all αl B < n B(1 )+. As under monoolization grou A s equilibrium sharing rule is strictly increasing in α B, it follows that αb L < αs B imlies αf A < αs A. Given that only grou A is active, α equilibrium aggregate effort is equal to n A (n A 1) A V which is strictly increasing in n 2 A α A, hence aggregate effort is strictly greater in the simultaneous case for any 1 and αb L < n B(1 )+. When the suorts of the two equilibria overla (i.e., for αb L n B(1 )+ ), α L B can be either greater or smaller than αb S. Following the same reasoning as before, we have that aggregate effort in the sequential case can be either greater or smaller than in the simultaneous case deending on the secific value of α B one considers in each game. If > 1 both grous are active in both the simultaneous and sequential cases. Equilibrium aggregate effort in the simultaneous case is given by EA S + ES B = n An B (1 n A n B )+[n A +n B +2(n A 3)n A n B +2n A n 2 B] [2n B 1+n A (2+n B (n A +n B 5))] 2 n A [2n B ( 1) ] n B V while equilibrium aggregate effort in the sequential case where the small grou is the leader is given by EA F + EL B = n B(n A +n B 1) n B (n A +n B 3) 2n B V. Therefore, equilibrium aggregate effort is strictly greater in the simultaneous case if and only if (n A n B )[(n A n B 1)n B ( 1)+]V 2n B [n A (2n B (1 )+)+n B ] > 0. Given that the denominator is ositive, this requires that the numerator is also ositive, hence that [(n A n B 1)n B ( 1)+] > 0, which is true if and only if > 1. Consequently, aggregate effort in the sequential case is strictly smaller than in the simultaneous case for any > 1. Proof of Proosition 7. Clearly, if the small grou is inactive, the GSP cannot occur. If > 1, the GSP arises if and only if 21