The collective action problem: Within-group cooperation and between-group competition in a repeated rent-seeking game

Games and Economic Behavior 74 (2012) 68–82

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The collective action problem: Within-group cooperation and between-group competition in a repeated rent-seeking game ✩ Guillaume Cheikbossian ∗ Université de Montpellier 1 and Toulouse School of Economics (GREMAQ), France

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 27 March 2008 Available online 19 May 2011

This paper analyzes the ability of group members to cooperate in rent-seeking activities in a context of between-group competition. For this purpose, we develop an infinitely repeated rent-seeking game between two groups of different size. We first investigate Nash reversion strategies to support cooperative behavior in both groups, before analyzing double-edge trigger strategies which support cooperation in one group only. These last strategies have the property that cheating on the agreement in the cooperative group is followed by non-cooperation in this group and cooperation in the rival group. The main conclusion is that the set of parameters for which cooperation can be sustained within the larger group as a subgame perfect outcome is as large as that for which cooperation can be sustained in the smaller group. Hence, in contrast to Olson’s (1965) celebrated thesis, but in accordance with many informal and formal observations, larger groups can be as effective as smaller groups in furthering their interests. © 2011 Elsevier Inc. All rights reserved.

JEL classification: D72 D74 C72 C73 Keywords: Collective action Rent-seeking Within-group cooperation Between-group competition Repeated game

1. Introduction In his pioneering work, Olson (1965) argues that the free-rider or collective action problem makes larger groups less effective than smaller groups in furthering their interests. His analysis of the difficulties facing large groups seems to remain relevant in a dynamic setting. Indeed, despite the potential of trigger strategies to induce mutual cooperation in a repeated game (Axelrod, 1981), scholars of collective action doubt that such strategies can be effective in large groups (e.g., Hardin, 1982; Olson, 1982; Sandler, 1992; Taylor, 1982). Bendor and Mookherjee (1987) confirm this presumption and formally show that trigger strategies are in fact less effective in sustaining cooperation in the repeated game when group size is large or when there is imperfect monitoring. In this paper, we also analyze the logic of collective action in a repeated game setting. But, in contrast to most collective action theory, we develop a model which allows us to explicitly analyze the interactions between intragroup collective action problems and the conflict between groups. Indeed, as most social activities involve opposing groups (e.g., nations,

✩ I thank Léonard Dudley, Patrick Gonzalez, Bruno Jullien, Laurent Linnemer, Nicolas Marceau, Javier Ortega, Wilfried Sand-Zantman, Nicolas Treich, Etienne Wasmer and especially Philippe Mahenc and Thomas Mariotti for valuable comments and discussions. The paper has also benefited from seminar presentations at Université de Montpellier, Université de Toulouse, Université de Montréal, UQAM (Montréal), Université Laval (Québec), the Canadian Economic Association (Montréal) and the ESEM (Vienna) annual conferences. Finally, I am grateful to two referees and an Advisory Editor, whose detailed comments greatly improved the presentation of this paper. The usual disclaimer applies. Correspondence to: Université des Sciences Sociales, Manufacture des Tabacs – Bât S, 21 Allée de Brienne, 31000 Toulouse, France. Fax: +33 0 5 61 22 55 63. E-mail address: [email protected].

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

69

firms, departments), the benefits of collective action within a group may come at the expense of other groups’ interests and, at the same time, intergroup competition may affect the incentives for within-group cooperation. We build a simple rent-seeking game à la Tullock (1980) and Nitzan (1991), where two unequally-sized groups compete for a private prize divisible both at the group level and at the individual level. The sum of individual efforts contributed in a group determines the proportion of the rent allocated to each group and, moreover, each group’s share is equally distributed among group members. As a result, in the one-shot game, the more populous group spends less on rent-seeking activities and receives a lower share of the rent, as predicted by Olson’s theory. In the repeated game, cooperation within one group, or both, can be maintained through the use of an appropriate trigger strategy if players are sufficiently patient. In the context of this paper, the ease of cooperation in a given group will then be measured by the lowest discount factor supporting the optimal level of group effort as a subgame perfect outcome. To be more precise, we construct two types of Stationary Subgame Perfect Equilibria (SSPE), one which supports cooperation within the two groups, and the other which supports cooperation within one group only. For the first type of equilibrium, we assume the members of both groups to follow Nash Reversion Strategies (NRSs) to support cooperative behavior within their respective groups. These strategies prescribe that any deviation from the cooperative path in a given group is met with permanent reversion to the non-cooperative outcome within the same group, irrespective of the behavior of the members of the rival group. Yet the ability of a given group to maintain within-group cooperation—as implied by the minimum discount factor—depends on whether the group faces a cooperative or non-cooperative group. The reason is that group members’ payoffs are contingent on the opponents’ behaviors through intergroup competition. In particular, we show that within-group cooperation is more difficult to sustain when facing a cooperative group, and that this is even more so for the smaller group. As a result, cooperation within both groups generally cannot be sustained as a SSPE under (bilateral) NRSs, except for a small range of discount factors close to 1. We then focus on another type of SSPE, which supports cooperation within either the smaller group or the larger group. To this end, we construct Double-Edge Trigger Strategies (DETSs) which have the property that cheating on the agreement in the cooperative group is followed by perpetual non-cooperation in this group and cooperation in the rival group. Off the equilibrium path, the members of the rival group abide by the (new) cooperative agreement until cheating is detected, in which case they also revert back to the non-cooperative outcome forever. It is shown that the range of discount factors supporting cooperation within the larger group as a SSPE through DETSs is larger than that supporting cooperation within the smaller group only. Though the framework here is highly stylized, it provides new insights into the collective action problem in the presence of conflicting groups. In a repeated game setting, we show that intergroup competition fosters within-group cooperation and that larger groups are more effective in overcoming their collective action problems. The key factor driving these findings is that each group faces an external threat, which is endogenously determined by the opponents’ behaviors. When there is reversion to the single-period non-cooperative outcome within a given group, the free-rider problem within that group is exploited by the rival group’s members who best-respond by increasing their rent-seeking effort. This makes the penalty for cheating—on the cooperative agreement within the group—more costly than if group members were playing an isolated, single-group game. The argument is especially strong when group size is large and when the opponents are cooperative or threaten to cooperate with each other. These conclusions are supported by a growing body of experimental studies in social psychology indicating that intergroup competition enhances group productivity by improving coordination within the group (see, e.g., Bornstein et al., 1990, 2002; Erev et al., 1993; Bornstein, 2003). Regarding the effect of group size on overall group performance, another interesting empirical regularity is that government transfers targeted at specific groups are, in general, increasing in the size of the groups (see, e.g., Congleton and Shugart, 1990 for pensions and retirement benefits; Kristov et al., 1992 for social insurance benefits; and Potters and Sloof, 1996 for a comprehensive survey on empirical analysis of interest groups’ influence). This paper contributes to the literature on collective rent-seeking or group contests (see, e.g., Katz et al., 1990; Nitzan, 1991; Riaz et al., 1995; and for a recent survey see Konrad, 2009). While most of the analysis concentrates on the level of rent dissipation in a static setting, the current paper examines the ability of competing groups to overcome their free-riding problem in a repeated game setting. More closely related, in its focus, is the work by Esteban and Ray (2001). In a static game, they show that if the individual cost of contributing to group action is sufficiently convex, then larger groups are more successful. The reason is that the higher level of individual effort contributed in smaller groups (due to a lower freerider problem) is not sufficient to counterbalance the lower number of contributors. Our work complements that of Esteban and Ray (2001) by showing that larger groups can also be more effective in overcoming their collective action problems in a repeated game. Our analysis is also related to the literature on tacit cooperation. Leininger and Yang (1994) analyze dynamic rent-seeking games in which two players move sequentially over a finite or infinite period of time. They show that the dynamics of the rent-seeking process may involve implicit collusion between the two players with less rent dissipation than in a static game. Pecorino (1998) focuses on the effect of an increase in the number of firms on their ability to overcome free-riding in lobbying for tariffs in a repeated game setting, and concludes that maintaining cooperation does not necessarily become

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

more difficult as the number of firms rises.1 Again, the distinctive feature of our work is to analyze the interactions between intergroup competition and intragroup cooperation in a repeated game setting. 2. The model 2.1. The stage game We start by specifying the details of the stage game G. There are N players divided in two groups A and B which have n A and n B immobile members, respectively. The government implements a policy generating a divisible rent Y , and decides on the division of this pie among the two groups in response to the rent-seeking pressures and lobbying activities of the members of the groups. Let (i , j ), for all i = 1, 2, . . . , n j and j = A , B, denote the member i of group j. e i j ∈ + is the rent-seeking expendi-

n j

ture/effort, expended at unit cost, by member (i , j ), and E j = i =1 e i j ∈ + is the sum of rent-seeking efforts of group j members. Following much of the contest literature, we assume that the share allocated to group j, for j = A , B, is given by the contest success function p j : 2+ → [0, 1], which has the Logit form



p j (E A , E B ) =

E j /( E A + E B ),

if ( E A , E B ) = (0, 0),

1 /2 ,

if ( E A , E B ) = (0, 0).

(1)

Hence, a group gets a share of the total rent that depends on the sum of expenditures of its members relative to the total expenditure.2 When nobody invests anything, the rent is equally shared between the two groups. Individual contributions cannot be observed, and so the rent appropriated by each group is equally shared among its members. We also assume that each player (i , j ) is endowed with a twice differentiable and additively separable utility function πi j : + × [0, 1] → + . The payoff to player (i , j ), for all i = 1, 2, . . . , n j and j = A , B, is

πi j (e i j , E A , E B ) =

p j ( E A , E B )Y nj

− ei j .

(2)

We first analyze the one-period equilibrium outcome in which neither group exhibits any cooperation in among its members. Observe first that there is no equilibrium where nobody invests anything. The relevant best reply of player (i , j ) given all other players’ efforts is, therefore, given by the following first-order condition3

nk

e ik

Y

i =1 = 1, n n [ i =j 1 e i j + i =k 1 e ik ]2 n j

k = j .

(3)

Due to the public-good nature of this problem, first-order conditions only determine group expenditures. We look, however, for the symmetric equilibrium such that all members in a group provide the same level of effort. Let e ∗∗ , j = A , B, be the j common equilibrium level of individual effort in group j when there is non-cooperation within both groups. The conditions for an equilibrium are then

nk ek∗∗

[n j e ∗∗ j

Y

+ nk ek∗∗ ]2 n j

= 1,

j = A , B , j = k.

(4)

The solution to this system of equations is

e ∗∗ j =

nk

Y

(n j + nk )2 n j

,

j = A , B , j = k.

(5)

∗∗ Let p ∗∗ ≡ p j ( E ∗∗ A , E B ), j = A , B, denote the equilibrium share allocated to group j when there is non-cooperation within j both groups, and where E ∗∗ = n j e ∗∗ (for j = A , B). Substituting (5) into (1), we obtain j j

p ∗∗ j =

nk n j + nk

,

j = k.

(6)

1 Lambson (1984, 1987) had previously shown that, in general, collusion cannot be maintained in a repeated game oligopoly setting as the number of firms rises to infinity. 2 In the working paper version (Cheikbossian, 2009), we show that the results of this section as well as Lemma 2 in the subsequent section still hold for p j ( E A , E B ) = ( E j )r /[( E A )r + ( E B )r ], where r ∈ [0, 1] is the marginal return to rent-seeking expenditures. This function is a particular case of p j ( E A , E B ) = f j ( E j )/( f A ( E A ) + f B ( E B )), where f j (0) = 0, f j (.) > 0 and f j (.)  0, which has been shown by Skaperdas (1996) to have “good” axiomatic properties as a contest success function. 3 Note that the marginal return to an additional unit of individual rent-seeking effort (as well as to an additional unit of group rent-seeking effort) is decreasing in effort. Hence, each player’s problem is strictly concave and the first-order conditions are both necessary and sufficient for characterizing the best-response functions of the players.

G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

71

The group with fewer members gets the larger share of the total rent. Thus, in accordance with the traditional collective action theory, when the collective prize is private and divisible among group members, the negative size effect due to freeriding dominates the positive size effect due to a larger number of contributors. It is also worth pointing out that the larger group not only ends up with the lower share of the rent but must also divide this share by a larger number of claimants. ≡ The payoff for each member of group j, when there is non-cooperation within both groups, is denoted by π ∗∗ j ∗∗ , E ∗∗ ). Substituting (5) and (6) into (2), we obtain π j (e ∗∗ , E B A j

nk (n j + nk − 1) Y

π ∗∗ j =

(n j + nk )2

nj

,

j = A , B , j = k.

(7)

In the absence of within-group cooperation, the members of the smaller group have a higher payoff than the members of the bigger group. In addition, individual payoff is falling in the number of fellow members and increasing in the size of the rival group. 2.2. The efficient level of group effort We now consider a situation where there can be full within-group cooperation. In other words, the members of at least one group jointly choose their contributions so as to maximize their aggregate welfare. Assume first that there is cooperation within each of the two groups. In the cooperative equilibrium, the members of n j group j, for j = A , B, jointly maximize π (e , E j , E k ) = p j ( E j , E k )Y − E j with respect to E j taking the collective i =1 i j i j action of group k = j, i.e. E k , as given. The optimal level of aggregate expenditure of group j must then satisfy the following first-order condition

Ek

[ E j + E k ]2

Y = 1,

j = A , B ; k = j .

(8)

Let  e j , j = A , B, be the common level of individual effort in group j, when there is cooperation within both groups. It is easily checked that the solution of the above system is

 e j = Y /4n j ,

j = A, B .

(9)

Let  p j ≡ p j( E A, E B ), j = A , B, denote the equilibrium share allocated to group j, when there is cooperation within both

groups, and where  E j = n j e j (for j = A , B ). Substituting (9) into (1), we obtain:  pA =  p B = 12 . Hence, each group obtains half of the total rent in equilibrium. Because individual preferences are represented by a utility function that is additively separable and linear in Y and in rent-seeking activities, within-group cooperation makes each group act as if it were a single agent.

 j ≡ π j ( e j, E A, E B ). The payoff for each member of group j, when there is cooperation within both groups, is denoted by π  Using the fact that p j = 1/2 and substituting (9) into (2), we obtain

j = π

Y 4n j

,

j = A, B .

(10)

Next, consider a situation in which there is cooperation in group j only. In other words: (i) the members of group j jointly  n j π (e , E j , E k ) = [ E j /( E j + ni=k 1 e ik )]Y − E j with respect to E j , taking the decisions of group k members maximize i =1 i j i j

n

n

k k (for k = j) as given; (ii) each member of group k = j maximizes πik (e ik , E j , E k ) = [ i = e /( E j + i = e )](Y /nk ) − e ik 1 ik 1 ik with respect to e ik , taking all other players’ decisions in group k as well as the collective action of group j, i.e. E j , as given. The optimal level of aggregate expenditure of group j must satisfy the following first-order condition

nk [E j +

e ik

i =1  nk

e ]2 i =1 ik

Y = 1,

k = j .

(11)

The optimal expenditure of a non-cooperative member of group k = j must satisfy the following first-order condition

Ej

[E j +

nk

e i =1 ik

Y

]2

nk

= 1,

j = k.

We focus on symmetric equilibria where all members of the non-cooperative group make the same level simplify notations, we adopt the following convention. A single tilde over the variable of interest for group with a single star superscript to the same variable for group k = j, indicates that there is cooperation within non-cooperation within group k. Hence, let  e j and ek∗ be respectively the cooperative and non-cooperative level effort within group j (for j = A , B) and k = j. Solving the above system, we have

(12) of effort. To j associated group j and of individual

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

 ej =

nk

Y

ek∗ =

(nk

1

Y

+ 1)2

nk

j = k,

,

(nk + 1)2 n j

(13) (14)

.

The equilibrium shares of the rent allocated to groups j and k, for j = A , B and j = k, when there is cooperation within p j ≡ p j ( E j , E k∗ ) and pk∗ ≡ pk ( E k∗ ,  E j) = 1 −  p j, group j and non-cooperation within group k, are denoted respectively by  ∗ ∗ E j = n j e j and E k = nk ek . Substituting (13) and (14) into (1), we then obtain where 

 pj =

nk nk + 1

,

j = k.

(15)

Note that the share of the rent allocated to group j tends towards 1 for nk large. j ≡ π j ( Finally, let π e j, E j , E k∗ ) and πk∗ ≡ πk (ek∗ , E k∗ ,  E j ) be respectively the equilibrium payoff to each member of group j and k, for j = A , B and j = k, when there is cooperation in group j and non-cooperation in group k. Substituting (13) and (15) into (2), we obtain

 j = π

nk nk + 1

2

Y nj

,

j = k.

(16)

Substituting (14) and (15) into (2), we also obtain

πk∗ =

nk

Y

(nk + 1)2 nk

.

(17)

By comparing the utility of each individual in each group under different scenarios, we can establish the following lemma, which will prove useful later.

 j and π ∗ < π ∗∗ for j = A , B.4 j > π Lemma 1. If n j  2, then π j j The members of a given group, no matter whether they cooperate or not, have a higher (respectively, lower) payoff when facing a non-cooperative (respectively, cooperative) rival group. In other words, cooperation of the others is always detrimental to the members of a given group. First, the gains from cooperation are larger when facing a non-cooperative rival group. Second, non-cooperation is less costly when facing a rival group which is also vulnerable to the free-rider problem. 3. Within-group cooperation and between-group competition 3.1. Preliminaries We now define the discounted infinitely repeated game denoted G ∞ (δ), where δ ∈ (0, 1) is the common discount factor per period. While individual behavior cannot be observed by others, the aggregate level of contributions made by each group in each period is however perfectly observed by any member of the two groups. Let E j ,t ∈ + , for j = A , B, be the aggregate level of contributions of group j members in period t. Hence, a publicly observable (finite) history in period t  1 1 is ht = (( E A ,0 , E B ,0 ), ( E A ,1 , E B ,1 ), . . . , ( E A ,t −1 , E B ,t −1 )) or ht = {( E A ,s , E B ,s )}ts− =0 . In fact, each player (i , j ) can also observe his

1 t −1 own actions {e i j ,s }ts− =0 , so that player (i , j )’s history in period t is given by h i j ,t = {(e i j ,s , E j \i ,s , E k,s )}s=0 , where e i j ,s + E j \i ,s = E j ,s . However, in the following, we will focus on trigger strategies in which any deviation—from the aggregate level of contributions maximizing group welfare—is met with symmetric punishment within the group, irrespective of the identity of the deviator(s). This requires only the use of publicly observable histories. Therefore, in order to lighten the notation, we will write players’ strategies as functions of observable histories, without affecting equilibrium outcomes. Let H t be the set of t-period publicly observable histories. We further define ∞ the initial history to be the null set, H 0 ≡ {∅}, and H to be the set of all possible publicly observable histories, H ≡ t =0 H t . A pure strategy for player (i , j ) in G ∞ (δ), for all i = 1, 2, . . . , n j and j = A , B, is a mapping from the set of all possible histories into the set of effort levels, σi j : H → + . Let σ j ≡ (σ1 j , . . . , σn j j ), for j = A , B, be the strategy profile of group j members. Any strategy profile σ ≡ (σ A , σ B ) generates the path of aggregate contributions { E A ,t (σ ), E B ,t (σ )}t∞ =0 defined inductively by: ( E A ,0 (σ ), E B ,0 (σ )) ≡ σ (∅) and ( E A ,t (σ ), E B ,t (σ )) ≡ σ (( E A ,0 (σ ), E B ,0 (σ )), . . . , ( E A ,t −1 (σ ), E B ,t −1 (σ ))) for all t  1. In period t, ( E A ,t (σ ), E B ,t (σ )) yields a flow payoff of πi j ( E A ,t (σ ), E B ,t (σ )) to player (i , j ). An outcome path { E A ,t (σ ), E B ,t (σ )}t∞ =0 thus implies an infinite stream of stage-game payoffs {πi j ( E A ,t (σ ), E B ,t (σ ))}t∞ =0 for player (i , j ). The discounted payoff to player (i , j ) from the infinite

4

All the proofs are in Appendix A.

G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

73

∞

t ∞ sequence of stage-game payoffs {πi j ,t }t∞ t =0 δ πi j ,t , so that her payoff in G (δ) obtained with the strategy =0 is given by profile σ is given, for all i = 1, 2, . . . , n j and j = A , B, by

πiδj (σ ) =

∞ 



δt πi j E A ,t (σ ), E B ,t (σ ) .

(18)

t =0

σ is a Nash equilibrium in G ∞ (δ) if, for all i = 1, . . . , n j and j = A , B, σi j is a best response to σ j\i = (σ1 j , . . . , σi −1 j , σi +1 j , . . . , σn j j ) and to σk for k = j. And it is a subgame perfect equilibrium in G ∞ (δ) if after every history h ∈ H , σ |h (i.e., the continuation of σ after h) is a Nash equilibrium in the corresponding subgame. We will restrict attention A strategy profile

to stationary subgame perfect equilibria (SSPE), i.e., equilibria in which after any history, a stationary profile of actions is played thereafter, and which also satisfy the additional requirement of symmetry within groups, in the sense that all players of the same group play the same action at every history. 3.2. Nash Reversion Strategies (NRSs) We suppose that the members of both groups use Nash Reversion Strategies (NRSs) to support cooperative behavior within their respective groups. NRSs prescribe that any deviation from the cooperative path in group j is met with permanent reversion to the one-period symmetric non-cooperative outcome within this group.5 Hence, group j members’ decisions to cooperate (or not) only depend on the history of play of the group to which they belong. For instance, if no deviation has occurred within group j, then group j members continue to cooperate regardless of whether a deviation has occurred within group k = j. Yet group j members optimally respond in every period to the level of collective action of group k. Therefore, (bilateral) NRSs must specify how player (i , j ) behaves following a deviation within group k = j.

1 ht = {(  E j ,s ,  E k,s )}ts− We use the following notations. (i)  =0 is the history featuring cooperation within the two groups

ht −1 ; ( E j ,t −1 ,  E k,t −1 )) with E j ,t −1 =  E j is the history featuring cooperation within the through period t − 1. (ii) ht = ( two groups through period t − 2, followed by a deviation within one’s own group—i.e. group j—in period t − 1; the corresponding history which involves a deviation within the rival group—i.e. group k = j—in period t − 1 is denoted by

 1 ht = (  ht −1 ; (  E j ,t −1 , E k,t −1 )) with E k,t −1 =  E k . (iii)  hj ,t = {(  E j ,s , E k∗,s )}ts− =τ is the history featuring cooperation within group j and non-cooperation within group k = j from period τ  1 to period t − 1 (with t  2); the corresponding history featuring 1 = {( E ∗j ,s ,  E k,s )}ts− cooperation within group k and non-cooperation within group j = k, over the same period, is denoted h∗ =τ . j ,t  ∗ (Note that these notational conventions imply that  h = h for j = k.) NRSs can then be expressed as, for all i = 1, 2, . . . , n j j ,t

and j = A , B,

σi j (ht ) =

⎧  ⎪ ej ⎪ ⎪ ⎪ ⎪ ⎨ ej

k,t

if t = 0 or if ht =  ht and t  1;

if ht =  ht and t  1 or if ht = ( hτ ; hj ,t ),

τ  1 and t  2;

⎪ ⎪ e ∗j if ht = ht and t  1 or if ht = (hτ ; h∗ ), τ  1 and t  2; ⎪ j ,t ⎪ ⎪ ∗∗ ⎩ ej otherwise.

(19)

In words, group j members, for j = A , B, cooperate with each other and contribute the cooperative level of effort  e j (given by (9)) in the first period and in all periods t  1 if the cooperative agreement within the two groups has been observed in all preceding periods. If a deviation is observed within the rival group, i.e. group k = j, then all group k members permanently revert to the non-cooperative level of effort, and group j members still continue to cooperate with each other and contribute  e j (given by (13)) as long as no deviation within group j is observed. Symmetrically, if a deviation is detected within one’s own group, i.e. group j, then all group j members permanently revert to the non-cooperative level of effort and contribute e ∗j (given by (14)) as long as no deviation within group k = j is observed. If there have been deviations within both groups, then the members of the two groups are stuck at the one-shot non-cooperative equilibrium and contribute e ∗∗ j (given by (7)), for j = A , B. We first establish the following lemma. Lemma 2. If n j  3, then for j = A , B and for every period in which all group j members cooperate with each other and produce the joint-maximizing level of effort, the best possible deviation for each member of group j is to cut her contribution to 0, irrespective of the behavior of the members of group k = j. Histories in which there has been no shirking within group j can be grouped into two classes: Those which involve a past defection within group k = j, and those which do not involve such a defection. 5

In his seminal work, Friedman (1971) used Nash reversion as punishment.

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

Consider first histories featuring no shirking in either group, and that player (i , j ) is considering deviating from NRSs. d Let  p j ≡ p j( E j i ,  E k ) be the group j’s share of the rent when player (i , j ) optimally deviates from NRSs, and where

d  E j i = (n j − 1) e j . By Lemma 2 and (1), we have  p j = [(n j − 1) e j ]/[(n j − 1) e j + nk ek ], for j = k. The corresponding deviation d  j ≡ π j (0,  E j i ,  E k ). Using (9), we then have payoff is denoted π

j = π pj d

Y

d

nj

=

nj − 1 Y 2n j − 1 n j

(20)

.

NRSs prescribe that, following a deviation within group j, group j members permanently revert to the non-cooperative level of effort, while group k members continue to cooperate with each other. It follows that no member of group j has an incentive to deviate from within-group cooperation if and only if

j + π d

δ 1  j, π ∗j  π 1−δ 1−δ

(21)

 j (given by (10)) and π ∗ (given by (17)) are individual payoffs in group j under respectively cooperation and nonwhere π j cooperation within group j (and cooperation within group k = j). The lowest value of the discount parameter above which cooperation can be sustained within group j, for j = A , B, is then given by

 d   d j π j − π∗ , j − π δ Cj (n j ) = π j

(22)

where the C superscript indicates that the opponents are cooperative. This critical discount factor is independent of nk as the rival group acts as a single entity under cooperation. Consider now histories featuring a past defection within group k, so that group k members are no longer cooperative. Suppose further that no deviation has yet occurred within group j = k but that player (i , j ) is considering deviating from pdj ≡ p j ( E j i , E k∗ ) be the group j’s share of the rent when member (i , j ) optimally deviates from the cooperative NRSs. Let  level of effort, and where  E j i = (n j − 1) e j . By Lemma 2 and (1), we have  pdj = [(n j − 1) e j ]/[(n j − 1) e j + nk ek∗ ], for j = k.

dj ≡ π j (0,  E j i , E k∗ ). Using (13) and (14), we then have The corresponding deviation payoff is denoted π

dj =  π pdj

Y nj

=

nk (n j − 1)

Y

nk (n j − 1) + n j n j

,

j = k.

(23)

Again NRSs prescribe that, following a deviation within group j, group j members permanently revert to the noncooperative level of effort. Group k members also do not cooperate with each other (because of a past defection within that group) and, hence, no member of group j = k has an incentive to deviate from within-group cooperation if and only if

dj + π

δ 1−δ

π ∗∗ j 

1 1−δ

j , π

(24)

j (given by (16)) and π ∗∗ where π (given by (7)) are individual payoffs in group j under respectively cooperation and nonj cooperation within group j (and non-cooperation within group k = j). The lowest value of the discount parameter above which cooperation can be sustained within group j, for j = A , B, is then given by

 d   d j − π ∗∗ j − π j π , δ Nj (n j , nk ) = π j

(25)

when the N superscript indicates that the opponents are not cooperative. Substitute (10), (17) and (20) into (22) to get

δ Cj (n j ) =

(2n j − 3)(n j + 1)2 4[n2j (n j − 1) − 1]

(26)

.

Substitute (7), (16) and (23) into (25) to get

δ Nj (n j , nk ) =

(n j + nk )2 [nk (n j − 2) + (n j − 1)] , (nk + 1)2 [n j (n j + nk )(n j − 2) + nk (n j − 1) + n j ]

j = k.

(27)

NRSs defined by (19), for j = A , B, are subgame perfect equilibrium strategies if and only if there are no profitable one-shot deviations for each member of either group both along the equilibrium path and on the path following a deviation within the rival group. Hence, cooperation within the two groups can be sustained as a SSPE under NRSs if and only if

  δ  max δ NA (n A , n B ), δ BN (n A , n B ), δ CA (n A ), δ BC (n B ) .

(28)

We then need to determine which of the different values of the critical discount factor is the highest. To this end, we now study how the ability of group j to maintain a cooperative outcome, as measured by δ Cj (n j ) or δ Nj (n j , nk ), is affected

G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

75

by an increase in its size. It depends on how payoffs under defection, cooperation and non-cooperation evolve as the group grows larger. Observe first that an increase in size diminishes the per-capita value of the prize and that this affects equally the different payoffs. Indeed, one can see that the individual payoffs, under cooperation (given by (10) and (16)), non-cooperation (given by (17) and (7)) and defection (given by (20) and (23)), when there is respectively cooperation or non-cooperation within the rival group, are the product of two terms, one being the per-capita value of the total rent, i.e., Y /n j . We can thus ignore the direct effect of an increase in size on the per-capita value of the prize and, hence, evaluate the impact of an increase in n j on δ Cj (n j ) and δ Nj (n j , nk ) in terms of group j’s payoff (given by multiplying individual payoff by group size). There are several effects. First, an increase in n j does not affect group j’s payoff under cooperation whether there is cooperation or non-cooperation within group k = j, as one can see from (10) and (16). This is because group j acts as a single entity under cooperation. Second, the severity of the penalty for cheating is increasing in n j because the larger the size of group j is, the more pervasive the free-rider problem and the lower the relative effectiveness of group j, both when there is cooperation and non-cooperation within group k = j.6 This effect causes δ Cj (n j ) and δ Nj (n j , nk ) to be decreasing in n j . The remaining effect is that the temptation to cheat increases with increasing group size because group j’s share of the rent under defection is growing closer to group j’s share of the rent under cooperation with an increase in n j , both when there is cooperation and non-cooperation within group k = j, as one can observe from (20) and (23).7 This last effect causes δ Cj (n j ) and δ Nj (n j , nk ) to be increasing in n j . Hence, an increase in membership has conflicting effects on the ability of a given group to maintain cooperation through the threat of reverting to non-cooperation within the group; a point already stressed by Pecorino (1998) in a single-group game. The impact of an increase in nk on δ Nj (n j , nk ) (for j = k) also involves several effects. First, the penalty for cheating becomes less severe because of the increased free-rider problem within the rival group as a result of its larger size.8 Second, the temptation to cheat rises with nk because group j’s share of the rent under defection is increasing in nk , as one can see from (23). These two effects cause δ Nj (n j , nk ) to be increasing in nk . However, the benefits from cooperation for group j are increasing in the size of group k = j because, again, the larger the size of the rival group is, the greater its collective action problem and the higher the relative effectiveness of group j. This last effect causes δ Nj (n j , nk ) to be decreasing in nk . Hence, an increase in size of the non-cooperative rival group has also conflicting effects on δ Nj (n j , nk ). However, treating group size as a continuous variable, Lemma 3 shows that there are monotonicity results for the effect of nk on δ Nj (n j , nk ) and for the effect of n j on both δ Cj (n j ) and δ Nj (n j , nk ). Lemma 3. If n j  4 (for j = A , B), then: (i) (ii)

dδ Cj (n j ) dn j

< 0 and

∂δ Nj (n j ,nk ) ∂ nk

∂δ Nj (n j ,nk ) ∂n j

< 0 for j = A , B;

< 0 for j = A , B and j = k.

These results contrast strongly with the general presumption that maintaining cooperation becomes more difficult as the collectivity increases (see, e.g., Hardin, 1982; Olson, 1982; Sandler, 1992; Taylor, 1982). A key factor driving result (i) of Lemma 3 is that the severity of the penalty for cheating is much larger when the group is competing against another strategic group, than if it were playing a single-group game. Indeed, when there is reversion to the non-cooperative outcome within a group, the free-rider problem within that group is exploited by the competing group which responds by increasing its rent-seeking effort so as to extract a larger share of the rent to the detriment of the first group. This can explain why the severity of the penalty for cheating increases more rapidly than the temptation to cheat as group size rises, so that cooperation is less difficult to maintain as the number of fellow members rises, both when there is cooperation and non-cooperation within the rival group. Within-group cooperation is also less difficult to maintain as the size of the noncooperative rival group rises (as shown by (ii) of Lemma 3). Again, when the rival group grows larger, defection is more tempting and the penalty for cheating also becomes less severe. But, there is much to be gained from cooperation because the group’s share of the rent approaches 1 as the rival group grows larger (as shown by (15)) as a result of its increased collective action problem. Keeping constant the size of each group, we obtain: Lemma 4. If n j  4 (for j = A , B), then δ Cj (n j ) > δ Nj (n j , nk ) for j = A , B.

6

Using (7) and (17), one can easily verify that Π ∗∗ = n j π ∗∗ and Π ∗j = n j π ∗j —i.e., group j’s payoff under non-cooperation when there is respectively j j

non-cooperation and cooperation within group k = j—are indeed both decreasing in n j . 7 It follows that the deviator’s payoff multiplied by group size—which actually does not correspond, in this case, to group j’s payoff because the deviator has not the same payoff than non-deviators—is increasing in n j . 8 Π ∗∗ = n j π ∗∗ with π ∗∗ given by (7)—i.e. group j’s payoff under non-cooperation within the two groups—is indeed increasing in nk for k = j. j j j

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In other words, cooperation can be more easily sustained when facing a non-cooperative rival group, which confirms the presumption that a weaker rival group reinforces the cohesion and strength of the group. The Nash punishment is less severe when facing a non-cooperative rival group because, in that case, both groups suffer from the collective action problem. But the gains from cooperation are also much larger and, furthermore, the incentive to cheat is reduced because the cooperative level of effort is lower with a non-cooperative rival group. Lemma 4 shows that these two effects dominate the lower penalty for cheating when there is non-cooperation compared to cooperation within the rival group. Finally, Lemma 5 ranks the different values of the critical discount factor. Lemma 5. If n j  4 (for j = A , B) and n A  n B , then C C 0 < δ BN (n A , n B )  δ N A (n A , n B ) < δ A (n A )  δ B (n B ) < 1.

(29)

The inequality δ CA (n A )  δ BC (n B ) directly follows from (i) of Lemma 3. If there is cooperation within the two groups, each group gets half of the total rent in equilibrium. But, when there is cooperation within the rival group, the larger group has a lower share of the rent than the smaller group if fellow members behave non-cooperatively. Hence, the punishment being harsher, cooperation is less difficult to sustain in the larger group, i.e. group A, even though defection is more tempting in this group. However, when there is non-cooperation within the rival group, cooperation is more difficult to sustain in the larger group than in the smaller group. Indeed, while the Nash punishment is less severe for the smaller group, the gains from cooperation are also much higher. This is because the share allocated to the cooperative group is increasing in the size of the non-cooperative rival group. Furthermore, defection is less attractive for the smaller group and then δ BN (n A , n B )  δ NA (n A , n B ). Using (28) and (29), we are now ready to state the following result. Proposition 1. Assume n j  4 (for j = A , B) and n A  n B . Then NRSs defined by (19), for j = A and j = B, form a SSPE of G ∞ (δ) if and only if δ  δ BC (n B ). In every period along the equilibrium path, the members of each group cooperate with each other (within their respective groups), and each group obtains 1/2 of the total rent. From (29), δ  δ BC (n B ) implies that δ  δ CA (n A ). Hence, for any δ  δ BC (n B ), no individual in either group has an incentive to deviate from the cooperative level of effort. If a deviation has taken place within group k in the previous period, then the members of group j = k still have an incentive to continue cooperating because, by Lemma 5, δ  δ BC (n B ) implies that δ > δ Nj (n A , n B ), for j = A , B. It follows that for any δ ∈ [δ BC (n B ), 1), perpetual cooperation within both groups can be sustained as a SSPE through (bilateral) NRSs defined by (19), provided any deviation within a given group is followed by this group reverting to non-cooperative play. Hence, on the equilibrium path, the two groups neutralize each other, and each group obtains half of the total rent in every period. If however δ < δ BC (n B ), then perpetual cooperation within both groups cannot be sustained as a SSPE under (bilateral) NRSs. In particular, it is interesting to note the following. Remark 1. Assume n j  4 (for j = A , B) and n A  n B . After any history featuring cooperation within both groups in all previous periods, all members of the smaller group (i.e. group B), but no members of the larger group (i.e. group A), have a profitable one-shot deviation from (bilateral) NRSs defined by (19), for any δ ∈ [δ CA (n A ), δ BC (n B )). For any δ  δ CA (n A ), group A members have an incentive to maintain cooperation independently of whether there is cooperation or non-cooperation within group B. In turn, all members of group B have an incentive to deviate from withingroup cooperation if δ < δ BC (n B ). Again, when facing a cooperative group, cooperation can be more easily sustained in the larger group because reversion to non-cooperative behavior is more costly for this group due to a more severe free-rider problem in the non-cooperative outcome within the group. 3.3. Double-Edge Trigger Strategies (DETSs) We now focus on situations where cooperative behavior can be supported as a SSPE within one group only. We could consider the strategy profile in which the members of one group—say group j—unilaterally play Nash-reversion strategies, while the members of group k = j, are assumed never to cooperate with each other. However, Lemma 1 reveals that reversion to non-cooperation within group j—due to a defection within this group—makes cooperation more profitable for any member of group k. For this reason, it is only natural to investigate whether the members of the rival group can support, in their turn, cooperative behavior through the same threat of reverting back to non-cooperative play. If it is the case, then defection in group j triggers both permanent reversion to non-cooperative behavior within group j and permanent cooperation within group k. This would in turn enhance the incentives to cooperate in group j. To account for this double edge threat to the cooperative group, we thus introduce Double-Edge Trigger Strategies (DETSs), which constitute a plausible explanation for the stability of within-group cooperation in the intergroup competition.

G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

77

1 We now use the following notations. (i)  h j ,t = {( E j ,s , E k∗,s )}ts− =0 is the history featuring cooperation within group j and

h j ,t −1 ; ( E j ,t −1 , E k∗,t −1 )) with E j ,t −1 =  E j is the history non-cooperation within group k = j through period t − 1. (ii) h j ,t = ( featuring cooperation within group j and non-cooperation within group k = j through period t − 2, followed by a deviation 1 = {( E ∗j ,s ,  E k,s )}ts− within group j in period t − 1. (iii) h∗ =τ is the history featuring non-cooperation within group j and j ,t

= hk ,t .) cooperation within group k = j from period τ  1 to period t − 1 (with t  2). (Again, these notations imply that h∗ j ,t The DETS for player (i , j ) can then be expressed as, for all i = 1, 2, . . . , n j ,

σi j (ht ) =

⎧ ej ⎪ ⎨

if t = 0 or if ht =  h j ,t and t  1;

⎪ ⎩

otherwise,

e∗ j

e ∗∗ j

if ht = h j ,t and t  1 or if ht = (h j ,τ ; h∗ ), j ,t

τ  1 and t  2;

(30)

and the DETS for player (i , k) with k = j can be expressed as, for all i = 1, 2, . . . , nk ,

⎧ ∗  ⎪ ⎨ ek if t = 0 or if ht = h j ,t and t  1; σik (ht ) =  ek if ht = h j ,t and t  1 or if ht = (h j ,τ ; h∗ ), τ  1 and t  2; j ,t ⎪ ⎩ ∗∗ ek otherwise.

(31)

In words, in the first period, group j members cooperate with each other and contribute  e j (given by (13)) while group k members do not cooperate and then contribute ek∗ (given by (14)). This cooperative phase in group j lasts as long as group j members abide by the cooperative agreement. If a member of group j deviates, then all group j members permanently revert to the non-cooperative level of effort e ∗j , and all members of group k = j start cooperating and then contribute the ek . This outcome path (i.e. cooperation within group k and non-cooperation within group j) lasts cooperative level of effort  as long as no deviation from the cooperative level of effort is observed within group k. If, however, a deviation is also observed within group k, then all group k members revert back to the non-cooperative outcome, so that the members of and ek∗∗ (given by (5)). groups j and k are stuck at the one-shot non-cooperative equilibrium and contribute respectively e ∗∗ j Observe that DETSs allow for within-group cooperation after a phase of non-cooperation but only if group members never cooperated before. DETSs are subgame perfect equilibrium strategies of the infinitely repeated game if and only if a single-period deviation from the strategy (and sticking to it subsequently) after any history, is not profitable for any member of either group. Suppose first that no deviation has occurred in the past. DETSs then prescribe that group j members cooperate while the members of group k = j do not cooperate. As long as all players follow this prescription, each member of group j collects j given by (16) in each period. In contrast, suppose a member of group j (optimally) deviates from DETSs in a payoff of π dj given by (23). DETSs prescribe that, in the subsequent periods, period t . She then obtains, in that period, a payoff of π group j members do not cooperate while group k members cooperate with each other. Hence, from period t + 1 onwards, each member of group j collects a payoff of π ∗j given by (17). Thus, a necessary condition for maintaining cooperation within group j under DETSs is

dj + π

δ 1 j , π ∗j  π 1−δ 1−δ

j = k.

(32)

The lowest value of the discount parameter satisfying this inequality is then given by

 d   d j − π ∗j , j − π j π δ NC j (n j , nk ) = π

(33)

where the NC superscript indicates that non-cooperation is followed by cooperation in the other group in case of a deviation within group j. Clearly, because cooperation of the others is always detrimental to the members of a given group, i.e. π ∗j < π ∗∗ (Lemma 1), δ NC (n j , nk ) is lower than δ Nj (n j , nk ) (given by (25)) which was calculated assuming perpetual nonj j cooperation within the rival group. To ensure subgame perfection, we must also verify that no player of group k = j has an incentive to deviate from cooperation along the out-of-equilibrium path following a deviation within group j. From the analysis of the previous section, we know that, when there is non-cooperation within group j, it is not profitable for any member of group k to deviate from cooperation (provided any deviation is met with permanent reversion to the non-cooperative outcome within the group) as long as δ  δkN (n j , nk ). Therefore, cooperation within group j (and non-cooperation within group k) can be sustained as a SSPE under DETSs if and only if

  N δ  max δ NC j (n j , nk ), δk (n j , nk ) ,

j = k.

(34)

Suppose first that the members of the smaller group, i.e. group B, cooperate in the first period and follow the strategy described in (30), while the members of the larger group, i.e. group A, do not cooperate in the first period and follow the strategy described in (31). Since DETSs involve harsher punishments for the group which begins by cooperating than when there is permanent non-cooperation within the rival group, we have that δ BNC (n A , n B ) < δ BN (n A , n B ). From (29), we also have NC N that δ BN (n A , n B )  δ N A (n A , n B ). This implies that δ B (n A , n B ) < δ A (n A , n B ), and so, from (34), the following proposition holds.

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

Proposition 2. Assume n j  4 (for j = A , B) and n A  n B . Then DETSs defined by (30) and (31), with j = B and k = A, form a SSPE of G ∞ (δ) if and only if δ  δ N A (n A , n B ). In every period along the equilibrium path, group B members cooperate and collectively obtain p B = n A /(1 + n A ) of the rent, while group A members do not cooperate and collectively obtain a proportion (1 −  p B ) of a proportion  the rent.

δ BNC (n A , n B ) is not relevant to the outcome of the game. Indeed, if cooperation can be maintained within group A

(following a deviation within group B resulting in non-cooperation within that group), it can also be maintained within N N group B (when there is non-cooperation within group A) because δ N A (n A , n B )  δ B (n A , n B ). Then, for any δ  δ A (n A , n B ), each member of group B finds it more profitable to pursue cooperation than to deviate, while the members of group A never cooperate. If however δ < δ N A (n A , n B ), then defection of one group B member (triggering reversion to the noncooperative outcome within that group) cannot credibly trigger perpetual cooperation within group A and, therefore, DETSs do not satisfy the subgame perfection requirement. This leads to the following remark. Remark 2. Assume n j  4 (for j = A , B) and n A  n B . If δ ∈ [δ BN (n A , n B ), δ N A (n A , n B )), then cooperation within group B cannot be sustained as a SSPE under DETSs defined by (30) and (31) with j = B and k = A, while it could be sustained as a SSPE if group B members were unilaterally playing Nash-reversion strategies and group A members were assumed never to cooperate. The situation where group B members unilaterally play Nash-reversion strategies and face a non-cooperative group corresponds to the analysis (undertaken in the previous section) of the subgame following a deviation from cooperation within group A. Again, in that situation, no member of group B has an incentive to deviate from cooperation whenever δ  δ BN (n A , n B ). The ‘result’ reported in Remark 2 is somewhat counterintuitive because cooperation of the others is always detrimental to the members of a given group. The reason is that—off the equilibrium path—cooperative behavior is more difficult to support in the larger group than in the smaller group when facing a non-cooperative group. Hence, the threat of cooperation within the rival group is not effective for the members of the smaller group. The out-of-equilibrium “perfection” constraint implied by DETSs restrains the range of discount factors for which cooperation can be supported as a SSPE in the smaller group, compared to the situation where group B members unilaterally adopt Nash-reversion strategies and face (by assumption) a non-cooperative group. We now analyze the symmetric situation where the members of the larger group, i.e. group A, cooperate in the first period and follow the strategy described in (30) while the members of the smaller group, i.e. group B, do not cooperate in the first period and follow the strategy described in (31). Again, since DETSs involve harsher punishments for the group which begins by cooperating than when there is permanent non-cooperation within the rival group, we have δ NC A (n A , n B ) < δ NA (n A , n B ). From (29), we also have δ BN (n A , n B )  δ NA (n A , n B ). To make use of (34), we then need to compare δ NC A (n A , n B ) and δ BN (n A , n B ) and determine which of these critical values is smaller and which is higher. Indeed, and in contrast to the symmetric case analyzed above, DETSs can be used to support cooperative behavior in group A for some discount parameters below δ N A (n A , n B ) if cooperation within group B can be supported when there is permanent reversion to the non-cooperative outcome within group A, i.e., if δ  δ BN (n A , n B ). Substitute (16), (17) and (23) into (33) to get

δ NC A (n A , n B ) =

n B (n A + 1)2 [n B (n A − 2) + (n A − 1)]

(n B + 1)2 [n2A (n A n B − 1) − n B ]

(35)

.

N Now δ NC A (n A , n B ) can be compared to δ B (n A , n B ) given by (27) to yield the following lemma.

Lemma 6. Assume n j  4 (for j = A , B), and let n A = γ n B with

δ NC A (n A

γ  1. Then, there exists γ¯  1 such that δ BN (n A = γ n B , n B ) 

= γ n B , n B ) as γ  γ¯ .

Therefore, the critical discount factor above which cooperation within group A can be sustained as a SSPE under DETSs crucially depends on the degree of group size asymmetry.9 By Lemma 6 and (34), we can then establish the following proposition. Proposition 3. Assume n j  4 (for j = A , B), and n A = γ n B with γ  1. If γ  γ¯ (respectively, γ > γ¯ ), then DETSs defined by (30) and (31), with j = A and k = B, form a SSPE of G ∞ (δ) if and only if δ  δ BN (n A , n B ) (respectively, δ  δ NC A (n A , n B )). In every period p A = n B /(1 + n B ) of the rent, while along the equilibrium path, group A members cooperate and collectively obtain a proportion  p A ) of the rent. group B members do not cooperate and collectively obtain a proportion (1 −  9 One cannot obtain an explicit solution for γ¯ . However, one can find numerically that this value is relatively large, and that it is increasing in the size of the reference group (i.e. group B). For example, when n B equals 4, 8 and 10, we obtain a value of γ¯ approximately equal to 10, 100 and 10 000.

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79

This shows the existence of a situation symmetric to that depicted in Proposition 2, except that the range of discount factors supporting cooperation within the larger group as a SSPE through DETSs is larger than that supporting cooperation within the smaller group only. Indeed, both δ BN (n A , n B ) and δ NC A (n A , n B ) are lower than the minimum discount factor supporting cooperation within the smaller (which is given by δ N ( n A , n B )). A In the previous section (on NRSs), we have shown that cooperation is generally easier within large groups. This is the case when we analyze the implications of increasing group size (Lemma 3), but also when making across-groups comparisons on the equilibrium path with cooperation within the two groups (i.e., by Lemma 5, we have δ CA (n A )  δ BC (n B )). There is one exception. The smaller group can more easily sustain cooperation in case of permanent reversion to the non-cooperative equilibrium within the rival group (i.e., by Lemma 5, we also have δ BN (n A , n B )  δ N A (n A , n B )). But, this greater ability of the smaller group, in this particular case, turns again to the advantage of the members of the larger group because, under DETSs, they play on the expectation that any deviation will trigger perpetual cooperation within the smaller group. Note, however, that when the size of the prize is constant, as is the case in this paper, large groups are disadvantaged in terms of individual payoff. But it might well be possible that larger groups are more effective both in terms of aggregate levels of collective action and in terms of per-capita payoffs. In the context of this paper, when there is cooperation within the larger group and non-cooperation within the smaller group, individual pay-off is also higher in the larger group if group √ size asymmetry is not too strong. By (16) and (17), this is case if n A < n B . 4. Concluding remarks In this article, we develop an analysis of collective action that exhibits two central features. First, two groups of different size compete for an exogenous rent and, second, the members of each group can sustain within-group cooperation through the use of trigger strategies in a repeated game structure. While none of these features is new in the collective action literature, the innovation of this paper is in putting them together in a simple framework. In contrast to the Olson paradox, we then show that larger groups can be more successful than smaller groups in achieving collective goals. Indeed, in a context of competition between groups, larger groups stand to loose the most if their members abandon cooperation and free-ride on each other’s efforts. The simplicity of the framework analyzed in this paper is attractive but might be criticized on several fronts. First, we have focused on equilibria such that full cooperation within one group or both can be sustained as a subgame perfect outcome. Alternatively, we could have considered, for a fixed value of the discount factor, the maximal level of effort that members in a group can achieve. It is, however, important to remember that in our analysis, the effectiveness of a group is determined not by its absolute efficiency but by its efficiency relative to that of the other group (through the intergroup competition); hence by its relative efficiency at circumventing free-riding. Therefore, as long as the extent of free-riding—as measured by the divergence between the non-cooperative level and the actual level of collective action—is increasing in group size, the qualitative results of our rent-seeking game should continue to hold. Second, we assumed that any deviation from the cooperative path is punished with Nash-reversion within the corresponding group (and possibly with cooperation within the rival group). A natural extension of the present analysis would be to consider two-phase punishment schemes à la Abreu (1986). With Abreu’s stick-and-carrot strategies, the members of each group would conform to a “stick” phase—in which group members would contribute less than in the one-shot game—because of the “carrot” of a subsequent return to cooperation. We believe that the analysis of optimal punishment schemes in a setting such as ours is an interesting question for future research. Appendix A A.1. Proof of Lemma 1

 j given by (10) for nk = 1, and is increasing in nk for nk > 1. We then have j given by (16) is equal to π Given n j  2, π  j for any n j  2 and nk  2. Similarly, given n j  2, π ∗∗ given by (7) is equal to π ∗ (given by replacing nk by n j j > π π j j in (17)) for n j = 1, and is increasing in nk for nk > 1. Hence, we have π ∗∗ > π ∗j for any nk  2 and n j  2. j A.2. Proof of Lemma 2 Fix the aggregate level of effort of group k, E k  0. Given E k , the optimal level of aggregate effort of group j = k is given E j ( E k ) = arg max E j 0 {[ E j /( E j + E k )]Y − E j }.  E j ( E k ) must satisfy the first-order condition by 

Ek

( E j ( E k ) + E k )2

Y = 1.

(A.1)

Let  e j ( E k ) be the common level of individual effort in group j as a function of E k , i.e.,  e j (Ek ) =  E j ( E k )/n j . (A.1) implies that

√

 e j (Ek ) =

Ek Y − Ek

for any 0  E k  Y .

nj

,

j = k,

(A.2)

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

Let edij ≡ edij ( E j \i ( E k ), E k ) be the effort level of the agent who defects and where  E j \i ( E k ) is the optimal level of effort of group j, excluding player (i , j )’s contribution. The deviator chooses edij to maximize E j \i ( E k ), E k )Y /n j ] − edij , where p j (edij ,  E j \i , E k ) = [ p j (edij , 

edij +(n j −1) e j (Ek ) edij +(n j −1) e j ( E k )+ E k

πidj (edij ,  E j \i ( E k ), E k ) =

. Using (A.2), the first-order condition to this prob-

lem implies that



Ek

edij

Y

= 1, n −1 √ 2 + nj j ( E k Y − E k ) + E k n j

j = k.

(A.3)

The left-hand term is decreasing in edij . The individual who defects will then cut her contribution to 0 if

∂ πidj ( E j \i ( E k ), E k )  ∂ edij



edij =0



0, i.e., if

Ek

Y

 1,  n j −1 √ 2 nj ( E Y − E ) + E k k k nj

j = k.

(A.4)

Simplifying the above expression, we have that

nj

Y  1, √ √ √ [(n j − 1)( Y − E k ) + n j E k ]2 This reduces to

√

Y [(n j − 1) −

√

nj ] +

√

j = k.

(A.5)

E k  0, which is satisfied for any E k ∈ [0, Y ] and any n j  3.

A.3. Proof of Lemma 3 The derivative of δ Cj (n j ), given by (26), with respect to n j is given by

dδ Cj (n j ) dn j

=−

(3n j − 2)(n j − 2)(n2j − 1) 4[n2j (n j − 1) − 1]2

< 0,

(A.6)

which is indeed negative for any n j  2, and for j = A , B. The derivative of δ Nj (n j , nk ), given by (27), with respect to n j is given (for j = k) by

 ∂δ Nj (n j , nk ) ∂n j

(n j + nk ) =−

n j nk [n j (nk + 1)(n j − 5) + nk2 (n j − 4) + 7nk + 8] + n j (n j − 1)2 + nk (3nk2 − 4nk − 3)



(nk + 1)2 [n j (n j + nk )(n j − 2) + nk (n j − 1) + n j ]2

< 0.

(A.7)

It can be easily verified that the term in brackets [.], in the numerator, is strictly positive for any nk  2 and any n j  4, implying that ∂δ Nj (n j , nk )/∂ n j < 0, for j = A , B. Finally, the derivative of δ Nj (n j , nk ) with respect to nk for j = k is given by

⎡ ∂δ Nj (n j , nk ) ∂ nk

⎢ (n j + nk )(n j − 1) ⎣ =−

⎤

nk2 (n3j − 2n2j − n j + 3)

⎥ + nk (n4j − 3n3j + 7n2j − 8n j + 1) ⎦ + (n4j − 2n3j + 3n2j − 3n j )

(nk + 1)3 [n j (n j + nk )(n j − 2) + nk (n j − 1) + n j ]2

< 0.

(A.8)

Again, it can be easily verified that the term in brackets [.], in the numerator, is strictly positive for any n j  4 implying that ∂δ Nj (n j , nk )/∂ nk < 0, for j = A , B and j = k. A.4. Proof of Lemma 4 First, δ CA (n A ) is decreasing in n A implying that δ CA (n A ) reaches its minimum when n A goes to infinity. We have limn A →∞ δ CA (n A ) = 1/2. Second, δ N A (n A , n B ) is also decreasing in n A and hence, given n B , it reaches a maximum when C the size of group A is at its minimum, i.e., when n A = 4. A sufficient condition for the inequality δ N A (n A , n B ) < δ A (n A ) to (n +4)2 [2n +3]

N N B B be satisfied is then δ N A (n A = 4, n B ) < 1/2. We have δ A (n A = 4, n B ) = (n B +1)2 (11n B +36) . The inequality δ A (n A = 4, n B ) < 1/2 3 2 then reduces to 7n B + 20n B − 29n B − 60 > 0, which is always satisfied for any n B  4. By symmetry, we also have δ BC (n B ) > δ BN (n A , n B ), as stated in Lemma 4.

G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

81

A.5. Proof of Lemma 5 Since δ Cj (n j ) (for j = A , B) is decreasing in n j , we have δ CA (n A )  δ BC (n B ) for any n A  n B . From Lemma 4, we also have

δ NA (n A , n B ) < δ CA (n A ). Finally, let (n A , n B ) = δ NA (n A , n B ) − δ BN (n A , n B ). We have

(n A , n B ) =

1



D (n A , n B )

n A n B [n2A (n A − 3) − n2B (n B − 3) + (n A − n B )(2n A n B + 5)] + n3A (n A − 4) − n3B (n B − 4) + n A (n A + 2) − n B (n B + 2)

 > 0,

(A.9)

N where D (n A , n B ) is the product of the denominators of δ N A (n A , n B ) and δ B (n A , n B ). The term in brackets {.} is strictly positive N for any n A > n B (and equal to 0 for n A = n B ). Hence, we also have δ A (n A , n B )  δ BN (n A , n B ) which proves Lemma 5.

A.6. Proof of Lemma 6 Keeping n B unchanged, we first show that Φ(n A , n B ) ≡ δ BN (n A , n B ) − δ NC A (n A , n B ) is decreasing in n A over [n B , ∞). The derivative of Φ(n A , n B ) with respect to n A is given by

9 i ∂Φ(n A , n B ) i =0 λi (n B )n A = , 2 3 ∂n A (n B + 1) (n A + 1) [n B (n A + n B )(n B − 2) + n A (n B − 1) + n B ]2 [n2A (n A n B − 1) − n B ]2

(A.10)

where



 λ0 (n B ) = −(n B − 1)n4B n3B n2B + 3n B − 8 + (7n B + 4)(n B − 1) , 

 λ1 (n B ) = −(n B − 1)n3B n4B (n B + 11)(n B − 2) + 27n2B + 4n B − 2 (n B − 1) ,   λ2 (n B ) = −n2B n6B (16n B − 53) + n4B (79n B − 77) + n2B (42n B − 17) + 2(5n B + 2) ,   λ3 (n B ) = n B 2n8B (n B − 11) + n6B (73n B − 97) + n4B (53n B − 32) + 5n2B (11n B − 6) − 2(4n B − 1) ,   λ4 (n B ) = n B n8B (2n B − 29) + n6B (82n B − 47) − n4B (24n B + 59) + n2B (79n B − 3) + 3(n B + 3) ,    

  λ5 (n B ) = −n B n6B n B (15n B − 24) − 29 + n2B n B 6n2B − 13 − 19 + 87n4B − 25n B − 13 ,   2

2





7 6 3 2 2 2 λ6 (n B ) = − n10 B + n B 3n B − 55 + 2n B 7n B − 4 + 3n B 20n B − 3 + n B 20n B − 13 − (2n B + 1) ,  





   λ7 (n B ) = − n7B n2B (n B − 1) − 9 + 2n5B n B 4n2B − 9 − 7 + n2B 36n2B − 7 + n B 25n2B − 4 − 3 ,       λ8 (n B ) = −n B n6B n B (2n B − 3) − 1 + n B n B (8n B − 5) − 5 + 3 , 

 λ9 (n B ) = −n B n5B (n B + 2)(n B − 3) + 6n4B + (2n B + 1) 5n2B − 5n B − 1 .

(A.11)

We want to show that the numerator of (A.10) is negative. It is easily checked that λi (n B ) is strictly negative for i = {0, 1, 2, 5, 6, 7, 8, 9}, and for any n B  4. However, both λ3 (n B ) and λ4 (n B ) are of ambiguous sign. But, we can show that λ3 (n B )n3A + λ4 (n B )n4A + λ6 (n B )n6A + λ7 (n B )n7A  0 or equivalently that Ψ (n A , n B ) ≡ λ3 (n B ) + λ4 (n B )n A + λ6 (n B )n3A + λ7 (n B )n4A  0. First, observe that





 λ3 (n B ) − λ4 (n B ) = n B n6B 7n2B − 9n B − 50 + n2B 77n3B − 24n B − 27 + 27n4B − 11n B − 7 .

(A.12)

This expression is positive (i.e. λ3 (n B )  λ4 (n B )) for any n B  4. (In particular, the polynomial 7n2B − 9n B − 50 has one positive root n˜ B ≈ 3, 19, and is positive for any n B  n˜ B .) It follows that λ3 (n B ) + λ4 (n B )n A  (1 + n A )λ3 (n B ). We also have that λ6 (n B )n3A + λ7 (n B )n4A  (λ6 (n B ) + λ7 (n B ))n3A since λ7 (n B )  0. Hence, a sufficient condition for Ψ (n A , n B ) to be negative is that Γ (n A , n B ) ≡ (1 + n A )λ3 (n B ) + (λ6 (n B ) + λ7 (n B ))n3A  0. The derivative of Γ (n A , n B ) with respect to n A is given by ∂Γ (n A , n B )/∂ n A = λ3 (n B ) + (λ6 (n B ) + λ7 (n B ))2n2A . This derivative is negative since Λ(n B ) ≡ λ3 (n B ) + (λ6 (n B ) + λ7 (n B )) is negative (and recalling that both λ6 (n B ) and λ7 (n B ) are negative). We indeed have

  Λ(n B ) = − n8B (24n B − 51) + n6B (33n B − 79) + n4B (78n B + 1) + 2n2B (23n B − 6) + 4(2n B + 1)  0.

(A.13)

Therefore, Γ (n A , n B ) is decreasing in n A and reaches a maximum in n A = n B . It follows that a sufficient condition for Γ (n A , n B ) to be negative is that Γ (n A = n B , n B )  0. We indeed have



Γ (n A = n B , n B ) = −n B

9 8 5 2 2 2 2n12 B + n B (2n B − 44) + n B (20n B − 77) + 5n B (14n B − 1) 3 2 2 + n B [n B (100n B − 43) − 31] + 2(17n B + 3n B − 1)



 0.

(A.14)

This expression being negative, it follows that Γ (n A , n B )  0 for any n A  n B . This in turn implies that Ψ (n A , n B ) is negative, which in turn implies that the numerator of (A.10) is negative. As a result, Φ(n A , n B ) is decreasing in n A for any n A  n B  4. We now show that Φ(n A , n B ) is positive in n A = n B , while it is negative when n A goes to infinity (keeping n B unchanged). Using (27) and (35), we have

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G. Cheikbossian / Games and Economic Behavior 74 (2012) 68–82

Φ(n A = n B , n B ) =

(n2B − n B − 1)[n2B (2n B − 3) − (4n B + 3)] (n B + 1)2 (2n B − 1)(n3B − n B − 1)

> 0.

(A.15)

This term is indeed positive since n2B  (4n B + 3)/(2n B − 3) for any n B  4. (The right-hand term of the inequality is decreasing in n B and reaches a maximum in n B = 4, in which case it is equal to 19/5 < 4 < n2B .)

1 N Furthermore, in the limit, we have limn A →∞ δ NC A (n A , n B ) = 1+n B and limn A →∞ δ B (n A , n B ) = that

lim Φ(n A , n B ) = −

n A →∞

1

[1 + n B ][n B (n B − 1) − 1]

< 0.

n B −2 , n B (n B −1)−1

which implies

(A.16)

Therefore, letting n A = γ n B with γ  1 and keeping n B unchanged, there exists γ¯ such that Φ(n A = γ n B , n B ) = 0, and ¯ ¯ hence Φ(n A , n B ) is positive (i.e. δ BN (n A , n B )  δ NC A (n A , n B )) for any γ  γ , while it is negative for any γ  γ . References Abreu, D., 1986. Extremal equilibria of oligopolistic supergames. J. Econ. Theory 39, 191–225. Axelrod, R., 1981. The emergence of cooperation among egoists. Amer. Polit. Sci. Rev. 75, 306–318. Bendor, J., Mookherjee, D., 1987. Institutional structure and the logic of ongoing collective action. Amer. Polit. Sci. Rev. 81 (1), 129–154. Bornstein, G., 2003. Intergroup conflict: individual, group and collective interests. Personal. Soc. Psychol. Rev. 7 (2), 129–145. Bornstein, G., Erev, I., Rosen, O., 1990. Intergroup competition as a structural solution for social dilemma. Soc. Behav. 5, 247–260. Bornstein, G., Gneezy, U., Nagel, R., 2002. The effect of intergroup competition on group coordination: an experimental study. Games Econ. Behav. 41, 1–25. Cheikbossian, G., 2009. The collective action problem: within-group cooperation and between-group competition in a repeated rent-seeking game. SSRN Working Paper. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1115825. Congleton, R.D., Shugart, W.F., 1990. The growth of social security: electoral push or electoral pull. Econ. Inquiry 28, 109–132. Erev, I., Bornstein, G., Galili, R., 1993. Constructive intergroup competition as a solution to the free rider problem: a field experiment. J. Exp. Soc. Psychol. 29, 463–478. Esteban, J., Ray, D., 2001. Collective action and the group size paradox. Amer. Polit. Sci. Rev. 95, 663–672. Friedman, J.W., 1971. A non-cooperative outcome for supergames. Rev. Econ. Stud. 38, 1–12. Hardin, R., 1982. Collective Action. John Hopkins University Press, Baltimore. Katz, E., Nitzan, S., Rosenberg, J., 1990. Rent-seeking for pure public goods. Public Choice 65, 49–60. Konrad, K., 2009. Strategy and Dynamics in Contests. Oxford University Press. Kristov, L., Lindert, P., McClelland, R., 1992. Pressure groups and redistribution. J. Public Econ. 48, 135–163. Lambson, V.E., 1984. Self-enforcing collusion in large dynamics markets. J. Econ. Theory 34 (2), 282–291. Lambson, V.E., 1987. Dynamic behaviour in large markets for differentiated products. Rev. Econ. Stud. 54, 293–300. Leininger, W., Yang, C.-L., 1994. Dynamic rent-seeking games. Games Econ. Behav. 7, 406–427. Nitzan, S., 1991. Collective rent dissipation. Econ. J. 101, 1522–1534. Olson, M., 1965. The Logic of Collective Action. Harvard University Press, Cambridge, MA. Olson, M., 1982. The Rise and Decline of Nations. Yale University Press, New Haven. Pecorino, P., 1998. Is there a free-rider problem in lobbying? Endogenous tariffs, trigger strategies, and the number of firms. Amer. Econ. Rev. 88, 652–660. Potters, J., Sloof, R., 1996. Interest groups: a survey of empirical models that try to assess their influence. Europ. J. Polit. Economy 12, 403–442. Riaz, K., Shogren, J., Johnson, S.R., 1995. A general model of rent-seeking for pure public goods. Public Choice 82, 243–259. Sandler, T., 1992. Collective Action: Theory and Applications. University of Michigan Press, Ann Arbor. Skaperdas, S., 1996. Contest success functions. Econ. Theory 7, 283–290. Taylor, M., 1982. Community, Anarchy and Liberty. Cambridge University Press, Cambridge. Tullock, G., 1980. Efficient rent-seeking. In: Buchanan, J.M., Tollison, R.D., Tullock, G. (Eds.), Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station.