Group cooperation against a hegemon

Abstract

In this paper, I study the ability of a group of citizens to cooperate against a hegemon in a repeated contest game and where group members and the hegemon have different valuations of the prize. I first consider that group members use grim trigger strategies (GTSs) to support cooperative behavior and show that full cooperation within the group is more easily sustained as a stationary subgame perfect (Nash) equilibrium (SSPE) as either group size or the heterogeneity in the valuation of the prize increases. In turn, I show that full cooperation within the group can also be sustained as a weakly renegotiation-proof equilibrium (WRPE). However, an increase in group size makes it more difficult to sustain within-group cooperation, but an increase in the relative valuation of the prize by group members still facilitates group cooperation.

Keywords

Collective action group cooperation renegotiation repeated game trigger strategies

1. Introduction

According to the conventional rational choice theory (Olson, 1965; Tullock, 1971), participation to mass political action, such as protest or rebellion, is a paradox because the group’s success has the characteristics of a (pure) public good whereas participation to collective action induces private costs. In other words, there is a free-rider problem that should discourage people from participating in collective political action. For instance, regarding revolutions, Tullock (1971: 92) writes:

“Under these circumstances, the public good remains in the equation, but has very slight weight unless the individual feels that his participation or non-participation will have a major influence on the outcome. Since most participants in revolution should have no such illusions, it would appear that the public good aspects of a revolution are of relatively little importance in the decision to participate. They should, therefore, be of relatively little importance in determining the outcome of the revolution.”

Both Olson (1965) and Tullock (1971) thus consider that citizens will not participate in mass political action unless there exist “selective incentives” or personal material rewards. The traditional theory of collective action has given rise to numerous conceptual and empirical papers. For instance, Finkel et al. (1989) proposed two explanations of individual participation in a collective protest. The first is that citizens tend to overestimate the effect of their own contributions to the probability of group success.¹ The second type of explanation, what they call the “collective rationality model,” is that citizens’ decisions to participate in collective political action is based on the belief that the participation of all group members is necessary for group success or on the ethical belief that there is a moral obligation to participate in collective action if others are participating. Using three surveys conducted in the Federal Republic of Germany in 1987 and 1988, they show that the first explanation is supported for legal and illegal protests, whereas the “collective rationality” explanation is supported only for legal protests.

Subsequently, scholars have emphasized that collective action does not (only) rely on Olsonian “selective incentives,” but on whether people strongly identify with a group and, thus, feel the moral obligation to do their part in collective action (see, e.g., Goldstone, 1994; Moore, 1995; Oberschall, 1994; Opp, 2012). Furthermore, the social identity theory originally formulated by social psychologists Tajfel and Turner (1979) has pointed out that individuals derive a “social identity” from the group to which they belong even though the group is not formed according to some intrinsic characteristics but by random assignment (Tajfel, 1970). In turn, social identity can foster individual participation in the collective action of one’s own group. Indeed, a large number of experimental studies by social psychologists show, in simple team games, that intergroup competition enhances group effort above the level corresponding to that chosen by a purely self-interested individual (see, e.g., Bornstein, 1992, 2003; Bornstein et al., 2002; Rapoport and Bornstein, 1987, 1989). Economists have also conducted experimental studies with monetary incentives and generally conclude that subjects over-contribute to group effort in group contests or conflicts, compared with the predictions implied by the Homo economicus paradigm (see, e.g., Abbink et al., 2010, 2012).

Another solution to the free-rider problem in collective political action that remains in the tradition of the cost–benefit analysis, without referring to notions such as ethical norms or social identity, is based on strategies of reciprocity when the situation is repeated over time (Axelrod, 1981; Taylor, 1976). The idea is that mutual cooperation in political collective action can be achieved with the use of trigger strategies where cooperation is rewarded with cooperation of the others and defection punished with defection. In other words, cooperation in the repeated game can result from (Nash) non-cooperative behavior where players take their decisions of participating or not to collective action on the basis of a standard cost–benefit analysis. Yet, political scientists doubt that such decentralized strategies of reciprocity can be effective in large groups owing to the problem of monitoring (Hardin, 1982; Olson, 1982; Taylor, 1982). Bendor and Mookherjee (1987) formalized this conjecture by analyzing an n-person repeated prisoner’s dilemma. They indeed formally showed that cooperation cannot be sustained in large groups with imperfect monitoring, which brings us back to the conclusions of Olson (1965) and Tullock (1971).²

A limitation of previous models of participation to collective political action is that they consider a single group in isolation. However, protests and, a fortiori, revolutions aim at challenging the established political power or even at overthrowing it. In the present analysis, I consider that collective political action is undertaken to counteract the power of a hegemon and that the situation is repeated over time. More specifically, I model the conflict between a group of citizens, referred as the challenger group, and the hegemon as a contest game for a prize (or a rent) that has the characteristics of a public good. The challenger group faces the collective action or free-rider problem: each citizen would prefer the others to incur the cost of political activity to oust the hegemon and obtain the prize/rent. The hegemon does not have any collective action problem to solve but must also bear a cost for countering the political action of the challenger group in order to keep the prize/rent.

The probability of success for each political actor, the challenger group and the hegemon, is given by the lottery contest success function proposed by Tullock (1980) where each player’s probability of winning is the ratio of its effort to the total effort. Hence, the probability of success of the challenger group is increasing in the level of its collective effort, which is given by the sum of group members’ efforts. Thus, collective effort is a public good to the members of the challenger group, thus giving rise to the free-rider problem. In this context, I first investigate the effectiveness of grim trigger strategies (GTSs) à la Friedman (1971), for sustaining full cooperation within the group. These strategies prescribe that any deviation from the cooperative level of effort is met with permanent reversion to the non-cooperative outcome within the group. Cooperation within the challenger group can be achieved as a subgame perfect Nash equilibrium if the one-period net gain of deviating from the cooperative agreement is lower than the discounted value of avoiding a (permanent) breakdown of cooperation within the group. The difficulty of supporting cooperation within the group can thus be measured by the lowest discount factor supporting the optimal level of group effort as a subgame perfect outcome. I show that cooperation within the group is more easily sustained as either the relative valuation of the prize by group members or group size increases. This last result is in contrast to the intuition that group size is detrimental to the effectiveness of trigger strategies to support cooperation because of stronger incentives to defect in large groups. The explanation is that in the contest game, the severity of the Nash punishment increases rapidly as group size grows because the free-rider problem within the challenger group makes the hegemon more powerful in the contest.

However, GTSs may not seem realistic for at least two reasons. The first is that the punishment is infinite and unforgiving, which is rather extreme. The second reason, pointed out by Barrett (1999), is that the punishment does not correspond to a collectively rational strategy. Although GTSs are subgame perfect implying that group members have no incentive to deviate unilaterally from the Nash punishment, group members do have an incentive to deviate collectively from the punishment by renegotiating the agreement and returning back (immediately) to the fully cooperative outcome. Therefore, I also investigate whether full cooperation can be sustained with punishments that last only one period and that satisfy both the criterion of subgame perfection and that of robustness to renegotiation. In order to be (weakly) robust to renegotiation, a strategy profile must be such that some players are at least as well off with the punishment as with renegotiation. A strategy profile that satisfies the two requirements of subgame perfection and renegotiation-proofness is a weakly renegotiation-proof equilibrium (WRPE), in the sense of Farrell and Maskin (1989). We thus consider a strategy profile proposed by Asheim et al. (2006) called penance-p, which prescribes that a deviation from cooperation triggers a punishment period with a subset of group members continuing to exert the cooperative level of effort and the remaining group members, the punishing members, exerting no effort at all. I show that full cooperation can be sustained as a WRPE with penance-p even in large groups and that an increase in the relative valuation of the prize by group members still facilitates group cooperation. However, and in contrast to GTSs, within-group cooperation becomes more difficult to sustain as group size grows owing to the renegotiation-proofness criterion. Thus, in a dynamic setting, the requirement of collective rationality applied to punishment strategies bring us back to the conjecture that larger groups are disadvantaged in maintaining cooperation.

My analysis is related to the literature on tacit cooperation in repeated games applied to public economics. McMillan (1979) was the first to apply GTSs to a supergame model of private provision of a public good but without investigating the effect of an increase in the number of agents. Pecorino (1999) focuses on the effect of an increase in the number of contributors on the minimum value of the discount parameter above which full cooperation in the provision of a (pure) public good can be sustained. He cannot derive monotonicity results but shows that cooperation can be maintained when the number of contributors goes to infinity for several specifications of the individual payoff function. Haag and Lagunoff (2007) analyzed a collective action game played by a collection of heterogeneous individuals in terms of time preferences. They showed that homogeneous and larger groups are more cooperative on average for a large class of collective action games.

The present analysis is also related to the literature on international agreements. Barrett (1999) developed a model of international cooperation where each country has a binary choice between cooperating or defecting and showed that cooperation can be supported as a WRPE for a limited number of countries.³ In a similar model, Asheim et al. (2006) showed that two international agreements can do better than a single one for involving a larger number of countries in an agreement. Still in a model where players have a binary choice, Froyn and Hovi (2008) showed that full participation can be sustained as WRPE provided that only a limited number of countries are permitted to punish a defection. Finally, Asheim and Holtsmark (2009) showed that this result carries over a more general model where players or countries have a continuum of (emission) choices. However, they considered a very simple model because all players have dominant strategies and can all participate in the cooperative agreement. In contrast to all above-mentioned works that analyze tacit cooperation in isolated groups (of players or countries), the contribution of the present analysis is to investigate the problem of sustaining group cooperation against an active hegemon.

Finally, my work is related to the literature on group contests (see, e.g., Katz et al., 1990; Nitzan, 1991; Riaz et al., 1995; and for a survey, see Konrad, 2009). In particular, Esteban and Ray (2001) considered a contest between two groups of different sizes for the allocation of a prize with a varying mix of public and private characteristics. It is a static game and all players in the two groups decide independently of the others, including in-group members, of their contributions to group effort. They showed that if the individual cost of contributing to group action is sufficiently convex, then the larger group is more successful. The reason is that the higher level of individual effort contributed in the smaller group, owing to lower free-rider incentives, is not sufficient, owing to the convexity of the cost function, to counterbalance the lower number of contributors (see also Cheikbossian and Fayat, 2018). In another paper (Cheikbossian, 2012), I also considered a contest game between two groups of unequal size and where all individuals have the same valuation of the prize. I then investigated the effectiveness of Nash punishment for sustaining cooperation in one group or both, and showed 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. Again, in the present analysis, I consider a conflict between one group facing a collective action problem and a hegemon, with both sides having different valuations of the prize/rent. More importantly, I also consider repeated game strategies that comply with a collective rationality requirement of renegotiation-proofness.

2. The model

2.1. The stage game

I start by specifying the details of the stage game G. There are $n + 1$ risk-neutral agents, indexed by $i = 0, 1, 2, \dots, n$ who compete for a public prize. Agent $0$ is a unitary entity, called the hegemon, and has a valuation $V_{I}$ for the prize. Agents $i = 1, 2, \dots, n$ are grouped into a common entity, called the challenger group, and join their efforts to challenge the hegemon in the award of the prize. All group members have the same valuation $V_{G}$ of the prize. We suppose that $V_{G} \geq V_{I}$ because the prize is a pure public good and, thus, $V_{G}$ is essentially the sum of the (identical) valuations of a group of n agents whereas $V_{I}$ is the valuation of a unique agent. These valuations are publicly known. Let $x_{i} \in ℜ_{+}$ be the contest expenditure/effort expended at unit cost by group member i, and $X = \sum_{i = 1}^{n} x_{i} \in ℜ_{+}$ be the sum of contest efforts of group members. The contest expenditure/effort of the hegemon is denoted $Y \in ℜ_{+}$ . Following much of the contest literature, I assume that the probability of winning the prize for group members is given by a contest success function $p : ℜ_{+}^{2} \to [0, 1]$ , which has the ratio form (Tullock, 1980), i.e.,

p (X, Y) = {\begin{matrix} X / (X + Y) & if (X, Y) \neq (0, 0) \\ 1 / 2 & if (X, Y) = (0, 0) \end{matrix}

(1)

The preferences of all players are represented by an additively separable utility function, i.e.,

v_{i} = p (X, Y) V_{G} - x_{i}

(2)

for member $i = 1, 2, \dots, n$ of the challenger group, and

v_{0} = (1 - p (X, Y)) V_{I} - Y

(3)

for the hegemon.

I first analyze the one-shot equilibrium outcome in which group members do not cooperate in the contest. Let $X_{- i} = \sum_{j \neq i} x_{j}$ be the sum of contest efforts of the group members excluding that of member i. Maximizing $v_{i}$ with respect to $x_{i}$ subject to the non-negativity constraint $x_{i} \geq 0$ yields group member i’s best response to $X_{- i}$ and to Y, i.e., $r_{i} (X_{- i}, Y)$ , which is given by the following first-order condition⁴

\frac{Y}{{(x_{i} + X_{- i} + Y)}^{2}} V_{G} \leq 1

(4)

with equality for $x_{i} > 0 .$

Similarly, maximizing $v_{0}$ with respect to Y subject to the non-negativity constraint $Y \geq 0$ , yields the best-response function of the hegemon to the collective effort X of the challenger group, i.e., $R_{0} (X)$ , which is given by the following first-order condition

\frac{X}{{(X + Y)}^{2}} V_{I} \leq 1

(5)

with equality for $Y > 0$ .

Owing to the public-good nature of collective effort, first-order conditions only determine group effort. I look, however, for an interior symmetric equilibrium such that all group members make the same level of individual effort. The symmetric equilibrium is thus a couple $(x^{N}, Y^{N})$ such that $R_{0} (X^{N}) = Y^{N}$ and $r_{i} (X_{- i}^{N}, Y^{N}) = x^{N}$ for all $i = 1, 2, \dots, n$ (and where $X_{- i}^{N} + x^{N} = X^{N} = n x^{N}$ ).

Solving the system given by (4) and (5) with equality and letting $λ$ be the ratio of the valuations of the prize, i.e., $λ = V_{G} / V_{I} \geq 1$ , we have that

x^{N} = \frac{λ}{n {[1 + λ]}^{2}} V_{G}

(6)

whereas $Y^{N} = X^{N} / λ$ .

The equilibrium winning probability for the challenger group is thus given by

p (X^{N}, Y^{N}) = \frac{λ}{1 + λ}

(7)

whereas that of the hegemon is $1 - p (X^{N}, Y^{N})$ . Note that these probabilities are independent of the size of the challenger group. Indeed, with the lottery contest success function and linear cost functions, the increased free-rider problem induced by a larger group size is exactly compensated by the larger number of contributors (see Katz et al., 1990; Ursprung, 1990).

Under non-cooperation, each group member has thus the following utility

v^{N} = \frac{λ [n (1 + λ) - 1]}{n {[1 + λ]}^{2}} V_{G}

(8)

whereas that of the hegemon is given by

v_{0}^{N} = \frac{V_{I}}{{[1 + λ]}^{2}}

(9)

2.2 The optimal solution

Consider now a situation where there can be full cooperation within the challenger group. In other words, group members can jointly choose individual efforts so as to maximize the aggregate welfare of their group, i.e., they choose X so as to maximize $\sum_{i} v_{i} = np (X, Y) - X$ , given Y. The collective best response of the challenger group to Y, i.e., $R (Y)$ , is given by the following first-order condition

\frac{nY}{{(X + Y)}^{2}} V_{G} \leq 1

(10)

with equality for $X > 0$ .

Similarly, the best-response function of the hegemon to the collective effort X of the challenger group, i.e., $R_{0} (X)$ , is given by the following first-order condition

\frac{X}{{(X + Y)}^{2}} V_{I} \leq 1

(11)

with equality for $Y > 0$ .

An interior equilibrium is a couple of efforts $(X^{C}, Y^{C})$ such that $R (Y^{C}) = X^{C}$ and $R_{0} (X^{C}) = Y^{C}$ . Assuming that the members of the challenger group share equally the collective effort, and denoting by $x^{C} = X^{C} / n$ this common individual effort under within-group cooperation, I obtain

x^{C} = \frac{λ n}{{[1 + λ n]}^{2}} V_{G}

(12)

and $Y^{C} = x^{C} / λ$ .

As a result the equilibrium winning probability for the challenger group is given by

p (X^{C}, Y^{C}) = \frac{λ n}{1 + λ n}

(13)

whereas that of the hegemon is $1 - p (X^{C}, Y^{C})$ .

Under cooperation within the challenger group, each group member thus has the following utility

v^{C} = {[\frac{λ n}{1 + λ n}]}^{2} V_{G}

(14)

whereas that of the hegemon is given by

v_{0}^{C} = \frac{V_{I}}{{[1 + λ n]}^{2}}

(15)

Of course, group members obtain a higher utility level when they fully cooperate than when they decide independently of each other of their contributions to collective effort, i.e., $v^{C} > v^{N}$ . Conversely, the hegemon obtains a lower utility level when he faces a cooperative group than when he faces a non-cooperative group, i.e., $v_{0}^{C} < v_{N}^{C}$ .

3. The repeated game

3.1. Preliminaries

The $n + 1$ agents play an infinitely repeated game with discounting. Time is discrete and dates are denoted by $t = 0, 1, 2, \dots$ . Let $G^{\infty} (δ)$ be the repeated game obtained by repeating G infinitely often, and where $δ \in (0, 1)$ is the discount parameter per period for each player. Let $Y (t) \in ℜ_{+}$ and $x (t) \equiv (x_{1} (t), x_{2} (t), \dots, x_{n} (t)) \in ℜ_{+}^{n}$ be, respectively, the effort of the hegemon and the vector of individual efforts within the challenger group in period t. A publicly observable history in period $t \geq 1$ is thus $h (t) = ((x (0), Y (0)), (x (1), Y (1)), \dots, (x (t - 1), Y (t - 1)))$ or $h (t) = {(x (s), Y (s))}_{s = 0}^{t - 1}$ . The initial history is the null set, $H_{0} \equiv {\emptyset}$ , and $H_{t}$ the set of t-period histories. A pure strategy for group member i in $G^{\infty} (δ)$ , for $i = 1, 2, \dots, n$ , is an infinite sequence of functions $σ_{i} = {σ_{i}^{t}}_{t = 1}^{\infty}$ , where $σ_{i}^{t} : H_{t} \to ℜ_{+}$ is a mapping from the set of all possible public histories up to $t - 1$ (included) into the set of effort levels. Similarly, a pure strategy in $G^{\infty} (δ)$ for the hegemon is defined on the same set, i.e., $σ_{0} = {σ_{0}^{t}}_{t = 1}^{\infty}$ , where $σ_{0}^{t} : H_{t} \to ℜ_{+}$ . The strategy profile is denoted by $σ \equiv (σ_{0}, σ_{1}, σ_{2}, \dots, σ_{n})$ .

Any strategy profile $σ$ generates an outcome path $S = {x (t), Y (t)}_{t = 0}^{\infty}$ , defined inductively by ${x (0), Y (0)} = σ (\emptyset)$ and $(x (t), Y (t)) = (σ_{0}^{t}, σ_{1}^{t}, \dots, σ_{n}^{t})$ for all $t \geq 1$ . A path of contest efforts S thus implies an infinite stream of stage-game payoffs ${v_{i} (x (t), Y (t))}_{t = 0}^{\infty}$ for group member $i = 1, 2, \dots, n$ and ${v_{0} (x (t), Y (t))}_{t = 0}^{\infty}$ for the hegemon. Let $v_{i} (t) \equiv$ $v_{i} (x (t), Y (t))$ for $i = 0, 1, 2, \dots, n$ , with $i = 0$ for the hegemon and $i = 1, 2, \dots, n$ for the members of the challenger group. The average discounted payoff of player i for the outcome path $S = {x (t), Y (t)}_{t = 0}^{\infty}$ is given by

V_{i}^{δ} (S) = (1 - δ) \sum_{t = 0}^{\infty} δ^{t} v_{i} (t)

(16)

Thus, the average discounted payoff of player i, for $i = 0, 1, 2, \dots, n$ in $G^{\infty} (δ)$ obtained with the strategy profile $σ$ is $V_{i}^{δ} (σ) = V_{i}^{δ} (S (σ))$ . $σ$ is a Nash equilibrium if $σ_{0}$ is a best response to $(σ_{1}, σ_{2}, \dots, σ_{n})$ and if $σ_{i}$ , for $i = 1, \dots, n$ , is a best response to $σ_{- i} = (σ_{0}, σ_{1}, \dots, σ_{i - 1}, σ_{i + 1}, \dots, σ_{n})$ . In addition, it is a subgame perfect Nash equilibrium in $G^{\infty} (δ)$ if after every history, the continuation of $σ$ is a Nash equilibrium in the corresponding subgame. I will restrict attention to stationary subgame perfect (Nash) equilibria (SSPE), i.e., equilibria in which after any history, a stationary profile of strategies is played thereafter, and which also satisfy the additional requirement of symmetry within the challenger group, in the sense that all group members make the same level of effort at every history.

3.2. GTSs

I first consider that the members of the challenger group use GTSs à la Friedman (1971) in order to support cooperative behavior. GTSs prescribe that group members cooperate in the first period and in all subsequent periods if all members cooperated in the previous period. If any member deviated in the previous period, then all group members revert to the single-shot Nash equilibrium forever, independently of the behavior of the hegemon. Finally, it is worth pointing out that GTSs do not require perfect observation of individual efforts but only perfect observation of the aggregate level of contributions made in the previous periods.

Group members’ actions must be mutual best responses and must also be individual best responses to the action of the hegemon, in case of non-cooperation within the group, or must constitute a collective best-response function to the action of the hegemon, in case of cooperation within the group. Furthermore, it is supposed that the hegemon plays in every period his static best response to the group members’ strategies, depending on whether they cooperate or not, regardless of the history (i.e., $σ_{Y}^{t} = R_{0} (x (t))$ for all t). To put this informally, in every period, the challenger group and the hegemon play non-cooperatively against each other whereas cooperation within the challenger group can be maintained depending on the history of the game.

In general, in infinitely repeated games, the set of SSPE is very large. I thus focus on the “best” SSPE from the viewpoint of group members, that is the one sustaining the joint-payoff maximizing level of contest effort $x^{C}$ .⁵ Formally, I thus focus on the strategy profile characterized by: $σ_{i}^{t} = x^{C}$ for $i = 1, 2, \dots, n$ , and $σ_{0}^{t} = R_{0} (X^{C}) = Y^{C}$ if $t = 0$ or if $h (t) = {(x^{C} (s), Y^{C} (s))}_{s = 0}^{t - 1}$ for $t \geq 1$ ; $σ_{i}^{t} = x^{N}$ for $i = 1, 2, \dots, n$ and $σ_{0}^{t} = R_{0} (X^{N}) = Y^{N}$ otherwise. This strategy profile is denoted by $\tilde{σ}$ .

Let $x^{D}$ be the effort level of a group member who considers defecting from the cooperative phase while other members play $\tilde{σ}$ . The payoff of the deviator would thus be given by

v^{D} = \frac{(n - 1) x^{C} + x^{D}}{(n - 1) x^{C} + x^{D} + Y^{C}} V_{G} - x^{D}

(17)

where $x^{C}$ is given by (12) and where $x^{C} = λ Y^{C}$ . The deviator chooses $x^{D}$ so as to maximize $v^{D}$ . The first-order condition to this problem implies that

\frac{\partial v^{D}}{\partial x^{D}} = \frac{Y^{C}}{{[(n - 1) x^{C} + x^{D} + Y^{C}]}^{2}} V_{G} - 1 \leq 0

(18)

In any period in which all group members contribute the joint-maximizing level of effort $x^{C}$ , each member’s best possible deviation from $\tilde{σ}$ is, thus, given by

x^{D} = {\begin{matrix} \frac{\sqrt{n} (1 + λ n) - n [1 + λ (n - 1)]}{{(1 + λ n)}^{2}} V_{G} & if n = 2 \\ 0 & if n \geq 3 \end{matrix}

(19)

Thus, each member’s best possible deviation from $\tilde{σ}$ is to cut her contribution to $0$ for any $n \geq 3$ , independently of $λ$ . In contrast, we have $x^{D} > 0$ for $n = 2$ , independently of $λ$ .⁶

Substituting (19) into (17) and using (12), I obtain the (best) deviation payoff for any group member, that is

v^{D} = {\begin{matrix} \frac{4 λ^{2} + (3 - 2 \sqrt{2}) (1 + 2 λ)}{{(1 + 2 λ)}^{2}} V_{G} & if n = 2 \\ \frac{λ (n - 1)}{1 + λ (n - 1)} V_{G} & if n \geq 3 \end{matrix}

(20)

The one-period net gain to deviating from the cooperative agreement within the group is $v^{D} - v^{C}$ , whereas the per-period net benefit to maintaining cooperation within the group is $v^{C} - v^{N}$ . Thus, the discounted value of avoiding a (permanent) breakdown of cooperation within the group is given by $[δ / (1 - δ)] (v^{C} - v^{N})$ . Therefore, no group members have an incentive to deviate from the cooperative agreement if and only if

v^{D} - v^{C} \leq \frac{δ}{1 - δ} [v^{C} - v^{N}]

(21)

Focusing on situations where self-enforcement is a binding constraint on the abilities of group members to cooperate, the critical value of the discount parameter above which cooperation can be sustained as a SSPE through GTSs is then $δ = (v^{D} - v^{C}) / (v^{D} - v^{N})$ . Substituting (8), (14), and (20) into this expression, I obtain

δ^{N} (2) = \frac{2 (3 - 2 \sqrt{2}) (1 + 2 λ) (1 + λ)^{2}}{8 (2 - \sqrt{2}) λ^{3} + 4 (8 - 5 \sqrt{2}) λ^{2} + (23 - 16 \sqrt{2}) λ + 2 (3 - 2 \sqrt{2})}

(22)

if $n = 2$ , and

δ^{N} (n) |_{n \geq 3} = \frac{n {(1 + λ)}^{2} [λ n (n - 2) + (n - 1)]}{{(1 + λ n)}^{2} [λ (n^{2} - n - 1) + {(n - 1)}^{2}]}

(23)

if $n \geq 3$ .

I have the following result.⁷

Result 1: (i) $δ^{N} (n)$ is decreasing innfor any $n \geq 3$ independently of $λ$ ; (ii) $δ^{N} (n)$ is decreasing in $λ$ for any $n \geq 2$ independently ofn.

The sustainability of within-group cooperation depends on the effect of increasing n or $λ$ on the one-period net gain of deviating from the cooperative agreement, given by $v^{D} - v^{C}$ , relative to its effect on the net benefit of maintaining within-group cooperation, given by $v^{C} - v^{N}$ . I focus on the situation where $n \geq 3$ . I have

v^{C} - v^{N} = \frac{λ (n - 1) [λ n^{2} (1 + λ) - (1 + λ n)]}{n {(1 + λ)}^{2} {(1 + λ n)}^{2}} V_{G}

(24)

and

v^{D} - v^{C} = \frac{λ [λ n (n - 2) + (n - 1)]}{[1 + λ (n - 1)] {(1 + λ n)}^{2}} V_{G}

(25)

As shown in the Proof of Result 1 (in the Appendix), the net benefit of cooperation, given by $v^{C} - v^{N}$ , is increasing in n. This reflects the fact that an increase in group size exacerbates the free-rider problem, which, in turn, makes the punishment threat relatively more effective. As regards the effect of group size on the incentive to deviate, that is on the net benefit of defection, given by $v^{D} - v^{C}$ , it is decreasing in n for any $n \geq 4$ . The incentive to deviate from the cooperative outcome is determined by the benefit of withdrawing his/her individual contribution and by the resulting negative effect on the probability of success of the group. However, this probability is concave in aggregate effort. Hence, the decrease in the probability of success owing to an individual defection becomes less pronounced as n increases, which would reinforce the incentive to defect. However, the concavity of the probability of success also implies that, when n increases, it increases the optimal level of aggregate effort but by a lower extent than the size of the group. In turn, an individual contribution and, thus, the benefit of withdrawing his/her contribution also decreases with n, which would relax the incentive to deviate. It turns out that the decrease in the benefit of withdrawing his/her individual contribution dominates that of the associated cost in terms of probability of success. As a result, the net benefit of defection decreases with group size and this effect goes in the same direction than the increase in the punishment threat for facilitating the sustainability of group cooperation when $n \geq 4$ . When group size grows from three to four members, the incentive to deviate increases, but this is more than compensated by the increase in the net benefit of cooperation for making cooperation easier to sustain.

This result is in contrast to the conjecture that decentralized strategies of reciprocity may be less effective at sustaining cooperation in larger groups because of the larger incentives to defect. In the present analysis, where group members face a hegemon, I have that an increase in group size unambiguously facilitates group cooperation. Furthermore, cooperation remains an equilibrium in very large groups if their members place any weight at all on the future because, then, the critical discount factor $δ^{N} (n) |_{n \geq 3}$ converges to zero as n goes to infinity.

In addition, an increase in the valuation of the prize by group members compared with that of the hegemon, i.e., an increase in $λ$ , makes within-group cooperation less difficult to sustain.⁸ As shown in the Appendix, an increase in $λ$ increases the net benefit of cooperation and, thus, the effectiveness of the punishment threat as does an increase in group size. However, it also increases the net benefit of deviating from the cooperative outcome. That is, as $λ$ increases, the benefit of withdrawing his/her individual contribution becomes more important than the cost in terms of probability of success. As a result the net benefit of defection increases, but this effect is always dominated by the increased punishment threat as $λ$ rises, which, in turn, makes within-group cooperation easier to sustain.

3.3. Renegotiation

Although infinite reversion to non-cooperation within the group is subgame perfect, it seems not very realistic because it is unforgiving and because it hurts the punishers as well as the deviator. Furthermore, it is hard to believe that agents remain forever in a Pareto-dominated equilibrium without renegotiating back to a cooperative outcome. In other words, the punishment scheme does not satisfy a criterion of “collective rationality.” I thus now consider another strategy profile that is robust to renegotiation. Specifically, I analyze whether full cooperation can be sustained as a WRPE, in the sense of Farrell and Maskin (1989). To be WRPE, a strategy profile must satisfy two requirements: (i) that of subgame perfection, which requires that no player can gain by a one-period deviation after any history; (ii) after any history, there must not exist two continuation equilibria such that all players are better off in one continuation equilibrium than in the other. Otherwise, the players would renegotiate from one to the other.

A strategy profile satisfying these two requirements is penance-p, which has been proposed by Asheim et al. (2006) and Froyn and Hovi (2008) in the context of a n-person prisoner’s dilemma. Our contribution here is to extend the use of this concept to a game of group cooperation with continuous payoffs and strategic interactions between players and, more importantly, with one player, the hegemon, not participating to the agreement but fighting against the cooperative group. Specifically, the penance-p subgame perfect strategy profile, denoted by $\tilde{σ} (p)$ , specifies: (i) that a group member contributes the joint-maximizing level of effort except if another group member has been the sole deviator from penance-p in the previous period; (ii) that if a deviation occurs, then p group members, the punishing members, stop contributing to collective action, whereas the $n - p$ other group members continue to act cooperatively. This means that they produce the joint-maximizing level of group effort of a group consisting of $n - p$ members, whereas the p punishing members do not contribute at all to collective action but still benefit from the collective action of their group. Punishments last only one period and so this strategy profile is robust to renegotiation if not all group members strictly gain by restarted full cooperation instead of carrying out the punishment. Finally, it is important to note that, in contrast to GTSs, penance-p requires perfect monitoring, i.e., that everyone can observe, without error, individual levels of effort made in the previous period.

Again, I focus on the “best” WRPE from the viewpoint of group members, that is the one sustaining the joint-payoff maximizing level of contest effort $x^{C}$ given by (12). Recall that $x (t) \equiv (x_{1} (t), x_{2} (t), \dots, x_{n} (t)) \in ℜ_{+}^{n}$ . Thus, the penance-p strategy profile can be formally described as follows

σ_{i}^{t} = {\begin{matrix} x^{C} & if t = 0 or if (x (t - 1), Y (t - 1)) = (x^{C}, Y^{C}) \\ x^{P} & if x_{i} (t - 1) \neq x^{C} or if x_{j} (t - 1) \neq x^{C} for j \neq i and i \notin P_{j} \\ 0 & if x_{j} (t - 1) \neq x^{C} for j \neq i and i \in P_{j} \end{matrix}

(26)

for $i = 1, 2, \dots, n$ .

$x^{P}$ is the joint-payoff maximizing level of contest effort of a group of size $(n - p)$ , and where $P_{j}$ is the set of punishing members when member j deviates from $\tilde{σ} (p)$ . As in the previous section, the hegemon is assumed to play in every period his static best response to the group members’ strategies, depending on whether they correspond to a cooperative period or punishment period, regardless of the history, that is $σ_{0}^{t} = R_{0} (x (t))$ for all t.

3.3.1. The subgame-perfection requirement

Consider first the requirement of subgame perfection. There are two kinds of histories to check: when no deviation has taken place in the previous round $t - 1$ and when a deviation did occur in $t - 1$ .

No deviation in period $t - 1 .$ In this history, penance-p prescribes that all group members continue to cooperate. If, however, one member deviates in period t and reverts to penance-p in $t + 1$ , then his/her deviation payoff is $v^{D}$ given by (20) in period t, whereas in period $t + 1$ he/she obtains the payoff of group cooperation with $n - p$ contributing members. Substituting n by $n - p$ into (14), I obtain

v^{P} = \frac{{[λ (n - p)]}^{2}}{{[1 + λ (n - p)]}^{2}} V_{G}

(27)

In period $t + 2$ onward, full cooperation is restored so that member i obtains $v^{C}$ , given by (14) in that period and in any future period. Therefore, any group member has no incentive to unilaterally deviate from the cooperative outcome unless he/she can obtain a larger discounted sum of payoffs in period t and $t + 1$ by deviating, that is if and only if $v^{D} + δ v^{P} \leq (1 + δ) v^{C}$ or $δ \geq$ $δ^{P}$ with $δ^{P} \equiv (v^{D} - v^{C}) / (v^{C} - v^{P})$ .

Recall that the best deviation payoff $v^{D}$ from the cooperative phase depends on whether $n = 2$ or $n \geq 3$ . If $n = 2$ , then $p = 1$ . Indeed p must be always strictly positive, otherwise defection would never be punished and so each member would always defect. It must also be lower than two, otherwise the strategy profile would correspond to Nash reversion and the renegotiation-proofness requirement would not be satisfied. Then, using (14), the first expression in (20), and (27), I obtain for $n = 2$ and $p = 1,$

δ^{P} (2, 1) = \frac{(3 - 2 \sqrt{2}) (1 + 2 λ) (1 + λ)^{2}}{λ^{2} (3 + 4 λ)}

(28)

Now consider that $n \geq 3$ . In this case, the member who deviates from the cooperative phase cuts her contribution to $0$ (see (19)), so that the best deviation payoff is given by the second expression in (20). Also using (14) and (27), I then obtain the following threshold value of the discount parameter

δ^{P} (n, p) |_{n \geq 3} = \frac{{[1 + λ (n - p)]}^{2} [λ n (n - 2) + (n - 1)]}{λ p [1 + λ (n - 1)] [2 λ n (n - p) + 2 n - p]}

(29)

We now have to verify the subgame-perfect requirement, when a single deviation has taken place in period $t - 1$ .

A deviation in period $t - 1 .$ In this history, penance-p prescribes that in period t, there are ppunishing members who do not contribute to the collective action of the group. I first determine under which condition the $n - p$ group members (including the member who deviated in $t - 1$ ), that are prescribed to cooperate to the benefit of the whole group by contributing $x^{P}$ , have no incentive to deviate from penance-p in period t.

If $n = 2$ (and, thus, $p = 1$ ) or $n - p = 1$ , the best deviation payoff from the punishment phase, denoted $v^{DP}$ is equal to $v^{P}$ because $v^{P}$ is the optimal payoff of the unique member who exerts contest efforts in t (because he/she deviated in $t - 1$ ). Thus, the incentive compatibility constraint on the punishment path is always satisfied if $n - p = 1$ .

Now suppose that $(n - p) > 1$ and let $x^{DP}$ be the effort level of a contributing member who deviates from the punishment in period t. The best deviation payoff of a contributing member in the punishment phase who decides to deviate depends on whether $n - p = 2$ or $n - p \geq 3$ and is given by (20) with n replaced by $n - p$ for $n \geq 3$ , that is,

v^{DP} = {\begin{matrix} \frac{4 λ^{2} + (3 - 2 \sqrt{2}) (1 + 2 λ)}{{(1 + 2 λ)}^{2}} V_{G} & if n - p = 2 \\ \frac{λ (n - p - 1)}{1 + λ (n - p - 1)} V_{G} & if n - p \geq 3 \end{matrix}

(30)

Therefore, each of the $n - p$ contributors in period t has no incentive to unilaterally deviate from penance-p unless he/she can obtain a larger discounted sum of payoffs in period t and $t + 1$ by deviating, that is, if and only if $v^{DP} + δ v^{P} \leq v^{P} + δ v^{C}$ or $δ \geq δ^{D P}$ with $δ^{DP} \equiv (v^{DP} - v^{P}) / (v^{C} - v^{P})$ .

Thus, using (14), (27), and (30), the minimum discount factor above which no contributing member in the punishment phase has an incentive to deviate is given by

δ^{DP} (n) |_{n - p = 2} = \frac{(3 - 2 \sqrt{2)} (1 + 2 λ) (1 + λ n)^{2}}{(n - 2) [n + 2 + 4 λ n] λ^{2}}

(31)

if $n - p = 2$ and

δ^{DP} (n, p) |_{n - p \geq 3} = \frac{{(1 + λ n)}^{2} [λ (n - p) (n - p - 2) + (n - p - 1)]}{λ p [λ (n - p - 1) + 1] [2 λ n (n - p) + 2 n - p]}

(32)

if $n - p \geq 3$ .

I now establish the following result that will prove useful in the rest of the analysis.

Result 2: The minimum discount factors $δ^{P} (n, p) |_{n \geq 3}$ and $δ^{DP} (n, p) |_{n - p \geq 3}$ above which the strategy profile $\tilde{σ} (p)$ satisfies the subgame-perfect requirements are both decreasing in the number of punishing members p.

I also need to verify that the punishing members who do not contribute at all to collective action in the punishment period have no incentives to deviate by producing a positive level of effort. If the p punishing members adhere to the punishment in period t and do not participate to collective action, then the payoff of each of them is given by the probability of success (multiplied by $V_{G}$ ) of a cooperative group consisting of $n - p$ members, that is, from (13),

v^{PA} = \frac{λ (n - p)}{1 + λ (n - p)} V_{G}

(33)

If $n - p \geq 3$ , then each group member’s best possible deviation is to cut her contribution to $0$ , which corresponds exactly to what is prescribed by penance-p for the p punishing members. Thus, in this case, the punishing members do not have any incentives to deviate in the punishment phase, all the more it will trigger the punishment again in $t + 1$ .

If $n - p \leq 2$ , then the best deviation for a contributing member is to make a positive level of effort. However, a non-contributing member still cannot do better than not contributing at all to collective action independently of n and p. Indeed, let $x^{A}$ be the effort level of a punishing member who considers defecting in the punishment period t. His/her payoff would be given by

v^{A} = \frac{(n - p) x^{P} + x^{A}}{(n - p) x^{P} + x^{A} + Y^{P}} V_{G} - x^{A}

(34)

where $x^{P} = λ Y^{P}$ . Here $\partial v^{A} / \partial x^{A}$ is decreasing in $x^{A}$ and, thus, a punishing member does not have any incentive to deviate from penance-p by producing a positive level of effort if and only if $(\partial v^{A} / \partial x^{A}) |_{x^{A} = 0} \leq 0$ , that is, if $V_{G} / {[1 + λ (n - p)]}^{2} \leq Y^{P}$ . As $Y^{P} = x^{P} / λ = [(n - p) / {[1 + λ (n - p)]}^{2}] V_{G}$ , this inequality is always satisfied for $n - p \geq 1$ .

3.3.2. The renegotiation-proofness requirement

I now turn to the renegotiation-proofness requirement. It implies an upper bound on the number of punishing players. Indeed, full cooperation is weakly renegotiation-proof if not all group members are strictly worse off with the punishment than with renegotiation (i.e., than returning immediately to cooperation without punishment). Suppose member i deviates in $t - 1$ . Clearly, in period t, the $n - p$ members (including the defecting member of period $t - 1$ ) who contribute to group effort are worse off with the punishment than with renegotiation. Thus, weak-renegotiation proofness requires that the p punishing members are at least as well off with punishment as with renegotiation.

If the p punishing members adhere to the punishment in period t and do not participate to collective action, then the payoff of each of them is $v^{PA}$ given by (33). If, however, full cooperation is restored immediately, then all members obtain $v^{C}$ given by (14). In either cases, all group members obtain $v^{C}$ in $t + 1$ onward. Thus, weak renegotiation-proofness requires that $v^{PA} \geq v^{C}$ or $p \leq \bar{p}$ , with

\bar{p} \equiv \frac{n (1 + λ n)}{1 + 2 λ n}

(35)

Intuitively, the number of punishing members must not be too large, otherwise, the effective level of collective action in the punishment path is too low to prevent renegotiating back to the fully cooperative outcome. One can also easily verify that $\bar{p} \in (n / 2, (n + 1) / 2)$ for any $λ > 1$ . Thus, the maximal number of punishing members satisfying the renegotiation-proofness requirement will be given by

\bar{p} = n / 2 if n is even or \bar{p} = (n - 1) / 2 if n is odd

(36)

From Result 2, I have that $δ^{P} (n, p) |_{n \geq 3}$ is decreasing in the number of punishing members. Thus, substituting (36) into (29), the lowest values of the discount factor satisfying the incentive compatibility constraint on the cooperative phase are given by

δ^{P} (n) = | \begin{matrix} \frac{{[2 + λ (n + 1)]}^{2} [λ n (n - 2) + (n - 1)]}{λ (n - 1) [1 + λ (n - 1)] [1 + 3 n + 2 λ n (n + 1)]} if n = 3, 5, 7, . . . & if n = 3, 5, 7, \dots \\ \frac{{[2 + λ n]}^{2} [λ n (n - 2) + (n - 1)]}{λ n^{2} (3 + 2 λ n) [1 + λ (n - 1)]} if n = 4, 6, 8, . . . & if n = 4, 6, 8, \dots \end{matrix}

(37)

$δ^{DP} (n, p) |_{n - p \geq 3}$ given by (29) is also decreasing in the number of punishing members provided that $n - p \geq 3$ . It can be easily verified that this inequality reduces to $n \geq 5$ if n is odd with $\bar{p} = (n - 1) / 2$ ; whereas it reduces to $n \geq 6$ if n is even with $\bar{p} = n / 2$ . Thus, under these conditions and substituting (36) into (32), the lowest values of the discount factor satisfying the incentive compatibility constraint on the punishment phase are given by

δ^{DP} (n) = | \begin{matrix} \frac{2 {[1 + λ n]}^{2} [λ (n + 1) (n - 3) + 2 (n - 1)]}{λ (n - 1) [2 + λ (n - 1)] [1 + 3 n + 2 λ n (n + 1)]} if n = 5, 7, 9 \dots \\ \frac{2 {[1 + λ n]}^{2} [λ n (n - 4) + 2 (n - 2)]}{λ n^{2} (3 + 2 λ n) [2 + λ (n - 2)]} if n = 6, 8, 10 . \end{matrix}

(38)

Thus, provided that $n \geq 5$ , penance-p strategies with full cooperation within the challenger group is a WRPE if and only if $δ \geq$ Max ${δ^{P} (n), δ^{DP} (n)} .$

Let us now consider that $n \leq 4$ . Again, with a group of two members, subgame-perfection only requires that each member has no incentive to deviate from the cooperative path, i.e., $δ \geq δ^{P} (2, 1)$ , where $δ^{P} (2, 1)$ is given by (28). The renegotiation-proofness requirement is also satisfied because $p = 1$ . Now, for a group of three members, the renegotiation-proofness requirement also implies that $p = 1$ . Subgame-perfection requires that $δ \geq Max {δ^{P} (3), δ^{DP} (3) |_{n - p = 2}}$ , where $δ^{P} (3)$ and $δ^{DP} (3) |_{n - p = 2}$ are given by (37) and (31), respectively. For $n = 4$ , the renegotiation-proofness requirement is such that $p \leq 2$ . As $δ^{P} (n, p) |_{n \geq 3}$ is decreasing in p, $δ^{P} (4)$ most relaxes the incentive compatibility constraint on the cooperative phase. In this case, penance-p strategies with full cooperation within the challenger group is a WRPE if and only if $δ \geq$ Max ${δ^{P} (4), δ^{DP} (4) |_{n - p = 2}, δ^{DP} (4, 1) |_{n - p \geq 3}}$ , where $δ^{P} (4)$ , $δ^{DP} (4) |_{n - p = 2}$ , and $δ^{DP} (4, 1) |_{n - p \geq 3}$ are given by (37), (31), and (32), respectively.

I have the following result.

Result 3: The strategy profile $\tilde{σ} (\bar{p})$ with full cooperation within the challenger group is a WRPE if and only if:

(i) $δ \geq δ^{P} (2, 1)$ when $n = 2$ ;

(ii) $δ \geq δ^{P} (n)$ when $n = {3, 4}$ ;

(iii) $δ \geq δ^{p} (5)$ when $n = 5$ and $λ \leq \tilde{λ}$ or $δ \geq δ^{DP} (5)$ when $n = 5$ and $λ \geq \tilde{λ}$ , with $\tilde{λ} \approx 1.13;$

(iv) $δ \geq δ^{DP} (n)$ when $n \geq 6$ .

As mentioned previously, Froyn and Hovi (2008) presented a model where players have a binary choice between cooperating and defecting, and Asheim and Holtsmark (2009) presented a model with continuous choices, but without strategic interactions between players, and both found that full cooperation can be sustained with a WRPE through penance strategies. I show that this result extends to a more general model with strategic interactions between players and in a context where full cooperation within a group is undertaken to counter the action of a hegemon. Furthermore, in contrast to the above-mentioned analysis, the number of players determines which incentive compatibility constraint is more stringent. Observe that the inequality $δ^{DP} (n) \geq δ^{P} (n)$ reduces to $v^{DP} - v^{P} \geq v^{D} - v^{C}$ because $δ^{DP} (n) \equiv (v^{DP} - v^{P}) / (v^{C} - v^{P})$ and $δ^{P} (n) \equiv (v^{D} - v^{C}) / (v^{C} - v^{P})$ . Here $v^{DP} - v^{P}$ represents the net benefit of deviation for a contributing member on the punishment path, whereas $v^{D} - v^{C}$ represents the net benefit of deviation for each member on the cooperative path. When group size is larger than five, the best possible deviation from either the cooperative phase or the punishment phase is to cut his/her contribution to zero. In this case, as shown and explained in the previous subsection, the net benefit of defection decreases with group size, that is, with the number of contributing members. It follows that there are more incentives to deviate from the punishment path than from the fully cooperative outcome (and, hence, $δ^{DP} (n) \geq δ^{P} (n)$ ). This result is reversed when the challenger group has few members (less than five). In this case, the best possible deviation from the punishment path is to make a positive level of effort because contributing members are not enough to make it profitable full defection. In turns, it makes defection from the punishment path less attractive than defection from the cooperative path.

I now characterize the impact of an increase in group size n, or in the relative valuation of the prize by group members given by $λ$ , on the difficulty of sustaining within-group cooperation, as measured by the lowest discount factor supporting the optimal level of group effort as a WRPE. For $n \leq 5$ , I have the following result.

Result 4: Suppose that $n \leq 5$ , then:

(i) $δ^{P} (2, 1) < δ^{P} (4) < δ^{P} (3);$

(ii) $δ^{P} (5) > δ^{P} (4)$ and $δ^{DP} (5) > δ^{P} (4)$ ;

(iii) $δ^{P} (2, 1), δ^{P} (n)$ for $n = {3, 4, 5}$ and $δ^{DP} (5)$ are all decreasing in $λ$ .

For a group of less than five members, the incentive compatibility constraint along the cooperative path is more stringent than that along the punishment path. Furthermore, more group members is likely to make full cooperation harder to sustain as a WRPE except when group size increases from three to four members. In addition, whether a group of five members has more or less difficulty to sustain within-group cooperation as a WRPE compared with a group of three members is indeterminate and depends on the exact value of $λ$ . Intuitively, a group of three members is that for which the punishment is relatively the least severe because, in that case, the number of punishing members in proportion of group size, that is, $p / n = 1 / 3$ , is the smallest compared with a group of two, four, or five members. However, an increase in the valuation of the prize by group members compared with that of the hegemon, i.e., an increase in $λ$ , unambiguously makes within-group cooperation less difficult to sustain as a WRPE. The question now is whether these results remain valid for larger groups with an arbitrary number of members. For $n \geq 5$ , I have the following result.

Result 5: Suppose that $n \geq 5$ , then:

(i) $δ^{DP} (n + 1) < δ^{P} (n)$ ifnis odd;

(ii) $δ^{DP} (n)$ is increasing innindependently of $λ$ ;

(iii) $δ^{DP} (n)$ is decreasing in $λ$ independently ofn.

For a group of more than five members, the binding incentive compatibility constraint is that corresponding to the punishment path. The intuition for point (i) is the same as before. If the challenger group has an odd number of members, then integrating one more member facilitates within-group cooperation because this allows increasing the number of punishing members also by one, thus increasing the proportion of group members punishing a deviation. Nevertheless, as stated in point (ii), an increase in group size generally raises the threshold value of the minimum discount factor above which within-group cooperation on the efficient level of group effort can be sustained as a WRPE. As mentioned previously, the net benefit of defection from the punishment path, that is, $v^{DP} - v^{P}$ , is decreasing in the number of contributing members, in the same way as does the net benefit of defection from cooperation with group size. However, and in contrast to the situation where group members use GTSs, the punishment threat also becomes less effective as group size increases owing to the renegotiation-proofness requirement. In fact, Result 5 shows that $v^{C} - v^{P}$ is decreasing even more with group size than $v^{DP} - v^{P}$ , thus increasing $δ^{DP} (n)$ and making full cooperation more difficult to sustain as a WRPE as group size increases. In other words, the constraint of renegotiation-proofness brings us back to the intuition that maintaining cooperation becomes more difficult as the size of the collectivity increases (see, e.g., Hardin, 1982; Olson, 1982; Sandler, 1992; Taylor, 1982). Yet, full cooperation can still be maintained in arbitrarily large groups provided the discount factor of group members is sufficiently high.

As for the effect of the heterogeneity in the valuation of the prize on the difficulty of sustaining full cooperation as a WRPE (point (iii)), I obtain the same result than that obtained when group members use simple GTSs. The cooperative level of individual effort, whether there are n or only $(n - p)$ contributing members, increases with $λ$ and, hence, withdrawing his/her individual contribution becomes increasingly tempting even though a defection also decreases the probability of winning the public prize. However, the net benefit of maintaining full cooperation relative to the punishment phase also increases with $λ$ , thus making the punishment threat more effective. As shown by Result 5, the renegotiation-proofness requirement with a limiting number of punishing members does not alter the increased effectiveness of the punishment threat for making within-group cooperation easier to sustain, as the relative valuation of the prize by group members increases.

4. Concluding remarks

In a repeated contest game, I have analyzed the ability of group members to cooperate in order to challenge a hegemon for the award of a public prize. The results have shown that group cooperation can be more easily sustained through GTSs as the relative valuation of the prize by group members or group size increases. This last result is in contrast to the intuition that decentralized strategies of reciprocity may be ineffective to induce mutual cooperation in large groups. However, as already mentioned, one may doubt that participants to collective political action use such extreme strategies to induce mutual cooperation. Indeed, once a deviation has occurred, the deviating members and the punishing members (who also suffer from the punishment) would be tempted to renegotiate a cooperative outcome which would make infinite reversion to the Nash punishment not credible (Farrell and Maskin, 1989). Furthermore, in the context of governance of commons, it has been observed that punishments are targeted at members who defect from the rule (provided that monitoring is possible) and that many commons have been robust against occasional and temporary deviations (see, e.g., Ostrom, 1990).

This motivates the analysis of the effectiveness of weak renegotiation-proof strategies for supporting cooperation in political action against a hegemon. These strategies require perfect monitoring (and the identification of deviators) and satisfy a criterion of collective rationality. Following a deviation from cooperation, the punished and punishing members agree to carry out the one-period punishment in return of full cooperation in the subsequent periods. I then show that group cooperation can still be maintained in very large groups and that a greater heterogeneity in the valuation of the prize between the members of the challenger group and the hegemon makes within-group cooperation easier to sustain. However, in contrast to GTSs, group cooperation becomes more difficult to sustain as group size grows. In particular, the critical discount factor above which full cooperation can be maintained converges to zero with GTSs, whereas with weak renegotiation-proof strategies it converges to one as group size grows to infinity.

Clearly, the renegotiation-proofness requirement weakens the punishment threat by limiting the number of punishing members (and, hence, the severity of the punishment). However, the implicit assumption behind this requirement is that strategies must be robust against perfect and costless renegotiation. In reality, collective renegotiation induces some costs, for instance the time spent in coordination meetings, and costly renegotiation can, in turn, reinforce the ability of group members to sustain cooperation (by making the punishment more costly). Furthermore, renegotiation costs are likely to be increasing in group size and this effect should restore (at least partially) the advantage of larger groups in an ongoing political conflict with a hegemon.

Supplemental Material

Response-Referee1-FINAL – Supplemental material for Group cooperation against a hegemon

Supplemental material, Response-Referee1-FINAL for Group cooperation against a hegemon by Guillaume Cheikbossian in Journal of Theoretical Politics

Supplemental Material

Response-Referee2-FINAL – Supplemental material for Group cooperation against a hegemon

Supplemental material, Response-Referee2-FINAL for Group cooperation against a hegemon by Guillaume Cheikbossian in Journal of Theoretical Politics

Footnotes

Appendix

Acknowledgements

I thank participants of the Annual Conference of the Association for Public Economic Theory in Rio de Janeiro (APET), the conference “Contests: Theory and Evidence” at the University of East Anglia (Norwich), and seminar participants in Montpellier and Toulouse School of Economics. I am also very grateful to two anonymous reviewers and to the Editor of this journal for their comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Guillaume Cheikbossian

Notes

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