Conflict technology in cooperation: The group size paradox revisited Sung-Ha Hwang∗ Korea Advanced Institute of Science and Technology (KAIST)

Abstract This paper studies the implications of punishment-induced conflict in a pubic goods game. It shows, under plausible assumptions, how larger group size sometimes enhances punishing behavior in social dilemmas and hence supports higher levels of cooperation. Unlike existing approaches that focus on uncoordinated punishment, I consider punishment as a coordinated activity that may be resisted by those being punished and study the implications of punishment-induced conflict situations. Developing a conflict model of punishment and combining it with a standard public good game, I show that coordinated punishment can yield the concentration effect of punishment, leading to a larger group advantage; that is, the larger the group, the easier it becomes to organize cooperation. The key idea is that when punishers coordinate their punishment, punishers as a coalition successfully divide defectors and punish each defector one by one. Surprisingly, even when coordination among punishers decays as group size increases, as long as the rate of decaying remains relatively slow, the larger group advantage still obtains. Keywords: Collective action, group size, conflict, coordinated punishment, evolutionary games, stochastic stability, Lanchester’s equations JEL Classification Numbers: D74 (Conflict); D03 (Behavioral Economics); H41 (Public Goods)

∗

Corresponding author; March 15, 2017 Email address: [email protected] (Sung-Ha Hwang)

What limit [of the State] would you propose? I [Socrates] would allow the State to increase so far as is consistent with unity; that, I think, is the proper limit. Very good, he [Adeimantus] said. Here then, I said, is another order which will have to be conveyed to our guardians: Let our city be accounted neither large nor small, but one and self-sufficing (The Republic, Book IV, Plato). 1. Introduction Formal or informal sanction often provides effective ways of upholding social norms in communities where individuals can free ride on fellow members (Axelrod, 1986; Ostrom et al., 1992; Ostrom, 1998; Sethi, 1996). Existing literature reports that punishment of norm violators sometimes induces conflicts between enforcers and those being punished. In 1947, “Meidung”− a form of ostracism or shunning − among the Old Order Amish led to a serious conflict between an individual and the Old Order Amish church, whose resolution eventually relied on the legal system (Gruter, 1986). I study the implications of conflict technology in a cooperative setting, focusing on the group size effect, and explore the possibility that punishment can be more effective in larger groups. Discussions on optimal group size date back to at least the ancient Greek philosophers, as the opening quote shows. Group size assumes an important role in various forms of collective action, such as the provision of public goods in a community, the organization of strikes, and revolutionary activities to overthrow corrupt and inefficient governments. A standard argument by Olson (1965) asserts that large groups face more difficulties in achieving common goals compared to small groups, because of aggravated free-rider problems. However, the possibility that larger group size facilitates collective actions has been widely recognized by many social scientists (DeNardo, 1985; Marwell and Oliver, 1993). Isaac and Walker (1988) found no support for the argument that increases in group size increases free-riding behavior. Moreover, a significant number of empirical studies report that group size is positively related to group performance. For example, Haan and Kooreman (2002) find evidence, using data from a candy bar honor system in 166 firms in the Netherlands, that free riding decreases with group size. Similarly, in experimental laboratories, Carpenter (2007) finds that larger groups contribute at higher rates than smaller groups since punishment of norm violators does not diminish appreciably in large groups. This inconsistency between theory and empirical evidence is called “the paradox of group size” (see Marwell and Oliver (1993)). This paper shows how coordinated punishment and the associated punishment technology help to solve “the paradox of group size.” While punishment implemented in experimental labs and theoretical models is typically individualistic (see Section 2), punishment commonly observed in social organizations is often coordinated via gossip, shame, and ostracism (Boehm, 1982; Gruter, 1986; Francis, 1985). Unlike monetary punishment, which is based on 1

individual decisions, non-monetary punishment is conducted and coordinated by punishers, who impose sanctions on norm violators. Ostracism is executed by a consensus of community members, and gossiping and ridicule require the public to join the process of punishment (Mahdi, 1986; Wiessner, 2005; Cinyabuguma et al., 2005). I develop a conflict model describing punishment processes and combine it with an existing evolutionary model of a public goods game among cooperators, defectors, and punishers (Sethi and Somanathan (2006), Bowles and Choi (2012)). When punishers coordinate in implementing punishment, they can concentrate on punishing and attacking defectors. Thus when punishers coordinate their punishment and defectors counteract individually, punishers as a coalition successfully divide defectors and punish each defector one by one. As a result, in a larger group, fewer punishers may successfully implement punishment. In other words, in larger groups, punishers may enjoy the advantage of the concentration principle, and hence cooperation is easier to organize in larger groups. Some existing evolutionary models of public good games with punishment also address “collective” punishment (see, for example, Chapter 11 in Bowles (2004)). However, these models do not explicitly pursue the punishment technology associated with it. As a result, group size does not play a role in these models.1 The coordinated nature of punishment and its implications for cooperation have recently received attention from evolutionary biologists and anthropologists as well (see Bingham (2000); Boyd et al. (2010)). For example, Boyd et al. (2010) show that coordinated punishment sustained cooperation in early human society even when punishment was rare. However, they simply assume the advantage of a large number of punishers in sustaining cooperation by adopting a specific functional form for the cost of punishment. Thus, the mechanism of how coordinated punishment leads to a larger group advantage is not clearly spelled out, nor have the conditions for the group size effect of the punishment technology been identified, in these studies. Using the proposed model, I show that a larger group advantage obtains under plausible conditions (Proposition 2). Then, using stochastic evolutionary game theory, I also show that as group size increases, the state in which all individuals choose to be punishers is stochastically stable (see, for example, Young (1998) for stochastic stability). Thus, the concentration effect of punishment can lead to a larger group advantage in a public goods game (Proposition 2 and Proposition 3-(1)) and cooperation can persist in the long run. My results regarding the effects of group size on cooperation are robust in the following two senses. First, as long as punishers are even slightly better coordinated than defectors 1

Olcinia and Calabuig (2015) also study coordinated punishment in a team trust game and show two long-run outcomes; (1) the full-cooperation outcome and (2) the no-cooperation outcome. However, they too did not study the effect of group size.

2

in punishment conflict, the concentration effect and the resulting larger group advantage obtain (Proposition 2, Proposition 3-(1)). Second, and more surprisingly, even as the degree of coordination among punishers decreases as group size increases (as long as it deceases relatively slowly), the larger group advantage still exists and persists (Proposition 3-(2)). Considering the abundant literature on the coordinated nature of punishment in various human organizations (see Section 2), (slightly) better coordination among punishers compared to defectors is empirically plausible. Hence, slightly relaxing the existing assumption about punishment in an empirically plausible way yields surprising implications for the paradox of group size. The proposed conflict model generalizes a military conflict model called Lanchester’s equations—differential equations that describe the time evolutions of two competing armies when they concentrate on attacking each other (Lanchester, 1917) (see Section 3.2).2 The next section provides empirical evidence for coordinated punishment. Section 3 introduces a conflict model and a public good game with punishment. Section 4 presents the main analysis, and Section 5 relaxes the assumptions of the main model and studies the larger group effect in various settings. The final section concludes the paper. 2. Hypothesis of coordinated punishment: evidence In this section, I briefly review the literature on punishment, focusing on coordinated punishment (for more extensive surveys, see Ledyard (1995); Chaudhuri (2011)). Monetary punishment is one of extensively studied form of punishment in economics, especially in the laboratory experimental setup. It includes fine, penalty and reduction in one’s payoffs in public good games experiments. Implementation of monetary punishment is commonly assumed to be based on individual levels (Fehr and G¨achter (2000); Bowles and Gintis (2004); Sethi and Somanathan (2006); Carpenter et al. (2009)). For example, in the public goods experiment designed by Fehr and G¨achter (2000) group member i can reduce group member j’s payoff by some proportion, and the total punishment inflicted on member i is the sum of all punishment amounts by all other members. Recently, there has been an increasing recognition that sustaining high contribution to the public good is possible via other mechanisms—broadly categorized as non-monetary punishment (e.g., moral suasion in Ledyard (1995)). In a well-known study, Masclet et al. (2003) consider informal sanctions such as peer pressure as a method to effectively deter deviations from socially acceptable behaviors. They distinguish between two hypotheses, the 2

To study the concentration effect of aircraft during World War I, Lanchester proposed these equations, whose empirical validity has been confirmed by previous studies (Engel, 1954; Samz, 1972). My model is a general model based on Lanchester’s model of conflict with parameters capturing the degree of coordination among punishers.

3

direct punishment hypothesis (under which monetary punishment can be used to sustain contributions in public good games) and the indirect punishment hypothesis (under which non-monetary punishment, such as the expression of disapproval, serves as an informal sanctioning mechanism against norm violators). They find that both the direct and indirect punishment hypotheses are supported (see also G¨achter and Fehr (1999)). In a similar vein, Rege and Telle (2004) study social approval and framing on cooperation in public good situations and find that social approval incentives increase the contribution levels. Cinyabuguma et al. (2005), studying ostracism in a laboratory experiment, allow the opportunity to expel free-riders by majority rule in a public goods game and find that the average contribution to the public good is higher in the expulsion treatment (see also Hirshleifer and Rasmusen (1989), G¨ uth et al. (2007), Maier-Rigaud et al. (2010)). Wu et al. (2016) compare the effectiveness of gossip and monetary punishment and find that the gossip option increased both cooperation and individual earning, whereas the monetary punishment option had no overall effect (see also Piazza and Bering (2008); Feinberg et al. (2014)). One of important characteristics of non-monetary punishment such as gossiping, ostracism, and shunning is coordinated among those who implement punishment. Examples of such punishment are more extensively reported in the literature of anthropology, sociology and psychology. Wiessner (2005), an anthropologist, collected data from 308 conversations among Ju’/hoani (!Kung) bushmen and found that “most punishment takes place in conversations, or at least it begins as talk”(p.121). She reports that out of 192 actions that threatened community stability and harmony, 170 actions were punished by coalitions (groups of three or more people who jointly engaged in punishment) rather than individuals (see also Wiessner (2014)). In a study on the Pathan hill tribes of Afghanistan, Mahdi (1986) observes that “Pathan society ostracism functions simultaneously to deter behavior that violates customary legal norms, to punish specific acts that are culturally defined as improper, and to unify the primary reference group that individuals depend for protection and economic support” (p.295). Other examples of non-monetary and coordinated punishment can be found in the punishment of whistle-blowers in political organizations and strike-breakers in labor unions. Francis (1985), in his study of a 1984-85 strike in coal-mining communities in South Wales, writes: “To isolate those who supported the ‘scab union’, cinemas and shops were boycotted, there were expulsions from football teams, bands and choirs and ‘scabs’ were compelled to sing on their own in chapel services. ‘Scabs’ witnessed their own death in communities which no longer accepted them”(p.269). In sum, Boehm (1982) emphasizes the functioning of collective punishment or sanctioning in maintaining social norms: Group sanction emerged as the most powerful instrument for regulation of in4

dividually assertive behaviors, particularly those which very obviously disrupted cooperation or disturbed social equilibrium needed for group stability. (Boehm, 1982, p.146) Section 3 develops a formal model of public goods provision in which coordinated punishment plays a key role sustaining cooperation. 3. Cooperation, Punishment, and Conflict 3.1. A public goods game with coordinated punishment This subsection introduces a model of a public goods game with coordinated punishment, focusing on the basic assumptions and features of payoff specifications. A detailed introduction and description of punishment conflict is presented in Subsection 3.2. Consider a community consisting of n individuals, indexed by i = 1, · · · , n, who may contribute to a public project. They may adopt three strategies: punishing, defecting, and cooperating. Punishers contribute and participate in punishment, Defectors do not contribute but are subject to punishment and counter-punish it, and Cooperators contribute and do not participate in punishment. Suppose that the total numbers of Punishers, Defectors, and Cooperators in the community are respectively denoted by x, y, and z. When Punishers and Cooperators contribute, they pay the cost of contribution c and the aggregate contribution in the community is thus x + y, which yields a total benefit b(x + y) shared equally by the community members. Therefore, the net benefit of the contribution is nb (x + y) − c, where b/n < c < b, so it is individually rational not to contribute (b/n < c), but contribution by all is socially desirable (c < b). During the punishment process, which will be presented in more detail in Subsection 3.2, punishment conflict occurs between punishers and defectors and the outcome of the conflict depends on the process of conflict. Based on the outcome of the punishment conflict, either punishers or defectors incur some costs related to the conflict. Defectors pay the cost, dY , when the punishment is successful, whereas punishers incur the cost, dX , in the case of unsuccessful punishment. Let IS be an indicator function of successful punishment—a function taking the value of 1 in the case of successful punishment and 0 otherwise. For defectors, then, the cost of being punished is dY IS . For punishers, the cost of unsuccessful punishment, with analogous notation, is dX IF , where IF is an indicator function of unsuccessful punishment. In Subsection 3.2, I will define the precise meaning of “successful” punishment using a conflict model developed there and derive the following functions indicating successful and

5

unsuccessful punishment:

IS (x, y) :=

 1

if

1 x1+ρ 1+ρ

>

1 y 1+θ 1+θ

0

if

1 x1+ρ 1+ρ

≤

1 y 1+θ 1+θ

and

IF (y, x) := 1 − IS (x, y),

(1)

where ρ and θ are the degrees of coordination among punishers and defectors, respectively. Recalling that the total numbers of Punishers, Defectors, and Cooperators are x, y, and P z, i ei is equal to x + z. Putting together all the ingredients discussed so far, the payoffs for Punisher (X), Defector (Y ), and Cooperator (Z) are as follows: b (x + z) − c − dX IF +  n b πY (x, y, z) := (x + z) − dY IS n b πZ (x, y, z) := (x + z) − c, n

πX (x, y, z) :=

(2)

For equations in (2), I assume that the individuals who participate in punishment may obtain a small utility bonus,  ≈ 0. A large volume of behavioral and experimental literature reports that individuals are willing to punish others, even incurring personal costs (Andreoni, 1990; Casari and Luini, 2012; Carpenter et al., 2009). People punish those who exploit the cooperative behavior of others since they value punishment ethically for its own sake (Bowles and Gintis (2011)). In the context of collective action, Wood (2003) describes the feeling of participating in a good cause as the “pleasure of agents”. Roland and Xie (2016) write that “participation in collective action can help the individual stand out,” feeling the gratification of higher emotion, and hence obtaining a positive “social” payoff to participation (see also Gorodnichenko and Roland (2015)). Not-participating in punishment itself is regarded as “second-order” free-riding (Panchanathan and Boyd, 2004), and thus punishers who do not free-ride punishment may obtain a psychological positive reward, while non-punishing cooperators do not. Alternatively, the assumption of a small utility bonus can be replaced by the assumption that a (small) fraction of individuals in the population are committed punishers. I explain this in more details in Subsection 5.2. One example of dY and dX is as follows: when the outcome of the punishment process determines whether defectors or punishers are to be expelled from the community (as in the example of “ostracism”), the party being expelled experiences costs of exclusion from the benefits or the productive activities of the community. There may be some costs associated with punishment and counter-punishment, regardless of the outcome of conflict, since actual physical or psychological conflict may exhaust some real resources. I examine this alternative

6

assumption in Subsection 5.1. 3.2. Conflict technology in punishment The key idea of the conflict model, as explained, is that when punishers coordinate their punishment and defectors counteract individually, punishers as a coalition successfully divide defectors and punish each defector one by one. In a conflict model, punishers implement punishment and defectors counter-punish or resist it. Specifically, punishers and defectors devote some resources to weaken their opponents by reducing and offsetting the opponents’ resources. A standard textbook in psychology defines punishment as “a decrease in the strength of behavior due to its consequences”(see p.208 in Chance (2009); see also Holth (2005)). In the model, when punishers exert efforts (hence devote resources) to reduce the strength of defectors, the resources (representing the strength of defectors) decreases. The form of punishment can be physical, such as knockouts resulting from actual fighting and ensuing elimination from the community (Knauft, 1987). More importantly, as reviewed in Section 2, punishment can be psychological, involving gossiping and intentional shunning and the outcome can be psychological exhaustion, for example, from the barrage of criticism based on talking (Wiessner, 2005). Scabs, who experienced “psychological” death in their community in the study by Francis (1985) (in Section 2) may have been subject to this kind of punishment. Another important aspect of the current conflict model of punishment is that defectors can resist and counteract. This feature is also in line with the recent experimental findings that counter-punishment and anti-social punishment by norm-violators indeed occur and sometimes lead to the reduced effectiveness of punishment (Nikiforakis and Normann, 2008; Herrmann et al., 2008; Cinyabuguma et al., 2006; Denant-Boemont, 2007). A vendettalike cycle can occur between punishers and those who resist punishment, especially when the punishment is motivated by emotions such as anger (Hopfensitz and Reuben, 2009). Retaliation of punishing behaviors—often called perverse punishment—was persistent in the experiments performed by Cinyabuguma et al. (2006). To study the process of punishment conflict, let xt and yt be the amounts of resources in conflict devoted by punishers and defectors, respectively, at time t. Throughout the paper, I assume that one individual possesses one unit of resources, and hence, the amounts of punishers’ and defectors’ resources are equal to the numbers of punishers and defectors, respectively. However, this assumption can be easily relaxed. First, consider changes in the amount of defectors’ resources during conflict. The reduction in the effective amount of defectors’ resources after a conflict period of ∆t, ∆y = yt+∆t − yt , depends on how punishers implement punishment, which in turn depends on the 7

degree of coordination among punishers. With a high degree of coordination, punishers as a coalition can successfully concentrate on reducing the resources of defectors. In general, the degree of coordination determines the number of punishers who can punish defectors simultaneously during a short period of time, ∆t. If a(x) is the number of simultaneous punishers, and hence the amount of punishers’ resources, the decrease in the amount of defectors’ resources is given by ∆y = yt+∆t − yt = −a(x).

(3)

Similarly, the amount of punishers’ resources decreases because defectors resist and counter-punish. If ∆x = xt+∆t − xt and b(y) is the number of defectors who counter-punish simultaneously, the decrease in the amount of punishers’ resource is ∆x = xt+∆t − xt = −b(y).

(4)

Taking ∆t → 0 in equations (3) and (4) yields the following differential equation system: dy dx = −b(y), = −a(x) (5) dt dt where a(·) and b(·) are nonnegative and nondecreasing functions. This heuristic derivation of the differential equations of (5) from intuitive equations (3) and (4) can be rigorously justified based on the stochastic model of conflict (for example, see Darling and Norris (2008)). The equations in (5) jointly model the time evolutions of the effective amounts of resources of punishers and defectors during conflict. For example, when the punishers can coordinate perfectly, all of them, numbering x, are simultaneous punishers: a(x) = x. When the defectors behave individually, only one defector counter-punishes simultaneously: b(y) = 1. The analytic advantage of (5) is that this system admits of a special function H(x, y), called a “constant of motion”: a function whose values are constant along a solution of the system (see Hirsch et al., 2004), ˆ

ˆ

x

a(s)ds −

H(x, y) = 0

y

b(s)ds.

(6)

0

This special property of function H is easily verified by taking the time derivative of H(x(t), y(t)), where x(t) and y(t) denote the solutions of the equations in (5); that is, d H(x(t), y(t)) = 0 for all t. This implies that the value of H(x(t), y(t)) remains constant dt as time passes by, and thus each level set of H, a curve along which the function H has a

8

Defectors

a

c

100

80

b

60

40

20

0 0

20

40

60

d

80

100

Punishers

Figure 1: Solution curves for the differential equations. Each line corresponds to a solution of the differential equation. If a system starts from a, it reaches b. By contrast, a system starting from c reaches d. a(x) = xρ , b(y) = y θ ,ρ = 1, θ = 0.8

constant value, depicts each trajectory of a solution of (5). The punishment is defined to be successful if punishers exhaust the defectors’ resources, and unsuccessful, otherwise. Figure 1 illustrates how the initial amounts of resources of punishers and defectors are related to the outcome of one stage conflict through the function, H. First, each curve in Figure 1 corresponds to the level set of H(x, y)—the set of all x and y that give the same value of H. Thus if the system starts from, say, point a in Figure 1, then the solution (x(t), y(t)) starting from point a remains in the curve originating from point a in Figure 1 (since dtd H(x(t), y(t)) = 0) and eventually reaches point point b. The outcome is then the failure of punishment, since x(T1 ) = 0, y(T1 ) > 0 at time T1 when it hits the vertical axis. Similarly, if the system starts from point c instead, the final state will be at point d, whose outcome corresponds to the success of punishment, since x(T2 ) > 0, y(T2 ) = 0 at time T2 when it hits the horizontal axis. In this way, using the function H(x, y) and the corresponding level sets of H(x, y), the initial state of the conflict dynamic is uniquely associated with the final state, giving the outcome of punishment. From (6), the remaining amount of the winning side’s resources at the end of conflict can be computed. To do this, let T be the time when one of two sides’ resources are exhausted. From the property of the H function, H(x(T ), y(T )) = H(x(0), y(0)) holds, where x(0) and y(0) are the initial amounts of punishers’ and defectors’ resources, respectively. Observe that except in the case of a tie, the positive number of one side (e.g., x(T ) > 0 ) at time T means that the other side’s resources are completely exhausted (i.e., y(T ) = 0). Figure 1 again illustrate how each initial point can be uniquely associated with either (x(T ) > 0, y(T ) = 0) or (x(T ) = 0, y(T ) > 0), except for the case of a tie. Thus, if x(T ) > 0, the remaining

9

amounts of punishers’ resources, x(T ), satisfies ˆ

ˆ

x(T )

y(0)

a(s)ds −

a(s)ds = 0

ˆ

x(0)

0

b(s)ds.

(7)

0

Now, consider a situation where the degree of coordination in the punishment conflict differs between punishers and defectors. To model this situation, I introduce parameters representing the degree of coordination by punishers, ρ ∈ [0, 1], and the degree of coordination by defectors θ ∈ [0, 1]. Suppose that the number of simultaneously attacking punishers, a(s), and the number of simultaneously counter-attacking defectors, b(s), are a(s) = sρ

b(s) = sθ .

where ρ, θ > 0. Thus, at the lowest degree of coordination (ρ = 0), punishers behave individually and confront defectors on a one-to-one basis, and at the highest degree of coordination (ρ = 1), punishers coordinate perfectly. When 0 < ρ < 1, the degree of coordination among punishers is limited, and thus some fraction of punishers, as a coalition, implement punishment. A similar interpretation is possible for the parameter θ. Then, using the conditions in (6) and (7), I derive IS and IF as follows:

IS (x, y) :=

 1

if

1 x1+ρ 1+ρ

>

1 y 1+θ 1+θ

0

if

1 x1+ρ 1+ρ

≤

1 y 1+θ 1+θ

and

IF (x, y) := 1 − IS (x, y)

(8)

as in equation (1)3 . In Appendix A, I provide a micro-foundation of the multi-stage conflict model yielding the expression in (8). The so-called Lanchester’s equation is a special case of (5), when a(s) = b(s) = s (Lanchester, 1917). Lanchester developed these differential equations to study the concentration effect of aircraft during World War I. Lanchester’s model has been reinterpreted and extended to conflicts among social animals such as fire ants (Adams and Mesterton-Gibbons, 2003; Plowes and Adams, 2005) and early human lethal conflict (Bingham, 2000; Boyd et al., 2010). In economics, Hirshleifer (1991a,b) uses the Lanchester model to justify some forms of contest success functions. Engel (1954) and Samz (1972) test the empirical validity of Lanchester’s theory, using actual combat data from Iwo Jima during World War II. The conflict model in this paper generalizes these existing models, accommodating various kinds and degrees of coordination during conflict. Another way of viewing punishment conflict and its effect on payoffs (2), dX IF and dY IS , 3

Here, I assume that when

1 1+ρ 1+ρ x

=

1 1+θ , 1+θ y

the outcome is unsuccessful punishment.

10

is to regard the conflict situation as a contest, in which punishers and defectors spend their resources to win a prize (Konrad, 2009). While in the existing contest models, participants compete for a prize, the punishment conflict in the model results in penalties, dX and dY (or a “negative” prize), so contestants compete to avoid a negative prize. 4. Do larger groups alleviate collective action problems? The payoffs in equation (2) together with the indicator function of successful punishment in (1) define the complete payoffs for each type (or strategy), which in turn depend only on the numbers of agents choosing each type. A population state (x, y, z) consists of the numbers of Punishers (x), Defectors (y), and Cooperators (y) and the set of all population states is denoted by ∆n := {(x, y, z) : x + y + z = n, x ≥ 0, y ≥ 0, z ≥ 0} . A Nash equilibrium for this game is defined as a state in which no individual can obtain a strictly higher payoff by deviating from the current strategy given others’ strategy choices (see Definition 6 in Appendix A). When there exists a small utility bonus for punishers ( > 0), the game specified by (2) admits two kinds of equilibrium: (i) the all-Defector state or (ii) the all-Punisher state. The all-Defector state is the equilibrium in the original public good game. When the punishment is effective (dY > c − nb − ), the all-Punisher state is also a Nash equilibrium. The condition dY > c − nb −  is consistent with the conditions identified in other studies (for example, see Sethi and Somanathan (2006)) and is in line with numerous experimental findings that punishment indeed sustains cooperation in public good games. When  = 0, the payoffs for punishers and cooperators are the same in the absence of defectors, and this yields the third kind of equilibrium in which punishers and cooperators coexist (see Figure 4 in Section 5.2). This motivates the following proposition holds. Proposition 1. Suppose that ρ, θ > 0 and the punishment is effective, dY > c − nb − . (1) When  > 0, an all-Defector state (0, n, 0) and an all-Punisher state (n, 0, 0) are two Nash equilibria. (2) When  = 0, an all-Defector state (0, n, 0) and Punisher-Cooperator states {(x, 0, n − x) : x ≥ 3} are Nash equilibria. Proof. See Appendix B.1. How does group size affect agents’ motivation to switch from a defecting strategy to a punishing strategy? To study the effect of group size on the payoffs of the underlying game 11

Panel A (n = 18) Punishers

Cooperators

Defectors

Panel B (n = 90)

Panel C (n = 180)

Punishers

Punishers

Cooperators

Defectors

Defectors

Cooperators

Figure 2: The state in which punishment is likely to be successful. Each point in the triangles uniquely corresponds to one population state, composed of numbers of each strategy, through the Bary centric coordinate. For example, the point located on the bottom left vertex corresponds to a population state in which all individuals choose the defector strategy. The dotted points are population states of successful punishment. b = 2, c = 1, s = 2, d = 2, ρ = 1, θ = 0.

in equation (2), let n BX := {(x, y, z) ∈ ∆ : πX (x + 1, y − 1, z) − πY (x, y, z) > 0}

be a set of all states for which agents benefit by switching from being a defector to a punisher. Suppose that punishers are better coordinated in punishment than defectors (ρ > θ). Then, for any given group size, there exists a concentration effect for punishers. Because of concentration, a smaller number of punishers is needed to eliminate a given fraction of defectors as group size increases, and this generates a larger group effect. Thus, the larger n the group, the larger the area of BX in ∆ becomes; it would be better off by switching from being defectors to punishers at almost all states (see Figure 2). In the statement of the following proposition, |A| denotes the number of elements in set A. Proposition 2 (The Larger Group Advantage). Suppose that  = 0 or  > 0 and ρ > θ. n Suppose that punishment is effective dY > c − b/n for all n. As n increases, the area of BX becomes larger: that is, n |BX | =1 lim n→∞ |∆n | Proof. See Appendix B.2. n Figure 2 illustrates Proposition 2. As the region of BX , consisting of the dotted points, expands, being a defector becomes less favorable in almost all states. Does the all-Punisher equilibrium persist in the long run? Since there are multiple Nash equilibria in the game specified by (2), it is not obvious which equilibrium would be more

12

Punishers

  

Resistance from all-Punisher state: n − x

    

Resistance from all-Defector state: x

A

B

Cooperators

Defectors

Figure 3: Resistances from the two absorbing states. The thick arrows show the direction in which agents benefit by deviating. Thus, region A is the basin of attraction for the all-Punisher state, and region B for the all-Defector state.

robust and persist in the long run. To address this question more precisely, I study the long-run stable states, using equilibrium selection approaches in evolutionary game theory (Kandori et al., 1993; Young, 1993). Specifically, I consider the following dynamics during each period: D1 An individual is drawn from the population at random. D2 With probability (1 − δ) for 0 < δ < 1, the chosen individual plays the best response at a given state. D3 With probability δ, the individual makes a mistake in choosing a strategy, and hence any strategy is chosen at random. The stochastic updating rule specified by D1−D3 is called a perturbed best-response rule (See Young, 1998). The mistakes and errors specified by D3 are based on the (empirically plausible) behavioral assumption that when individuals select a strategy, they are subject to mistakes or errors, or they experiment with idiosyncratic strategies (Naidu, Hwang, and Bowles, 2010). In the following analysis, I focus on a case where two strict Nash equilibria, the all-Defector state and the all-Punisher state, exist; that is, I assume that  > 0. I discuss the degenerate case,  = 0, with multiple non-strict Nash equilibria, in Section 5.2. It is easy to see that a Markov chain specified by D1−D3 is irreducible, and hence there exists a unique stationary distribution µδ . As δ becomes small, the stationary distribution µδ converges to a point mass on the state. The state that receives a positive weight at δ → 0 is called a stochastically stable state. When the perturbation vanishes in the long run, the system remains near this state almost all the time. Because of this, the stochastically stable state is regarded as a long-run equilibrium of the system (see Young (1998) for more details). The standard procedure to identify a stochastically stable state is to first find absorbing 13

states in the dynamics without mistakes (called unperturbed dynamics) and then calculate special numbers, called resistance, which measure how hard it is for the system to escape from each absorbing state (see Appendix B.3 for more precise definitions). The resistance is, intuitively, the number of mistakes (non-best responding choices) necessary to escape from one absorbing state (see Figure 3). Since a state with the greatest resistance is hardest to escape and is more likely to persist in the long run, the stochastically stable state corresponds to the state with the greatest resistance (Young, 1998). The system starting near the all-Punisher state (region A in Figure 3) reaches the allPunisher state by the best responses of agents. Similarly, the system starting near the all-Defector state (region B in Figure 3) reaches the all-Defector state by the best responses of agents. Thus, the unperturbed system for the current dynamics admits two absorbing states, corresponding to the all-Defectors state and the all-Punisher state. In the literature, these regions are often called “basins of attraction” of the two absorbing states. Thus, the n corresponds to the basin of attraction of the all-Punishers equilibrium, whereas the set BX n corresponds to the basin of attraction of the all-Defector equilibrium. complement of BX Based on this characterization of the basins of attraction, it can be shown that transitions with minimum mistakes escaping the all-Punisher state and the all-Defector state occur along the edge between the all-Punisher state and the all-Defector state (see Figure 3 again). More precisely, let the minimum number of Punishers, x¯, that ensures higher payoffs for them than for Defectors, in the absence of Cooperators, be x¯ := min {x : πX (x + 1, n − (x + 1), 0) > πY (x, n − x, 0)} . When the number of Punishers is greater than x¯, the payoffs for Punishers are greater than those for Defectors and vice versa. Then, the resistance from the all-Punisher state to the all-Defector state, rP D , and the resistance from the all-Punisher state to the all-Defector state, rDP , can be shown to be rP D = n − x¯ and rDP = x¯

(9)

(see Appendix B.3). Therefore, the all-Punisher state is stochastically stable if and only if x¯ <

n . 2

Thus, as Proposition 2 shows, when ρ > θ in equation (1), limn→∞ x¯/n = 0 holds, which implies that the all-Punisher state is stochastically stable in a large population (Proposition 3 (1)). 14

So far, I have assumed that the degree of coordination among punishers, ρ, does not depend on group size; however, as group size increases, coordination among punishers would become more difficult, and the advantage from it may therefore disappear. To examine this possibility, first write ρ(n) ∼ an , when limn→∞ ρ(n)/an = c for some constant c. Suppose that ρ(n) ∼ 1/n. The minimum fraction of Punishers in the population required to eliminate the fraction of Defectors q can be computed as p∼

q nρ(n)/(1+ρ(n))

and since nρ(n)/(1+ρ(n)) → 1 as n → ∞, the minimum fraction of Punishers in the population does not decrease according to group size. However, it turns out that the all-Punisher state is still a stochastically stable state in this case. The intuition behind this result is that the concentration effect is of the order of group size n, and thus, as long as coordination decays at a lower rate than (or the same rate as) the order of concentration, the larger group advantage still obtains. This leads to the following proposition: Proposition 3. Suppose that  > 0 and dY > c − . (1) Suppose that ρ and θ are independent of n. If ρ > θ, the all-Punisher state is stochastically stable. (2) If ρ(n) ∼ n1 and θ = 0, the all-Punisher state is stochastically stable. Proof. See Appendix B.3. The sufficient condition for the larger group advantage is that the actual number of punishers is more than a threshold number of punishers ensuring successful punishment. Proposition 2 shows that the region of the population states that ensures this sufficient condition enlarges as group size increases (see Figure 2 again). However, the question whether there exist a sufficiently large number of agents who are willing to be punishers and whether being punishers is a viable strategy in the long-run is not obvious. Proposition 3 demonstrates that the all-Punisher equilibrium sustains as the long-run equilibrium as the size of basin of attraction in which agents are better off by being punishers enlarges, irrespectively of whether ρ behaves as a constant or decreases in the order of n. Thus, Proposition 3 corroborates how robust the larger group advantage is in two ways. First, as Proposition 3 (1) shows, a (slightly) better coordination among punishers is sufficient for the larger group advantage. Furthermore, as Proposition 3 (2) shows, even if the coordination among punishers decays, the larger group advantage may still obtain. This occurs when the coordination among punishers decreases at the same rate as does group size.

15

The result of Proposition 3 (i) can be compared with the result in Young and Foster (1991), who show that the all-Defector state is stochastically stable. The main reason is that in the absence of defectors, the mix states of punishers and cooperators are neutrally stable and cooperators and punisher are equally doing well. The mix states of punishers and cooperators are then easier to escape than the all-Defector state. The authors also show the possibility that the all-Defector state is not stochastically stable when there exist punishers who are disguised as defectors (called, disguised punishers). In contrast with this approach, the assumption of the small positive value ensures that the all-punisher state is a strict Nash equilibrium and adding the coordinated punishment gives the results in Proposition 3. I discuss the assumption of no utility bonus in more detail in Subsection 5.2. Gale et al. (1995) also provide an alternative approach to show that mix states of cooperation and punishment are asymptotically stable. They consider the Ultimatum Game and show that a weakly dominated equilibrium, as well as a unique subgame perfect equilibrium, can be asymptotically stable in the presence of noise in learning processes. In particular, if responders are noisy enough relative to proposers, a sufficiently large number responders can reject low offers, so making low offers is not the best response of proposers. In the current model, even without the small utility bonus, if cooperators punish defectors at random, the same mechanism might work and a mix state of cooperators and punishers can be asymptotically stable. 5. Variations: alternative assumptions In this section, I examine alternative assumptions and explore whether the main result in Proposition 3 still hold or not. 5.1. Positive costs of punishment Another plausible assumption about punishment is that punishers pay some costs irrespectively of the success of punishment. Typically in the laboratory experimental setting, subjects always incur some costs whenever they participate in punishment activities (Fehr and G¨achter, 2000). To explore the group size effect in this setting, suppose that each punisher pays unconditional costs of punishment, which linearly depends on the population fraction of defectors. This assumption means that the cost of punishment is proportional to the number of defectors, implying that the larger the number of defectors, the more costly

16

is the punishment (Sethi and Somanathan, 2006). Thus new payoff functions are given by y b (x + z) − c − dX IF +  − γ n n b πY (x, y, z) := (x + z) − dY IS n b πZ (x, y, z) := (x + z) − c. n

(10)

πX (x, y, z) :=

Note that in the absence of defectors, a punisher does not pay a cost, while when almost all agent are defectors (i.e., when ny ≈ 1), each punisher pays a cost of γ. In this setting, the larger group advantage still exists as long as the cost of punishment, γ, is relatively small. To see the underlying logic, suppose that the punishment conflict is successful. Then the payoff advantage for punishers relative to defectors is (approximately) bounded below by πX − πY ≥ −c +  − γ + dY

(11)

and thus if γ is not large (i.e, dY ≥ c−+γ), then punishers have an advantage over defectors (πX > πY ) as equation (11) shows. This discussion leads to the following statement. Proposition 4. Suppose that  > 0 and dY > c −  + γ. (1) If ρ, θ are independent of n and ρ > θ, the all-Punisher state is stochastically stable. (2) If ρ(n) ∼ n1 and θ = 0, the all-Punisher state is stochastically stable. Proof. See Appendix B.3. When the outcome of conflict is subject to stochastic influence and randomness, as in many applications of contests, punishers may be subject to some cost even in the case of successful punishment. To explore this possibility, consider the following probability functions with the successful and unsuccessful punishments replaced, respectively, with IS and IF : PS (x, y) :=

1 exp(β 1+ρ x1+ρ ) 1 1 exp(β 1+ρ x1+ρ ) + exp(β 1+θ y 1+θ )

,

PF (y, x) :=

1 exp(β 1+θ y 1+θ ) 1 1 exp(β 1+ρ x1+ρ ) + exp(β 1+θ y 1+θ )

.

These function are called contest success functions, used in the literature on conflict and contest (Konrad, 2009). The payoff for punishers in this case is given by πX (x, y, z) :=

b (x + z) − c − dX PF + . n

and the punishers incur some cost even with the successful punishment(i.e., 17

1 x1+ρ 1+ρ

>

Panel A: e > 0

Panel B: e = 0

Punisher

Defector

Punisher

Cooperator

Cooperator

Defector

Figure 4: The location of Nash equilibria when n = 4 The states circled in the dotted lines are Nash equilibria. The arrows along the edges show the direction of deviation. For edges with no arrows, no deviation motivation exists. Panel A shows the case where  > 0. Panel B shows the case where  = 0. 1 y 1+θ ). 1+θ

Indeed, since PF (y, x) > 0 for all x, y, punishers are subject to positive costs regardless of the outcome of punishment. It is also easy to see that as β → ∞, PS and PF converge to IS and IF ; hence, these functions can be regarded as generalizations of IS and IF . Furthermore, at the all-Punisher state PF (0, n) =

1 1 exp(β 1+ρ n1+ρ )

+1

,

and as n → ∞, PF (n, 0) → 0. Thus if n is sufficiently large, dX PF <  and the all-Punisher state is a strict Nash as in Proposition 1 (i) and the same results in Propositions 3 and 4 1 1 x1+ρ − 1+θ y 1+θ ), PF (y, x) := 1 − PS (x, y), where F hold true. In general, if PS (x, y) := F ( 1+ρ is a distribution function that is increasing and satisfying F (−∞) = 0 and F (∞) = 1, the same conclusion holds. 5.2. No small utility of being punishers Next I consider the situation in which punishers do not experience the small utility bonus, . As Proposition 1 shows, in the absence of the utility bonus, there are many Nash equilibria consisting of punishers and cooperators (see Figure 4). As discussed after Proposition 3, this is because, in the absence of a defector in the population, punishers and cooperators are indistinguishable in terms of their behaviors, and both obtain the same utility. Then, the Nash equilibrium consisting of a mix of punishers and defectors becomes a neutrally stable state under the deterministic evolutionary dynamics. Under the stochastic evolutionary dynamics, whether this kind of Nash equilibrium is an absorbing state in the unperturbed process depends on the specific rule of updating. Concretely, if agents choose types randomly when multiple best responses exist, then the Nash equilibrium of a mixed population cannot be an 18

absorbing state, and the all-Defector state is the unique absorbing state in the unperturbed process. In this case, the all-Defector state is automatically stochastically stable since it is a unique absorbing state. Does the large group advantage completely dissipate even in the absence of the small utility bonus? As Proposition 2 shows, there still exists the larger group advantage in the absence of the small utility bonus in the sense that the size of the basin of attraction of the all-Defector equilibrium approaches zero as the group size increases. Thus, there may be circumstances under which the all-Punisher state is stochastically stable in the absence of the small utility bonus. To answer this, consider that a (small) fraction of agents are committed to being punishers even in the absence of defectors. For example, Ostrom et al. (1992) report experimental evidence that appropriators in common pool resources develop credible commitments in many cases without relying on external authorities. Under this assumption, it can be shown that the large group advantage still obtains. This similarly follows from the fact that when punishers are better coordinated in the punishment process, the threshold fraction of punishers ensuring that successful punishment goes to zero and thus the fraction of committed punishers is sufficient to prevent the evolution of defectors in the population. Proposition 5. Suppose that  = 0, ρ > θ. Suppose also that the fraction of committed punishers in the population, in the absence of defectors, is p¯ > 0. Then, for sufficiently large n, the states consisting of punishers and cooperators are stochastically stable. Proof. See Appendix B.3. 6. Conclusion Having developed a conflict model incorporating various forms of punishment technology, I studied the implications of coordinated punishment by combining this model with the existing public goods game. I showed that as long as punishers coordinate (even slightly) better than defectors, the coordinated punishment might produce a concentration effect, leading to more effective punishment in relatively larger groups. This implies that cooperation is sometimes easier to organize in larger groups. Since Olson’s (1965) seminal book, the Logic of Collective Action, the group size hypothesis has been examined in various studies. Theoretically, the relationship between group size and provision of collective goods has been studied by various researchers (Chamberlin, 1974; McGuire, 1974; Oliver and Marwell, 1988; Sandler, 1992; Marwell and Oliver, 1993; Agrawal and Goyal, 2001; Esteban and Ray, 2001; Pecorino, 2009). However, the literature provides few satisfactory solutions to the group size paradox. For example, Esteban and Ray (2001) 19

consider competition between groups and show that under some conditions of marginal costs, the probability of winning is greater for a larger group than a smaller group. However, the situation they consider is neither an original Olson setting nor the common settings in which members of one community engages in collective actions (see related discussion in Hwang (2009)). Thus, no existing theoretical study satisfactorily explains why an increase in group size might facilitate collective action. Unlike the existing approaches, my model provides a plausible explanation for the larger group advantage commonly observed by social scientists. For example, the political scientist James DeNardo, in the book Power in Numbers, writes : Regardless of the political context, there always seems to be power in numbers. No less an authority than Mao Tse-tung insists that “the richest source of power lies in the masses of the people,” and it is nearly impossible to imagine political circumstances where the disruptiveness of dissident activity would diminish as its scope increased (DeNardo, 1985, p.35). In addition to the three strategies considered in the paper, there can be other strategies. For example, Sethi (1996) considers a “bully” who defects but punishes cooperators and shows that the bully strategy can be an evolutionarily stable strategy. To consider the effect of punishing cooperators, suppose that defectors punish cooperators as well. Under this situation, it is easy to see that there is no change in equilibrium, and hence Proposition 1 remains true. Moreover, whether the punishment is successful or not depends only on the numbers of punishers and defectors; hence, the states (or regions) in which punishment is likely to be successful remain invariant (Figure 2). Thus, under the situation where defectors are allowed to punish cooperators, the larger group advantage still holds and the main results remain true. In this paper, I study the model of n-person prisoner’s dilemma games in which each agent has a binary choice of whether to contribute or not. One of the popular models studied extensively in laboratory experiments is a voluntary contribution mechanism (VCM) game in which the contribution level can be chosen continuously. A natural question is whether the larger group advantage still obtains in the VCM game. When the strategy set is continuous, the analysis similar to the one in this paper is difficult because it involves new techniques dealing with probability measures over a continuum space. However, it can be conjectured that the larger group advantage still holds in this setting because of the following reason. Suppose that there is a threshold level of contribution such that contributors with higher than the threshold contribution level punish those with lower than the threshold level. The situation is then similar to the one in the main model, and if punishers are better coordinated 20

than those subject to punishment, because of the concentration effect of punishment, the larger group advantage may still obtain. I leave more precise analysis to future work. As is shown, whether the larger group advantage exists or not depends on the degree of coordination among punishers and defectors. Even though I provide some empirical evidence for coordinated punishment in Section 2, the relative degree of coordination can be more precisely examined experimentally in the laboratory or field. In an experimental setup in which non-monetary punishment is available, the designer can allow subjects to form coalitions before the punishment stage and test whether punishers or defectors are willing to form a coalition to punish or to counter-punish. For example, Kosfeld et al. (2009) study an institution formation game in which subjects decide whether or not to participate in a sanctioning institution before a public goods game is played. Similarly, a designer may introduce sanctioning and anti-sanction institutions and study the willingness of subject to participate in each institution. Moreover, one can examine how the degrees of coordination among punishers and defectors vary with group size. The proposed model can be extended to a model of optimal group sizes in various organizations, as in the opening quote. Examples include working teams and special committees in firms and companies. When monitoring among team members is an important factor in enhancing the productivity of workers on teams, monitoring technology can be studied similarly to punishment technology. In addition, depending on the situation, the public benefit function, assumed linear in the current paper, can actually be concave or convex. Combined with public good benefit technology, punishment technology may help solve problems such as designing an optimal community size for successful governance of commons.

Acknowledgements. I would like to thank the associated editor, two anonymous referees, Samuel Bowles and participants in the behavioral working group seminar in Santa Fe Institute for their comments.

21

Appendix A. Micro-foundation of the conflict model: Divide and punish I consider a simple situation in which punishers are perfectly coordinated, a(s) = s, and defectors are individualistic, b(s) = 1, for simplicity; but a similar deviation is possible for a(s) = sρ and b(s) = sθ . Suppose that there are initially x0 punishers and y0 defectors and the conflict outcome is the successful elimination of defectors. Then, a single defector confronts a coalition of perfectly coordinated punishers during the punishment confrontation and since there are y0 defectors, y0 encounters occur between a single defector and the coalition of punishers (see Figure A.5). Thus, there is a series of y0 confrontations between the coalition of punishers and a single defector. Let T1 , T2 , · · · , Ty0 be the successive end times of each confrontation (see Figure A.5). Then, one easily sees, applying equation (7) successively, that the remaining numbers of punishers at each time, x(T1 ), x(T2 ), · · · , x(Ty0 ), must satisfy  ´ ´ x0 ´1 x(T1 )   sds = sds − 1ds,  0 0 0   ´ ´ ´  x(T ) x(T ) 1 2  sds = 0 1 sds − 0 1ds 0 . y0 times ..   .      ´ x(Ty0 ) sds = ´ x(Ty0 −1 ) sds − ´ 1 1ds 0 0 0

(A.1)

Then, summing the equations in (A.1) yields ˆ

ˆ

x(Ty0 )

x0

sds − y0 , or equivalently

sds = 0

0

1 1 x(Ty0 )2 = x20 − y0 . 2 2

Recalling that the punishment is defined to be successful if, after a series of confrontations, punishers exhaust the defectors’ resources, and to be unsuccessful, otherwise. Thus, in this 0 Punishers’ Resources Defectors’ Resources

T1

T2

T3

x0 → x(T1 ) x(T1 ) → x(T2 ) x(T2 ) → x(T3 ) 1 → 0 1 → 0 1 → 0

··· ···

Ty0 −1

Ty0

x(Ty0 −1 )→x(Ty0 ) 1 → 0

Figure A.5: A series of confrontations between punishers and defectors. There are y0 confrontations in total. Here, T1 ,T2 ,· · · ,Ty0 denotes the end times of each confrontation. During the first confrontation (from 0 to T1 ), x0 punishers confront one defector. As a result of the first confrontation, x(T1 ) punishers remain and are to join the second confrontation. As a result of the second confrontation, x(T2 ) punishers remain and are to join the third confrontation, and so on. This process goes on until y0 defectors are eliminated.

22

example, the indicator function of successful punishment can be given by

IS (x, y) :=

 1

if 12 x2 > y

0

if 12 x2 ≤ y

(A.2)

where, in the case of a tie, unsuccessful punishment is assumed to be the outcome. B. Proofs First, I define a Nash equilibrium for the game as follows. Definition 6. (x∗ , y ∗ , z ∗ ) is a Nash equilibrium if x∗ > 0 implies πX (x∗ , y ∗ , z ∗ ) ≥ πY (x∗ − 1, y ∗ + 1, z ∗ ) and πX (x∗ , y ∗ , z ∗ ) ≥ πZ (x∗ − 1, y ∗ , z ∗ − 1), y ∗ > 0 implies πY (x∗ , y ∗ , z ∗ ) ≥ πX (x∗ + 1, y ∗ − 1, z ∗ ) and πY (x∗ , y ∗ , z ∗ ) ≥ πZ (x∗ , y ∗ − 1, z ∗ + 1), z ∗ > 0 implies πZ (x∗ , y ∗ , z ∗ ) ≥ πX (x∗ + 1, y ∗ , z ∗ − 1) and πZ (x∗ , y ∗ , z ∗ ) ≥ πY (x∗ , y ∗ + 1, z ∗ − 1) B.1. Proof of Proposition 1 I provide the proof of Proposition 1. First, I show that a Nash equilibrium occurs at the vertices of the simplex. Observe that IS (x, y) is non-decreasing in x and is non-increasing in y IF (y, x) is non-decreasing in y and is non-increasing in x. Case 1: x > 0 and y > 0. Consider πX (x, y, z) − πY (x − 1, y + 1, z) ≥ 0 and πY (x, y, z) − πX (x + 1, y − 1, z) ≥ 0. Then, b − c − dX IF (y, x) + dY IS (x − 1, y + 1) +  ≥ 0 n b c − − dY IS (x, y) + dX IF (y − 1, x + 1) −  ≥ 0. n Thus, b − c + dY IS (x − 1, y + 1) +  ≥ dX IF (y, x) > dX IF (y − 1, x + 1) n b ≥ − c + dY IS (x, y) +  n and IS (x − 1, y + 1) > IS (x, y) which is a contradiction.

23

Case 2: x = 0, y > 0 and z > 0. Next, consider πY (x, y, z) − πZ (x, y − 1, z + 1) ≥ 0, πZ (x, y, z) − πY (x, y + 1, z − 1) ≥ 0. Then, b b c − − dX IS (x, y) ≥ 0, − c + dX IS (x, y + 1) > 0 n n and thus, IS (x, y + 1) > IS (x, y) which is a contradiction. Case 3: x > 0, y = 0, and z > 0. Finally, suppose that πX (x, y, z) − πZ (x − 1, y, z + 1) ≥ 0, πZ (x, y, z) − πX (x + 1, y, z − 1) ≥ 0 Then this implies dX IF (0, x + 1) −  ≥ 0 which in turn implies that  = 0, another contradiction. This shows that the only possible Nash equilibrium can occur at (n, 0, 0), (0, n, 0), or (0, 0, n). First, I show that the all-Cooperator equilibrium is impossible. To see this, first note that 1 πZ (0, 0, n) = b−c, πY (0, 1, n−1) = (1− )b−dY IS (0, 1), πX (1, 0, n−1) = b−c−dX IF (0, 1)+ n Thus, πZ (0, 0, n) − πY (0, 1, n − 1) =

b − c + dY IS (0, 1) < 0 n

Next, consider the all-Defector state: πY (0, n, 0) = −dY IS (0, n) = 0, πX (1, n−1, 0) =

b b −c+dX IF (n−1, 1)+, πZ (0, n, −1, 1) = −c n n

Thus, for  < c − nb , πY (0, n, 0) − πX (1, n − 1, 0) = c −

b + dX −  > 0 n

and πY (0, n, 0) − πZ (0, n − 1, 1) = −dY IS (0, n) − ( The all-Defector state is a Nash equilibrium. 24

b b − c) = c − > 0. n n

Now consider the condition for the all-Punisher state to be a Nash equilibrium. To see this, note that πX (n, 0, 0) = b − c − dX IF (0, n) + , πZ (n − 1, 0, 1) = b − c, n−1 b − c − dY IS (n − 1, 1) πY (n − 1, 1, 0) = n Observe that

1 1 (n − 1)1+ρ − >0 1+ρ 1+θ

for n > 2 since 1/(1 + ρ)21+ρ > 1 >

1 . 1+θ

Then,

πX (n, 0, 0) − πZ (n − 1, 0, 1) = −dX IF (0, n) +  =  > 0

(B.1)

Moreover, b − c − dX IF (0, n) + dY IS (n − 1, 1) + . n b = − c + dY +  n

πX (n, 0, 0) − πY (n − 1, 1, 0) =

Then, if dY > c − b/n − , the all-Punisher state is a Nash equilibrium. Finally, suppose that  = 0. Then b − c − dX IF (0, k) + dY IS (k − 1, 1) n πX (k, 0, n − k) − πZ (k − 1, 0, n − (k − 1)) = −dX IF (0, k) πX (k, 0, n − k) − πY (k − 1, 1, n − k) =

πZ (k, 0, n − k) − πX (k + 1, 0, n − (k + 1)) = −dX IF (0, k + 1) b πZ (k, 0, n − k) − πY (k, 1, n − (k + 1)) = + dY IS (k, 1) n

(B.2) (B.3) (B.4) (B.5)

Thus IF (0, k) = 0 and IF (0, k + 1) = 0 for k ≥ 1 and IS (k − 1, 1) = 1 and IS (k, 1) = 1 if k ≥ 3. Therefore, the desired result follows.  B.2. Proof of Proposition 2 For notational convenience, define ι(s) := 1 if s > 0, ι(s) = 1/2 if s = 0, and ι(s) = 0 if s < 0 for s ∈ R. Let VXn := {(x, y, z) ∈ ∆n :

1 1 1+θ x1+ρ > y }. 1+ρ 1+θ

25

Then first observe that 1 1+θ 1 1 1 x1+ρ > y =⇒ (x + 1)1+ρ > (y − 1)1+θ . 1+ρ 1+θ 1+ρ 1+θ Also, if (x, y, z) ∈ VXn , πX (x + 1, y − 1, z) − πY (x, y, z) b 1 1 1 1 1+θ = − c − dX ι( (y − 1)1+θ > (x + 1)1+ρ ) + dY ι( x1+ρ > y ) n 1+θ 1+ρ 1+ρ 1+θ b = − c + dY > 0. n Let r ∈ (0, 1) be fixed and x˜ ∈ R such that 1 1 (˜ x)1+ρ = (n − x˜ − dnre)1+θ . 1+ρ 1+θ where dnre is the smallest integer that exceeds nr. If nx˜ → 0 as n → ∞, then hence the desired result follows. Let pˆn := nx˜ . Then from (B.6), 0 ≤ pˆ1+ρ ≤ n

pˆ1+ρ n1+θ 1 + ρ n = (1 − pˆn − dnre/n)1+θ n1+ρ 1 + θ

(B.6) n| |VX |∆n |

→ 1 and

(B.7)

1+θ

If ρ > θ, then nn1+ρ → 0 as n → ∞ and thus pˆn → 0 as n → ∞. From this the result of Proposition 2 follows. B.3. Determination of resistances and conditions for stochastic stability B.3.1. Cost of transitions and resistances Here I briefly review the concepts of costs and resistances for stochastic evolutionary games (see Young (1998) for more details). Let σ = (x, y, z) ∈ ∆n be a state and Pδ (σ, σ 0 ) be the transition probability from σ to σ 0 specifying the stochastic dynamic defined by D1D3 in Section 4. We define the cost of transition between two states, σ and σ 0 , as follows: ln Pδ (σ, σ 0 ) . δ→0 ln δ

c(σ, σ 0 ) := lim

Observe that if σ 0 is obtained by the best response of an agent from σ, then c(σ, σ 0 ) = 0. Next we define a path γ to be a sequence of states induced by consecutive transitions by

26

agents: γ = (σ1 , σ2 , · · · , σT ). We define the cost for a path, γ = (σ1 , σ2 , · · · , σT ), as c(γ) :=

T −1 X

c(σt , σt+1 ).

t=1

From Proposition 1 (1), we see that the all-Defector state and the all-Punisher state are two absorbing states, denoted by ED and EP , respectively. We define the basins of attractions for these two absorbing states as follows: BP :={σ ∈ ∆n : c(γ) = 0 for some γ = (σ1 , · · · , σT ) such that σ1 = σ and σT = EP } (B.8) BD :={σ ∈ ∆n : c(γ) = 0 for some γ = (σ1 , · · · , σT ) such that σ1 = σ and σT = ED } (B.9) Further, the resistances from EP to ED and from ED to EP are defined to be rP D := min{c(γ) : γ = (σ1 , · · · , σT ), σ1 = EP and σT 6∈ BP } rDP := min{c(γ) : γ = (σ1 , · · · , σT ), σ1 = ED and σT 6∈ BD } B.3.2. Computation of rP D Recall that x¯ := min {x : πX (x + 1, n − (x + 1), 0) > πY (x, n − x, 0)} . Define φ(x) := πX (x+1, n−(x+1), 0)−πY (x, n−x, 0). Then by definition we have φ(¯ x) > 0 and φ(¯ x − 1) ≤ 0. Thus −φ(¯ x − 1) = πY (¯ x − 1, n − (¯ x − 1), 0) − πX (¯ x, n − x, 0) ≥ 0. Thus at state (¯ x, n − x¯, 0), both being a punisher and being a defector are best responses (see Panel A in Figure B.6). Let A := {(x, y, z) ∈ ∆n : x ≥ x¯} (see again Panel A in Figure B.6). Recall that IS (x, y) is non-decreasing in x and is non-increasing in y IF (y, x) is non-decreasing in y and is non-increasing in x.

27

Panel A

Panel B Punisher

Punisher

D A



(x , n − x , 0)

( np  , 0, z )

C

(x z , n − x z − z , z )

Defector

Cooperator

Defector

Cooperator

Figure B.6: Illustrations of areas

First, I will show that A ⊂ BP . Let (x, y, z) ∈ A. Then since x ≥ x¯, y ≤ n − x¯ holds and b − c − dX IF (y − 1, x + 1) + dY IS (x, y) +  n b ≥ − c − dX IF (n − x¯ − 1, x¯ + 1) + dY IS (¯ x, n − x¯) +  n =πX (¯ x + 1, n − (¯ x + 1), 0) − πY (¯ x, n − x¯, 0) + 

πX (x + 1, y − 1, z) − πY (x, y, z) =

>0.

(B.10)

Thus the cost of transition from (x, y, z) ∈ A to (x + 1, y − 1, z) is zero. Also we find that πX (x + 1, 0, n − (x + 1)) − πZ (x, 0, n − x) = dX IF (0, x + 1) +  =  > 0

(B.11)

Combining (B.10) and (B.11) shows that A ⊂ BP . Consider all paths escaping A; i.e, γ = (σ1 , · · · , σT ) where σ1 = EP and σT 6∈ A. By D3 in Section 4, the cost of transition between states is constant (called a uniform mistake model) and thus the cost of a path is simply given by the length of the path in the simplex. Thus, the minimum cost path for escaping is either on the boundary, {x ∈ ∆n : z = 0} or the boundary {x ∈ ∆n : y = 0}. Let γ ∗ be the straight line path from EP to (¯ x, n − x¯, 0). Then γ ∗ escapes BP as well as A. Since every path that escapes BP must escape A (because A ⊂ BP ) and γ ∗ is the minimum cost path for escaping A, γ ∗ must be the minimum cost path for escaping BP . This shows that rP D = c(γ ∗ ) = n − x¯

28

B.3.3. Computation of rDP To compute rDP , I proceed in the same way as before. Let C := {(x, y, z) ∈ ∆n : y ≥ n − x¯} (see Panel A in Figure B.6). Let (x, y, z) ∈ C. Then since y ≥ n − x¯, x ≤ x¯ holds and b + c + dX IF (y, x) − dY IS (x − 1, y + 1) −  n b ≥ − + c + dX IF (n − x¯, x¯) − dY IS (¯ x − 1, n − x¯ + 1) −  n = πY (¯ x − 1, n − x¯ + 1, 0) − πX (¯ x, n − x¯, 0)

πY (x − 1, y + 1, z) − πX (x, y, z) = −

≥0 and πY (0, y + 1, n − (y + 1)) − πZ (0, y, n − y) = c −

b >0 n

Again, these show that C ⊂ BD and the same argument as stated before shows that rDP = x¯ B.3.4. Conditions for stochastic stability Recall that x¯ satisfies x¯ := min {x ∈ N : πX (x + 1, n − (x + 1), 0) > πY (x, n − x, 0)} . Define xˆ = min{x ∈ R : πX (x + 1, n − (x + 1), 0) > πY (x, n − x, 0)} 1 1 x˜ such that (˜ x)1+ρ − (n − x˜)1+θ = 0 1+ρ 1+θ

29

First it is easy to see x¯ − 1 < xˆ < x¯. Then, πX (˜ x + 2, n − (˜ x + 2), 0) − πY (˜ x + 1, n − x˜ + 1, 0) 1 1 b (n − (˜ x + 2))1+θ − (˜ x + 2)1+ρ ) = − c − dX ι( n 1+θ 1+ρ 1 1 + dY ι( (˜ x + 1)1+ρ − (n − x˜ − 1)1+θ ) +  1+ρ 1+θ b 1 1 > − c − dX ι( (n − x˜ − 1)1+θ − (˜ x + 1)1+ρ ) n 1+θ 1+ρ 1 1 + dY ι( (˜ x + 1)1+ρ − (n − x˜ − 1)1+θ ) +  1+ρ 1+θ b = − c + dY +  > 0 n Thus, xˆ < x˜ + 1. If x˜ <

n−4 , 2

(B.12)

then from x¯ − 1 < xˆ < x˜ + 1

x¯ <

n 2

hold. From this, the following sufficient conditions for stochastic stability holds. If x˜ <

n−4 , the all-Punisher state is stochastically stable 2

(B.13)

Proof Proposition 3. For (1), observe that pˆn =

n−4 x˜ < n 2n

(B.14)

Since pˆn → 0 and n−4 → 21 as n → ∞, there exist n ¯ such that for all n ≥ n ¯ , the condition 2n 1 in (B.13) holds. Thus, the desired result follows. For (2), let η(x, y; ρ) := 1+ρ x1+ρ − y and η(˜ x, n − x˜, n1 ) = 0. By taking n to infinity, it follows that lim η(

n→∞

n−4 n+4 1 , ; ) = ∞. 2 2 n

(B.15)

Since η(x, n−x, n1 ) is increasing with respect to x, (B.15) implies that there exists n ¯ such that for all n > n ¯ , condition (B.13) holds. This implies that the all-Punisher state is stochastically stable.

30

Proof Proposition 4. This follows by modifying the estimation in (B.12) as follows: πX (˜ x + 2, n − x˜ − 2, 0) − πY (˜ x + 1, n − x˜ − 1, 0) 1 1 b (n − x˜ − 1)1+θ − (˜ x + 1)1+ρ ) > − c − dX ι( n 1+θ 1+ρ 1 x˜ + 1 1 (˜ x + 1)1+ρ − (n − x˜ − 1)1+θ ) − γ(1 − ) + dY ι( 1+ρ 1+θ n b = − c + dY +  − γ > 0 n

(B.16)

Proof Proposition 5. Suppose that, in the absence of defectors, the fraction of committed punishers in the population is p¯. Define E := {(x, 0, n − x) : x ≥ dn¯ pe}, where dn¯ pe is the smallest integer that exceeds n¯ p. Choose n such that dn¯ pe > 3. Then Proposition 1 implies that E is the absorbing set of states consisting of punishers and cooperators in the unperturbed process. Moreover, BP is clearly a basin of attraction for E (See Figure 4). Let z¯ := n − dn¯ pe and x¯z := min {x ∈ N : πX (x + 1, n − (x + 1) − z¯, z¯) > πY (x, n − x − z¯, z¯)} . Then I will show that the minimum cost path for escaping E is given by a straight line from (dn¯ pe, 0, z¯) to (¯ xz , n − x¯z − z¯, z¯) (see Panel A in Figure B.6). Let D := {(x, y, z) : x ≥ x¯z , y ≤ n − x¯z − z¯} (see Panel A in Figure B.6) and I will show that D ⊂ BP . Let (x, y, z) ∈ D. b − c − dX IF (y − 1, x + 1) + dY IS (x, y) n b ≥ − c − dX IF (n − (¯ xz + 1) − z¯, x¯z + 1) + dY IS (¯ xz , n − x¯z − z¯) n =πX (¯ xz + 1, n − (¯ xz + 1) − z¯, z¯) − πY (¯ xz , n − x¯z − z¯, z¯)

πX (x + 1, y − 1, z) − πY (x, y, z) =

>0.

(B.17)

Thus, D ⊂ BP . Let γ ∗ be a straight line path from (dn¯ pe, 0, z¯) to (¯ xz , n − x¯z − z¯, z¯). Then since D ⊂ BP , the minimum cost path for escaping starts from (¯ xz , n − x¯z − z¯, z¯). Then, the ∗ same argument as in Subsection B.3.2 shows that γ is indeed the minimum cost escaping path and thus rED = dn¯ pe − x¯z 31

The resistance from D to E is the same as in Proposition 3 (1) and thus rDE = x¯ and E is stochastically stable if dn¯ pe x¯z x¯ rDE = − > n n n n Then since x¯nz → 0 (see the proof of Proposition B.2) and such that for all n > n ¯ , the inequality in (B.18) holds.

32

(B.18) x ¯ n

→ 0 as n → ∞, there exists n ¯

References Adams, E. S. and M. Mesterton-Gibbons (2003). Lanchester’s attrition models and fights among social animals. Behavioral Ecology 14 (5), 719–723. Agrawal, A. and S. Goyal (2001). Group size and collective action: Third party monitoring in common-pool resources. Comparative Political Studies 34, 63–93. Andreoni, J. (1990). Impure altruism and donations to public goods: A theory of warm glow giving. Economic Journal 100, 464–77. Axelrod, R. (1986). An evolutionary approach to norms. American Political Science Review 80, 1095–1111. Bingham, P. (2000). Human evolution and human history: A complete history. Evolutionary Anthropology 9 (6), 248–257. Boehm, C. (1982). The evolutionary development of morality as an effect of dominance behavior and conflict interference. Journal of Social and Biological Structures 5, 413–421. Bowles, S. (2004). Microeconomics: Behavior, Institutions, and Evolution. Princeton: Princeton Univ. Press. Bowles, S. and J.-K. Choi (2012). The Holocene revolution: the co-evolution of agriculture technologies and private property institutions. Santa Fe Institute. Bowles, S. and H. Gintis (2004). The evolution of strong reciprocity: Cooperation in heterogeneous populations. Theoretical Population Biology 65, 17–28. Bowles, S. and H. Gintis (2011). A Cooperative Species: Human Reciprocity and its Evolution. Princeton Univ. Press. Boyd, R., H. Gintis, and S. Bowles (2010). Coordinated punishment of defectors sustains cooperation and can proliferate when rare. Science 328, 617–620. Carpenter, J., S. Bowles, H. Gintis, and S.-H. Hwang (2009). Strong reciprocity and team production: Theory and evidence. Journal of Economic Behavior and Organization 71 (2), 221–232. Carpenter, J. P. (2007). Punishing free-riders: How group size affects mutual monitoring and the provision of public good. Games and Economics Behavior 60, 31–51.

33

Casari, M. and L. Luini (2012). Peer punishment in teams: Expressive or instrumental choice. Experimental Economics 15, 241–259. Chamberlin, J. (1974). Provision of collective goods as a function of group size. American Political Science Review 68, 707–716. Chance, P. (2009). Learning and Behavior: Active Learning Edition. Wadsworth. Chaudhuri, A. (2011). Sustaining cooperation in laboratory public goods experiments: A selective survey of the literature. Experimental Economics 14, 47–83. Cinyabuguma, M., T. Page, and L. Putterman (2005). Cooperation under the threat of expulsion in a public goods experiment. Journal of Public Economics 89, 1421–1431. Cinyabuguma, M., T. Page, and L. Putterman (2006). Can second-order punishment deter perverse punishment? Experimental Economics 9, 265–279. Darling, R. and J. Norris (2008). Differential equation approximations for Markov chains. Probability Surveys 5, 37–79. Denant-Boemont, L, M. D. N. C. (2007). Punishment, counterpunishment and sanction enforcement in a social dilemma experiment. Economic Theory 33, 145–167. DeNardo, J. (1985). Power in Numbers: The Political Strategy of Protest and Rebellion. Princeton: Princeton Univ. Press. Engel, J. H. (1954). A verification of Lanchester’s law. Journal of the Operational Research Society of America 2, 163–171. Esteban, J. and D. Ray (2001). Collective action and the group size paradox. American Political Science Review 95 (3), 663–672. Fehr, E. and S. G¨achter (2000). Cooperation and punishment in public goods experiments. American Economic Review 90 (4), 980–94. Feinberg, M., R. Willer, and M. Schultz (2014). Gossip and ostracism promote cooperation in groups. Psychological Science 25, 656–664. Francis, H. (1985). The law, oral tradition and the mining community. Journal of Law and Society 12 (3), 267–71. G¨achter, S. and E. Fehr (1999). Collective action as a social exchange. Journal of Economic Behavior and Organization. 34

Gale, J., K. Binmore, and L. Samuelson (1995). Learning to be imperfect: the ultimatum game. Games and Economic Behavior 8, 56–90. Gorodnichenko, Y. and G. Roland (2015). Culture, institutions and democratization. Gruter, M. (1986). Ostracism on trial: The limits of individual rights. Ethology and sociobiology 7, 271–279. G¨ uth, W., M. Vittoria Levati, M. Sutter, and E. van der Heijden (2007). Leading by example with and without exclusion power in voluntary contribution experiments. Journal of Public Economic. Haan, M. and P. Kooreman (2002). Free riding and provision of candy bars. Journal of Public Economics 83, 277–291. Herrmann, B., C. Thoni, and S. G¨achter (2008). Antisocial punishment across societies. Science 319 (7), 1362–7. Hirsch, M., S. Smale, and R. Devaney (2004). Differential Equations, Dynamical System and an Introduction to Chaos. Elsevier. Hirshleifer, D. and E. Rasmusen (1989). Cooperation in a repeated prisoners dilemma with ostracism. Journal of Economic Behavior and Organization 12, 87–106. Hirshleifer, J. (1991a). The paradox of power. Economics and Politics 3 (3), 177–200. Hirshleifer, J. (1991b). The technology of conflict as an economic activity. American Economic Review Papers and Proceedings 81 (2), 130–134. Holth, P. (2005). Two definitions of punishment. The Behavior Analyst Today 6, 43–47. Hopfensitz, A. and E. Reuben (2009). The importance of emotions for the effectiveness of social punishment. Economic Journal 119 (540), 1534–1559. Hwang, S.-H. (2009). Three Essays on Conflict and Cooperation. Dissertation, University of Massachusetts Amherst. Isaac, R. M. and J. M. Walker (1988). Group size effects in public goods provision: the voluntary contributions mechanism. Quartely Journal of Economics 103, 179–199. Kandori, M., G. J. Mailath, and R. Rob (1993). Learning, mutation, and long run equilibria in games. Econometrica 61 (1), 29–56.

35

Knauft, B. M. (1987). Reconsidering violence in simple human societies: Homicide among the gebusi of new guinea. Current Anthropology 28, 457–500. Konrad, K. A. (2009). Strategy and Dynamics in Contests. Oxford: Oxford University Press. Kosfeld, M., A. Okada, and A. Riedl (2009). Institution formation in public goods games. American Economic Review 99, 1335–1355. Lanchester, F. (1917). Aircraft in Warfare. New York: D. Appleton and Company. Ledyard, O. (1995). Public goods: some experimental results. In J. Kagel and A. Roth (Eds.), Handbook of experimental economics, Chapter Chap. 2. Princeton Univ. Press. Mahdi, N. Q. (1986). Pukhtunawali: Ostracism and honor among the Pathan hill tribes. Ethology and Sociobiology 7, 295–304. Maier-Rigaud, F., P. Martinsson, and G. Staffiero (2010). Ostracism and the provision of a public good: experimental evidence. Journal of Economic Behavior and Organization 73, 387–395. Marwell, G. and P. Oliver (1993). The Critical Mass in Collective Action: A Micro-Social Theory. Cambridge: Cambridge Univ. Press. Masclet, D., C. Noussair, S. Tucker, and M.-C. Villeval (2003). Monetary and non-monetary punishment in the voluntary contributions mechanism. American Economic Review 93 (1), 366–80. McGuire, M. (1974). Group size, group homogeneity and the aggregate provision of a pure public good under Cournot behavior. Public Choice 18, 107–126. Naidu, S., S.-H. Hwang, and S. Bowles (2010). Evolutionary bargaining with intentional idiosyncratic play. Economics Letters 109, 31–33. Nikiforakis, N. and H.-T. Normann (2008). A comparative statics analysis of punishment in public-good experiments. Experimental Economics. Olcinia, G. and V. Calabuig (2015). Coordinated punishment and the evolution of cooperation. Journal of Public Economic Theory 17, 147–173. Oliver, P. and G. Marwell (1988). The paradox of group size in collective action. American Sociological Review 53, 1–8.

36

Olson, M. (1965). The Logic of Collective Action. Cambridge, MA: Harvard University Press. Ostrom, E. (1998). A behavioral approach to the rational choice theory of collective actions. American Political Science Review 92, 1. Ostrom, E., J. Walker, and R. Gardener (1992). Covenants with and without a sword: Self governance is possible. American Political Science Review 86, 404–416. Panchanathan, K. and R. Boyd (2004). Indirect reciprocity can stabilize cooperation without the second-order free rider problem. Nature 432, 499–502. Pecorino, P. (2009). Public goods, group size, and the degree of rivalry. Public Choice 138, 161–169. Piazza, J. and J. M. Bering (2008). Concerns about reputation via gossip promote generous allocation in an economic game. Evolution and Human Behavior 29, 172–178. Plato. The Republic. Plowes, N. J. R. and E. S. Adams (2005). An empirical test of Lanchester’s square law: Mortality during battles of the fire ant solenopsis invicta. Proceedings of the Royal Society B 272, 1809–1814. Rege, M. and K. Telle (2004). The impact of social approval and framing on cooperation in public good situations. Journal of Public Economic 88, 1625–1644. Roland, G. and Y. Xie (2016). Culture and collective action. In T. Besley (Ed.), Contemporary Issues in Development Economics, pp. 44–60. Palgrave McMillan. Samz, R. (1972). Some comments on Engel’s “verification of Lancehser’s law”. Operational Research 20 (1), 49–52. Sandler, T. (1992). Collective Action: Theory and Applications. Ann Arbor: University of Michigan Press. Sethi, R. (1996). Evolutionary stability and social norms. Journal of Economic Behavior and Organization 29, 113–140. Sethi, R. and E. Somanathan (2006). A simple model of collective action. Economic Development and Cultural Change 54 (3), 725–747.

37

Wiessner, P. (2005). Norm enforcement among the Ju/’hoansi bushmen: A case of strong reciprocity? Human Nature 16 (2), 115–45. Wiessner, P. (2014). Embers of soceity: Firelight talk afmong the Ju/’hoansi bushmen. Proceedings of the National Academy of Sciences 111 (39), 14027–14035. Wood, E. (2003). Insurgent Collective Action and Civil War in El Salvador. Cambridge: Cambridge University Press. Wu, J., D. Balliet, and P. A. M. V. Lange (2016). Gossip versus punishment: the efficiency of reputation to promote and maintain cooperation. Scientific Report 6. doi:10.1038/srep23919. Young, H. P. (1993). The evolution of conventions. Econometrica 61 (1), 57–84. Young, H. P. and D. Foster (1991). Cooperation in the short and the long run. Games and Economic Behavior 3, 145–156. Young, P. (1998). Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton: Princeton Univ. Press.

38