Article

An alternative mechanism through which economic inequality facilitates collective action: Wealth disparities as a sign of cooperativeness

Journal of Theoretical Politics 24(4) 461–484 ©The Author(s) 2012 Reprints and permission: sagepub.co.uk/journalsPermissions.nav DOI:10.1177/0951629812448236 jtp.sagepub.com

Tim Johnson Atkinson Graduate School of Management, Willamette University, USA

Oleg Smirnov Department of Political Science, Stony Brook University, USA

Abstract Past models treat economic inequality as an exogenous condition that can provide individuals a dominant incentive to produce collective goods unilaterally. Here we part with that tradition so as to treat economic inequality and collective action as endogenous, and to examine whether economic inequality can foster collective action even when all individuals can gain from freeriding. Using evolutionary game theory and computer simulations, we study whether cooperation can evolve when agents play multiple, one-shot prisoner’s dilemma (PD) games per generation and employ strategies that condition cooperative play on their game partners’ wealth holdings. In this game environment, we find that collective action succeeds via a strategy in which players choose to cooperate when joining a PD with an economic equal and defect when partnered with a player possessing wealth holdings unequal to their own. These results signal an alternative avenue through which economic inequality can influence the viability of collective action.

Keywords Coequals; collective action; cooperation; economic inequality; prisoner’s dilemma

1. Introduction The relationship between economic inequality and collective action has captured scholars’ attention since Mancur Olson (1965) first recognized the challenge of joint activity. Early in The Logic of Collective Action, Olson established that economic inequality could spark the production of collective goods: if a group member holds a disproportionate Corresponding author: Oleg Smirnov, SBS-7, Stony Brook, New York, 11794, USA. Email: [email protected]

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amount of assets that would benefit from a collective good, then that member will find it in her interest to produce some portion of the collective good unilaterally. Others have expanded on this insight and, drawing on Olson’s framework, have shown that economic inequality can have a positive, negative, or null effect on the production of collective goods (see Andreoni, 1989; Baland and Platteau, 1997; Bardhan et al., 2007; Bergstrom et al., 1986; Cornes and Sandler, 1996; Heckathorn 1993, 1996; Johnson and Libecap, 1982; Marwell et al., 1988; Oliver et al., 1985; Warr, 1983). While this line of research illuminates how economic inequality compels individuals to furnish collective goods when they have an incentive to do so unilaterally, it does not explain collective action in scenarios where individuals have a dominant incentive to free-ride on others’ cooperation. That is, in the framework originating with Olson (1965), the percentage of a group’s wealth held by a given individual determines the benefits that individual draws from the collective good (see, most recently, Bardhan et al., 2007). This means that individuals possessing considerable wealth may have, in some circumstances, an incentive to produce some or all of a collective good, even if poorer members of the community will free-ride on those efforts (Olson, 1965). Viewed in those terms, the past literature explains how economic inequality—or, perhaps more precisely, group members’ relative concentration of wealth—can promote some degree of collective good production. It does not, however, show that economic inequality can help individuals mutually cooperate in collective action scenarios in which individuals have a dominant incentive to free-ride. In addition to this limitation, the past literature also treats economic inequality as exogenous. Scholars, beginning with Olson (1965), assume some level of inequality— either by fiat or by an exogenously imposed redistribution scheme—and then examine how this inequality influences an individual’s incentive to contribute to collective action (see Bergstrom et al., 1986). In real collective action scenarios, however, economic inequality arises endogenously with collective action (Fowler et al., 2005). That is, freeriding necessarily creates wealth disparities (Fowler et al., 2005), and those wealth disparities may influence the strategies that individuals use in subsequent collective action scenarios (Dawes et al., 2007). Accordingly, modeling economic inequality as exogenous is only appropriate in a one-time collective action scenario. Recognizing these limitations, and seeking to overcome them, we deviate from the past literature by studying a set of models in which (a) individuals have a dominant incentive to free-ride on others’ cooperation, and (b) economic inequality is treated as endogenous. At the core of our analysis is an evolutionary game theory model in which a population of agents play multiple, one-shot prisoner’s dilemma (PD) games with various partners over a given time period.1 No agent plays with the same partner for two or more games in a row; this feature distinguishes our model from iterated prisoner’s dilemma (IPD) models. We study the one-shot PD since past scholars have identified it as the most basic form of collective action—any solution to the simple prisoner’s dilemma we study can be generalized to more complicated multi-player models of collective action (Hardin, 1982). In the one-shot PD games we study, players use a strategy determined at the beginning of a time period, which, via interaction with another player’s strategy, secures them a one-shot PD game payoff. Once this payoff is obtained, it is added to the player’s earnings from past one-shot games, thereby creating an aggregate payoff—the player’s ‘wealth holdings’—that other players can observe and respond to in subsequent

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one-shot games. After all games in a time period have ended, players update their PD strategies by assessing the payoffs that other strategies obtain and switch their strategy, if necessary, to one that has obtained greater wealth in past play over the time period. In other words, more successful strategies are more likely to be replicated over time, regardless of the mechanism governing selection, but agents only alter their strategies at the end of a time period. Using this framework, we discover that collective action succeeds—even in the presence of rationally self-interested individuals—when agents (a) possess sufficiently accurate information about others’ wealth holdings and (b) employ a strategy, called COEQUALS,2 in which they cooperate only with game partners who possess wealth holdings equal to their own.3 The success of this strategy is due to the fact that, when economic inequality and collective action are modeled endogenously, wealth disparities provide information about an individual’s cooperativeness. Specifically, since payoffs are—in part—a function of an agent’s strategy, an agent is more likely to have the same wealth holdings as an agent using its same strategy than to have the same wealth holdings as a player using a different strategy. Thus, wealth holdings provide information that allows players adopting the COEQUALS strategy to avoid free-riders and to direct their cooperation toward individuals who will more likely reciprocate it. In serving this purpose, the information contained in wealth disparities provides a mechanism— different from that which Olson (1965) proposed—through which economic inequality can contribute to successful collective action. Moreover, our models advance the study of cooperation by providing a novel and theoretically justifiable information mechanism through which cooperation can evolve. Unlike information mechanisms that previous authors have proposed as signals of cooperativeness—e.g., tags (Riolo et al., 2001) or image scores (Nowak and Sigmund, 1998), both of which require theoreticians to add new constructs to the PD model— wealth disparities derive from basic features of the PD. A simple evolutionary game theoretic model of agents playing multiple, one-shot PD games per generation would implicitly include wealth holdings as a means of tabulating agents’ fitness. Our model simply makes that information available to players, thus transforming a fundamental feature of evolutionary game models into an information mechanism. This feature of a player’s wealth holdings also makes those holdings an information mechanism that is less vulnerable to counterfeiting. Whereas past information mechanisms serve no other purpose but to display cooperativeness (thus making them a clear target for forgery), wealth holdings—as a tangible product of past game play—signal cooperativeness and determine fitness. This makes wealth holdings independently verifiable and, thus, less susceptible to counterfeiting. Moreover, unlike arbitrary markers (Riolo et al., 2001), wealth holdings cannot change via mutation. As a result, there is no clear way—absent supplying agents with the ability to falsify wealth holdings (a possibility we consider below)—for a defector displaying fake wealth holdings to emerge in the population. However, even though there are theoretical reasons why wealth holdings would be difficult to fake, our analysis shows that the information contained in wealth disparities between players is robust even when noise is added to players’ wealth holdings. Adding this noise to wealth holdings is substantively equivalent to either (a) creating the possibility that defectors can falsify their wealth holdings strategically, (b) assuming that players may not accurately perceive others’ wealth holdings in all instances,

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or (c) allowing players to offer some leeway in the rules they use to condition their behavior on wealth holdings. Computer simulations designed to study the robustness of the analytic model’s findings show that even when this noise is added to the model, the COEQUALS strategy persists in the population. Its evolutionary success, however, is inversely proportional to the size of the range of equality—as the range of equality approaches infinity, COEQUALS becomes a strategy that indiscriminately cooperates with other players. This result underlines the value of information contained in disparities between players’ wealth holdings. By providing insight into players’ strategies, wealth holdings allow cooperation to evolve even in a Spartan environment consisting entirely of one-shot games that do not afford players punishment opportunities. With the exception of kin altruism, we know of no cooperative strategy other than COEQUALS that can evolve in this environment (see Nowak, 2006 for a discussion of existing models). Beyond these theoretical insights, the results we report illuminate empirical phenomena ranging from the decline of civic participation under conditions of inequality to the norms and institutions groups use to foster collective action. Before discussing these and other implications, we present, in the next section of this paper, our core analytic model. Following that presentation, we describe the methods used to analyze the model and report the results of that analysis. These results raise further questions that cannot be answered through analytic methods. As a result, we construct a computer model and report how this model provides further insight into the interplay between economic inequality and collective action. With these computational results, as well as the results from our analytic model, we discuss the implications of our analysis and put forward future means to extend and empirically test our theory. Following that, we conclude.

2. An analytic model of collective action and economic inequality We propose a standard evolutionary game theoretic model in which an infinitely large population of individuals play a series of one-shot PD games over the course of a given time period, which we also refer to as a ‘generation.’4 One can regard this period as any relevant time span over which collective action could occur. For example, the span might represent, for members of a labor union, the length of time over which their contract with management holds. For farmers, the period might represent the season, from seeding to harvest, in which they use irrigation. A precinct person might consider it the time between elections. In sum, the time period we model can represent any meaningful term over which collective action can transpire. During this time period, individuals are randomly paired to play a number of one-shot PD games. In other words, upon each pairing there is, for each player, an equal probability of drawing any other player as a game partner. While this feature makes games one-shot, we note that the one-shot character of these PD games is further preserved by the fact that players have no information that could identify a past PD partner. This fact will become clear shortly. PD games are defined by the classic payoff structure T > R > P > S and 2R > T + S. That is, in the PD, two agents must simultaneously choose either to cooperate or defect. When both cooperate, each receives R, the game payoff that yields the

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greatest social welfare; when one player defects while the other cooperates, the defecting player earns T—the game’s highest payoff—and the cooperating player earns S—the game’s lowest payoff. Thus, whether to obtain additional resources or to defend against exploitation, self-interested players defect; when both do so, the suboptimal outcome of mutual defection, P, results. At the end of each game, a player’s payoff is added to her previous earnings (in the first round of a time-span a player’s payoff is zero) and this aggregate payoff—labeled her ‘wealth holdings’—can be observed by future PD game partners. At the end of the lifespan, individuals are subject to selection governed by the standard replicator dynamics (Taylor and Jonker, 1978). Note that the replicator dynamics, although traditionally used to model natural selection by means of the differential transmission of genetic information, can also capture non-genetic processes of population level behavioral change, such as imitation or social learning (Bendor and Swistak, 1997). In our baseline analytic model, three types of players exist in the studied population: cooperators (C) always cooperate in PD play, defectors (D) always defect, and COEQUALS (A) cooperate only when playing with players who possess equal wealth holdings. Past scholarship justifies our inclusion of these three strategies in the baseline model. Including so-called ‘naïve’ cooperation—the strategy employed by cooperators— is justified because it appears in laboratory and everyday social encounters (Sally, 1995) and serves as an informative benchmark in theoretical models of cooperative evolution (Nowak 2006). Including defectors, on the other hand, is legitimate because they play the strategy posited by conventional models of strategic, self-interested choice (Hardin, 1982); in so doing, defectors provide the main challenge for players aiming to foster successful collective action. The strategy COEQUALS obtains its theoretical merit by fusing the egalitarian motives that individuals use when assessing wealth distributions (Dawes et al., 2007) with the similarity-driven assortment that occurs in community activities (Alesina and La Ferrara, 2000). These theoretical justifications limit our analytic focus to these three strategies; however, in the computer model we report later, we include a wider range of strategies and, in so doing, study every possible strategy that makes choices contingent on other players’ wealth holdings. As the reader may notice, economic inequality enters our model in a manner different from the way in which it enters the model studied by Olson (1965) and subsequent scholars. In the Olson (1965) model, agents examine what portion of society’s wealth they possess and ask themselves ‘Would it be profitable for me to contribute to a collective good, even if others do not do so?’. In our model, agents adopting the COEQUALS strategy examine their game partner’s wealth, compare it to their own, and then ask themselves ‘Should I cooperate with my game partner, given the extent to which our wealth holdings differ?’. Accordingly, whereas Olson’s approach limits cooperative actions—by assumption—to players in the upper reaches of the wealth distribution, who find it profitable to furnish collective goods regardless of whether others cooperate, our framework leaves open the possibility that individuals in any segment of the wealth distribution might act cooperatively. Indeed, in our model, successful collective action does occur among individuals in lower reaches of the wealth distribution—a possibility that the Olson (1965) framework denies by fiat. The proportions of the population consisting of cooperators, defectors, and COEQUALS are, respectively, c, d, and a, such that c + d + a = 1. Again, observe

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that population proportions do not change between each PD game; selection takes place at the end of a generation when all one-shot PD games have been played. Since agents using the strategy COEQUALS make decisions contingent upon their partner’s wealth holdings, we introduce additional notation to account for this feature of that strategy. Let pi represent the probability that an individual following the strategy COEQUALS has the same wealth holdings as a cooperator in game i of a time period; likewise, let zi represent the probability that COEQUALS has the same wealth holdings as a defector in game i and qi represent the probability that COEQUALS has the same wealth holdings as another player using COEQUALS in game i of a time period. The payoff structure for the whole game—consisting of n one-shot PD interactions—is therefore:

COOP

⎛ COOP DEF COEQ

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ n ⎝  i=1

DEF

nR

nS

nT (Rpi + T (1 − pi ))

nP n  i=1

(Szi + P (1 − zi ))

n 

COEQ (Rpi + S (1 − pi ))



⎟ ⎟ ⎟ (Tzi + P (1 − zi )) ⎟ ⎟ i=1 ⎟ n ⎠  (Rqi + P (1 − qi ))

i=1 n 

i=1

(1) Examining the payoffs that a player using COEQUALS earns over the course of a time period—the third row in the payoff matrix above—reveals several aspects of the strategy. The lowest left cell in the matrix indicates that players using COEQUALS engage in both cooperative and exploitative relations with cooperators; as a result, those following the COEQUALS strategy potentially enjoy the PD’s top two payoffs. The middle cell of the third row shows that COEQUALS can guard against exploitation by defecting on those defectors that possess different wealth holdings. The lower rightmost cell shows that an individual employing the strategy COEQUALS cannot exploit another organism that employs COEQUALS: players using COEQUALS either cooperate with each other or mutually defect. Consider an illustrative example of the model in which individuals play two games in a generation (see Appendix 1 for details). In this case, COEQUALS is always exploited by defectors in the first game of a generation—when all players have a fitness cue equal to zero—but it is never exploited in the second game of a generation, since COEQUALS and defection strategies never have the same wealth holdings at the outset of the second game. Thus, when players play two or more games per generation, defectors’ exploitation of COEQUALS in the first game of the generation may be eclipsed by COEQUALS’ ability to use economic inequality to foster successful cooperation. Indeed, if it were computationally possible to calculate the universe of possible wealth holdings which players could have at each game of a generation (conditional on their strategy), then an exact probability of COEQUALS cooperating with each type (including itself) could be specified. Unfortunately, these calculations are laborious even when the number of rounds is equal to 2 (see Appendix) and grow intractable after even a modest number of rounds. As a result, we resort to the above mentioned parameters, pi , zi , and qi , which give the probabilities of a COEQUALS player cooperating, respectively, with a player adopting a cooperator, defector, or COEQUALS strategy.

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T

DE F

C O EQ

Figure 1. Replicator dynamics of the multiple one-shot prisoner’s dilemma games with cooperators, defectors, and COEQUALS

With this model specified, we can get an initial sense of the model’s predictions by studying the population dynamics resulting from the interaction of all strategies. Figure 1 presents these population dynamics for a sample set of parameters (n = 2, T = 1, R = 0.9, P = 0.1, S = 0). The simplex displayed in Figure 1 conveys information about the composition of the modeled population at a given state and the future composition of the population at that state due to selective forces. Specifically, each point in the simplex, on the edges of the simplex, and at the vertices of the simplex represents a possible composition of types in the modeled population. The arrows within the simplex point to the composition of the population in the next generation, thereby indicating the evolutionary trajectory of the population. Substantively, the graphic suggests that both defectors and COEQUALS resist invasion from a group of ‘mutant’ players that change their strategy by pure chance. The graphic also suggests that, for some parameter settings, the basin of attraction for COEQUALS is large: it takes a relatively small number of COEQUALS to invade a population of defectors, while a large number of defectors are required to prevent the population from moving toward a state in which all individuals adopt the strategy COEQUALS. A more general analysis provides the exact conditions in which strategies resist invasion. In our model COEQUALS resists an invasion of mutant cooperators and defectors when, respectively, (a) πA > πC and πA > πD and (b) COEQUALS is the incumbent population, a = 1 − εC − εD , and other types are mutants—viz., c = εC and d = εD —constituting a very small percentage of the population. Examination of the payoff structure in matrix (1) indicates that πA = πC for εD = 0 and πA > πC for εD > 0. Thus, COEQUALS is neutrally stable against cooperators in the absence of mutant defectors and evolutionarily stable when mutant defectors are present in the population. The condition required for the incumbent population of COEQUALS to resist invasion of

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mutant defectors, πA > πD , holds when n

(Rqi + P (1 − qi )) >

i=1

n

(Tzi + P (1 − zi ))

(2)

i=1



n

(R − P) qi >

i=1

n

(T − P) zi

i=1

Thus, while (R − P) will always be less than (T − P), the probability that two individuals using the strategy COEQUALS have the same wealth holdings is much higher than the probability that an organism using COEQUALS has the same wealth holdings as a defector, i.e., qi >> zi , i = 2, 3, ..., n.5 In fact, since COEQUALS is the incumbent population, this probability is approaching one, q1,2,...,n → 1. If game payoffs are drawn from a continuous set of rational numbers, the probability that a COEQUALS strategy has the same aggregate payoff as a defector is asymptotically zero. Thus, the difference in payoffs after the first round will persist in subsequent rounds since defectors will never receive either R or S, which are the only two payoffs available to COEQUALS in the first PD game of a generation: that is, z1 = 0 and z2,3,...,n → 0. Thus, it follows that COEQUALS is an evolutionary stable strategy for R>

T + (n − 1) P n

(3)

T −P R−P Condition (3) is identical to that which enables TIT-FOR-TAT (TFT) to remain stable against ‘always defect’ (ALLD) in the repeated PD (Nowak, 2006); moreover, when n = 2, the formula yields the exact condition required for WIN–STAY, LOSE–SHIFT (WSLS) to avoid invasion by ALLD in the repeated PD (Nowak, 2006). Notice, however, that we model one-shot interactions with various game partners—a social environment, which is distinct from that in which TFT and WSLS were studied (Axelrod, 1984; Nowak and Sigmund, 1993). Analysis of the model also shows that defectors resist mutant invasion. When defectors hold incumbency—that is, d = 1 − εC − εA , εC → 0, and εA → 0—the game payoffs n  ((Szi + P (1 − zi ))), suggesting that πD > πA > πC . are πC = nS, πD = nP, and πA = ⇔n>

i=1

COEQUALS can only invade the population of defectors if its mutant proportion is large. Though convention forbids us from considering COEQUALS a ‘mutant’ strategy in this case, the required proportion of COEQUALS needed to invade a population of defectors remains a potentially interesting solution to the collective action problem. That is, by examining how large an unexpected, mass shift of behavior must be in order to unseat a socially sub-optimal state of the world, we can discern how dramatically social mores would need to change in order to produce socially beneficial outcomes (see, e.g., Mackie, 1996). To perform this analysis, we solve for c∗ —defined as the minimal proportion of COEQUALS needed to invade the population of defectors—and find that c∗ >

P−S nR − (T + S + P (n − 2))

(4)

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Inequality (4) indicates that a relatively small proportion of COEQUALS can invade a population of defectors when values of R, S, and n are large relative to values of T and P. For example, when T = 4, R = 3, P = 2 , S = 1, and n = 11, COEQUALS must constitute only c∗ > 0.1 to successfully invade a population of defectors. In our illustrative phase diagram (Figure 1), this threshold proportion is c∗ > 0.125. Furthermore, this analysis suggests that the robustness of COEQUALS critically depends on (a) the existence of mutant defectors and (b) the accurate detection of wealth holdings. Mutant defectors promote the fitness of COEQUALS by enabling the strategy to stave off invasion by cooperators. When defectors are extinct, cooperators and COEQUALS have identical wealth holdings, raising the possibility of genetic drift; any mutant defection, however, halts drift by enabling COEQUALS to obtain greater wealth holdings than cooperators. Accurate detection of wealth holdings—a phenomenon we explore in more detail in the next section—also increases the fitness of COEQUALS. When wealth holdings are inaccurately transmitted or perceived, the likelihood of COEQUALS cooperating with a defector increases, and this reduces the strategy’s fitness. Most importantly, inaccurate detection of wealth holdings leads to mutual defection among COEQUALS themselves. This outcome results because agents can no longer use economic inequality to inform themselves about the cooperativeness of their game partner. Note, however, that this does not mean that collective action suffers when noise is added to the model; it may be that COEQUALS, by cooperating only with equals, forgoes beneficial opportunities to engage in collective action with cooperators. In the next section of this paper we explore how making inequality less informative, by adding noise to wealth holdings, influences the prospects of collective action.

3. A computational model of collective action and economic inequality Although the analytic model presented above shows that economic inequality can foster collective action by helping COEQUALS distinguish potential cooperators from defectors, several questions remain. First, when wealth holdings cannot be accurately assessed, how badly does COEQUALS—and, perhaps, the success of collective action—suffer? Second, does the information contained in economic inequality only help players employing COEQUALS, or do other strategies that act contingently on wealth holdings also obtain beneficial outcomes due to economic inequality’s informativeness? Below we answer those questions with a computer model that extends the analytic model presented in the previous section. We turn to computer modeling in this situation because amendments to our model make it analytically challenging. That is, although the computer model still shares the central features of the analytic model (players still play a number of one-shot PD games over a given time period and payoffs from these games aggregate so as to create a wealth holding for each player), the computer model differs by adding new strategies and allowing imperfect detection of others’ wealth holdings. Both of these amendments provide further insight into the dynamic interplay of economic inequality and collective action. Incorporating a broader array of strategies into the model allows us to discern whether or not other strategies that act contingently on their partner’s wealth holdings can foster

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collective action. One type, which we label the BOTTOM DWELLER (BD), compares its partner’s wealth holdings to its own and cooperates if the partner’s wealth holdings are less than its own; if its partner’s wealth holdings are equal to or greater than its own, it defects. Players adopting the LADDER CLIMBER (LC) strategy, on the other hand, compare their wealth holdings to those of their partner and, if their partner’s wealth holdings are greater than their own, cooperate; otherwise, they defect. Last, ANTI-COEQUALS (AC) inverts the COEQUALS method of game play. Instead of cooperating with those who have the same wealth holdings, it defects with them and cooperates with those who have different wealth holdings. Adding the above strategies allows us to examine whether COEQUALS benefits from its specific form of payoff contingent decision making (i.e. cooperating with equals) or whether any form of payoff contingent decision making is beneficial. Altering agents’ ability to perceive other players’ PD earnings also allows us to better understand the choice mechanism that distinguishes COEQUALS from other payoff contingent strategies. In the analytic model, we study COEQUALS in an environment where the connection between strategies and observed payoffs is undisturbed. As the analytic model indicates, that environment is ideally suited for a strategy that cooperates only with equals. Yet, in some everyday collective action scenarios, a ‘clean’ link between strategies and payoffs may not exist. Perceptual error, for instance, might cloud a player’s ability to assess wealth holdings; alternatively, individuals might adopt a slightly more tolerant variant of COEQUALS that elects to cooperate with those whose wealth holdings differ slightly from their own. Such possibilities make it necessary to study how random errors alter the efficacy of COEQUALS.

3.1. General features of the computational model In order to implement the modifications discussed above, we created a computational model that closely resembled our analytic model (see Appendix 2 for the computer code of the simulations in R). In the computational model, a population of N = 200 agents played a number, n, of one-shot PD games per generation. Agents were randomly paired so that regular, repeated play was impossible; moreover, agents could not identify past partners, as is evident in the description of strategies provided in the previous section. We further imposed a technical restriction so that no agent could interact with others more than once during a single one-shot game. All simulation parameters were drawn from uniform distributions. The parameter space that we examined can be inferred from the descriptive statistics in Table 1. Among the simulation parameters that we vary is the number of games played in a generation, n. To implement random variation of payoffs, we followed computational modeling conventions (see Nowak, 2006) and captured all PD payoffs with two parameters, b and c, such that T = b, R = b − c, P = 0, and S = −c. The mutation rate, mr, at which a given strategy might randomly switch to another strategy, irrespective of fitness, was also varied across simulation runs, thereby allowing us to understand the effect of strategy variation on the interplay of inequality and collective action. The simulation was run 1000 times and each simulation run lasted 500 generations, g, to ensure dynamic convergence. For each simulation, we recorded the proportion of each type at g = 500. We also collected information about the frequency with which COEQUALS cooperated with all types in the population throughout all 500 generations.

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Table 1. Descriptive statistics for simulation with additional types Variable

Mean

Std.Dev.

Min

Max

Benefit, b Cost, c Number of rounds, n Mutation rate, mr

0.798 0.207 5.912 0.031

0.113 0.112 2.535 0.014

0.6 0.01 2 0.01

1 0.399 10 0.05

Proportion of Cooperators (at g=500) Proportion of Defectors (at g=500) Proportion of COEQUALS (at g=500) Proportion of Anti-COEQUALS (at g=500) Proportion of Bottom-Dwellers (at g=500) Proportion of Ladder-Climbers (at g=500)

0.071 0.265 0.477 0.044 0.065 0.078

0.104 0.285 0.353 0.065 0.084 0.108

0 0 0 0 0 0

0.7 0.978 0.997 0.682 0.581 0.751

0.354

0.217

0.065

0.927

0.000

0.000

0

0

0.439

0.187

0.153

0.958

0.011

0.009

0

0.037

0.004

0.004

0

0.014

0.004

0.004

0

0.015

Proportion of games in which COEQUALS cooperated with Cooperators Proportion of games in which COEQUALS cooperated with Defectors Proportion of games in which COEQUALS cooperated with COEQUALS Proportion of games in which COEQUALS cooperated with Anti-COEQUALS Proportion of games in which COEQUALS cooperated with Bottom-Dwellers Proportion of games in which COEQUALS cooperated with Ladder-Climbers

Note: Descriptive statistics for n>2 (since players start with the same fitness and COEQUALS cooperate with all types), 1000 runs of the simulation, all three treatments combined.

Moreover, we implemented three treatment conditions on top of the above procedures to assess the degree to which the initial distribution of agent types influenced simulation results. In one treatment condition we set the initial proportion of defectors in the population to unity; we then examined how other strategies, via mutation, might penetrate this non-cooperative state of the world. In a second treatment condition, we seeded the population fully with COEQUALS and, again, examined whether or not other strategies might succeed in this environment. Last, we implemented a baseline treatment condition in which the population started with a random distribution of types, drawn from a uniform distribution. As the results reported below indicate, these treatment conditions, along with some of the other parameters varied in the simulation, carried significant consequences for the viability of both cooperation and COEQUALS.

3.2. The effect of additional strategies on collective action and COEQUALS As noted above, the addition of new strategies to the model allows us to determine whether or not payoff contingent strategies generally perform better than conventional strategies, or whether the specific payoff contingent mechanism employed

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Table 2. Logistic regression analysis for simulation with multiple types and perfect information Simulation parameters

DV: Proportion of COEQUALS in the population at g=500

DV: COEQUALS cooperating with Cooperators

DV: COEQUALS cooperating with COEQUALS

Starting population: 100% Defectors Starting population: 100% COEQUALS Benefit, b Cost, c Number of games, n Mutation rate, mr Constant AIC Observations Standard error in parentheses

–0.109 (0.182)

0.062 (0.172)

0.066 (0.162)

0.263 (0.189)

0.305 (0.178)

0.184 (0.169)

3.147 (0.680) –10.003 (0.774) 0.321 (0.032) –18.283 (5.340) –1.955 (0.604) 816.31 1000

0.746 (0.635) –2.996 (0.643) –0.257 (0.030) –33.384 (5.133) 1.753 (0.579) 834.84 1000

0.215 (0.599) –1.070 (0.603) –0.228 (0.028) –29.467 (4.820) 1.954 (0.548) 960.38 1000

by COEQUALS (i.e. cooperate with equals) is responsible for COEQUALS’ success. Table 1 shows that even when other strategies make decisions contingent on other players’ wealth holdings, COEQUALS persists at higher rates than any other strategy in the population. This suggests that payoff contingent decision making, per se, does not determine COEQUALS’ success. Rather, the manner in which COEQUALS processes information about wealth holdings (i.e., positing that inequality between others’ income and one’s own signals defection) appears to play a pivotal role in fostering its success. When players use lower or higher wealth holdings to predict cooperation, the strategy they use—along with collective action generally—fails to prosper. Also, as is evident in the small proportion of ANTI-COEQUALS that exist at g = 500, the information conveyed by equal wealth holdings asymmetrically benefits those who cooperate with their equals. This is because partners with the same payoffs likely use the same strategy, meaning that, on average, one punishes one’s own strategy by defecting on an equal and rewards one’s own strategy by cooperating with an equal. Further evidence of this can be found in Table 1: after the first round (in which COEQUALS is always exploited by defectors), COEQUALS will never cooperate with a defector. Identifying equals effectively identifies cooperators. Cautious cooperation, however, does not come without a cost. That is, although players employing COEQUALS cooperate far more frequently with cooperators than they do with those enlisting other strategies (except those enlisting COEQUALS), COEQUALS exploits cooperators in the majority of encounters (see Table 1). In so doing, COEQUALS passes up opportunities for successful collective action. The regression results reported in Table 2 provide further insight into this finding. The proportion of cooperative encounters that COEQUALS has with cooperators (as well as with other players employing COEQUALS) decreases with the number of one-shot games played in a generation (see the second and third columns of Table 2).

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Table 3. Mean wealth holdings for all strategies, by game, relative to COEQUALS Game

Coop

Def

COEQ

Anti-C

BD

LC

1 2 3 4 5 6 7 8 9 10

0 –0.1 –0.5 –1.1 –1.9 –3.0 –4.2 –5.6 –7.2 –9.0

0.2 0.3 0.3 0.2 0.2 0 0 –0.1 –0.2 –0.2

0 0 0 0 0 0 0 0 0 0

0.2 0.2 –0.2 -0.8 -1.5 –2.6 –3.8 –5.2 –6.9 –8.7

0.2 0.2 –0.2 –0.4 –0.9 –1.5 –2.2 –3.0 –3.9 –4.9

0.2 0.2 0.1 –0.2 –0.6 –1.2 –1.8 –2.2 –3.2 –4.1

Note: Numbers in bold indicate the smallest absolute difference, among strategies, between a given strategy mean wealth holdings and the mean wealth holdings of COEQUALS. The numbers are based upon 400 Monte-Carlo simulation runs for each treatment. Table 1 describes the simulation parameters.

This decline in cooperation appears to be due to the fact that COEQUALS’ wealth holdings diverge from those of cooperators over the course of a generation (see Table 3). With each one-shot game, the chances of a given COEQUALS agent encountering a cooperator with the same wealth holdings decreases, and this leads the COEQUALS agent to defect on cooperators more often in later games in a generation (see Figure 2). Collective action, in other words, declines with economic inequality in the simulation. The defection that disrupts collective action, however, helps COEQUALS, which appears to obtain greater wealth holdings relative to other players when more games are played in a generation (see Figure 3). Thus, while COEQUALS’ choice mechanism appears to be more successful than other choice mechanisms that strategies can use to make payoff contingent decisions, this success has a dark side. A player employing COEQUALS forgoes many potentially cooperative encounters and often exploits cooperators in order to guard against his own exploitation. The cost of achieving successful collective action may ultimately be modest levels of unilateral defection. Also, the simulations show that the likelihood of COEQUALS cooperating with other types is not constant. Rather, the probability of COEQUALS cooperating with other types depends on the frequency of COEQUALS in the population. Table 4 shows that as the proportion of COEQUALS in the population increases from 16.7% (under the uniform starting distribution of six types) to 75%, players adopting COEQUALS are more likely to cooperate with both naive cooperators and others using COEQUALS. Notice, however, that the result holds only for the first few one-shot games of the generation.

3.3. The effect of errors on collective action and COEQUALS The previous section showed that much of COEQUALS’ success is due to its ability to guard against exploitation: COEQUALS is never exploited by a defector after the first game of a generation (see Table 1). COEQUALS avoids defection by making use of the fact that wealth holdings are correlated with strategy use and, thus, serve as a good

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Figure 2. Mean proportion of COEQUALS cooperating with COOPERATORS and COEQUALS for a different number of one-shot PD games per generation Note: Mean proportion of COEQUALS cooperating with COOPERATORS (pccoop) and mean proportion of COEQUALS cooperating with COEQUALS (pccoeq). Horizontal axis represents the number of one-shot

0

.2

Mean (COEQUALS) .4 .6

.8

games per generation.

2

3

4

5

6

7

8

9

10

Figure 3. Mean proportion of COEQUALS in the population for a different number of one-shot PD games per generation Note: Horizontal axis represents the number of one-shot games per generation.

proxy for another player’s cooperativeness. What happens, however, when players using COEQUALS begin to cooperate with individuals whose wealth holdings deviate slightly from their own, thereby ignoring—to some degree—the correlation between strategy use and wealth holdings? To answer this question, we relax the assumption that players can perfectly detect other players’ wealth holdings.6 Specifically, we introduce a new type of COEQUALS which cooperates with other agents, so long as those agents’ wealth holdings fall within

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Table 4. Frequency dependent nature of cooperation by COEQUALS Game number

2 3 4 5 6 7 8 9 10

Proportion of COEQUALS cooperating with all types

Proportion of COEQUALS cooperating with COEQUALS

Uniform distribution

75% COEQUALS

Uniform distribution

75% COEQUALS

0.186 0.048 0.037 0.017 0.012 0.009 0.010 0.007 0.008

0.228 0.112 0.059 0.018 0.011 0.008 0.006 0.004 0.003

0.550 0.221 0.130 0.090 0.063 0.051 0.056 0.039 0.046

0.692 0.348 0.157 0.088 0.055 0.040 0.030 0.022 0.019

Note: The numbers are based upon 400 Monte-Carlo simulation runs for each treatment. Table 1 describes the simulation parameters.

a specified range. This behavior can be interpreted either as succumbing to perceptual error or, alternatively, as exhibiting a tolerance for cooperating with others who have slightly different wealth. Consider an agent i who uses COEQUALS and possesses wealth holdings Wi (CE), while another player j possesses wealth holdings Wj . In our previous models, players employing COEQUALS follow the algorithm: if Wi (CE) = Wj , then cooperate; otherwise, defect. Players employing the new type of COEQUALS, however, will cooperate with j so long as Wj lies within the interval [Wi (CE) − ψ, Wi (CE) + ψ], where ψ is the product of an exogenously defined proportion—Error—of the average wealth holding in the population: ψ = Average Population Wealth Holdings * Error. For example, if the average wealth holding in the population is 10 and Error is exogenously set at 0.1, then ψ = 1, meaning that a COEQUALS agent will cooperate with players whose wealth holdings lie within the interval [Wi (CE) − 1, Wi (CE) + 1] . In order to understand the effect of imperfect assessment of wealth holdings or, equally, an agent’s tolerance for cooperating with non-equals, we vary the Error parameter in our simulation. The descriptive statistics reported in Table 5 show the effect of modifying the model to allow the inaccurate assessment of wealth holdings. Relative to simulation runs in which wealth holdings are perfectly discernible (see Table 1), the runs reported in Table 5 suggest that the inaccurate assessment of wealth holdings causes COEQUALS to cooperate with defectors in roughly 16% of all plays after the second round. This finding accounts, at least in part, for the fact that the proportion of COEQUALS in the population is substantially lower in these simulation runs than in runs in which wealth holdings are perfectly perceived (see Table 1).7 Yet, although perceptual errors degrade the fitness of COEQUALS, they appear to have a positive—or, at the very least, a null—effect on collective action generally. Table 6 reports regression results that speak to this finding. The results show that when

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Table 5. Descriptive statistics for simulation with noise Variable

Mean

Std. Dev. Min

Max

Benefit, b Cost, c Number of games, n Mutation rate, mr Error (in percentage of wealth holdings)

0.794 0.206 6.020 0.030 0.100

0.115 0.114 2.568 0.014 0.058

0.6 0.011 2 0.01 0

1 0.4 10 0.05 0.2

Average Proportion of Cooperators from g=400 to g=500 Average Proportion of Defectors from g=400 to g=500 Average Proportion of COEQUALS from g=400 to g=500

0.071 0.739 0.190

0.077 0.273 0.219

0.006 0.008 0.008

0.599 0.983 0.951

Proportion of COEQUALS cooperating with Cooperators Proportion of COEQUALS cooperating with Defectors Proportion of COEQUALS cooperating with COEQUALS

0.373 0.158 0.488

0.192 0.161 0.175

0.076 0 0.163

0.941 0.864 0.951

Table 6. Logistic regression analysis for the runs with errors Simulation parameters

DV: Average proportion of COEQUALS in the population between g=400 and and g=500

Perception error –13.989 (1.679) Starting population: –0.123 (0.219) 100% Defectors Starting population: –0.052 (0.221) 100% COEQUALS Benefit, b 2.003 (0.775) Cost, c –6.530 (0.846) Number of games, n 0.134 (0.036) Mutation rate, mr 13.707 (6.409) –1.851 (0.746) Constant AIC 515.96 Observations 1000 Standard error in parentheses

DV: COEQUALS cooperating with Cooperators

DV: COEQUALS cooperating with COEQUALS

2.021 (1.187) 0.323 (0.170)

1.015 (1.147) 0.392 (0.163)

0.430 (0.172)

0.426 (0.166)

–0.046 (0.597) –0.255 (0.608) –0.232 (0.028) –12.781 (4.889) 0.843 (0.575) 950.69 1000

–0.238 (0.578) 2.032 (0.589) –0.196 (0.027) –16.637 (4.716) 1.020 (0.558) 1059.00 1000

the size of Error increases, the average proportion of COEQUALS between g=400 and g=500 drops significantly. Yet the regression analysis also shows that greater Error increases the number of instances in which COEQUALS cooperates with both cooperators and other players using COEQUALS. In other words, perceptual error appears to hurt COEQUALS by making COEQUALS more cooperative. The increased exploitation of COEQUALS due to perceptual error clearly exposes the compromise resting at the strategy’s core. By aiming cooperation only at equals, players adopting COEQUALS guard against free-riding, but they also sacrifice opportunities

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for successful cooperation. Often this cautious defection results in the exploitation of cooperators who possess different payoffs; that exploitation, in turn, hurts the chances of future collective action. In order to limit the exploitation of others and support collective action, players adopting COEQUALS must allow themselves to be exploited with greater frequency.

4. Conclusion Taken together, the findings from our analytic and computer models raise novel theoretical insights about the relationship between economic inequality and collective action. While previous models hold that economic inequality influences collective action by altering individuals’ incentives to provide collective goods unilaterally (e.g., Bardhan et al., 2007; Bergstrom et al., 1986; Olson, 1965), our analysis suggests that economic inequality—in the form of wealth disparities—can provide information about a game partner’s cooperativeness. In so doing, it can allow the strategy COEQUALS to engage in successful collective action even when incentives to free-ride remain. The COEQUALS strategy dictates that individuals employing it should cooperate with any player who possesses the same level of wealth holdings as the COEQUALS player; otherwise, the adopter of the strategy should defect. This strategy exploits the correlation between strategy use and prior accumulative past payoffs in order to target cooperation to those using the COEQUALS strategy. By introducing this novel strategy, as well as by treating wealth disparities and PD game strategies as part of a dynamic endogenous process, our analysis provides a new understanding of how economic inequality might influence collective action. Along with those implications, our analysis offers new theoretical insights into the study of cooperative evolution (Axelrod, 1984; Nowak, 2006). Numerous studies have examined how information mechanisms such as reputation (Nowak and Sigmund, 1998) and tags (Riolo et al., 2001) support the evolution of cooperation. Reputation systems, for instance, allow potentially cooperative organisms to evaluate a partner’s past actions in order to determine whether cooperation in the present PD encounter will be exploited. Arbitrary tags (Axelrod et al., 2004; Riolo et al., 2001) also provide agents with information about the cooperativeness of their game partners. Although successful at fostering cooperation, these information mechanisms appear biologically rare. Reputation systems demand that organisms possess sophisticated cognitive machinery that allows them to maintain information about all (or many) players’ reputations (Nowak and Sigmund, 1998) and few non-human animals possess the cognitive equipment to perform such tasks. Likewise, tag-based systems are unlikely to emerge, because animal signals generally evolve so that the form of the signal is adaptively coupled with its content (Maynard Smith and Harper, 1995). Given these implausible features of previously proposed information mechanisms, wealth holdings—or perhaps, in the non-human animal world, ‘fitness cues’—might serve as a theoretically meaningful information mechanism that signals cooperativeness. Empirical analysis could readily test this possibility. Accumulated past PD payoffs could be interpreted as modeling any phenotypic attribute—e.g., body size (Richner, 1989)—that correlates with fitness but is not actively signaled; thus, researchers could test our results by examining if cooperation occurs among organisms

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that are equally matched on phenotypic attributes correlated with fitness, or whether it occurs among organisms whose ‘fitness cues’ differ. These theoretical insights, moreover, illuminate previously observed empirical phenomena. For one, our findings offer an alternative theoretical account of prominent findings concerning the effect of economic inequality on collective action and civic participation. Scholars have shown that (a) cooperative environmental conservation (Shanmugaratnam, 1996) and civic participation (Putnam, 2000) decline with greater levels of inequality, but (b) the collective action that does succeed under conditions of high inequality occurs among individuals of relatively similar economic standing (Alesina and La Ferrara, 2000). These results correspond with our theoretical observations: as inequality grows (read: there is greater variability in wealth holdings, not a greater concentration of wealth), it becomes more difficult for any one individual to find an equal with which to cooperate (thus reducing instances of collective action), but the cooperation that occurs under such conditions results from the joint action of equals.8 Our findings also shed light on norms and institutions common in groups seeking to act collectively. For instance, trade union stewards often foster cooperation by reminding fellow members that those who accept union benefits but do not contribute to union causes receive greater total benefits than those who toe the union line (Prosten, 1997). Not only do such practices inform union members about the possibility of free-riding, but they also hint at a means by which members can attempt to detect free-riders: if a team of workers chewing baloney notice that one of their coworkers pulls caviar out of his lunch pail, they might come to believe that their pal is violating union rules and moonlighting as an independent, non-union contractor. Systematic investigation of such phenomena, in light of our findings, might contribute considerably to an understanding of workplace norms and institutions. Finally, our models provide insight into the burgeoning literature on trust (Fukuyama, 1995; Putnam, 1993; Yamagishi, 1998). Experimental studies indicate that cooperation breeds trust and not vice versa (Yamagishi et al., 2005). Thus, in addition to showing that PD game play conditioned on wealth equality can foster cooperation, our analysis suggests that parity in wealth holdings may serve as an impetus for trust. Testing this hypothesis in laboratory trust games (see Berg et al., 1995) and gauging its plausibility with observational data9 may provide new insights into the mechanisms that promote trust and trustworthiness. Such extensions suggest that our model holds implications for a diverse range of fields and can be readily evaluated through empirical testing. In sum, our model shows that when economic inequality and collective action are modeled endogenously, economic inequality transforms from a static state of the world into a source of information for participants in collective action. When used by players employing the strategy COEQUALS, information about economic inequality facilitates collective action by allowing players to distinguish between cooperative and non-cooperative game partners. This result holds even when COEQUALS competes with other strategies in which decisions are made based on game partners’ wealth holdings. The strategy’s success and the viability of collective action, however, are not one and the same. In order to avoid falling victim to free-riding, players adopting COEQUALS often forgo opportunities for successful collective action. In so doing, the COEQUALS strategy often leads players to exploit cooperative players, and this harms collective action. The addition

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of perceptual error to the model mitigates this effect and increases the frequency of COEQUALS’ cooperation with other types; but, in so doing, the viability of COEQUALS is compromised.

Appendix 1. Illustrative example involving two games per generation Here we present explicit calculations of the COEQUALS model in which individuals play two games per generation, which is the minimal number of games relevant to our theory. Recall that c is the proportion of cooperators in the population, d is the proportion of defectors, and a is the proportion of COEQUALS. After the first game, types earn the following payoffs: πC1 = R (c + a)+Sd, πD1 = T (c + a)+Pd, πA1 = R (c + a)+Sd. Since all individuals have the same aggregate payoff—viz. zero—at the start of the first game, COEQUALS cooperates with each type, thus earning the same payoff as cooperators; defectors, who exploit cooperators and COEQUALS, gain greater payoffs. In the second game of a generation, payoffs are more complex as, prior to choosing to cooperate or defect, COEQUALS compares its aggregate payoff with that of its partner. Since the payoff of COEQUALS is either R or S, it cannot share the same payoff as a defector at the beginning of the second game. It will thus defect and earn P with probability d. When facing a cooperator, however, four possible outcomes result (see Table A1). The margins of the table present the possible outcomes—and their probability of occurrence—in the first round. Cells show the possible outcomes of the second round and their respective probabilities. In two instances cooperators and COEQUALS have the same aggregate payoff, thus leading COEQUALS to cooperate and contribute to the mutual cooperation outcome, R, with probability d 2 + (c + a)2 . In the other two instances, cooperators and COEQUALS have different aggregate payoffs, causing COEQUALS to exploit cooperation with probability 2d (c + a). Table 7. Second round outcomes when COEQUALS is paired with a cooperator

COEQUALS obtained R in the first round with probability (c + a) COEQUALS obtained S in the first round with probability d

Cooperator obtained R in the first round with probability (c + a)

Cooperator obtained S in the first round with probability d

(R, R) with probability (c + a)2

(T, S) with probability (c + a) d

(T, S) with probability (c + a) d

(R, R) with probability d2

Similarly, when an organism playing COEQUALS is paired with another organism employing COEQUALS, four possibilities arise. If the two players have the same fitness (either [R, R] or [S, S]) they ooperate; otherwise they defect. These scenarios, along

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with standard play between C and D, yield the following payoffs in the second game of a generation:



(A1) πC2 = R c + a d 2 + (c + a)2 + Sd (1 + 2a (c + a)) πD2 = Tc + P (d + a)

πA2 = 2Tcd (c + a) + R (a + c) (c + a)2 + d 2 + Pd (2a (c + a) + 1)

(A2) (A3)

Notice that these are only payoffs in the second game. Total payoffs for the whole game are the combined earnings of the first and second one-shot PD game: πC = πC1 +πC2 , πD = πD1 + πD2 , πA = πA1 + πA2 . These equations can be evaluated to determine if any type resists invasion when individuals play two one-shot games in their lifespan. Having defined the payoffs of types in closed form, we can create replicator dynamics phase diagrams (Figure 1 in the main text).

Appendix 2. Computer code in R # Johnson and Smirnov. “Economic Inequality and Collective Action” # COEQUALS simulation: 6 types version. Monte Carlo runs. # Monte Carlo simulation parameters RUNS = 10; dataRUNS=array(0,dim=c(RUNS,17)); N=200; G=500; # starting the cycle of runs for (run in 1:RUNS) { # simulation parameters b=runif(1,0.6,1); c=runif(1,0.01,0.40); mr=as.integer(1+5*runif(1))*0.01; R=as.integer(2+9*runif(1)) # replicator dynamics arrays Lottery=array(0,dim=c(N,6)); Tot=array(0,dim=c(6)); P=array(0,dim=c(6)); AF=array(0,dim=c(6)) # population and data arrays pop=array(0,dim=c(N,3)); pop[1:N,1]=R*c; pop[1:N,3]=0 data=array(0,dim=c(G,6)); dataT=array(0,dim=c(G,6)) dataC=array(0,dim=c(G,6)); dataPC=array(0,dim=c(1,6)) # random choice: starting populations can be 100% defectors, 100% COEQUALS, or uniformly random treatment = as.integer(1+3*runif(1)) if (treatment==1) {pop[1:N,2]=2} if (treatment==2) {pop[1:N,2]=3} if (treatment==3) {pop[1:N,2]=as.integer(1+6*runif(N))} # cycles of generations, games per generation, and players for (g in 1:G) { for (r in 1:R) { for (x in 1:N) { # choice of behavior given type rules

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# types: 1-cooperators, 2-defectors, 3-COEQUALS, 4-anti-COEQUALS, 5-BD, 6-LC Coop1=0; Coop2=0; y=as.integer(1+N*runif(1)) if (pop[x,2]==1) Coop1=1; if (pop[y,2]==1) Coop2=1 if (pop[x,2]==3 & pop[x,3]==pop[y,3]) Coop1=1; if (pop[y,2]==3 & pop[y,3] ==pop[x,3]) Coop2=1 if (pop[x,2]==4 & pop[x,3]!=pop[y,3]) Coop1=1; if (pop[y,2]==4 & pop[y,3]! =pop[x,3]) Coop2=1 if (pop[x,2]==5 & pop[x,3]>pop[y,3]) Coop1=1; if (pop[y,2]==5 & pop[y,3] >pop[x,3]) Coop2=1 if (pop[x,2]==6 & pop[x,3]1) {if (pop[x,2]==3) {dataT[g,pop[y,2]]=dataT[g,pop[y,2]]+1 dataC[g,pop[y,2]]=dataC[g,pop[y,2]]+Coop1} if (pop[y,2]==3) {dataT[g,pop[x,2]]=dataT[g,pop[x,2]]+1 dataC[g,pop[x,2]]=dataC[g,pop[x,2]]+Coop2}} # payoff function pop[x,1]=pop[x,1] + Coop2*b - Coop1*c } # updating wealth holdings pop[1:N,3]=pop[1:N,3]+pop[1:N,1] } # average wealth holdings in the population AFP=sum(pop[1:N,3])/N # replicator dynamics; finding proportions of types for the next round for (i in 1:6) {Tot[i]=sum(pop[1:N,2]==i) P[i]=data[g,i]=Tot[i]/N if (Tot[i]>0) AF[i]=sum(pop[pop[1:N,2]==i,3])/Tot[i] # discrete replicator dynamics given the finite population size P[i]=P[i]*(AF[i]/AFP)} # stochastic replicator dynamics via transposed multinomial random number generation Lottery=t(rmultinom(N, size = 1, prob=c(P[1],P[2],P[3],P[4],P[5],P[6]))) pop[1:N,2]=max.col(Lottery) # random variation mutants=N*mr; if (mutants>1) {for (i in 1:mutants) { mutant=as.integer(1+N*runif(1)) pop[mutant,2]=as.integer(1+6*runif(1))}} pop[1:N,1]=1+R*c;pop[1:N,3]=0 } # saving data for (i in 1:6) {dataPC[1,i]=sum(dataC[,i])/sum(dataT[,i])}

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dataRUNS[run,1]=treatment;dataRUNS[run,2]=b;dataRUNS[run,3]=c;dataRUNS [run,4]=R; dataRUNS[run,5]=mr;dataRUNS[run,6:11]=P;dataRUNS[run,12:17]=dataPC } # saving data file write.csv(dataRUNS, file = "RUNSDATA.txt") done=1 Notes 1. Conventional, technical jargon labels this time period a ‘generation.’ Although we occasionally use that term, we avoid its constant use in order to emphasize that the models studied here focus on an evolutionary process that need not involve selection on genes. 2. The name ‘COEQUALS’ is shorthand for ‘CO-operate with EQUALS.’ 3. Later in the paper we report analyses that examine what happens when players adopting COEQUALS cooperate with players whose wealth holdings fall within a specified range of the COEQUALS player’s wealth holdings. 4. The finite population version of the model is examined in a computational model later in the article, leading to the same substantive results. 5. Except in the first game of a time period, when all players’ wealth holdings equal zero. 6. We also restrict the set of strategies we analyze to those contained in the analytic model. Bottom-dweller, ladder-climber, and anti-COEQUALS strategies are excluded from the simulation. This is due to the lack of selection on those strategies in the previous section and because the results remain substantively similar even with those strategies included. 7. The proportion of types in the population, as reported in Table 4, is an average calculated from the 400th generation to the 500th generation. We report the moving average 100 because, with errors, the simulation does not always converge to dominance of a single type, leading to fluctuation in population proportions in any single generation. 8. Some might contend that these results are achieved by assumption; that is, the COEQUALS strategy defects more when there is more variation in wealth holdings, thus there must be a decline in collective action with greater inequality. Indeed, it is evident that COEQUALS will defect more under conditions of high inequality, but it is not clear that collective action among equals would succeed under such conditions. Therefore, while increased defection is expected, the stability of collective action among equals is not. 9. Past work has examined how disparities in the concentration of wealth among community members influence trust and cooperation in society (e.g., Putnam, 1993). To our knowledge, however, no study examines the micro-level behaviors responsible for this phenomenon, nor does a study examine if wealth disparities between individuals influence trust. The hypothesis offered in this paper provides guidance for such a study.

References Alesina A and La Ferrara E (2000) Participation in heterogeneous communities. Quarterly Journal of Economics 115(3): 847–904. Andreoni J (1989) Giving with impure altruism: Applications to charity and Ricardian equivalence. Journal of Political Economy 97(6): 1447–1458. Axelrod R (1984) The Evolution of Cooperation. New York: Basic Books. Axelrod R, Hammond RA and Grafen A (2004) Altruism via kin-selection strategies that rely on arbitrary tags with which they coevolve. Evolution 58(8): 1833–1838.

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An alternative mechanism through which economic ...

Abstract. Past models treat economic inequality as an exogenous condition that can provide individuals a dominant incentive to produce collective goods unilaterally. Here we part with that tradition so as to treat economic inequality and collective action as endogenous, and to examine whether economic inequality can ...

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