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Efficient and inefficient social control in collective action.

1. Introduction

This paper analyzes the interdependence between social control mechanisms, social networks and collective action. We study how different forms of social control influence decisions of rational individuals and we examine how and under what conditions networks rationalize participation in collective action. Besides, we focus on the interactions between network effects and social control as we investigate which network properties form favorable conditions for the emergence of collective action when a selective incentive is the main driving force for cooperation, and which properties favor mass mobilization when conformity is an important form of social control. In order to capture the interrelations between social control, network structure and collective action, an integrative model is presented that is built on the n-person social dilemma model of collective action on one hand, and on local interaction games that capture social control mechanisms in network relations on the other hand. The integrated model provides a ground for a family of new models that can handle and help to explain a wide variety of social phenomena in which macro and micro interdependencies are interrelated.

Similarly to the majority of existing models, this study models collective action as an n-person social dilemma. (2) As narrow self-interest does not provide sufficient incentives for participation, a baseline economic model predicts collective action failure. Empirical studies, however, have found several examples of successful mobilization driven by social control mechanisms. (3) Theoretical developments, therefore, incorporated different forms of social control in modeling collective action and set down conditions for the possibility of mobilization. In its most widely used meaning, social control is a constraint on individual decision posed by the influence, as well as by the behavior, opinion, and expectations of relevant other individuals.

Several studies pointed out that the set of relevant others in practice is not the entire society, but only related others: family, friends, acquaintances, colleagues and neighbors. (4) Consequently, characteristics of the social network influence the impact of social control on collective action. Social network effects are important, because individuals tailor their behavior conditional on the behavior and expectation of relevant others.

For the integrated analysis of social control mechanisms and collective action, social control has to be conceptualized. As in previous studies, either in an internal or in an external form, social control will be modeled as a system of mutual rewards and punishments. Someone's decision whether to join a public protest or not is contingent on the decision of her friends. Besides, she gains additional satisfaction if her action meets the expectations or sympathy of friends. This holds also for their decisions and for their calculations on her expectation. This kind of strategic interdependence is modeled in local interaction games. (5) The existing literature on local interaction games, however, disregards the possibility of community interdependence. Local coordination games provide an illustrative basis for the influence of certain social control mechanisms, but neglect the macro effects of coordination. Successfully coordinated behavior often provides positive externalities and additional value for the entire community.

The modeling framework presented here incorporates dyadic social control mechanisms as strategic interactions and consequently social network effects in the single-shot n-person social dilemma model of collective action. We provide a simple game-theoretical analysis of the conditions under which individuals are better off by participation and we also examine equilibrium conditions for the emergence of collective action. The model assumes that players are rational and perfectly informed about the rules of the game. These restrictions do not mean that we disregard the impacts of bounded rationality, imperfect information or repeated interactions on mobilization in collective action. On the contrary, our restrictive assumptions may help to distinguish the different sources of success or failure of collective action.

As a main substantive goal of model building, we aim to present a theoretical rationale for some of the intriguing empirical results of the collective action literature. First, similar to the findings of Sandell and Stern we would like to demonstrate that collective action may exist even if rational actors prefer to free ride on all but few of their fellows in the group. (6) Second, the study is to present some new predictions about the possible effects of social control and network structure in a situation where individuals are rational, well informed and there is no iteration in collective action. We aim to show that there are network structures at which the effect of social control on cooperation is nonmonotonic. That is, at certain parameter values, stronger control may decrease the chance of mass collective action. This objective is in line with earlier findings that demonstrated reverse effects of social control. (7)

In the next section, we provide arguments for the necessity of including social control mechanisms in the explanation of collective action. We summarize previous findings and our research questions afterwards. We present our model and subsequently, we derive under what conditions social control would lead to an equilibrium in which collective action is established. This is followed by a discussion of the impacts of network properties on participation and by the analysis of interaction effects of social control and network characteristics. Implications and prospective directions are discussed in the concluding section.

2. Social control mechanisms and their operation in the network

Individual decisions are constrained by social control mechanisms and this is not otherwise also in collective action situations. The decision to participate (contribute) or not is affected by various forms of social control.

Social control in the form of selective incentives implicate that defectors are punished or contributors are rewarded. The incorporation of selective incentives into the models has been prevalent since the beginning of collective action research. Some studies also addressed the second order free rider problem that stems from the costs of providing selective incentives. (8)

Other models have emphasized the positive impact of cooperation of others on the individual's own decision. Conditional cooperation mechanisms work, for instance, in critical mass models. (9) The assumptions about fairness considerations in Sugden's and Gould's models also imply conditionally cooperative behavior. (10) Besides, in Oberschall's block-recruitment model cooperation is also conditional on cooperation of relevant others. (11)

Laboratory experiments of public good games confirm the importance of social control even among strangers. (12) From our point of view, the most important results of these experiments can be summarized as follows. First, contributors feel compensated for their effort and similarly, defectors seem to experience a smaller subjective gain because of a sense of guilt, embarrassment or punishment. Second, the perception of social approval/disapproval may have an impact on cooperation even if no monetary (or physical) incentives are available. It is sufficient if the peers express their approval/disapproval by symbolic points--the provision of which is costless. What is more, even the observation of the decisions or merely the presence of other players may be sufficient for higher level of contribution in a one-shot public good game. That is, the provision of effective social control might be costless in some situations. Third, experiments have provided evidence for the existence of altruistic punishment of defectors. These experiments show that individuals have intrinsic incentives to punish free riders, even at the cost of their own welfare. That is, the internal costs of watching an unpunished free riding friend may outweigh the cost of punishment itself. Fourth, if social control mechanisms operate, the level of contribution is positively correlated with the group size. That is, more people can impose stronger control. Fifth, the characteristics of social control also depend on the level of (expected) contributions of those actors who reward/punish the individual. A defector gets higher level of punishment from a cooperator than from another defector--or a cooperator gets higher reward from another cooperator than from a defector. That is, as the number of defectors increases, the social incentive for contribution decreases. Nonetheless, a contributor's positive effect on the individual's cooperation tends to be stronger than a defector's negative one.

These experimental results are mostly in line with the assumptions about social control in the most cited models of collective action. Olson's term of social selective incentives, for instance, can be interpreted in a way to match most of these observations. However, selective incentives and social control that is conditional on others' decisions are fundamentally different and hence one should distinguish at least two types of social control. The first simply rewards the cooperator or punishes the defector irrespective of the action taken by others. The second implicates that individuals compare their decisions with decisions of relevant others, and gain rewards if the decisions are similar. Following these arguments, we incorporate social control into our model in the form of two separate incentives. We will refer to the rewards for cooperators (punishments for the defectors) as selective incentives. We will refer to social control that rewards the similarity between decisions as behavioral confirmation or conformity. We are well aware of further social control mechanisms, for instance, people might participate in collective action precisely because they expect that others don't do anything, but for the sake of simplicity, we restrict our interest for the analysis of these two forms.

Similar to Hollander's analysis, we incorporate social control in the model as an automatic response to cooperation. (13) Nonetheless, providing social control may be costly and a second order free rider problem may emerge. Our model, however, is restricted to those situations in which provision of social control is costless, social control is internalized or the costs of not rewarding/punishing counterbalance the ones of giving approval/disapproval. As laboratory experiments show, this assumption fits the behavior of real-word individuals.

We will concentrate on selective incentives and conformity that operate in dyadic connections between individuals (friends, relatives, neighbors, etc). Those who contribute to the public good, get some (non-pecuniary) reward from their friends in the form of a positive social selective incentive. This form of social control is independent of the action taken by those who provide it. Behavioral confirmation or conformity, on the other hand, depends on the relations between the individual's choice and the decisions taken by her friends. The individual is rewarded if she does the same as her peers act.

Anonymous relations do not transmit these forms of social control as close contacts do; individual actions are relationally and structurally embedded. Intensive interpersonal ties are therefore the key routes of the spread of social control that facilitate mobilization in collective action. As the system of actors and interpersonal ties among them is referred to as the social network, macro properties of the social network are correlated with the success of mobilization in collective action.

The models of network effects in collective action, however, should not disregard the interdependence in the collective action situation that is inseparable from relational interdependencies. For instance, joining friends at a demonstration is partly a contribution to the success of collective action and partly a provision of conformity to all friends who participate. This does not require multiple decisions from the individual. For this reason we integrate global and local interdependences in a unified model that takes both into account simultaneously (Figure 1). On one hand, we model collective action as an n-person public good game that assumes global interdependence (left side of Figure 1). As we emphasized, this standard model does not distinguish between connected and unconnected pairs of actors, that is, between friends (relatives, neighbors, etc.) and strangers. On the other hand, our integrative model is based on a "network approach" that concerns dyadic relations as local interaction games (middle of Figure 1). Local interaction games deal with a network of dyads involved in two-person games, but they are unable to cope with a situation in which the entire set of players is involved in a public good game. Unlike these network models, we assume the presence of global interdependence between the players. Global interdependence links even those who are unconnected in the social network. We will refer to this as the structurally embedded public goods game.

[FIGURE 1 OMITTED]

3. The impact of network structure on collective action

In most of the models on collective action, social control is considered as a possible facilitating factor of cooperation. Nonetheless, conformity does not obviously help; it might also undermine collective action in large groups. Gould, for instance, showed that conformity may not foster but rather inhibit cooperation in large, but dense networks. (14) Conformity is captured in game-theoretical terms by modeling dyadic interdependencies as assurance games, in which mutual cooperation is payoff dominant equilibrium, but not a unique equilibrium. (15) Selective incentives, on the other hand, are defined as rewards (punishments) for cooperation (free riding).

Some studies on collective action that encountered effects of social control, also drew attention to the consequential effects of network structure on mobilization. (16) These studies emphasized mainly the effect of two network properties: density and centrality. Most analyses demonstrated that closed and dense social networks could produce more social capital for maintaining group norms, including norms of cooperation than atomized networks. The positive impact of density on cooperation is also supported by empirical evidence. (17) On the other hand, when large-scale cooperation is established, sparse networks might set more stable barriers against the spread of occasional defections. (18) Diani also concludes based on different empirical findings that dense networks support mobilization in small networks with strong identity, but the chances to find examples in large populations are low, while it may be possible to find activity in modular sections within them. (19) Gould's abovementioned analysis also provides examples when a highly dense network is capable to establish only a mediocre level of collective good production. How density will affect the success of collective action; depends on the rate at which normative pressure and enthusiasm about the prospects of mobilization encourage individuals to emulate the contribution of others and on the structural location of zealous actors.

Marwell, Oliver and Prahl emphasized that network centralization has a competing impact on collective action with density. (20) The key importance of central actors and the efficiency of centralized structures in mobilization have also been demonstrated by several other scholars. Gould emphasized that a volunteer can trigger the most contributions in a star-shaped structure. Central actors are less capable of spreading cooperation in highly dense networks with low degree variance. Moreover, Cummings and Cross have found in an empirical study of work groups that core-periphery and hierarchical network structures were negatively associated with performance. (21) This completely contradicts to empirical findings on the superiority of core-periphery structures for cooperation, (22) but also to theoretical predictions about the viability of cooperation in different network topologies that find that the evolution of cooperation is most prevalent in star-shaped, highly centralized networks. (23)

The effect of degree is prevalent in the study of Ohtsuki et al. that finds a general condition for the evolution of contribution conditional on the average degree in large random, small-world and scale-free networks. (24) The general condition prescribes that in an evolutionary perspective and fixed networks a sparse structure and low average degree is favorable for the evolution of cooperation. Another studies confirm that low density and heterogeneity support the evolution of cooperation also when the structure and behavior co-evolves. (25) When individuals are able to update their ties, the time scale parameter that determines how frequently the network is updated compared to the update in the cooperation strategy is among the most important determinants of the extent to which cooperation spreads. (26)

The seminal work of Granovetter highlighted the importance of bridging relations for getting important information and social capital. (27) These bridging ties might also be important to disseminate contribution norms between subgroups, as it is also suggested by Csermely. (28) Subgroups, however, are often reported to be highly reluctant to change their local behavioral code and to adopt established norms from outside. This also has the consequence that structures in which many ties need to be removed for collapsing into components; that are called cohesive structures; provide the ideal environment for high levels of cooperation.

As previous studies have demonstrated, structural effects are prevalent in collective action and they are caused by underlying social control mechanisms. In our analysis we will focus on the dependence of network effects on the type of social control that operates in dyadic relations. In particular, we examine whether selective incentives, when they reward contributors, would always foster collective action or under certain structural conditions they could also hinder mass collective action. As we compare macro effects of positive selective incentives and conformity, we demonstrate what they imply for social network effects in collective action. As far as network effects concerned, we emphasize which mechanisms of social control are responsible for the impact of network properties as density, minimum degree, network clustering, and bridging ties.

4. The structurally embedded public goods game

In this section, we introduce the model that integrates social control mechanisms and local interactions into the analysis of collective action. The fundaments of this model have also been introduced in the studies of Takacs, Janky and Flache. (29) We assume that a final set of players (N = {1,..., i,..., n}, where n>2) play an n-person public goods game with a linear production function. In this game, every individual has to decide whether to take part in the collective action or not. That is, each player has to choose between two alternatives: they can either participate in collective action (contribute to the provision of the public good) or not (defect). The action taken by the individual i [member of] N is denoted by [[sigma].sub.i], where [[sigma].sub.i]=1 is contribution and [[sigma].sub.i]=0 is defection. Contribution has a cost c, and this value is the same for everyone. Contribution means a provision of a unit of a public good [alpha] for all players and defection means no additional provision. Hence, the amount of public good that is provided equals to [alpha][n.summation over (j=1)][[sigma].sub.j] (there is a linear production function). We assume that narrow monetary interest does not provide sufficient incentives for cooperation, i.e. c > [alpha]. Although the cost of contribution is higher than the gain of the provision of one unit of the public good, we suppose that if there are enough contributors, the value of the public good provided to every individual is higher than the cost of contribution. In other words, there is a threshold number of contributors n* (1 < n* [less than or equal to] n), for which [alpha]n* > c.

This is a standard starting setup used by models of collective action. To incorporate structural effects in the model, we assume that players might be connected to each other by symmetric links. For the sake of generality, we do not specify what sort of relationships (friendship, kinship, or simply acquaintance) these links indicate. It is sufficient to claim that social control operates in these relationships. Social control can only be experienced between connected individuals, but it is also inevitable between them.

Hence, we consider a network of dyadic relations among the players involved in the standard n-person PG-game, in which nodes are individuals and edges are relationships. An undirected graph is considered, that is we assume that the transmission of social control is independent of the direction and strength of relationships or alternatively, every connection is mutual and equally important. We will denote the existence of a direct relationship between individuals / and j by [r.sub.ij] (i,j [member of]N, and i [not equal to] j), where [r.sub.ij] = 1 if there is a direct relationship between them, and [r.sub.ij] = 0 if they are not directly related (they are not adjacent in the network). As we discuss undirected graphs, [r.sub.ij] = [r.sub.ji], always holds. For the sake of simplicity, we will denote the total number of i's ties by [r.sub.i][n.summation over (j=1)] = [r.sub.i], where i [not equal to] j).

The existence of relational ties in combination with action profiles has payoff consequences. First, we suppose that every player receives rewards (punishments) for contribution (defection) from each of her friends. The amount of this selective incentive from a single tie is denoted by s, hence the total amount of selective incentives that i receives is [sr.sub.i]. We assume that actors always reward/punish those cooperators/defectors who are connected to them. One should note that, a selective incentive is an element of the rational calculation of the receiver, but is not of the one of the provider as it is free to produce and provided automatically.

In addition to this selective incentive, individuals might prefer to follow the behavioral patterns of related actors. The individual's deviation from a related player's choice implies lower payoff than the outcome where they behave in the same way. In the model, both the absolute number and the proportion of the deviators among the related actors matter. That is, we assume that conformity consists of two elements. The first form of conformity is received as a linear function of the number of friends with the same choice and we call it mass conformity. Formally, all related actors with the choice equivalent to 's decision increases 's payoff by [b.sub.1]. When mass conformity operates, an individual, who intends to participate in a demonstration, would like to be sure that there are enough friends in the crowd. On the other hand, in case this individual prefers to stay at home, she would like to be assured that many friends choose the same option. The second form of conformity is received as a linear function of the proportion of friends with the same choice, and we refer to it as proportional conformity. Proportional conformity is independent of the number of ties the given individual has. The coefficient for proportional conformity is denoted by [b.sub.2]. When proportional conformity operates, the individual prefers to follow the decision of the majority of her friends.

For the sake of simplicity, we assume that [alpha],c>0; [b.sub.1],[b.sub.2],s[greater than or equal to]0, as we present all of them as rewards in (1). However, this assumption could easily be relaxed in a subsequent analysis. Assuming social control in terms of punishments instead of rewards would lead to a slightly different model with similar results. Denote C and D two disjoint sets of the group N, such that C={N\D}. Moreover, let us denote [r.sub.ic] and [r.sub.id] the numbers of i's connections who are elements of sets C and D, respectively ([r.sub.ic] + [r.sub.id] = [r.sub.i]). If every member of C contributes and every member of D defects, then the payoffs of defection and contribution for i are the following:

[[pi].sub.i]([[sigma].sub.i] = 0) = [r.sub.id][b.sub.1] + [r.sub.id]/[r.sub.i] [b.sub.2] + [alpha] [n.summation over (j=1)] [[sigma].sub.j] (1)

[[pi].sub.i]([[sigma].sub.i] = 1) = [r.sub.i]s + [r.sub.ic][b.sub.1] + [r.sub.ic]/[r.sub.1] + [b.sub.2] + [alpha] ([n.summation over (j=1)] [[sigma].sub.j] + 1) - c

where j [member of] N\{i}. One can see that in a given network, social approval is the strongest if one cooperates in a cooperating social environment. Cooperation or defection in a group of defectors implicates weaker approval. Our model allows for the assumption that a community of defectors fosters individual's defection (this is the case if [b.sub.1] and [b.sub.2] are large comparing to s). Nonetheless, it is possible within this modeling framework that the defectors provide more approval for cooperators than for other defectors (in this case s is large relatively to [b.sub.1] and [b.sub.2]). Social approval is the weakest if one defects while her friends cooperate. From (1) it follows that the contribution of the individual i is rational if

[alpha] + [r.sub.i]s + ([r.sub.ic] - [r.sub.id]) ([b.sub.1] + [b.sub.2]/[r.sub.i]) [greater than or equal to] c. (2)

It implies that selective incentives foster contribution relative to the number of connections of the given individual. Conformity promotes contribution only when there are more contributing friends than defectors. In case the number of defecting friends exceeds the number of contributing friends, conformity drives towards defection. Mass conformity supports contribution to the extent of the difference between the numbers of contributing and defecting friends; while proportional conformity promotes contribution to the extent of the proportion of contributors among the related individuals.

In case the cost of contribution is too high we cannot expect any provision of the public good. If there are strong incentives for contribution, however, then collective action can be established. Defection is not a strictly dominant strategy of i anymore, if the individual's benefits from social control and provision of a unit of the public good exceed the cost of contribution at least in the case when all of those players contribute who are connected to the individual ([r.sub.i] = [r.sub.ic]). That is, if

[alpha] + [r.sub.i](s + [b.sub.1]) + [b.sub.2] [greater than or equal to] c. (3)

Moving a step further, contribution can be a dominant strategy of i, if the individual's benefits from social control and provision of a unit of the public good exceed the cost of contribution even in the case when all of those players defect who are connected to the individual ([r.sub.i]=[r.sub.id]). That is, if

[alpha] + [r.sub.i] (s - [b.sub.1]) - [b.sub.2] > c (4)

holds. Hence, under certain conditions, social control rationalizes unconditional cooperation in collective action. Due to the simplifications of the model, it is relatively easy to calculate an arbitrary actor's payoffs for a certain decision. One should note, however, that the costs and benefits of contribution and defection differ for players in different structural positions. Thus, it is misleading to conduct an equilibrium analysis similar to that is adopted in n-person games with a homogeneous set of players. In our model, the conditions for contribution at the individual level do not fully specify the macro level determinants for the emergence of collective action.

5. Possibility of collective action

After a brief analysis of individual decisions, let us now consider under which conditions collective action may emerge assuming an exogenously given network. Primarily, we search for situations in which collective action is equilibrium over the set of pure strategies. Foremost, we focus on the conditions for the emergence of full contribution in which each actor contributes to the provision of the public good. We also analyze the conditions for the existence of partial contribution equilibria, primarily because of their relevance for the predictions about the emergence of full contribution. In this way, we spare some additional assumptions and carry out a less extensive analysis. For the sake of simplicity, we assume that [r.sub.i]>0 holds for all i. One can see from (2) and (4) that a situation in which all actors defect (overall defection) is a Nash equilibrium if there is no i for whom contribution would be a dominant strategy. This means that c [greater than or equal to] [alpha] + [r.sub.i] (s - [b.sub.1]) - [b.sub.2] should hold for all i.

On the other hand, full contribution is Nash equilibrium if there is nobody for whom defection is a dominant strategy. In other words, in case (c - [alpha] - [b.sub.2]) / (s + [b.sub.1]) [less than or equal to] [r.sub.i] holds for all i ([r.sub.i] > 0, that is the graph is connected), then full contribution is a Nash equilibrium. That is, for full contribution being Nash equilibrium the network should have the property

min([r.sub.i]) [greater than or equal to] c - [alpha] - [b.sub.2]/s + [b.sub.1], (5)

where min([r.sub.i]) is the minimum number of ties individuals have in the group (minimum degree). A network parameter (here the minimum degree) on one side of the equation and social control parameters and other incentives on the other side simplifies the analysis of structural effects on collective action. In this case, we could make it transparent that as a necessary structural condition for overall mobilization in collective action; everyone has to be connected to the network to a certain extent. Individuals with few connections make overall participation impossible. As far as social control concerned, a stronger selective incentive and stronger conformity increase the chance of overall contribution. Moreover, in spite of the significant difference between their micro effects, selective incentives and mass conformity influence for the emergence of full contribution equilibrium exactly the same way.

In case the threshold number n* is not very high, partial contribution can also produce beneficial collective action. A partial contribution outcome, where all i [member of] C contributes and all j [member of] D defects, is a Nash equilibrium, if such C and D (C={N\D}) nonempty sets exist for which

[alpha] + [r.sub.i]s + ([r.sub.ic] - [r.sub.id])([b.sub.1] + [b.sub.2]/[r.sub.i]) [greater than or equal to] c for all i [member of] c and (6)

[alpha] + [r.sub.j]s + ([r.sub.jc] - [r.sub.jd])([b.sub.1] + [b.sub.2]/[r.sub.j]) [less than or equal to] c for all j [member of] D (7)

hold. In this case, there is no clear relation between a certain network property and partial contribution equilibrium that would be independent from the structural distribution of contribution choices.

The existence of partial contribution equilibrium is most likely in a segmented network. For instance, if there is a dyadic component that is isolated from the rest of the network, then partial contribution in which they defect is an equilibrium, given that full contribution is an equilibrium, if c [greater than or equal to] [alpha] + s - [b.sub.1] - [b.sub.2]. On the other hand, partial contribution equilibria do not exist in a network in which everyone is tied to everyone else.

No further formal analysis is necessary to see that a core-periphery structure is favorable for partial contribution. In this case, members of the core may cooperate, while individuals with few connections and in small components will be free riders. Thus, according to the model, when, for example, workers of a factory launch a wild cat strike and members of the major workshops participate, some new or part-time employees, members of small, peripheral units, and those who work individually outside the workshops may stay out of the strike without a negative impact on the contribution of the rest of workers. In the next section, we will show further structural determinants of the conditions of partial contribution equilibria.

If there is one Nash equilibrium over pure strategies of the game, we consider it as the expected outcome. In several cases, however, there are multiple equilibria. For equilibrium selection there are different approaches we could follow. Without choosing sides in the ongoing debate, the only assumption we make is that if there is a payoff dominant equilibrium, we consider it as the expected outcome of the game. (30) An equilibrium is payoff-dominant if it provides more (or equal) payoff for every player than any other equilibrium. In this study, we do not go into further analysis of what happens when none of the existing equilibria is payoff dominant.

If full contribution and overall defection are two Nash equilibria of the game and the number of players exceeds the threshold number n*, (31) then full contribution always dominates overall defection. In this case, it is likely that collective action emerges. Partial contribution equilibria, however, are not always dominated by full contribution. If there is a subset of players for whom collective defection provides higher rewards than collective contribution, then full contribution is not payoff dominant over the partial contribution equilibrium, in which this subset of players defects. In other words, full contribution is payoff dominant equilibrium in case it is a Nash equilibrium, and there are no partial contribution equilibria, or if they exist, in any set of possible defectors, contribution of the whole set provides higher payoffs for its members, than the equilibrium where they defect. That is, the existence of partial contribution equilibrium may undermine full contribution as a likely outcome.

The conditions for full contribution being Nash equilibrium are given in (5). Partial contribution equilibrium exists in which all i [member of] C cooperates and all j [member of] D defects (C={N\D}; C and D are non-empty sets), if equations (6) and (7) hold. In case both equilibria exist, full contribution is payoff dominant over partial contribution if rewards for all j [member of] D are higher in the former case. That is,

[n.sub.d] [alpha] + [r.sub.j]s + [r.sub.j][b.sub.1] + [b.sub.2] > c + [r.sub.jd] ([b.sub.1] + [b.sub.2]/[r.sub.j]) for all j [member of] D,

which is simplified to

[n.sub.d] [alpha] + [r.sub.j]s + [r.sub.jc]([b.sub.1] + [b.sub.2]/[r.sub.j]) > c for all j [member of] D,

where [n.sub.d] is the number of members of D. The smaller the number of defectors in the partial contribution equilibrium, the smaller the likelihood that full contribution dominates this partial contribution equilibrium. That is, full contribution is more likely to be undermined by some defectors if there are small and segregated subgroups in the community. These subgroups should be small enough not to have, even collectively, a significant effect on the public good. Moreover, they should be segregated not to be influenced too strongly by outside pressure. Note that if the entire community can be split into such small segregated subgroups (for instance, into components or bi-components, or it is a highly modular structure), then any level of contribution might be undermined by strategic considerations, even in case of strong social control and relatively high density of the network.

Granovetter's seminal study provides a classical example of this kind. He points to the failure of collective action in an ethnic Italian community in Boston that could be characterized by dense network and strong social ties. The source of cohesion in this community was the close-knit family network, in which every member knew and influenced one another. The emphasis on intra-family relations, however, resulted in the ignorance of other types of relations. Thus, one could observe high level of cohesion in any part of the community, although the lack of ties between families inhibited the provision of community-level public goods.

6. Social control, network properties, and collective action

Density and full contribution equilibrium

In this section we turn to a closer analysis of the effect of certain network properties and of their interactions with social control on collective action. First, consider the preconditions of full contribution equilibrium. Equation (5) reveals that the key structural property that is associated with the emergence of full contribution is the minimum degree of the network. Among other network characteristics, density positively correlates with the likelihood of full contribution equilibrium. The relation between density and full contribution being an equilibrium can be derived from the statistical relation between density (the number of edges in the graph) and the minimal number of individual relations (the minimum degree). The higher the density of a network, the smaller the likelihood is that there will be individuals with zero or few connections. It is easy to see that the general likelihood of min([r.sub.i])[less than or equal to]t (t<n-1) decreases, if density or the number of relations (r) increases (if n is given), which means that the likelihood of full contribution equilibrium increases by network density on average. The general likelihood min([r.sub.i])[less than or equal to]t (t<n-1) is equal to one, if r<(t+1)n/2 and it is zero, if r>(n-1)(n-2)/2+t. For the range in between, extensive calculations are necessary. In Figure 2 we only provide an illustration of the general likelihood that the minimum degree of a random graph with n nodes and r relations reaches a certain level (min([r.sub.i])=t). This likelihood is associated with some known properties of degree variance. Higher degree variance is associated with a smaller likelihood of full contribution equilibrium. The relationship between degree variance and full contribution equilibrium is weaker when density is very low or very high. As Figure 2 shows, density positively correlates with minimum degree, which supports the density-cooperation hypothesis.

[FIGURE 2 OMITTED]

The distributions are overlapping, which means that higher density does not necessarily mean a higher minimum degree and consequently a higher likelihood of full contribution equilibrium. In highly centralized networks, in which most relations lead to relatively few individuals, the minimum degree and the likelihood of full contribution equilibrium is low. This result is in contradiction with findings that emphasize the efficiency of centralized structures in mobilization for collective action. (32)

The model shows that high group centralization in itself does not strengthen the effect of social control in collective action. For example, in the factory, where workers are thinking of organizing a wild cat strike, a central actor will not be able to initiate collective action, if other workers have only connections to her and not to each other, since one tie is not likely to provide sufficient social benefits for them to participate. The central actor will be left alone with her enthusiasm. When the network of informal connections in the workshop resembles a star-shaped structure, workers in peripheral positions will not participate, because they would not be confirmed for their behavior by their peers. In a dense network, workers in relatively central structural positions will more likely have a key role in mobilization than in a sparse network as the colleagues in peripheral positions reassure each other that they have been convinced to participate. This also leads to a prediction that is competing with the theoretical results of Gould about the impact of density on the influence of central actors in collective action and with theoretical predictions of Hauert's study about the superiority of star-shaped structures in the evolution of cooperation. (33)

Payoff dominance: non-monotonic effects of social control

Similar incentives and structural characteristics foster the existence of a partial contribution equilibrium as of the emergence of full contribution (see equations 6, 7, and 8). The existence of partial contribution, however, might inhibit full contribution becoming payoff dominant. As a consequence, social control may have adverse effects, and the influence of network structure is strongly shaped by the relative importance of different types of social control. The emerging complexity cannot be interpreted as a purely technical problem and it has a clear substantial relevance. Contribution may be more stable, if everybody knows that any provision of the public good is possible only if everyone contributes to it. The possibility of partial contribution equilibrium means that some people reckon that others might contribute anyway, and therefore their incentives for contribution weaken. That is, if the conditions for contribution become more favorable for a subgroup of players, the rest of the group is tempted to become a free rider. Thus, strategic considerations may lead to the disappearance of large-scale collective action.

The model mostly predicts positive correlation between the strength of social control and collective action. Weak control is never favorable for collective action and extremely strong incentives always facilitate full contribution. In a certain range of parameters and in certain structural conditions, however, stronger social control might result in a lower likelihood of collective action. The double edge character of conformity is apparent at the micro level and therefore it is not surprising if it also appears in macro predictions. More surprising is that a stronger selective incentive may also undermine the emergence of collective action.

Let us illustrate the paradox effect of a positive selective incentive with a simple example. Figure 3 shows a network structure in a 5-person structurally embedded public goods game. For the sake of simplicity we focus on the change in the strength of selective incentive (s) and consider other parameters as given. Let the other parameters be c=3, a=1, [b.sub.1] = 1, and [b.sub.2] = 1. From equation (5) it follows that full contribution is an equilibrium outcome at any non-negative value of s. As we emphasized earlier, the full contribution equilibrium is always payoff dominant compared to the overall defection equilibrium, but not always when compared to partial contribution equilibria.

At the given parameter values, in a possible partial contribution equilibrium players A, B and C participate in collective action, while D and E defect. This equilibrium exists if C receives sufficient incentives for contribution, in spite of her connection to D and if D does not have sufficient incentives to turn to contribution. After substituting the parameter values into equation (6), it follows that C may cooperate in case of D's defection if s[greater than or equal to]2/9. Moreover, one can see from equation (7) that D might defect in this case if s[less than or equal to]1. There is also a third condition, the one that tells us whether the partial contribution equilibrium in which D and E defect is dominated by full contribution. Equation (8) shows that full contribution is not payoff dominant if s[less than or equal to]1. Considering this network and these parameter values, there is another partial contribution equilibrium in which A, B, and C defect while D and E participate in collective action. This equilibrium exists if 1[less than or equal to]s[less than or equal to]10/9. However, full contribution equilibrium is always payoff dominant in comparison to this equilibrium. Since there are no other partial contribution equilibria in this game, the full contribution outcome is a payoff dominant equilibrium except the cases at which 2/9[less than or equal to]s[less than or equal to]1. That is, a small value of s (0<s<2/9) is more favorable for mass collective action than a value almost equal to one.

[FIGURE 3 OMITTED]

This example demonstrates that in spite of the significant difference between the micro effects of a selective incentive and conformity on individuals' contribution, both types of social control may inhibit collective action under specific circumstances. The non-monotonic effect of a selective incentive shows that stronger social control is not always beneficial for mass collective action. As far as the network structure is concerned, counterproductive effects of control parameters show up when it is possible to divide the group into fairly segregated subsets. Adverse effects of social control are stronger if certain subsets have dense connections within while other subsets are not as cohesive. The phenomenon is even more likely if the latter subsets are relatively small.

Let us take our example about a wild cat strike in a factory. When normative pressure is low but significant (small selective incentives), workers of the major and most cohesive workshop participate in the strike only if their friends at peripheral units also join their demonstration. In this case, these friends do not risk the failure of the strike. However, when normative pressure becomes stronger, members of the major workshop sufficiently enforce each other to strike without the participation of peripheral units. In this case, workers at the periphery with connections to the central workshop do not have the same responsibility and they might stay out of the conflict and collectively free ride on the effort of the major workshop.

Interactions of network structure and social control

In the next step we consider the relationship between different forms of social control and the effect of network structure on collective action. In order to make the analysis as simple as possible, we inquire the marginal effects of s, [b.sub.1] and [b.sub.2], respectively, by assuming that the two other parameters are equal to zero.

If social control only means the operation of a positive selective incentive, then from (5) it follows that full contribution is a Nash equilibrium if

[r.sub.i] [greater than or equal to] c - [alpha]/s (9)

holds for all i. On the other hand, from (7) it follows that partial contribution equilibrium exists if there is a D subset of actors, in which

[r.sub.j] [less than or equal to] c - [alpha]/s (10)

holds for all j [member of] D. Equations (9) and (10) show that full contribution and partial contribution can only exist at the same time in case there is a subgroup D of individuals for whom the number of relations [r.sub.j] equals to (c-[alpha])/s. From equations (8) and (9) it follows that full contribution is always a payoff dominant equilibrium. Consequently, the structural determinants of full contribution being a payoff dominant equilibrium are equivalent to the conditions of Nash equilibrium. As we demonstrated before, the existence of full contribution equilibrium depends on the minimum degree of the network and therefore positively correlated with density and negatively correlated with degree variance and centrality measures.

Let us now consider the structural effects in case where only mass conformity ([b.sub.1]) operates and s and [b.sub.2] are equal to zero. In this case full contribution is Nash equilibrium if [r.sub.i] [greater than or equal to] (c - [alpha])/[b.sub.1] for all i. The existence of partial contribution equilibrium is much more likely than in the previous case as the conditions for this are given as:

[r.sub.ic] - [r.sub.id] [greater than or equal to] c - [alpha]/[b.sub.1] for all i [member of] c and (11)

[r.sub.jc] - [r.sub.jd] [less than or equal to] c - [alpha]/[b.sub.1] for all j [member of] D. (12)

Equations (11) and (12) show that for the existence of partial contribution equilibrium the difference between contributing and defecting friends for some individuals have to exceed a certain threshold, while for others it has to remain below this threshold. This happens most likely, if contributors and defectors are segregated in the network. Local confirmation pressure drives certain parts of the network towards contribution and other parts towards defection.

Consider any partial contribution equilibria where all i [member of] C cooperates and all j [member of] D defects. From equation (8) it follows that full contribution equilibrium is payoff dominant over partial contribution, if

min ([r.sub.jc]) > c - [n.sub.d][alpha]/[b.sub.1], where j [member of] D, (13)

and [n.sub.d] is the number of individuals (defectors) in D. It means that the necessary structural condition for full contribution equilibrium being payoff dominant over a given partial contribution equilibrium is the existence of contacts between each defector and a certain number of contributors in the latter equilibrium. In case there are defectors that are only connected to defectors in partial contribution, full contribution will not be payoff dominant compared to this equilibrium. Here again we have to emphasize the importance of universality; all defectors should be integrated to the required extent in order to achieve the benefits of full contribution. Density within the subset of defectors in this respect is irrelevant. What matters is the minimum degree of connectedness to the subset of contributors. Strong segregation of defectors inhibits full contribution in the community.

In case when only proportional conformity ([b.sub.2]) operates and s and [b.sub.1] are equal to zero, the conditions for full contribution being Nash equilibrium are completely independent of network characteristics. Nonetheless, the minimum degree should also be greater than zero in this case. If this presumption holds, then the existence of full contribution Nash equilibrium depends only on payoff parameters. That is, [b.sub.2] [greater than or equal to] c - [alpha] should hold.

In this case, segregation plays an even more important role for the emergence of partial contribution equilibria. The conditions for the existence of partial contribution equilibrium are:

[r.sub.ic] - [r.sub.id]/[r.sub.i] [greater than or equal to] c - [alpha]/[b.sub.2] for all i [member of] C and (14)

[r.sub.jc] - [r.sub.jd]/[r.sub.j] [less than or equal to] c - [alpha]/[b.sub.2] for all j [member of] D. (15)

Given that full contribution equilibrium exists, the necessary condition for partial contribution equilibrium is that for some individuals the proportion of contributors among their friends has to exceed a certain threshold and for other individuals it has to remain below this threshold. This is more likely to happen in clustered network structures. Dense subgroup structures increase, but overlapping dense structures decrease the chance of partial contribution equilibria. In a highly dense network it is less likely that a subgroup exists that is sufficiently isolated from others.

Full contribution is payoff dominant over partial contribution, if

min([r.sub.jc]/[r.sub.j]) > c - [n.sub.d][alpha]/[b.sub.2] (16)

where j [member of] D, and [n.sub.d] is the number of individuals (defectors) in D in the partial contribution equilibrium. It means that the necessary structural condition for full contribution being payoff dominant over a given partial contribution equilibrium is that the proportion of contributors among the connections of each defector should exceed a certain threshold in case of partial contribution. This is a requirement of minimum relative connectedness, unlike in the case of mass conformity, when it was a requirement of minimum absolute connectedness to contributors. Here the number of defecting friends also matters. Full contribution can be payoff dominant also when in the partial contribution equilibrium some defectors have only few contributing friends. On the other hand, they should also have only few defecting friends. This also means that if proportional conformity is highly relevant, then full contribution can be payoff dominant equilibrium also in highly centralized structures. For this, central actors have to be connected to diverse subgroups.

This short analysis of marginal effects showed that the minimum degree is a strong determinant of overall collective action in case selective incentives operate. Network clustering and segregation of defectors has a strong influence when conformity mechanisms are strong. The payoff dominance of full contribution equilibrium is not likely in centralized structures when mass conformity is strong, but it is possible in case proportional conformity is prevalent. Another often-cited network hypothesis, according to which bridging ties support the transmission of contribution incentives between subgroups, is not relevant if only selective incentives are at work. In case the segregated subgroups have dense networks the hypothesis may fail even if conformity plays a significant role. In this case, single bridging connections do not change defectors' incentives.

7. Discussion

This study investigated the effects of social control and network structure on the emergence of collective action in n-person communities. We analyzed network effects in single encounters, and highlighted interactions between social control and structural characteristics. To reach these objectives, an integrated framework of analysis has been used that combined the analysis of n-person games with local interaction games.

We discussed different social control mechanisms that are transmitted by interpersonal relationships. These mechanisms were incorporated in the standard n-person public goods game. Relationships and individuals were considered anonymous; there were no leaders, privileged actors, or binding coalitions. Social control mechanisms, namely selective incentives and forms of conformity were modeled as rewards that influence individual decisions through actors' relationships to relevant others. We demonstrated that as a consequence of social control, network topologies matter for the emergence of collective action.

Some results support widely accepted hypotheses about the facilitating factors of collective action. Besides, the analysis also shed some new light on the underlying mechanisms of social network effects in collective action. Results support the hypothesis that strong social control, on average, facilitates collective action. We also emphasized that it is not always necessary to provide a selective incentive for cooperation. Public good provision might be possible even in a large group where members match their behavior with a little subgroup of their friends. On the other hand, as a main achievement of this study, we demonstrated that under certain circumstances, stronger social control may inhibit overall contribution. Not only conformity, but also selective incentives might have adverse effects. This result provides indirect support to the "double edge of networks" hypothesis of Flache and it fits in the theoretical research line that demonstrates reverse effects of social control mechanisms. (34)

Among social network effects, we found that cohesion is a crucial determinant of mass collective action. As cohesion is associated with density, this is also in favor of the density-cooperation hypothesis. Density, however, is not the most useful indicator of cohesion. Our results showed that the minimum degree of the network and fragmentation are directly related to full contribution, which is in line with the concept of cohesion but has no close association with density itself. Density increases the chance of full contribution mainly because it is correlated with these measures. The model also showed that the impact of minimum degree on full contribution is correlated with the strength of selective incentives, while the lack of clear network clusters foster collective action if conformity plays a significant role in players' decisions. We also showed that clustering in a community might inhibit full contribution even when social control is relatively strong.

Nonetheless, we did not relax many of the model restrictions in this analysis. The floor is open, however, for generalizations as we tried to develop a flexible framework of analysis. That is, several restrictive assumptions of the model can be relaxed in subsequent research without shifting the basic building blocks of the model.

For instance, the analysis can be extended to cases in which a different production function is assumed for the public good provision. Basically, it is not even necessary to assume an increasing production function. Similar results can be produced for cases in which [alpha]<0, where we have the problem of sustaining a public bad. Similarly, instead of rewards of social control, punishments could be considered, for instance in the form of negative selective incentives for defectors. This modification, however, would not reshape model predictions radically. Another example is to relax the assumption of binary social relations (two individuals are either friends or not). We could assume that there are good friends and also mere acquaintances in the network by ordering weights to each tie. The strength of social control would then depend on the strength of the given tie.

As we investigated one-shot interactions with perfect and complete information and forward-looking, strategically rational individuals, a natural development is the consideration of repeated structurally embedded games. Furthermore, the rationality assumptions of the game theoretical approach could be regarded as serious shortcomings, although we are convinced that there are well-founded theoretical reasons for taking this type of actor-model as given. The presented equilibrium analysis also presumes perfect information of actors that is very likely an implausible assumption in large communities. One possible way to tackle this problem is to consider limited information and structurally constrained information flows. A game theoretical analysis in this direction is presented by Chwe. (35) Another possible way of relaxing the strict assumptions of the model is to consider boundedly rational actors. Backward-looking learning models fitted to collective action problems go in this direction. (36)

A critical assumption that can be relaxed in a subsequent analysis is the stability of the network, as it is done by Takacs, Janky, and Flache. (37) Individuals develop new relationships and sometimes abandon old ties, and this might have some consequences also for their decisions in collective action, especially if the structural change is a cause or a consequence of their behavior in the public context. This could be followed by a dynamic interrelated analysis of repeated collective action problems and structural dynamics. Most important, however, is that any model extension and theoretical development should also be fruitful for empirical research. Model predictions should also be tested in laboratory experiments as we already proposed in our earlier paper. (38)

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Bela Janky

Budapest University of Technology and Economics

Karoly Takacs

Corvinus University of Budapest

(1) We would like to thank Andreas Flache, Russell Hardin, Zoltan Szanto, Benedek Kovacs, and Attila Gulyas for their comments on an earlier version of the paper. The authors acknowledge support of the Hungarian Scientific Research Fund (OTKA, T 49432; T 76223; PD 76234).

(2) e.g., Mancur Olson Jr. The Logic of Collective Action. (Cambridge, MA: Harvard University Press, 1965.); Russel Hardin Collective Action. (Baltimore: John Hopkins University Press, 1982.)

(3) for an overview see Mario Diani, "Introduction: Social Movements, Contentious Actions, and Social Networks: From Metaphor to Substance?" in Social Movements and Networks. Relational Approaches to Collective Action ed. Mario Diani and Doug McAdam. (Oxford/New York: Oxford University Press, 2003.)

(4) e.g. Karl-Dieter Opp and Christiane Gern, "Dissident Groups, Personal Networks, and Spontaneous Cooperation--The East German Revolution of 1989" American Sociological Review, 58 (1993): 659-680.; Anthony Oberschall, "Rational Choice in Collective Protests" Rationality and Society, 6 (1994): 79100.

(5) e.g., Glenn Ellison "Learning, Local Interaction, and Coordination" Econometrica, 61 (1993): 1047-1071.

(6) Rickard Sandell and Charlotta Stern "Group Size and the Logic of Collective Action: A Network Analysis of a Swedish Temperance Movement 1896-1937" Rationality and Society, 10 (1998): 327-345.

(7) e.g., Andreas Flache The Double Edge of Networks. An Analysis of the Effect of Informal Networks on Cooperation in Social Dilemmas. (Amsterdam: Thela Thesis, 1996.)

(8) Pamela E. Oliver "Rewards and Punishments as Social approval for Collective Action: Theoretical Investigations" American Journal of Sociology, 85 (1980): 1356-1375.; Douglas D. Heckathorn, "Collective Action and the Second-Order Free-Rider Problem" Rationality and Society, 1 (1989): 78-100.

(9) e.g., Thomas C. Schelling Micromotives and Macrobehavior. (New York: W. W. Norton, 1978.); Marwell, Gerald and Pamela E. Oliver Critical Mass in Collective Action. (Cambridge: Cambridge University Press, 1993.)

(10) Robert Sugden "Reciprocity: The Supply of Public Goods through Voluntary Contributions" Economic Journal, 94 (1984): 772-787 Gould, Robert V. "Collective Action and Network Structure" American Sociological Review, 58 (1993): 182-96.

(11) Anthony Oberschall "Rational Choice in Collective Protests" Rationality and Society, 6 (1994): 79-100.

(12) e.g., Ernst Fehr and Simon Gachter "Cooperation and Punishment in Public Good Experiments" American Economic Review 90 (2000): 980-94; David Masclet,, Charles Noussair, Steven Tucker and Marie-Claire Villeval "Monetary and Nonmonetary Punishment in the Voluntary Contributions Mechanism" American Economic Review, 93 (2003): 366-380.,

(13) Heinz Hollander "A Social Exchange Approach to Voluntary Cooperation" American Economic Review, 80 (1990): 1157-67.

(14) Gould, "Collective Action and Network Structure"

(15) Oberschall, "Rational Choice in Collective Protests"

(16) e.g., Gerald Marwell and Pamela E. Oliver and Ralph Prahl "Social Networks and Collective Action: A Theory of the Critical Mass. III." American Journal of Sociology, 94 (1988): 502-534.; Hyojoung Kim and Peter S. Bearman "The Structure and Dynamics of Movement Participation" American Sociological Review, 62 (1997): 70-93.

(17) e.g., Robert V. Gould Insurgent Identities: Class, Community, and Protest in Paris from 1848 to the Commune (Chicago: University of Chicago Press, 1995.)

(18) Michael W. Macy "Chains of Cooperation: Threshold Effects in Collective Action" American Sociological Review, 56 (1991): 730-747.

(19) Mario Diani "Networks and Social Movements: A Research Programme" in Social Movements and Networks. Relational Approaches to Collective Action ed. Diani and McAdam. (Oxford/New York: Oxford University Press, 2003.), 308.

(20) Marwell, Oliver and Prahl "Social Networks and Collective Action: A Theory of the Critical Mass. III."

(21) Jonathon N.Cummings and Rob Cross "Structural Properties of Work Groups and Their Consequences for Performance" Social Networks, 25 (2003): 197-210.

(22) Jeffrey C. Johnson, James S. Boster and Lawrence A. Palinkas "Social Roles and the Evolution of Networks in Extreme and Isolated Environments" Journal of Mathematical Sociology, 27 (2003): 89-121.

(23) e.g., Hauert, Christoph Virtuallabs 2004. Database on-line. Available at http://www.univie.ac.at/virtuallabs/Moran/

(24) Hisashi Ohtsuki, Christoph Hauert, Erez Lieberman and Martin A. Nowak "A simple rule for the evolution of cooperation on graphs and social networks" Nature, 441(7092, 25 May 2006): 502-505.

(25) e.g. Francisco Santos, Marta D. Santos and Jorge M. Pacheco "Social diversity promotes the emergence of cooperation in public goods games" Nature 454 (2008): 213-216.

(26) Toshio Yamagishi and Nahoko Hayashi "Selective Play: Social Embeddedness of Social Dilemmas" in Frontiers in Social Dilemma Research ed. Wim B.G. Liebrand and David M. Messick. (Berlin: Springer, 1996). 326

(27) Mark Granovetter "The Strength of Weak Ties" American Journal of Sociology 78 (1973): 1360-80.

(28) Peter Csermely Weak Links. The Universal Key to the Stability of Networks and Complex Systems. (Berlin: Springer, 2009.)

(29) Karoly Takacs and Bela Janky "Smiling Contributions: Social Control in a Public Goods Game with Network Decline" Physica A, 378 (2007): 76-82., Karoly Takacs, Bela Janky and Andreas Flache "Collective Action and Network Change" Social Networks, 45 (2008): 177-189.

(30) One should note that from a purely theoretical and general viewpoint, payoff dominance cannot serve as a solution for the problem of equilibrium selection in games. If one approaches the problem of uniqueness from the perspective of our study, however, the concept of payoff dominance provides the most fruitful selection mechanism. Experimental findings are also ambivalent about whether subjects play the payoff dominant equilibrium or another outcome. In coordination games with multiple equilibria, for example, in case of few players and a salient payoff dominant equilibrium this outcome is played often, but more players and a higher resistance makes the rival risk-dominant equilibrium a more likely outcome. Ses e.g. Dean Corbae and John Duffy Experiments with Network Economies. (Mimeo, University of Pittsburgh, 2002.).

(31) As we defined above: [alpha][n.sup.*] > c. For the production of public bads, for which this assumption does not hold, the overall defection equilibrium is likely to be payoff dominant.

(32) Opp and Gern "Dissident Groups, Personal Networks, and Spontaneous Cooperation--The East German Revolution of 1989"; and Gould "Collective Action and Network Structure".

(33) Hauert Virtuallabs.

(34) Flache Double Edge of Networks.

(35) Michael Suk-Young Chwe, "Structure and Strategy in Collective Action" American Journal of Sociology, 105 (1999): 128-156.

(36) Michael W. Macy "Backward Looking Social Control" American Sociological Review, 58 (1993): 819-836.; Michael W. Macy and Andreas Flache "Learning Dynamics in Social Dilemmas" Proceedings of the National Academy of Sciences of the U.S.A, 99 (2002): 7229-7236.

(37) Takacs, Janky, and Flache "Collective Action and Network Change"

(38) Takacs and Janky "Smiling Contributions: Social Control in a Public Goods Game with Network Decline"
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Author:Janky, Bela; Takacs, Karoly
Publication:CEU Political Science Journal
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Date:Sep 1, 2010
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