Social Pressure in Networks Induces Public Good Provision

Why people participate in public good provision is one of the oldest questions in Economics. In the absence of enforcement mechanisms public goods would be under-provided. I develop a dynamic model of forward-looking agents in the presence of social pressure, which provides a potential enforcement mechanism. I show that social pressure is effective in generating provision in a public good game: after an agent starts contributing to the public good, other agents decide to contribute as well because of fear of being punished, what generates contagion in the network. In contrast to the previous literature, contagion happens fast, as part of the best response of fully rational individuals. The network topology has implications for whether the contagion starts and the extent to which it spreads. I find conditions under which an agent decides to be the first to contribute in order to generate contagion in the network, as well as conditions for contribution due to a self-fulfilling fear of social pressure.


Introduction
Why do public good provision and collective action take place? As Samuelson (1954) observed, given that collective action and public goods are by definition non-excludable, most individuals have no incentive to participate. One possibility is that people participate out of a fear of social sanctions. For example, large numbers of young British enlisted in WWI out of social pressure (Silbey, 2005, p. 115-6). However, this possibility has been largely neglected on theoretical studies of public good provision, despite the abundant evidence of its importance. 1 If social pressure is effective in enlisting people to go to war, which has an enormous private cost, why would it be any less effective in the provision of other public goods? Indeed, as the empirical literature in voting makes clear (Gerber et al., 2008;Funk, 2010;DellaVigna et al., 2016) social pressure plays a large role in inducing people to vote. I propose a model that is explicitly based on social pressure: individuals are embedded in a network, and they receive disutility when their friends contribute to the public good but they do not. Agents face a decision of whether to contribute to a public good: the benefit b is public, but the cost c is private and larger than the benefit, c > b. At the beginning of the game, nobody is contributing to the public good; the game is dynamic, and every period the game ends with constant probability. If it continues, an agent is selected randomly to revise her strategy.
This simple model captures two interesting features. The first one is that whenever an agent i contributes to the public good, there is an immediate contagion in the social network. Initially, friends of i find optimal to contribute, as they do not want to suffer social pressure. Crucially, they cannot revise their strategy until they are randomly chosen to do so. Friends of friends of i know that friends of i will contribute as soon as they are able to revise their strategy. They measure their potential expected disutility from social pressure (which increases in the probability that the game continues), and they compare it with their certain cost of contributing to the public good. If the former is greater than the latter, they also decide to contribute whenever they get to revise their strategy. But then agents who are at distance 3 in the network (that is, friends of friends of friends of i) will make the same reasoning, and so will the rest of agents in the network, generating a contagion in best responses by which most individuals in the network end up contributing. Unlike previous papers in the literature (Kandori et al., 1993;Young, 1993;Morris, 2000), contagion happens instantly (rather than over time), as part of the best response of the individuals in the network. This paper makes a theoretical contribution, by showing that social pressure explains why people would participate in revolts and other forms of collective action, without the need of having a private benefit from the success of the revolt, which is the assumption held by most collective action models (Granovetter, 1973;Palfrey and Rosenthal, 1984;Kuran, 1987;Chwe, 2001;Medina, 2007). There is a clear problem with the assumption made by the previous literature: while the benefit of the the revolt is a public good, each individual must bear a private cost of participation; but there is no reason why individuals would be willing to participate in collective action (Samuelson, 1954). The literature has attempted to produce reasons why individuals might privately profit from revolts, because they can loot or get a share of the benefits from revolting, but this does assumption seems dubious. 2 The paper contributes to this literature by providing a mechanism (social pressure), that would give individuals the incentive to contribute, even if the final outcome is a public good. 3 Leaders are important for collective action. Rosa Parks is regarded as a spearhead of the civil rights movement in the United States: in 1955, she refused to give up her seat on a bus to a white person; an act that sparked a movement that eventually led to major advances in civil rights in the United States (Theoharis, 2013). In terms of the model, I call the first individual to contribute the leader, and I show that under some conditions agents will decide to become leaders, generating a large contagion and hence a large contribution in the network as an outcome. One such conditions happens when the contagion generated by an agent i is so large that the expected benefit from inducing others to contribute is larger than the private cost of contributing. The contribution to the public good by the leader herself is negligible, but because she manages to spread the contagion to a large fraction of the population, she has incentives to contribute even when nobody has contributed before. It can also happen that, when several agents are afraid of social sanctions by others, those agents contribute (thereby punishing those who do not contribute); therefore the fear of social sanctions becomes self-fulfilling. The question of leadership is rich and complex, and there is a large literature that has addressed it in the context of teams (see Hermalin, 2012, for a survey). 4 Precisely because of this complexity, there are only a handful of papers that have consider leadership in a larger setting, where a single individual can tip the equilibrium of the whole society. For example, in a paper of technology/behavior diffusion, Morris (2000) takes the innovators as exogenous. But why would those individuals become innovators in the first place? Like me, Ellison (1997) considers the case in which a single player can generate contagion to the whole population. In his case a single "rational guy" can sometimes make the rest of the (myopic) players change to a Pareto-superior equilibrium. Corsetti et al. (2004) also consider the importance of a single player. However, both of these papers take leaders as exogenous. Acemoglu and Jackson (2012) consider a case where an individual can change a social norm, and that social norm will stick to at least some future generations. Leaders in their model are endogenous, but only a special class of individuals can become leaders. A contribution of the present paper is to analyze how and when an individual endogenously becomes the leader of a large group. This paper is connected to several important literatures. One of the most relevant papers is Miguel and Gugerty (2005), which has a model of public good provision with social sanctions, but (unlike in the present model) social sanctions are a function of the share of contributors in the whole population. Because of that, their model exhibits multiplicity of equilibria and a threshold such that once there are enough contributors, everyone find optimal to contribute. My contribution here is to have a dynamic model where social pressure is local, what generates a situation where contagion of contribution can happen even when everybody starts not contributing. 5 Karlan et al. (2009) consider a model where individuals that are not directly connected in the social network can nevertheless use links with other people as "collateral". The present paper is similar in spirit to the idea that a person can use her links in the social network as valuable assets, and apply this idea to public good provision. 6 Finally, this paper is related to the literature in social dynamics (Kandori et al., 1993;Young, 1993;Morris, 2000). Unlike the previous literature, I consider fully rational and forward-looking agents. This makes the present paper different in several features that I have mentioned above, mainly: contagion is fast, and it is started by leaders who do so in a calculated manner. A remarkable branch of this literature (Matsui and Matsuyama, 1995;Matsui and Oyama, 2006;Oyama et al., 2008Oyama et al., , 2016, considers agents who are forward-looking and can only revise their strategy as a Poisson process. However, unlike in my model, they consider a setting in which the action of any single agent does not affect the payoff for the rest of the population. Because of that, there is no contagion or leadership considerations. The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 analyzes 5 In support of this intuition, Besley et al. (1993); Besley and Coate (1995) have documented the importance of social pressure and social interactions in the context of savings associations. 6 The literature on public good provision has tried to analyze how different characteristics of the population (such as ethnic, racial and socioeconomic heterogeneity), affect the level of public good provided in equilibrium (Alesina et al., 1999;Alesina and La Ferrara, 2000;Vigdor, 2004). The results of this literature show that heterogeneity in the mentioned characteristics is associated with lower public good provision. Even though the present model does not include heterogeneity along those dimensions, I believe that it points to a causal mechanism: public good provision would indeed be lower if heterogeneity across ethnicity, socioeconomic status, etc. generates social norms with less sanctions, or makes it more likely that the network is fragmented in cliques that do not interact with each other (Example 2). Each of those cases would make social pressure not operational, and hence public good provision to decline. the contagion that takes place in the network, once people have started contributing. Section 4 analyzes under which conditions individuals become leaders by contributing when nobody has contributed before. Section 5 concludes. The Appendix provides proofs and details.

The model
There are is a finite number n of agents in the society, represented by set N . The structure of society is given by a friendship network g ∈ {0, 1} N ×N , where g ij = 1 if agents i and j are connected , and g ij = 0 otherwise. 7 The network is undirected (g ij = g ji ), that is, friendship is reciprocal. Moreover, the network is connected, i.e. any agent i ∈ N is a friend of at least another agent in N . Let N i denote the set of i's friends: N i = {j ∈ N : g ij = 1}. Each agent faces a binary choice: to contribute or not to contribute to the public good. Time is discrete, t = 1, 2, . . . , and each period only one agent (selected at random) can revise her strategy. Therefore, every agent must play the same strategy in every period until she has the opportunity to revise. The game can be summarized every period by a state S ∈ {0, 1} N , where s i denotes the action last chosen by agent i, and s i = 1 means the agent contributed. Initially S = {0} N , i.e. every agent starts the game not contributing. At the beginning of each period, the game continues with exogenous probability q (therefore, the game ends with probability 1 − q, in which case payoffs are realized). If the game continues, an agent i ∈ N is selected at random to play, from a i.i.d.
uniform distribution on the set of agents N . Agent i chooses an action from A i (S), where This means that once an agent has contributed, she must stick to that action for the rest of the game. We can think of this as the fact that once an individual has made a payment to the government, or participated in a violent demonstration, it is extremely difficult or impossible to undo such action. 8 Given this structure, the game is such that agents play one at a time, selected at random with replacement from the set N, until the game ends. When the game ends and the state is S, payoffs are realized, according to the following utility function: Agent i derives utility b from each agent who contributed and pays the private cost c if she herself contributed. Moreover, agent i suffers social pressure whenever she does not contribute to the public good: in that case she incurs disutility bψ for each friend who contributed. Therefore, the total social punishment is proportional to the importance of the public good b, the social pressure parameter ψ, and the number of friends who contributed. 9 To make this a game of public good provision, I assume that the private cost is larger than the benefit an agent enjoys by contributing.
With no social pressure (ψ = 0), it is easy to see that nobody would ever contribute in a subgame perfect equilibrium (SPE). 10 The key ingredient in the model that will generate contribution is social pressure. 11 In order to simplify the analysis, I assume that the cost of contributing is smaller than the punishment by a single friend. This ensures that whenever a friend of agent i contributes, it is optimal for agent i to contribute as well.
Given the probability q that the game continues, we can define Q as the probability that a given agent i will be able to play before the game ends. Because the game is symmetric, Q is the same for all agents. I assume that Q remains constant for any population size n. If this was not true, the results would depend artificially on the size of the population, as will become clear in section 3.
Assumption 3. The probability that the game continues q, as a function of n, is such that Q is constant.
The probability Q that an individual will be chosen before the game ends can be recursively 9 This assumes that agents value relationships "per se", for some emotional reason and not for example because having friends allows and individual to be connected to other agents in the network. This can be partially justified by the literature in psychology and behavioral economics (Gintis et al., 2003;Fehr and Gachter, 2002;de Quervain et al., 2004;Gürerk et al., 2006). Despite this evidence, it can also be considered as a "reduced form" utility from a more general model, where having friends signals some underlying disposition, like being honest.
10 Indeed, suppose that n − 1 agents have contributed, and the last agent has to decide whether to contribute. Because all other agents need to contribute in the future, there is no reputation loss for not contributing. Because c > b, she will choose not to. But that means that previous actions of other agents cannot change what the last agent will do, and so the second-to-last agent does not contribute either. By induction, no agent will ever contribute.
11 Social pressure and social sanctions can be modeled in various ways. In the Introduction I discussed Miguel and Gugerty (2005), where social pressure is inflicted presumably by the collectivity on the deviators, without taking into account the structure of society. I, on the other hand, view social pressure as derived precisely from the local interactions of agents.

Contagion
In this section I analyze how the fact that an agent contributes can affect how others behave in the network, and generate a "wave of contagion" that spreads contribution by best-response dynamics to a large fraction of the network. A very similar effect was analyzed in Morris (2000), in the context of myopic agents. The conditions I find are such that even when agents are forward-looking, they have as a dominant strategy to contribute.

Simple Case: An Illustration
Consider the case where a given agent i has contributed to the public good. Because, by Assumption 2 we have bψ > c, the best response for any agent j ∈ N i is to contribute. However, even if j desires to contribute, she still has to wait until she is selected to revise her strategy.
I define agents whose best response is to contribute but have yet not done so, as predisposed agents. Hence, all friends of i become predisposed after i contributes. The next question is: what will an agent k ∈ N i who is friend of j ∈ N i do when she gets a chance to play? If k decides to contribute, she obtains an extra payoff of b − c + ξ, where ξ denotes payoff from further contagion generated by k. However, if k does not contribute, her payoff V will depend on the different scenarios that could happen: • When j plays, hence contributing, and k gets to play before the game ends. In that case k will contribute (because bψ > c) obtaining an extra payoff bounded above by b − c + ξ. 12 • If k plays before j gets a chance to play, then k faces exactly the same choice, and hence the extra payoff is simply V • If j plays, hence contributing, and then k does not get a chance to play before the game ends, she will incur extra disutility −bψ • Finally, it might happen that the game ends before either k or j get to play, in which case k receives an extra payoff of 0 12 it is bounded above because k can only generate as much contagion as if it had originally contributed, hence ξ is an upper bound on the real contagion payoff at this point Therefore, j will become predisposed when the certain cost of contributing is smaller than the expected social punishment.
Definition 1. Let ρ m (h) be the probability that out of m agents in the network, exactly h get to revise their strategy before the game ends and before agent k can reviser her strategy. 13 Proposition 1. Agent k contributes when she has a predisposed friend if The intuition for Equation 3 is that the ratio of cost to benefit of contributing (which is higher than 1 by Assumption 1) cannot be too large in comparison with the expected social punishment is the probability that agent k's predisposed friend gets to revise her strategy before the game ends and before k can reviser her strategy. If condition 3 holds, the best response for agent k is to contribute, and so she becomes predisposed. But the argument only uses the fact that k has a predisposed friend, and hence this same argument can be repeated for every agent in the network who has a predisposed friend. This in turn implies that after a single agent i contributes, the whole network becomes predisposed. The reasoning is as follows. We already saw that once agent i contributes, agents in N i also become predisposed. When equation 3 holds, all agents who have a friend in N i will also become predisposed. But then all agents who have a friend of a friend in N i will also become predisposed, etc. The contagion will spread, and after a single agent contributes the whole connected component becomes predisposed. Note however that this argument is not exlusive to the case where Equation 3 holds: I turn now to analyze the more general case.

General Case
So far we only looked at those agents with a predisposed friend, but the same reasoning can be generalized to the case when an agent k has m friends who are predisposed (recall that an agent is predisposed if her best response is to contribute when she has the option to revise her strategy).
Proposition 2. If agent k has m predisposed friends, she contributes if . The derivation can be found in the Appendix.
The intuition for Proposition 2 is similar to that of Proposition 1: the agent will contribute if the cost/benefit ratio is smaller than the expected social punishment from her predisposed friends.
Note that the the left-hand side of equation 4 does not depend on m. The distribution ρ m (h) is increasing in m in the First Order Stochastic Dominance (FOSD) order (viewed as a marginal distribution on h), and hence the right-hand side is increasing in m. 14 Hence there exists a minimal m * that makes equation 4 hold, that is, such that for all m ≥ m * agents with at least m predisposed friends have as a unique best response to contribute when chosen to revise their strategy, therefore becoming themselves also predisposed.
Definition 2. The contagion threshold m * is the minimal number m of predisposed friends, that satisfies the sufficient conditions for an agent to contribute.
Note that m * is decreasing in b, ψ and Q, and increasing in c. Hence, greater values of b, ψ or Q, or lower values of c make contagion more likely. These comparative statics of m * with respect to the parameters of the model are intuitive: higher benefit, lower cost, or higher social pressure all induce more contagion. The fact that a higher Q (higher probability of revising before the game ends) induces more contagion is also clear: not only does this generate more possibilities for agents to contribute, but those possibilities are in turn anticipated by others, and that will make them also more likely to contribute.
Let's turn now to study how contagion spreads through the network. I use the notation from Morris (2000). Let S be a set of agents, I define operator Π 0 m (S) := S, and for k ≥ 1 I define Π r m recursively : It is clear that this is an increasing sequence of sets, and I define Π ∞ k (S) as the limit of that sequence. We can see then what happens when an agent i contributes when the contagion threshold is m * . By Assumption 2 bψ > c, so immediately all agents in N i become predisposed.
Then, all agents who have at least m * friends in N i also become predisposed, so the contagion spreads to the set Π 1 m * (N i ). Proceeding recursively, the contagion spreads through the network and in the end all agents in N ∞ m * (N i ) are predisposed. This contagion happens in the same period that agent i contributes, so when next period starts any agent in N ∞ m * (N i ) will contribute if selected to play. Hence we have the following result.
Proposition 3. When agent i contributes, all agents in Π ∞ m * (N i ) become predisposed.
Proof. I will prove it by induction. By Assumption 2, whenever i contributes all her friends become predisposed, and because Π 0 m * (N i ) = N i , this proves the case k = 0. Now, suppose the statement is true for k, what means that all individuals in Π k m * (N i ) are predisposed. Then, by the definition of m * , all agents with at least m * friends in the set Π k m * (N i ) will also become predisposed, what implies that individuals in Π k+1 We have seen how a contagion of best responses takes place in the network. This phenomenon is similar to the contagion in Morris (2000); Oyama and Takahashi (2011), and many other models of contagion, with two remarkable differences. In this paper, all agents are forward looking, and contagion takes place in anticipation to the actions of others. This is a contribution with respect to the previous literature, where the individuals are either not forward looking (or only some individuals are, as in Ellison, 1997), or they are unable to initiate a contagion (Matsui and Matsuyama, 1995). Second, contagion happens in the same period that agent i contributes: because players are forward-looking, contagion is instantaneous. This is in stark contrast with the papers in the literature of social dynamics, where contagion is supposed to happen in the long run or as the limit of some updating process. 15 Interestingly, both aspects of the current model are related. It is precisely because agents are rational and forward-looking that contagion spreads so fast: everyone anticipates that others will also contribute.
Note that despite the potential for multiple equilibria, agents who are affected by the contagion have as a dominant strategy to contribute, irrespectively of what non-predisposed agents do.
This comes at the expense of obtaining a lower bound on the actual contagion on the network.
The conditions I required here are stronger than necessary for a contagion to happen, but if they do happen, we can unambiguously claim that contagion takes place. As argued in the Introduction, I am interested in conditions that would generate collective action, hence the focus on sufficient conditions. Let's analyze how (and if) contagion happens in a couple of well-known networks.
Example 1 (Regular m-lattice 16 ). Consider that agents belong to Z d , where Z is the set of integers and d is the dimension of the lattice. Agents are friends if they are next to each other,i.e Ellison (2000); Kreindler and Young (2016), for a discussion about the speed of contagion. 16 Example from Morris (2000).
where x k i is the k-th coordinate for agent i. In a regular lattice, contagion does not propagate for m * > 1. The reason is that friends of i are not friends themselves, i.e. the clustering coefficient is 0. 17 Therefore, even if the friends of i become predisposed, their friends will not become predisposed for m * > 1.
Example 2 (Cliquish network). Consider network g such that 1. g is composed by a collection of cliques (a clique c is a subset of g, such that if i ∈ c, then g ij = 1 for all j ∈ c). In other words, all agents in a clique are friends with everybody else in that clique 2. Cliques are only connected via a single link, as can be observed in the figure below In a cliquish network, contagion does not happen for m * > 1. Consider for example m * = 2.
As soon as an agent i contributes, all agents in the clique become predisposed by Assumption 2. However, because cliques are only connected by a single link, contagion will not extend from a clique to the next one. Hence, in a cliquish network, contagion extends to the clique, but no further.
Until this point I have considered how contagion spreads when an agent contributes. In a sense, contagion exhibits strategic complements: the more likely an agent is to contribute, the more likely others contribute because of the fear of social pressure. However, why would an agent contribute in the first place? By not contributing she could wait and free-ride on others' contributions. What would then make an agent become a leader? I turn now to analyze these ideas in more detail.

Leadership
Hermalin (2012) defines a leader as someone with voluntary followers -as opposed to someone invested with authority, whom people are somewhat forced to obey. He suggests that leaders serve three main roles: they are judges, experts and coordinators. While the role of judges and experts are without a doubt important for leadership, I am especially interested in the third role: leaders as coordinators. There are some situations where the behavior of agents following early adopters of a technology or behavior might not be optimal, as in the case of herding (Banerjee, 1992). However, very often there are multiplicity of equilibria (as in the case of different conventions) and the optimal course of action is to coordinate on which equilibrium to choose. Hermalin (1998) recognized that even in this case, leaders might have an incentive to select one equilibrium over another, potentially misleading her followers. He analyzed when is it possible to lead by example, so that the leader is invested in the choice that she is advocating. In the present model there is no ex-ante conflict of interests between agents, as they all agree that more of the public good is better. However, agents can still lead by example, because by contributing they recruit their friends, and it is possible to generate a snowballing of social contagion, as we saw in the previous section.
Consider the case when nobody has ever contributed so far in the game, and i gets (randomly) chosen to play. By contributing, she spreads contagion to at least the set Π ∞ m * (N i ), and she hedges herself against future social pressure. However, she also incurs the private cost c, and if the game ends soon, she does not reap the benefits of the contagion she generated. If, on the other hand, i does not contribute, and someone else (say j) contributed afterwards, then i could obtain the public benefits of the contagion generated by j, without having paid the cost c (of course, in this case it could happen that i ends up suffering social pressure, if one of her friends contributes).
We see that there are two forces in play. One is the contagion, which exhibits strategic complementarities. But there is also a "volunteer's dilemma" of waiting for others to contribute in the first place, that exhibits strategic substitutes. Which force is stronger depends critically on the parameters of the game. Because of these competing forces, it is difficult to obtain a characterization of the set of equilibria. This phenomenon seems to happen in any game of strategic complements when it is modeled as a dynamic game that ends with a certain probability each period. For example, the literature on currency attacks, like Morris and Shin (1998), has used global games, a model with strategic complements. In these models there are no strategic substitutes; however, in real life situations like revolts and currency attacks we can imagine that some individuals may want to hold back in order to learn information or simply not to expose themselves until it is more convenient to do so, hence the volunteer's dilemma. I believe that this intuition is important in order to understand collective action as well, and I am not aware of any paper analyzing this particular phenomenon.
The concept I use is that of Subgame Perfect Equilibrium (SPE): σ is an SPE if it is a Nash equilibrium of every subgame; in other words, if whenever an agent i is chosen to play, she best responds to σ −i . 18 There are two main classes of equilibria which are useful to consider, because they represent the two extremes of the spectrum. Spearheaded equilibria are such that there is at least one agent i ∈ N , such that i contributes even if nobody else would ever contribute.
Agent i does not contribute out of social pressure (after all, if she did not contribute, it could happen that nobody ever would), but because she wants to generate a contagion in the network, so that many individuals end up contributing, and i can enjoy the public good generated by such contributions. Because of that, i is truly a leader, since her reasons to act are to induce others do so so. At the other extreme, we have social-pressure equilibria, where there is a set of agents Y such that they all fear that others will contribute, and that if they do not contribute they will be punished, and so they end up contributing as well. Therefore, the fear of punishment becomes self-fulfilling, and it is the reason why everybody in Y contributes. In this case, agents in Y are not concerned with inducing others to contribute; only with avoiding social punishment.

Spearheaded equilibria
I turn now to obtain sufficient conditions for a spearheaded equilibrium to occur. Notice that the key requisite is that there is at least one individual i who has enough incentives to contribute, even if nobody else would.
Proposition 4. If there exists an agent i such that where k = |Π ∞ m * (N i )|, then any SPE is a spearheaded equilibrium.
The intuition behind this result is as follows. The left-hand side of Equation 5 is just the usual ratio of cost to benefit, and the right-hand side is the number of agents who: 1) are predisposed by i, and 2) are expected to contribute before the game ends. When the benefit of mobilizing enough people is greater than the private cost, i contributes even if nobody else would contribute otherwise -and therefore in any SPE there must be at least one individual who contributes when chosen to revise her strategy. The right-hand side of Equation 5 is between 1 and k: when Q → 0, so that the game ends immediately, only i gets to contribute; when Q → 1, so that the game lasts indefinitely, all agents in Π ∞ m * (N i ) get to contribute. Therefore, for intermediate values of Q, the expected number of individuals in Π ∞ m * (N i ) who get to contribute is somewhere between 1 and 18 More details on SPE can be found in Fudenberg and Tirole (1991).
k: the more likely the game is to continue, the higher the right hand side of Equation 5, and the more likely a spearheaded equilibrium can exist. Crucially, the leader takes into account the fact that others will contribute after she does so, and hence forward-looking behavior is fundamental to leadership.
As it is clear from Proposition 4, agent i will be more likely to be a leader, the greater Π ∞ m * (N i ) is, i.e. the farther agent i can extend contagion. It would be interesting to characterize leaders in terms of characteristics usually considered in the networks literature. For example, being a hub according to degree centrality, i.e. an agent i that is connected to a disproportionately large number of other agents. Or eigenvector centrality, which assigns relative scores to all nodes in the network recursively, such that if node i is connected to high-scoring node j, then that connection contributes more to the score of i than connections to low-scoring nodes. However, neither of these characteristics fully characterize leadership in the present model. 19

Social pressure equilibria: Supermodularity and MPE
Because the condition in Proposition 4 might be too strong to be satisfied in certain networks, I will look at weaker conditions that still guarantee contribution. In order to do that, I will try to find conditions under which the game becomes supermodular, i.e. such that whenever an agent contributes, everybody is more likely to also contribute. In order to define supermodularity properly, we need to define Markov Perfect Equilibria. A strategy σ is a Markov strategy if σ only depends on the state S. Because I have made the assumption that once a player contributes, she cannot undo her contribution, restricting behavior to Markov strategies is quite intuitive. A Markov Perfect Equilibrium (MPE) σ is a Subgame Perfect Equilibrium in which players use Markov strategies.
Definition 3. The game is supermodular if all σ MPE are such that σ(S) is weakly increasing (with respect to the lattice order). The game is supermodular given state S 0 if all σ MPE are such that σ(S) is weakly increasing for all S ≥ S 0 .
19 Consider the case of hubs first. It is easy to see why being a hub does not necessarily imply being a leader. Consider the following example: the contagion threshold is m * = 2, node i is connected to a large number of agents and each j ∈ N i is connected to only one other agent k, such that k ∈ N j for j ∈ N i , j = j. In that case, when agent i plays 1, all agents in N i become predisposed, but contagion stops there, since by assumption no agent in the network is connected to two agents in N i . Hence, Π ∞ m * (N i ) = N i . Even if i is connected to a large number of agents, that might not be enough to have the condition in Proposition 4 hold. Eigenvector centrality does not necessarily imply leadership either. For example, consider a network with a few hubs of the type previously described, connected to each (but such that their friends are not friends of each other). Those hubs will have high eigenvector centrality, yet using the same reasoning, we can conclude that for each of them Π ∞ m * (N i ) = N i , so they will not be leaders.
The previous literature (Matsui and Matsuyama, 1995;Matsui and Oyama, 2006;Oyama et al., 2008Oyama et al., , 2016 has considered a similar problem than I do, in the sense that agents are forwardlooking and can only revise their strategy randomly (as a Poisson process). However, unlike in the present model, they consider a setting with an infinite number of agents, in such a way that the action of any single agent does not affect the payoff for the rest of the population. Oyama et al. (2008) prove that in such a setting, the game will be supermodular whenever the stage game is supermodular. That result does not carry over to the present setting, for the following reason: two agents might be willing to contribute if no one has already done so, in order to start a contagion process; however once the contagion has been started, neither of them has incentives to contribute. This becomes clear in the following (informal) example: in a network such that m * = 2, agents i and j can generate a contagion to a very large set of the population (so that it is worth to become a leader), but i ∈ Π ∞ 2 (N j ) and j ∈ Π ∞ 2 (N i ). In such a case, once i becomes a leader and starts contributing, j has no incentives to contribute anymore: contagion has already been initiated, and she is not likely to suffer from social pressure.
I look now at conditions that guarantee that there exists at least one MPE with contribution.
Note that proposition 4 does not use social pressure; I can find weaker conditions by exploiting the fear of social sanctions. In order to do that, I find a set of agents Y where they all are willing to become leaders, and each agent in Y contributes partly because of the fear of social pressure from the rest of agents in Y . I formalize this intuition in the following definition.
That is, Y is m-regular if all agents who belong to Y become predisposed when an agent from Y contributes and the contagion threshold is m: agents in Y contribute (even when nobody has done so) because the rest of Y is also willing to contribute, becoming a self-fulfilling prophecy.
Proposition 5. Let Y be a m * -regular set of agents, such that c b > 1 + ζ(N \Y ). Then, conditional on an agent from Y contributing, the game is supermodular.
Proof. Once agent j ∈ Y contributes, contagion extends to all agents in Y at least (by the assumption of Y being m * -regular). That means that all agents in Y become predisposed, and have "contribute" as best response. Hence, to check for the supermodularity of the game, we need to show that best responses of all players not in Y are weakly increasing in the state S. But agents not in Y have no incentive to become leaders, since b[1 + ζ(N \Y )] < c, that is, even if they could predispose the rest of the agents in the network, it would not be worth to contribute.
Hence, the only reason why they would contribute is because of social pressure, which is weakly increasing in the state.
Proposition 6. If there exists a m * -regular set Y , then there is a SPE where all agents in Y contribute whenever they play.
Proposition prop:sandia offers conditions under which a self-fulfilling fear of social pressure leads to individuals in Y to become leaders. This condition is interesting because it expands the contexts in which the public good can be provided: and the expected benefit b[1 + ζ(Π ∞ m * (N i ))] is high, then the prediction of the model is that we will have spearheaded equilibria. This would be the case for example with voting in a democracy, where the cost of going to the polls for an individual is relatively low. However, in the case of civil rights movements, the private cost of participating in protests and revolutions can be very high; in these cases, it seems plausible that the conditions in Proposition 4 are not met, and yet the conditions in Proposition 6 might still hold, what means that there might be some SPE where contribution happens, even in these cases with large private cost. The reasons why contribution happens sometimes and not some other times are difficult to pinpoint (as in every case of multiple equilibria), and can very well be a matter of coordination. In Jimenez-Gomez (2014), I analyze the conditions under which players are able to coordinate in revolting against a regime; however it is necessary to assume that each individual has an incentive to participate. Therefore, the present model provides some conditions under which it is incentive compatible to participate in collective action (due to social pressure), and I hope to contribute to this ongoing fundamental question in the literature.
Let's consider now contagion from the point of view of a principal who desires to prevent contribution from happening (for example, a dictator that wants to stop a demonstration from happening, or the Montgomery establishment who is against the bus boycott). The principal can impose an extra cost δ > 0, only to the first person to contribute. The intuition for this is that, while it is relatively easy for a government or an organized minority to retaliate against a single person, it is very hard to fight against a mass of individuals. For example, Rosa Parks and her husband suffered greatly as a consequence of her actions: they lost their jobs, developed health problems, and received hate call and mail persistently -until they left Montgomery due to this persecution, Theoharis (2009).
The key fact to observe is that δ affects the cost-to-benefit ratio for the first agent to contribute, which now becomes c+δ b . Note that, for spearheaded equilibria, the left hand side of Equation 5 increases in δ and, for δ large enough, no individual finds it optimal to become the first to contribute. If the principal has to pay an "intimidation cost" δ per player to which it wants to increase the cost, this is an inexpensive way to do so because, in equilibrium, the principal does not even need to pay the cost: the threat of incurring the extra cost δ is enough to prevent an agent from becoming a leader.
The same holds true for social-pressure equilibria. Note that in Equation 4 the left hand side is also increasing in δ, and therefore for high enough δ, we have that m * increases. For δ large enough, Y stops being m * -regular, for the new value of m * . In that case, agents in Y do not contribute, because the threat of social pressure is not high enough as compared to the private cost c + δ. Note that, once again, the principal only needs the threat of the higher cost c + δ, in order to stop contagion before it even starts.

Conclusion
I presented a model of collective action where the agents are forward looking and have concerns for social sanctions. In the model, agents can only revise their actions stochastically. I defined an agent as predisposed if she is willing to contribute to collective action whenever she can revise her strategy, and analyzed a contagion process through the network, by which agents become predisposed by best response dynamics. In particular, I found conditions under which an agent with m predisposed friends becomes predisposed herself, and defined the minimal such m * as the contagion threshold. Contagion in the literature of social dynamics is usually a slow process, which requires several generations of agents best-responding; in contrast, contagion in the present model happens immediately, due to the forward-looking nature of agents.
In the second part of the paper I studied leadership: when does an agent choose to contribute even when nobody has contributed so far. One condition that suffices for contribution happening in any SPE is that the expected gain from contributions by those affected by the contagion is larger than the private cost of contributing. This is the case when a single agent can use her influence on her friends, who in turn use their influence on theirs, etc. to generate a wave of contributions in the network, that eventually compensates for the private cost of contributing in the first place. The forward-looking behavior is critical: it is because agents are able to foresee the contagion that they are capable of generating that they decide to lead by contributing. I also found less demanding conditions, under which at least some SPE exhibit contribution: in these cases, agents start contributing not because of desire of generating contagion, but out of a self-fulfilling fear of being socially sanctioned. This paper contributes to the literature in social dynamics by considering a population fully composed by forward-looking rational agents. Because of the intrinsic difficulty of such analysis, a number of restrictive assumptions had to be made. A promising avenue of future research consists on improving these assumptions and finding weaker conditions under which still have meaningful results. Several extensions could be developed for the baseline model. I have assumed homogeneity in the benefits, costs, and punishments. Introducing heterogeneity would generate new interesting predictions; in particular, analyzing how heterogeneity in the parameters interacts with homophily (the tendency of people to have friends like themselves), when both heterogeneity and homophily take place along the same dimension (i.e. we could think that agents with lower cost of contribution tend to be friends with each other). A related question is how dispersion in those variables, as measured by Second Order Stochastic Dominance, affects contagion and leadership. It is not clear ex-ante that more dispersion in (for example) cost would lead unambiguously to more contagion, or more leadership.
In conclusion, this paper offers a simple explanation for why individuals participate in collective action in the presence of social pressure. The conclusions on this paper can be incorporated to the literature on political revolutions as a justification of why (and how) it is incentive compatible, under certain circumstances, for individuals to participate in collective action, giving a more solid microfoundation to reduced-form models of revolutions. This is true more generally for economic models of public good of provision.
. Moreover, we have that where γ(m) is the probability that the game ends before any of m agents (in this case, k and the other m − 1) gets to play: and hence Therefore, ρ m (h) is given by The probability that either of m players gets to play and player k also gets to play is given by Finally, given z, the probability that player k gets to play before any of m players do, and before the game ends, is or equivalently: Taking into account that q(n) = n n−1+ 1 Q , we find Distribution of η. Define η(k) as the random variable of the number of players that get to play before the game ends, from k out of the n total players, given that 1 has already played.

Proofs of the results
PROOF OF PROPOSITION 1. Let β = β(1), i.e. the probability that a given agent plays before another does and before the game ends. Recall that we have the following four scenarios: • With probability βQ, j plays, hence contributing, and then k gets to play before the game ends. In that case k will contribute obtaining an extra payoff b − c + ξ.
• With probability β, k plays before j gets a chance to play, and hence the extra payoff is simply V .
• With probability β(1 − Q), j plays, hence contributing, and then k does not get a chance to play before the game ends: she incurs extra disutility −bψ.
• With probability γ(2), the game ends before either k or j get to play, in which case k receives an extra payoff of 0.
Note that V as defined here is an upper bound on the actual payoff of player k, as it does not take into account the fact that some of her friends who are not predisposed might become so.
Because of that, we are finding sufficient conditions for the public good to be provided. Because of the argument above, V is given by Hence, a sufficient condition for individual k to decide to contribute is Therefore, making ξ = 0, a sufficient condition for player j contributing is c b < 1 + βψ, and since β = ρ 1 (1), this concludes the proof.
PROOF OF PROPOSITION 2. The payoff of contributing for an agent k remains the same independently of the number of predisposed friends m, namely b − c + ξ. The payoff for not contributing is bounded above by V (m), given by considering the probabilities and payoffs of three different events.
• If any of the m predisposed friends of k plays (hence contributing), but k gets to play again before the game ends, she will contribute (because bψ > c), and hence she obtains a payoff bounded above by b − c + ξ. This happens with probability α(m).
• If k plays before any of her friends gets to play, then k is again in the same situation, and hence she obtains payoff V (m). This happens with probability β(m).
• If h out of the m predisposed friends of k play (hence contributing), and k does not get to play again before the game ends, she will suffer disutility −hbψ. This happens with probability ρ m (h).
Solving for V (m), we find Agent k will contribute if b − c + ξ > V (m). When ξ = 0, we find sufficient conditions to contribute, which is exactly Equation 4.
PROOF OF PROPOSITION 4. If i plays and nobody has contributed or would ever do so, then the payoff for not contributing for i is simply 0. If she contributes, her payoff is at least bη(k) − c, where k = |Π ∞ m * (N i )|. Hence, when this is larger than 0, Player i prefers to contribute, and that is the definition of a spearheaded equilibrium. The right hand side of Equation 5 is simply the expression for η(k), what concludes the proof.
PROOF OF PROPOSITION 6. Let σ be the strategy profile where agents in Y always contribute (and all j ∈ Y best respond to N \j). Consider an agent i ∈ Y . Because agents in Y are predisposed, and i ∈ N ∞ m * (N j ) for all j ∈ Y , that means that i has at least m * predisposed friends, and by definition of m * , it is a best response for i to contribute.