Most analyses of collective action agree that overcoming the freerider problem requires organizing potential contributors, thus making their decisions interdependent. The potential for organizing depends on the social ties in the group, particularly on the overall density or frequency of ties, on the extent to which they are centralized in a few individuals, and on the costs of communicating and coordinating actions through these ties. Mathematical analysis and computer simulations extend a formal microsocial theory of interdependent collective action to treat social networks and organization costs. As expected, the overall density of social ties in a group improves its prospects for collective action. More significant, because less expected, are the findings that show that the centralization of network ties always has a positive effect on collective action and that the negative effect of costs on collective action declines as the group's resource or interest heterogeneity increases. These nonobvious results are due to the powerful effects of selectivity, the organizer's ability to concentrate organizing efforts on those individuals whose potential contributions are the largest. Social ties are important for collective action. For example, it is widely agreed that participants in social movement organizations are usually recruited through preexisting social ties and that mobilization is more likely when the members of the beneficiary population are linked by social ties than when they are not (e.g., Tilly 1978; OberschalI1973). But exactly how and why social ties are important is less well established. What kinds of ties are most important for collective action? What features 1 This material is based on work supported by the National Science Foundation under grant SES-8408131. We wish to thank Charles Halaby and Robert Mare for advice on the statistical analysis and David Weakliem for supplemental research assistance. Requests for reprints should be sent to Gerald Marwell, Department of Sociology, University of Wisconsin, Madison, Wisconsin 53706. © 1988 by The University of Chicago. All rights reserved. 0002-9602189/9403-0002$01.50  AJS Volume 94 Number 3 (November 1988): 502-34 Collective Action of social ties are especially relevant? Are social ties important because they make mobilization less costly, or because they connect more people? And how can we distinguish these explanations? This paper develops some analytic theory to begin to answer these questions by extending the "theory of the critical mass," our microsocial model of the process by which collective action emerges in groups. (See Oliver, Marwell, and Teixeira [1985] for an initial explication, and Oliver and Marwell [1988] for an application to the problem of "group size" in collective action.) We extend this theory here to include the overall frequency or density of social ties in a group, the extent to which the ties are centralized in a small number of group members, and the cost of using a tie for communication or coordination. We consider only the direct ties between potential organizers and potential participants, not the indirect ties that have often been investigated by social network theorists (Alba 1981; Arabie 1977; Burt 1978 Burt , 1980 Doreian 1974; Freeman 1979; Marsden 1981 Marsden , 1983 White, Boorman, and Breiger 1976) . We also do not distinguish the "strength" ofthe tie (Granovetter 1973) . In our conclusion, we discuss possible relations between our findings and other features of social networks. Like most formal models, ours requires some rather extreme simplifications. Even with these simplifications, the models are often too complex for analytic solutions, so we supplement our analysis with computer simulations, including Monte Carlo methods. Although we create the simulations, their outcomes often surprise us and force us to develop new theory to explain them. In fact, the most important finding of this research was wholly unanticipated. This is the finding that network centralization is beneficial for collective action, a finding that we show arises from the ability of organizers to be selective in choosing the targets of their organizing efforts. 2 Of course, our goal is not just to understand our artificial simulation, but to generate theory that may apply to natural groups and actions. Therefore, throughout the discussion we attempt to be clear about the scope conditions or limits of generality of our results, and in the conclusion we discuss their substantive applications. It will be helpful to ground our often technical discussion in an example. Many cities have passed ordinances requiring their employees to be city residents. Consider a number of cities that have just passed such 2 Although one reviewer paid us the compliment of believing that we knew about selectivity when we planned our simulations, we actually did not. We developed the concept of selectivity from struggling with simulation results that we did not understand. We believe there is value in showing readers some of the steps of that process, especially since many sociologists believe that simulations are incapable of producing any "new" knowledge that is not apparent in the initial assumptions. American Journal of Sociology an ordinance, effective next year. None now exempts current employees from the requirement, but an amendment providing such an exemption is possible if a critical mass of employees can be assembled to mount a visible campaign. Assume that each city happens to have 400 employees who live outside its limits and that the amount of effort required to gain passage of the amendment is the same in all cities. In the formal language of collective action theory, the 400 affected employees in each city are an interest group. The amendment is a collective good: all affected employees will benefit from it, regardless of whether they help to get it passed (Olson 1965) . 3 Most formal collective action theory assumes that individuals make isolated, independent decisions whether to contribute to collective action and predicts that individuals will usually try to "free-ride," that is, to reap the benefits of a collective good provided by the contributions of other group members while paying none of the price. Thus, each suburb-dwelling city employee hopes her similarly affected colleagues will spend their time picketing and lobbying for the amendment, while she goes bowling. However, since everyone has the same hope, no one pickets (and the bowling alleys are full). Of course, the assumption that individuals act in isolation is usually wrong. In most decision situations, people are at least generally aware of what others are doing, and often they have social relations that make influence, or even sanctions, possible. Empirically correct decision equations must take account of these interdependencies, and, when interdependency is considered, individuals can find it "rational" to participate in collective action (see Oliver, Marwell, and Teixeira 1985; Hardin 1982). 3 Although the sociological literature on formal collective action theory is relatively small (see Marwell and Oliver 1984; Marwell and Ames 1979; Oliver 1980; Oliver and Marwell1988; Oberschall1973, 1980 Fireman and Gamson 1979; Smith 1976) , the literatures in economics, political science, and psychology are much too large to review adequately here. The "public goods problem" is long-standing in economics, although Samuelson (1954) and ~ator (1958) are usually seen as the classic statements; Head (1974) provides a clear treatment of the fundamental economic issues, Schelling (1973) ties public goods to the more general issue of externalities, and Brubaker (1982) examines some empirical evidence in experimental economics. Olson (1965) is important for having exported the economic ideas to other social sciences but does not provide a definitive formal statement of the problem. Political scientists who have refined these ideas and applied them to collective action dilemmas include Chamberlin (1974), Frohlich and Oppenheimer (1970) and Frohlich et al. (1975 ), Hardin (1971 ), Schofield (1975 , and Van de Kragt, Orbell, and Dawes (1983) . Perhaps the most thorough and wide-ranging treatment of the issues of collective action is Hardin (1982). Psychologists generally explore experimentally the dynamics of social dilemmas in relatively small groups (Bonacich et al. 1976; Brewer 1985; Dawes 1980; Dawes, McTavish, and Shaklee 1977; Dawes and Orbell 1982; Kelley and Grzelak 1972; Messick and McClelland 1983; ). Collective Action Social networks matter when decisions are interdependent. But just how do they influence the prospects for collective action? What characteristics of their networks determine the ability of the affected employees to mobilize in support of the amendment? We here investigate three prominent features of social networks of interest groups: their density, their centralization (or concentration), and the cost of communicating across a tie. Most discussions of social networks and collective action consider rather stark contrasts, like that between the cities Alpha and Beta. Alpha is fairly isolated, and most of its affected employees live in a single suburb, Centauri. They attend Centauri's churches; their children attend Centauri's public schools; they belong to Centauri chapters of social and service clubs; they all are served by the same local telephone exchange. In contrast, Beta is part of an ethnically diverse two-state megalopolis, and its affected employees are scattered across a dozen different suburbs in two states and four counties. Thus, they rarely see one another after work. They go to different churches, send their children to different schools, read four or five different newspapers, and pay toll charges for telephone calls between many of the suburbs. We would find virtual unanimity among sociologists in predicting that the employees in Alpha are more likely than those in Beta to act collectively. Other things being equal, the large number of social interconnections in Alpha provides a solid basis for interdependence, while their virtual absence in Beta is surely a hindrance. It is practically a truism among social movements theorists that social networks are important for recruiting participants (see, e.g., Oberschall 1973; Tilly 1978; and Fireman and Gamson 1979) , and these theorists are supported by a fair amount of empirical evidence (see Snow, Zurcher, and Eckland-Olson [1980] and Walsh and Warland [1983] on participation in movement organizations, or Cohn [1985] on participation in a clerical workers' strike). The basic argument was important even to Marx, who reasoned that employees who are in regular contact with one another will develop a "habit of cooperation" and be more likely to act collectively than employees who work in isolation. Such generalizations, which are almost certainly true and of great practical importance, concern the overall number, or density, of social ties 4 in the group. 4 For network specialists, the term "density" often refers to the extent to which the people known by one person tend to know each other. We are using the term in the nonspecialist's sense of the number of actual social ties in a group divided by the number possible. This seems the only reasonable term for such a concept at the group 505 American Journal of Sociology Centralization But there is more to know about an interest group's social ties than simply how many there are. We also need to know how they are structuredmore specifically, the extent to whi~h they are centralized or concentrated in a few individuals rather than being spread more evenly across the whole group. Network centralization is the key independent variable for this analysis because of its complex and so far unexplored relationship to collective action. In contrast, although density is obviously important, its effects are generally just that-obvious. 5 To illustrate the effects of centralization, we may consider the cities Delta and Epsilon. The affected employees of Delta have the same total number, or density, of social ties as the employees of Epsilon. However, these ties are distributed quite differently. Delta has annexed all its potential suburbs. Its affected employees are therefore not suburbanites but residents of Delta's "exurbs." They are lightly sprinkled around a sparsely settled rural fringe. They shop in the city's many shopping centers and attend city churches. In fact, they share only one important local institution-the county primary and secondary schools, which serve the entire rural area. The one affected city employee who is on the county school board therefore accounts for a very high proportion (let us say 50%) of all the ties among affected employees. Social ties among affected employees in Epsilon are much less centralized. Epsilon is the central city of a small metropolis, and affected employees live in a number of contiguous suburbs with diffuse and indistinct social and economic boundaries. There are a few dozen churches that serve larger or smaller catchments, depending on the denomination. Service clubs have loose geographical bases and draw in residents of adjacent suburbs. There are eight municipalities, four school districts, and three suburban newspapers with distinct but partially overlapping territories. Different employees see each other in different contexts. To make a prediction about the prospects for Delta and Epsilon, we have to assess the effect of the different levels of centralization of network level. When group size is fixed, as it is in our example and simulations, there is an equivalence among the total number of ties in a group, the density (total ties divided by number of ties possible), and the average number of ties per member (total ties divided by number of members). Thus, we often use these three terms somewhat interchangeably, although, of course, when group size is not held constant, they are not equivalent. S Again, because we consider only direct asymmetric ties, our use of the term "centralization" is different from the more complex usage of network theorists, e.g., Freeman (1979) . We are referring simply to the tendency for a few people to account for most of the ties, while the network concept of centrality refers to the extent to which any particular person is an important link in the indirect network relations of others. Collective Action ties. We have to decide whether collective actions are more likely to thrive where a small number of people know many others (while most know almost no one), or where many people each know some others, but no one knows a large number. The centralization in the school board in Delta suggests the possibility of efficient coordination of action by the affected employee, who knows everyone else. But it also means that if the school board member is incompetent or is just not interested in spearheading the collective action, no alternative affected employee has much of a chance to get things going. On the other hand, the crisscrossing social ties and overlapping memberships in Epsilon evoke images of more diffuse potential loci for action. If one person does not show initiative, perhaps another one will. The problem in Epsilon is that no one person is in a social network position to gather together efficiently some major proportion of all the affected workers. There are advantages and disadvantages in either situation, and, in the absence of good theory, it is not obvious how these will balance out. Our third independent variable is organizing cost, that is, the cost of using one of the social ties for some purpose, such as asking the person to participate in a collective action. This variable is often confounded with network density in work on social ties. To some extent, we committed this confounding in our comparison of Alpha and Beta. Factors that make it cheaper to communicate across social ties, such as free local versus costly message-unit telephone rates, may also help determine whether such ties exist at all. Empirically, the two are often correlated. However, a theoretical distinction between the presence or absence of ties and the cost of using them is needed because each factor has a different effect on the prospects for collective action. To illustrate this issue, we need a three-way comparison, such as among cities Mu, Sigma, and Rho. Suppose Mu's affected employees average 10 ties to other affected employees and generally have to pay message-unit charges for calls to one another. Sigma's employees also average 10 ties, but all the suburban areas are in the same extended calling area. Finally, Rho's employees average 15 ties but have the same message-unit charges as Mu's. Clearly, we would expect the prospects for collective action to be worse in Mu than in either Sigma or Rho. But what prediction do we have about the prospects for success in Sigma compared with those in Rho? Another way of asking this is, If one were in Mu, would there be a bigger effect from increasing the number of employees who know each other or from devising a cheaper mode of communication among those who already have social ties? A real group of employees 507 American Journal of Sociology might have to decide if its efforts to generate collective action would be helped more by social gatherings and other efforts to increase the group's density of social ties or by finding ways to reduce the cost of communication among people who already know each other, perhaps by printing a directory of home addresses and telephone numbers. This is really a question about comparative rates of change, and it does not have a readily apparent answer. We expect the effects of the network variables to interact with the degree of resource and interest heterogeneity in the interest group. The fact that a good is collective or shared does not mean that all those individuals who are interested in it are the same. For example, although all present employees who live outside the city share an interest in an exemption, they differ in the magnitude of that interest, depending on such factors as how easy it would be for them to get other jobs, whether they own or rent their homes, and whether their spouses are employed. They may also vary in the amount of time and money (i.e., resources) objectively available to them. We have previously shown that groups that are quite heterogeneous differ in their patterns of collective action from groups that are more homogeneous (Oliver, Marwell, and Teixeira 1985) . The direct effects of group heterogeneity on collective action can be briefly summarized. When groups are homogeneous, everyone is interchangeable, and the collective action outcome is a simple function of how many people participate. In contrast, in a heterogeneous group, it matters who is organized as well as how many since one person may be willing and able to contribute much more than another. To understand the pure (ceteris paribus) heterogeneity effect, we may imagine that a group's resource and interest heterogeneity (i.e., standard deviation) could be varied while the mean is held constant. Changes in heterogeneity would have two effects, which we may call the "mean level" effect and the "conjunction of probabilities" effect. The mean level effect refers to the relation between the mean of a distribution of resources or interest and the level necessary for success. Briefly, if a distribution's mean is sufficiently high, more heterogeneity hinders collective action; if the mean is too low, heterogeneity helps.6 For example, if the mean amount of employees' free time is so low that no group members can spend three hours at a protest meeting, increasing the heterogeneity of free time could produce a few who could attend. They would be produced 6 For a variable for which high values are bad for collective action, heterogeneity is beneficial when the mean is high and harmful when the mean is low. Collective Action statistically by reducing further others' free time, but this will not affect collective action because those people could not participate in any case. Conversely, when the mean member has the three hours free already, increasing the heterogeneity reduces the proportion of the group that has three or more hours free by increasing the proportion below three. 7 The second effect, the "conjunction of probabilities," arises from the fact that the joint probability of two or more independent events must be lower than the marginal probability of anyone of them. Thus, increasing the heterogeneity of two or more variables at the same time tends to have a negative effect on the outcome. For example, the risk goes up that the employees who have the most time available (high resources) will not be the same ones who are most concerned about losing their jobs (high interest).8 Thus, the "mean level" principle (which produces a positive heterogeneity effect) and the "conjunction of probabilities" effect jointly create an optimization problem in which some heterogeneity generally improves outcomes but too much heterogeneity usually worsens them. Any formal analysis requires a series of definitions, simplifications, and assumptions that define its theoretical model of the complex empirical world. In this section, we describe the elements of our model. Networks are irrelevant to isolated independent actors. They affect collective action only when there is some "social resolution" to the collective action dilemma. A variety of assumptions modeling this "social resolution" are available, but the specific form of this assumption at the individ-7 This "mean level" principle requires a caveat for skewed distributions. In a skewed distribution with a lower bound of zero (like the lognormal), increases in heterogeneity come from a few people with very high values balanced by many people with values close to zero. If the mean is too low to support collective action, increases in heterogeneity first increase the proportion with a high enough value to act but then decrease that proportion as the distribution comes to consist of fewer and fewer individuals whose interest or resource level gets progressively higher, much higher than necessary, balanced by the vast majority whose interest is much too low. (This does not arise with symmetric distributions, for which increasing the heterogeneity is always beneficial if the mean is too low.) At the extreme, high interest heterogeneity would come from one person who would lose everything balanced by everyone else who would lose nothing. Unless this one person is also extraordinarily resourceful, there cannot be a critical mass for collective action. The terms of an all-or-none contract are that all parties to the contract contribute their agreed amounts, or no one does, and there is some mechanism for ensuring that the terms of the contract are carried out. Individuals are asked whether they will agree to bind themselves to the contract, with the proviso that they will actually make their contribution only if the sum of all contributions will exceed a specified level. We assume that, in this situation, individuals weigh their own costs against their personal payoffs from the total change in the collective good produced by all the contracted contributions. Individuals may therefore find it profitable to bind themselves to such a contract even when they would not make independent contributions (Oliver, Marwell, and Teixeira 1985) .9 Individuals' contributions to these contracts can be negotiated, and different parties to the contract can contribute different amounts. Formally negotiated all-or-none contracts are rather common resolutions to some problems, such as providing office microwaves. The model of the formal contract is a reasonable approximation of more informal processes relying on convention (Hardin 1982, pp. 155-230), normative pressure, or solidarity (Fireman and Gamson 1979). As common as formal or informal contracts are, they rarely happen spontaneously. They entail communication and enforcement costs that are never uniformly distributed across parties to the contract. Empirically, these kinds of contracts generally have well-defined points of origin, that is, some specific person (or small cadre) who incurs the organizing costs. In short, they have organizers. Organizers, just like everyone else, have available resources and an interest in the collective good, but they use their resources to organize a contract that will make others willing to contribute to the collective good. 9 We assume that people will contribute if their contributions will produce a profiti.e., if the total benefit exceeds their own costs-and that they ignore the possibility that someone else will provide the good. If actors do attend to the possibility that someone else would also find it profitable to contribute, we must have information about their subjective probability estimates about others' actions and employ a more complex model, which is beyond the scope of this paper. This gaming dilemma is acute when independent decisions are modeled, but it is less acute when social resolutions are modeled, for it is plausible to assume in such circumstances that all the highprobability actors are parties to the social resolution. Collective Action We assume that each person in the group has an asymmetric ego network (Burt 1980, p. 91) or "organizer network" of other group members he or she can organize. In this analysis, we ignore indirect links and consider only a direct asymmetric "send" link: A is linked to B if it is physically and socially possible for A to ask B to participate in a given collective action. We assume that organizers can recruit only those individuals with whom they have some prior social connection. It may be easy for a policeman to recruit other members of the force to support the amendment but socially impossible for him even to ask secretaries from the mayor's office to join the pickets. We assume asymmetry because we believe that stratification and other social realities often prevent a person from approaching someone who could approach him. 10 A police lieutenant might recruit patrolmen, who would be wary of initiating discussions with him. We treat network ties as present or absent without considering the intensity or nature of the connection; we treat the cost of communicating across a tie as a separate variable from the existence of the tie itself. These assumptions greatly simplify our treatment of networks. Let t; be the number of ties (i.e., the "out-degree") for individual i, and let T be the total number of ties in the group. If the total interest group size is N, the number of possible ties is simply N(N -1), and the overall interest group density, D, is TIN (N -1) ; the density, d;, of each individual's ego network is t;/(N -1). In general, therefore, the mean of the individual network densities equals the overall group density. Given a specific group density, the centralization of an interest group's social network may be defined as the standard deviation, at, of the number of ties across the ego networks of all group members. When all the group's ties are concentrated on one individual (and everyone else has none), the group's network is fully centralized, and at is very large. When all group members have the same number of ties, the network is completely decentralized, and at equals zero. The standard deviation of the ego-network densities, ad, equals aJ(N -1) and may be interpreted similarly. When N is treated as fixed, as it is in this article (where N = 400 for all groups), there is a direct equivalence between the standard deviation of ego-network densities and the standard deviation of numbers of ties since the former is just the latter divided by N -1. 10 These asymmetric ties may be mutual. The link of A to B does not preclude B's link to A; it just does not entail it. In the simulation, the two links (A to B and B to A) would have independent probabilities of occurrence. Our analysis assumes that the mutuality of ties has no effect, which is true in our simulation. American Journal of Sociology Organizing Costs Organizers mayor may not be able to communicate with all the group members they know, depending on communication costs. Face-to-face contacts take a certain amount of time per contact. Letters and telephone calls usually require less time per contact but more money. Local telephone calls cost less than long distance. Mass media (television, radio, newspapers) or diffuse media (posters, leaflets) have certain costs, which include the "waste" of communicating with people who are not potential contributors. It is much less expensive for employees to contact each other if they all work in the same building than if they are dispersed allover town, and less expensive if they all live in the same suburb than if they are spread thinly across the countryside. Interdepartmental mail is much cheaper to use than the U.S. mail. The surveillance or enforcement mechanisms for the contract may also have some material or social cost. The effects of different kinds of organizing costs are complex enough to require separate analysis. In the present work, we make the very simple assumption that organizing costs are exactly proportional to the number of parties to the all-or-none agreement and that this constant cost per participant is a characteristic of the interest group that does not vary with the organizer or the particular individuals involved. Thus, average costs may be higher in one city than another because of telephone cost differences, but Fred Jones and Ann Smith, who are both employed by the same city, are assumed to face identical costs per network member recruited. We assume that individuals' decisions about contributing to the all-ornone contract are determined by weighing the costs and benefits of such a contribution. We begin with a group of individuals defined by the fact that they will all be positively affected by some collective good, although they differ in their resources (r) and in the level of their interest in the good (I). Each person in that group is both a potential organizer and a potential contributor to a contract organized by someone else. The "production function" p(r) specifies the relation between inputs of resource contributions (r) and outputs of levels of the collective good (P). For dichotomous goods, such as whether the exemption amendment passes or not, p represents the probability of obtaining the good. times the probability of obtaining the good exceeds the size of her contribution. An algebraically equivalent expression for this condition is p(r) > r/ I. Although independent contributions to collective goods are rare, they are not impossible. Situations may exist in which there is at least one "large" potential contributor whose interest is sufficient to make him willing to provide the collective good himself. This occurs because collective goods often have high jointness of supply, which means that the cost of providing the good does not rise with the number who enjoy it (see Hardin 1982; Oliver and Marwell 1988) . If individuals are willing to contribute without organizing, there is no need for an organizer and no need for anyone to bear the organizing costs. (These large contributors might find it cheaper to organize others' interdependent contributions than to make their own independent contributions, but such considerations of alternative strategies are beyond the compass of this paper.) Thus, one scope condition for our theory of organizing is r/ I > p(r), that is, that individuals will not find it rational to make independent contributions. The individual decision rule for contributing to an interdependent allor-none contract makes one crucial change in the previous equation  A group member's decision to organize is more complex than the decision to contribute. Organizing will be profitable if the p(Ir) produced by the contributions of the other group members exceeds the organizers r/I. A group member's interest (I) is the same regardless of whether she is organizing or contributing. To find the cost of organizing, we make the simplifying assumption that organizing costs are linear with the number of contributors. If C is the cost per contributor and k is the number of contributors, then the organizers cost, or contribution, equals kC. If we let Irk represent the total contributions of the k individuals the organizer contacts, then the organizer's gain from organizing is Go = P(Irk)I -kC. The payoff will be greater than zero, and organizing will be profitable, if the potential organizer knows and can afford to contact a group of k individuals, such that P(Irk) > kC/I. The Process of Forming a Contract We simplify our model greatly by assuming that every group member has perfect information about everyone's interest and resource levels and that everyone's behavior is a perfectly determinate function of his own interests and resources. This means that when social worker Fred Jones, organizer, approaches patrolwoman Ann Smith, group member, with the proposition, "Will you agree to spend five hours fighting for an exemption if you are guaranteed that a total of fifty hours will be contributed to this project?" Fred knows the answer before he asks. He can mentally review a list of his acquaintances and determine whether he knows and can afford to contact a critical mass of people who are interested and resourceful enough that their combined contributions would produce a big enough impact to make everyone willing to contribute. He spends his resources to pay the organizing costs only if he is sure he will be successful. In this simplified world, there are no unsuccessful organizing drives. Group Outcomes and Multiple Organizers Two (or more) organizing campaigns going at the same time could reinforce each other, but they could also hurt each other if they compete for the same contributors. The complexities of multiple mobilizations are beyond the scope of this paper. We hope to address this issue in future analyses. Here, we will only identify the best contract possible in a given group. This implies that mobilization is preemptive, that is, that one organizer's efforts preclude another's. This is often a realistic assumption. Organizing has a decelerating production function (Oliver, Marwell, and Teixeira 1985) , and many people will not organize if they know someone else is (see Oliver 1984) . Preemption is especially common when the interest group already has some "official" representative (such as a city employees' union). In real life, the preemptive organizer is not necessarily the one who would organize the best possible contract. The probability that an interest group will contain at least one successful organizer is much higher than the probability that any particular individual can be a successful organizer. When preemption is assumed, the probability that at least one of N potential organizers will be successful is given by 1 -(1 -q)N, where q is the probability that any particular organizer will be successful. Very small probabilities of success for individual organizers produce very large probabilities of at least one success in a group. For example, in a 400-person group, if the probability of any individual's being a successful organizer is .01, the probability of there being at least one successful organizer is over. 98. For there to be a better than even chance of the group's having at least one success (i.e., the group 514 Collective Action probability is greater than .5), the individual probability can be as low as .002. Some of the effects on collective action of network density and centrality, organizing costs, and the heterogeneity of resources and interests may be shown with formal analysis. Their interactions, however, are more opaque. We need some kind of data for testing our ideas about these more complex relationships. Since no real data are available with the kind of information we need, this section presents an analysis of artificial data constructed using cQmputer simulation techniques. In the next section, we use the results of this analysis as a point of departure for a more general theoretical discussion. Our analysis builds from the simple to the complex. We first consider groups that differ only in their organizing costs. Next, we vary the density of network ties, and then their centralization. The results to that point are essentially analytic, although we create a data set to compare the relations among these variables with those from the simulation. The last step involves varying the dispersion of interest and resources around their means. This problem does not permit analytic solutions and is investigated with data produced by a rather complex simulation of the decision-making processes involved. We hold some factors constant. All groups have 400 members. The collective good is dichotomous: the exemption either passes or fails, with contributions of resources affecting the probability that the good will be provided. Individuals have resources that may be spent either for organizing or for making a contribution to a contract organized by another. Organizing costs are paid wholly by the organizer, are proportional to the number of group members who contribute to a contract (regardless of the sizes of their contributions), and do not vary among individuals in a group, although they vary between groups. As a further simplification, we do not permit organizers to make direct contributions to the collective good even if they have resources left over after organizing. A standard metric is created by setting the total resource contribution necessary to provide the good with certainty equal to one; that is, P(l) = 1. Organizing costs are assumed to have the same metric. An individual's interest level, I, is expressed in the same metric as the payoff he would experience if the good is provided with certainty, so that the 515 American Journal of Sociology individual's benefit from a particular provision level equals lp (!'r). If the production function is linear, the standardization means that the total amount of resources contributed equals the provision level; that is, p(Ir) = Ir. Rather than being linear, the production function we use in this analysis is slightly accelerating: p(Ir) = t1.2; 0 < t < 1. We have argued elsewhere that contractual resolutions to the collective dilemma are especially important when the production function is accelerating, that is, has increasing marginal returns (Oliver, Marwell, and Teixeira 1985) . In this case, individuals are positively interdependent, so the value of any individual contribution is greater, the more others contribute, and there are initial periods of low returns (i. e., start-up costs) that are difficult to overcome without interdependence. The particular choice of function is arbitrary. The total resource and interest levels of the group as a whole have an enormous influence on the prospects for collective action. Because we are interested in other independent variables, we must constrain these factors to ranges that neither guarantee nor prohibit contributions. We set the mean of the distribution at .06 for resources and .15 for interest. The interest mean is substantially higher than the resource mean so that many group members will meet the criterion p(Ir) > rll, which permits contracts that produce probabilities between zero and one. The group outcome is the best possible outcome across all possible organizers in the group. Each group member is a possible organizer, and we calculate the total contribution from his best possible contract, given the specific others to whom he has social ties. We then select the one organizer in the group whose contract commands the highest total contribution. His contract becomes the group's contract, and the group payoff becomes this total contribution raised to the 1.2 power. Since payoff and contribution are perfectly correlated, we use the latter in our regression analyses. In other analyses, we transform this continuous contribution! payoff variable in three ways: into the dichotomy between achieving certainty (contributions total 1.0 or more) and not; into the dichotomy between failing completely (contributions total zero) and not; and into the trichotomy among total failure, partial success, and achieving certainty. For each independent variable, we define a range of values of interest. The actual values assigned to each case on each variable are system-516 Collective Action atically generated or randomly selected from the uniform distribution defined by that range. These uniform distributions do not mean anything substantive; they are continuous analogs of the equal cell Ns in a standard fixed-effects analysis of variance design. Individuals' personal network sizes (or out-degrees), resources, and interests are distributed lognormally in each group (Aitchison and Brown 1963) . It is most plausible to assume these distributions are skewed, both because they are constrained to be nonnegative and because one would generally expect to find a large number of group members low and only a few high on each variable. In most groups, there are many more people who are very poor than are very rich. The choice of the lognormal, rather than some other skewed distribution, is largely practical. Interest and resource heterogeneity is operationalized as the standard deviations of lognormal distributions. Among groups, these range from .01 to .06 around the resource mean of .06, and range from .025 to .15 around the interest mean of .15. Organizing cost is defined as the cost per each member of a contract. It varies among groups from .001 to .0lD in increments of .001. Organizing is never costless, but its cost ranges from insignificant to prohibitive, given organizers' resource levels. Each of a group's 400 members faces the same organizing cost. The density and centralization of group networks are ope rationalized in a nonstandard way because these numbers must be integers. We operationalize a group network's density as the mode of a lognormal distribution of an individual's personal network sizes in the group, and the group's centralization or dispersion as the difference between the mode and the "largest possible" personal network, where the "largest possible" personal network is the one above which less than .0005 of a lognormal distribution with the given mode falls. In these data, the correlation between the mean and the mode of the distribution of personal network sizes is .99, while the correlation between the standard deviation and the measure of dispersion we used is .96. The range among groups for modal personal network size was set as low as possible, ranging from 1 to lD. Dispersion was chosen to range from 2 to 20, a range that seemed on its face to provide extensive variation among groups in the degree of centralization of network ties. The "largest possible" group size is the sum of mode and dispersion and therefore ranges from 3 to 30 with a flat bell shape. The range of modes is extremely unfavorable for successful contracts, but the range of "largest possible" sizes includes a good proportion that can achieve certainty. We used a computer algorithm for actual assignment of network sizes to individuals in the groups to produce the frequency distribution of integers that is the best approximation to the lognormal distribution 517
With regard to collective action research, we draw upon Marwell and Oliver's designation of collective action as actions surrounding "mutual interests and the possibility of benefits from coordinated action" REF .
Social Networks and Collective Action: A Theory of the Critical Mass. III
American Journal of Sociology
Sociology