Chapter 2 Rational-choice Models of Collective Action: A Generalization and Critical Assessment 2.1 Introduction A systematic study of collective action informed by game the- ory must deal with the problem of multiple equilibria. Equi- librium is the polar star of game theory: the whole purpose of a game-theoretic model is to arrive at an equilibrium that can serve as its testable prediction. Yet, when a game has mul- tiple equilibria, many possible outcomes are compatible with individual strategic rationality and the theory does not lead to a unique prediction. Multiple equilibria are ubiquitous in collective action the- ory. This may seem an exaggeration given that a large and prominent share of models of collective action, those based on the public goods approach introduced by Olson (1965), have indeed only one equilibrium and, hence, a unique and deter- ministic prediction. Although models involving multiple equi- libria have been a lasting and influential presence since the development of focal point arguments and tipping games by 21 22 Chapter 2: Rational-choice Models of Collective Action Schelling (1960, 1978), it would seem that they cannot claim any privileged status within the theory of collective action. The subdiscipline seems split on this matter, with different scholars often using different models to capture the same situa- tion. Tullock (1971) considers that the public goods argument, typical of the Olsonian approach, captures the essence of all known revolutions, and Muller and Opp (1986) follow some of these insights in their analysis of Peru’s guerrilla movement. But Wood (2002) uses a Schellingean model to analyze the Salvadoran insurgency. Kuran (1991) argues that the collapse of the Eastern European regimes obeyed the logic of mul- tiple equilibria, in keeping with Schelling’s original insight, but Finkel, Muller and Opp (1989) analyze German protest movements using an Olsonian theoretical template. To take a closely related example, the standard formulation of the “Paradox of Voting” is couched in terms of the analysis of public goods, but other treatments (e.g., that of Schuessler (2000)) adopt the perspective of tipping games. This abun- dance of collective action models in the literature is eloquent testimony of the creativity of contemporary political scien- tists but is in some ways an unhealthy development. Without a unifying framework, we do not have a firm basis for deciding which model is correct. This chapter will develop such a unifying model, allowing us to appreciate the extent to which collective action phe- nomena are indeed interactions with multiple equilibria. The single-equilibrium simplicity of the public goods models is the result of too many restrictive assumptions. Although logically consistent, these models are critically fragile: tiny changes in the assumptions destroy their fundamental conclusion. To prove this, in this chapter I will specify a model of col- lective action that subsumes both the single-equilibrium (Ol- sonian) approaches and the multiple-equilibria (Schellingean) ones. Instead of arguing about the relative merits of, say, the Prisoners’ Dilemma versus tipping games in a vacuum where each has an air of plausibility, this model shows what assump- tions are critical in each case so that we can make better modeling choices. Later, once the book’s basic technique is A General Model of the Collective Action Problem 23 in place, we will be in possession of a fully general and fully systematic framework of collective action: deriving implica- tions about its comparative statics will be largely a matter of choosing in each case the relevant specific model. 2.2 A General Model of the Collective Action Problem Whatever else we want it to represent, a model of collective action should capture the essence of situations that (a) involve more than one, preferrably many, agents who (b) have to de- cide whether to take a costly action that, in turn, (c) increases the likelihood of a goal that is (d) desirable by all the agents. Remove any of these, and the situation at hand stops being a collective action problem. At a minimum, a game-theoretic model must describe the agents, what they can do (i.e., their strategies) and what hap- pens to them when they do what they do (i.e., their pay- offs). From these three ingredients we obtain the basic decision problems that go into describing an equilibrium. Therefore, a model of collective action must include: a. Several agents. In this case, I will use i (occasionally j) to denote an arbitrary agent, where i = 1,...,N. N is a natural number subject only to the restriction that N > 2. b. A strategy space for each agent, representing her two possible actions (Cooperate or Defect). I will use the notation Ai = {Ci,Di} where Ci stands for “Player i cooperates” and Di for “Player i defects.” c. A construct to represent the likelihood of success and the fact that it increases as the number of agents who cooperate increase. d. A representation of the agents’ preferences. 24 Chapter 2: Rational-choice Models of Collective Action Representing the Likelihood of Success. Denote the share of agents who decide to cooperate by 0 < γ < 1, success of collective action by Succ and failure by Fail. The generic probability of an event E will be denoted as Pr(E) and the probability of event E conditional on event H as Pr(E|H). The function F will represent the probability of success, for every value of γ: Pr(Succ) = F (γ). I will assume that, all else being equal, higher levels of participation increase the likeli- hood of success so that F is increasing in γ. It also makes sense to assume that F (0) = 0, that is, that collective action cannot succeed without at least some agent cooperating. Often, the calculations will be easier, without any conceptual loss, if we assume that collective action is assured success if every agent participates, something that we can represent with F (1) = 1. Later I will consider what happens as we introduce further assumptions. Representing the Agents’ Preferences. Following the standard procedure of collective action models, I will assume that each individual agent cares only about the final outcome and whether she cooperated. To get an idea of the possibili- ties that these assumptions exclude, notice that in this model agents do not care about how the outcome is obtained, the ex- act identity of other cooperators or the payoffs of other players. There might be good reasons to introduce all these other fac- tors in a more general treatment, but I shall not do so here, mostly for two reasons. First, I want my discussion to stick to the standards of most models of collective action. Second, I believe that much can be accomplished within that stan- dard framework: the main ideas I want to get across need not depart from it. The agents’ payoffs will depend, then, only on the strategy they choose and the final outcome. Since there are two pos- sible strategies and two possible outcomes, this gives us four possible payoffs which may differ across agents: • If a player cooperates and collective action succeeds, her payoff is u(Ci, Succ) = w1i. A General Model of the Collective Action Problem 25 • If a player defects and collective action succeeds, her payoff is u(Di, Succ) = w2i. • If a player cooperates and collective action fails, her pay- off is u(Ci, Fail) = w3i. • If a player defects and collective action fails, her payoff is u(Di, Fail) = w4i. By assumption, participating in collective action is costly but, when successful, results in an outcome that is desirable for all the agents involved. So, the payoffs must satisfy the condition that, for every agent i: w2i > w4i > w3i and w1i > w4i > w3i. From now on, this general structure, the agents, their strategies, their payoff functions and the likelihood of success will be called Model 0. All the models that follow are special cases of this general setting. The central question of rational-choice models of collective action is under what conditions an individual will find it in her best self-interest to cooperate. In the language of game theory, this amounts to asking when the expected payoff from choosing Ci is larger than the expected payoff from choosing Di. The function vi(σi, σ−i) denotes the payoff for individual i from choosing an arbitrary strategy σi provided that the rest of the players are choosing σ−i. Since a player’s payoffs depend only on her strategy and the outcome, all that matters about σ−i is the aggregate level of turnout associated with it, γ(σ−i). So, unless it leads to confusion, I will drop the term σ−i and the player’s subindex, and denote the expected payoffs with the shorthand notation v(σi, γ) (especially in the nontechnical sections). With this notation, the decision problem of any given agent becomes: v(Ci, γ) = w1i Pr(Succ|Ci, γ) + w3i Pr(Fail|Ci, γ) (2.1) = w1i Pr(Succ|Ci, γ) + w3i(1 − Pr(Succ|Ci, γ)) = (w1i − w3i) Pr(Succ|Ci, γ) + w3i = (w1i − w3i)F (γ + 1/N) + w3i. (2.2) 26 Chapter 2: Rational-choice Models of Collective Action v(Di, γ) = w2i Pr(Succ|Di, γ) + w4i Pr(Fail|Di, γ) (2.3) = w2i Pr(Succ|Di, γ) + w4i(1 − Pr(Succ|Di, γ)) = (w2i − w4i) Pr(Succ|Di, γ) + w4i = (w2i − w4i)F (γ) + w4i. (2.4) Then, the agent will cooperate if: v(Ci, γ) ≥ v(Di, γ), (w1i − w3i)F (γ + 1/N) − (w2i − w4i)F (γ) ≥ w4i − w3i. (2.5) Equations 2.1 - 2.4 characterize the decision problem and Inequality 2.5 shows us the conditions under which its solution will be to cooperate.