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Throughout much of mankind's experience with elections, vote brokers – local elites who direct the voting decisions of a subset of the electorate – have been able to make or break political careers. In various polities, brokers have thrived in spite of the secret ballot, a surprising outcome given that vote secrecy would ostensibly allow citizens to pocket the inducements offered by such individuals and vote their consciences anyway. To address this puzzle, we develop a framework for understanding the persistence and demise of vote brokerage under the secret ballot. In our model, a broker contracts with voters using an outcome contingent contract: some fixed benefit is promised to all voters sharing one of several observable profiles should the broker's candidate win the election. Using this framework, we demonstrate that the existence of brokerage depends on the size of the electorate contained within the jurisdiction controlled by the broker, with large jurisdiction sizes tending to drive brokerage out of existence. Moreover, we detail the manner in which the strategies employed by brokers depend on their economic power, the size of social groups, and ideological polarization. Empirical evidence from Minas Gerais, Brazil is used to evaluate the performance of the model.

The theory of collective action has long since moved beyond the "free-rider problem" as originally stated by Olson (1965). It is now recognized that large-scale participation is possible even without selective incentives, but this has raised new questions. I discuss two types of contributions from the past few years. Some of the recent literature has remained close to the canonical game-theoretic framework, clarifying many analytical issues and proving important results in participation games that had remained elusive. We now know that the "turnout paradox" had been grossly overstated. Another set of contributions has expanded the original framework, incorporating results from psychology and social sciences with the aim of obtaining more realistic models with better empirical performance.

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The prevailing approaches to collective action in the rational choice paradigm often lead to implausible conclusions and tend to lack predictive power. This article introduces a method to overcome these difficulties. The method is based on the notion of stability sets of pure-strategy equilibria, already familiar from the literature on equilibrium selection and with close counterparts in evolutionary game theory. With the help of some simple examples, the article shows how this method can turn many intuitive insights into operational, testable hypotheses about phenomena of collective action.

This paper challenges the notion that voting games with purely instrumental players cannot account for high turnout (the ‘turnout paradox’). Although it has been known for over 25 years that such games can generate high-turnout equilibria, the said equilibria have been rejected on the grounds that they are fragile. This paper shows that this claim is incorrect because it is based on a computation of pivot probabilities that is not consistent with equilibrium analysis. Once the relevant computations are corrected, it becomes possible to obtain upper bounds on the cost of voting compatible with high-turnout equilibria showing that these equilibria are indeed quite robust. This paper concludes by extending the model to include continuous outcome functions, showing that while they preserve the high-turnout equilibria obtained before, they allow for better characterizations of the results

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