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We study the equilibria of the standard pivotal-voter participation game between two groups of voters of asymmetric sizes (majority and minority), as originally proposed by Palfrey and Rosenthal (Public Choice 41(1):7-53, 1983). We find a unique equilibrium wherein the minority votes with certainty and the majority votes with probability in (0, 1); we prove that this is the only equilibrium in which voters of only one group play a pure strategy, and we provide sufficient conditions for its existence. Equilibria where voters of both groups vote with probability in (0, 1) are analyzed numerically.1 In particular, PR analyze (1) identical group sizes and symmetry of strategies across groups and (2) aggregate probabilities of voting across individuals of different groups summing to 1. Besides (1) and (2), which are special cases of our analysis, they also study two alternative and tractable settings: one assuming a status quo (ties are broken in favor of one group, instead of randomly), and another, namely "k equilibria", in which the individuals of one group mix with identical probability, whereas among the individuals of the other group, k vote with probability 1 and the remaining with probability 0.

We exploit an original dataset from a referendum in Peru to study the influence of voting "arrangements" on electoral outcomes. The relative importance of these arrangements (e.g., ballot design) with respect to the fundamentals (e.g., ideology, candidates' quality) has not been measured. After controlling for a comprehensive set of politicians' characteristics, we estimate unbiased ballot order effects making use of the within party variation in outcomes. We estimate a non-linear probability model and we create counterfactuals to conclude that ballot design not only may have changed the electoral results but also has a greater importance than candidates' ideology, education, experience and party affiliation.

We study the effects of opinion polls on election results in proportional representation systems. Moderate voters have preferences over the vote shares received by the parties so that an agent's optimal voting decision might depend on the other agents' behavior. A voter's information about other voters' behavior can be improved through a series of opinion polls. We show that the mass of undecided voters decreases monotonically with the number of polls, but may not necessarily disappear. Voters who remain undecided have centrist ideologies. On average a series of polls brings the society closer to complete information even though specific polls may push the election result away from the complete information case.

We study how different forms of social capital lead to different distributions of multidimensional opinions by affecting the channels through which individuals communicate. We develop a model to compare and contrast the evolution of opinions between societies whose members communicate through bonding associations and societies where communication is through bridging associations. Both processes converge towards opinion distributions where there are groups within which there is consensus in all issues. Bridging processes converge to distributions that have, on average, fewer opinion groups and lower fractionalisation. We provide additional results that highlight the distinct characteristics of the two processes.