1 Fundamentals of Social Choice Theory

This paper offers a short introduction to some of the fundamental results of social choice theory. Topics include Nash implementability, monotonic social choice correspondences, the Muller-Satterthwaite impossibility theorem, anonymous and neutral social choice correspondences, sophisticated solutions of binary agendas, the top cycle of a tournament, the bipartisan set for two-party competition, and median voter theorems. The paper begins with a simple example to illustrate the importance of multiple equilibria in game-theoretic models of political institutions. 1. An introductory model of political institutions Mathematical models in social science are like fables or myths that we read to get insights into the social world in which we live. Our mathematical models are told in a specialized technical language that allows very precise descriptions of the motivations and choices of the various individuals in these stories. When we prove theorems in mathematical social science, we are making general statements about whole classes of such stories all at once. Here we focus on game-theoretic models of political institutions. So let us begin our study of political institutions by a simple game-theoretic model that tells a story of how political institutions may arise from games with multiple equilibria, such as those analyzed by Schelling (1960). Consider first the simple two-person "Battle of Sexes" game shown in Table 1. Player 2 f2 g2 Player 1 f1 0, 0 3, 6 g1 6, 3 0, 0 Table 1. The Battle of Sexes game. 1 The author is greatly indebted to Scott Ashworth for editorial comments that helped bring this paper into the new millennium.