Fundamentals of Social Choice Theory

This paper offers a short introduction to some of the fundamental results of social choice theory. Topices include: Nash implementability and the Muller-Satterthwaite impossibility theorem, anonymous and neutral social choice correspondences, two-party competition in tournaments, binary agendas and the top cycle, and median voter theorems. The paper begins with a simple example to illustrate the importance of multiple equilibria in gametheoretic models of political institutions. 1.1 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. In this book, 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. Consider first the simple two-person "Battle of Sexes" game shown in Table 1.1. Player 2 f g 2 2 Player 1 f 0,0 3,6 1 g 6,3 0,0 1 Table 1.1. The Battle of Sexes game. The two players in this game, who may be called player 1 and player 2, must independently choose one of two possible strategies: to defer (f ) or to grab (g ). If the players i i both grab or both defer then neither player gets anything; but if exactly one player grabs then he