Zero-Sum Games in Social Choice and Game Theory

Preferences Over Uncertain Outcomes 1 1.2 Game Theory 4 1.3 Social Choice Theory 10 2 preliminaries 19 2.1 Decision Theoretic Fundamentals 20 2.2 SSB Utility Theory 23 I zero-sum games 27 3 game theoretic fundamentals 29 3.1 Zero-Sum Games 29 3.2 Symmetric Zero-Sum Games 31 4 a proof of the minimax theorem 33 5 justification of maximin play 37 5.1 Independence, Rationality, Consistency, and Consequentialism 40 5.2 Characterization of Maximin Strategies 43 5.3 Concluding Remarks 47 6 random symmetric zero-sum games 49 6.1 The Distribution of Maximin Strategies 51 6.2 Concluding Remarks 55 II preference aggregation and social choice 57 7 social choice theoretic fundamentals 59 7.1 Social Welfare Functions and Social Choice Functions 59 7.2 Maximal Lotteries 62 8 arrovian preference aggregation 63 8.1 Arrovian Social Welfare Functions 66 8.2 Characterization of the Domain 67 8.3 Characterization of the Social Welfare Function 68 8.4 Interpretation of Results 70 8.5 Concluding Remarks 72 8.6 Characterization of the Domain: Proofs 74 8.7 Characterization of the Social Welfare Function: Proofs 80 9 relative utilitarian social choice 87 9.1 Relative Utilitarian Outcomes and Utilitarian Participation 88 9.2 Preferences based on Pairwise Comparisons 91 9.3 Concluding Remarks 92 xiii xiv contents 10 consistent social choice 95 10.1 Preliminaries 96 10.2 Population consistency and Composition consistency 99 10.3 Pure Social Choice Functions 101 10.4 Characterization of Maximal Lotteries 102 10.5 Concluding Remarks 104 10.6 Cloning Consistency Implies Neutrality 106 10.7 Pure Social Choice Functions: Proofs 107 10.8 Characterization of Maximal Lotteries: Proofs 109