Strategy and Politics Strategy and Politics: An Introduction to Game Theory is designed to introduce students to the application of game theory for modeling political processes. This accessible text examines the phenomena that power political machineries— elections, legislative and committee processes, and international conflict—and answers fundamental questions about their nature and function in a clear, acces- sible manner. Included at the end of each chapter is a set of exercises designed to allow students to practice the construction and analysis of political models. Although the text assumes only an elementary-level training in algebra, students who complete a course around this text will be equipped to read nearly all of the professional literature that makes use of game theoretic analysis. Emerson M.S. Niou is Professor of Political Science at Duke University. Peter C. Ordeshook is Professor of Political Science at California Institute of Technology. This page intentionally left blank Strategy and Politics An Introduction to Game Theory Emerson M.S. Niou Duke University Peter C. Ordeshook California Institute of Technology First published 2015 by Routledge 711 Third Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2015 Taylor & Francis The right of Emerson M.S. Niou and Peter C. Ordeshook to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Niou, Emerson M. S. Strategy and politics : an introduction to game theory / Emerson Niou, Peter C. Ordeshook. pages cm 1. Game theory. 2. Political science—Mathematical models. 3. Political science—Methodology I. Ordeshook, Peter C., 1942– II. Title. JA72.5.N56 2015 320.01′5193—dc23 2014034874 ISBN: 978-1-138-01948-5 (hbk) ISBN: 978-0-415-99542-9 (pbk) ISBN: 978-1-315-73515-3 (ebk) Typeset in Minion Pro by Apex CoVantage, LLC Contents 1 Politics as a Game 1 1.1 Decision Versus Game Theoretic Decision Making 1 1.2 Preferences, Risk and Utility 13 1.3 Economics Versus Politics and Spatial Preferences 24 1.4 Collective Versus Individual Choice 36 1.5 Key Ideas and Concepts 42 Exercises for Chapter 1 42 2 Extensive Forms, Voting Trees and Planning Ahead 44 2.1 Introduction 44 2.2 The Extensive Form 49 2.3 Voting Agendas 69 2.4 Games and Subgames 76 2.5 The Centipede Game: A Word of Caution 82 2.6 Key Ideas and Concepts 85 Exercises for Chapter 2 85 3 The Strategic Form and Nash Equilibria 89 3.1 Introduction 89 3.2 Strategies and Simultaneous Choice 90 3.3 Nash Equilibria 96 3.4 Mixed Strategies 106 3.5 Mixed Strategies and Domination 115 3.6 Finding Mixed Strategy Equilibria 122 3.7 Manipulation and Incentive Compatibility 126 3.8 Key Ideas and Concepts 132 Exercises for Chapter 3 132 vi Contents 4 Zero-Sum Games with Spatial Preferences 139 4.1 Introduction 139 4.2 Plott, McKelvey and the Core Results of Spatial Theory 145 4.3 Two-Candidate Elections and the Electoral College 153 4.4 Turnout and Responsible Political Parties 156 4.5 Multi-Candidate Elections 159 4.6 Candidate Objectives and Game-Theoretic Reasoning 163 4.7 The Strategy of Introducing New Issues 165 4.8 Elections with Uninformed Voters 168 4.9 Other Applications 174 4.10 Key Ideas and Concepts 175 Exercises for Chapter 4 176 5 The Prisoners’ Dilemma and Collective Action 180 5.1 The Prisoners’ Dilemma 180 5.2 Some Simple Dilemmas in Politics 183 5.3 Cooperation and the Problem of Collective Action 195 5.4 Escaping the Dilemma: Repetition and Reputation 200 5.5 Constitutional Design and A Recursive Game 211 5.6 Evolutionarily Stable Strategies and Corruption 219 5.7 Key Ideas and Concepts 229 Exercises for Chapter 5 229 6 Agendas and Voting Rules 233 6.1 Agendas and Voting 233 6.2 Two Special Voting Rules and Peculiar Results 239 6.3 Two Alternative Rules for Electing Presidents 251 6.4 Controlling the Issues Voted On 255 6.5 Referenda and Separability of Preferences 263 6.6 Key Ideas and Concepts 267 Exercises for Chapter 6 267 7 Games with Incomplete Information 271 7.1 Incomplete Information 271 7.2 A Simple Game of Incomplete Information 275 7.3 Bayes’s Law and Bayesian Equilibrium 281 7.4 A Game with Two-Sided Incomplete Information 289 7.5 Agendas Reconsidered 293 7.6 Reputation and the Chain-Store Paradox 300 Contents vii 7.7 Signaling, Deception and Mutually Assured Destruction 302 7.8 Economic Sanctions in International Affairs 310 7.9 Rationality Reconsidered 317 7.10 Key Ideas and Concepts 323 Exercises for Chapter 7 323 8 Cooperation and Coalitions 327 8.1 The Concept of a Coalition 327 8.2 Coalitions and Condorcet Winners 330 8.3 A Generalization—The Core 335 8.4 The Politics of Redistribution 340 8.5 The Core and Spatial Issues 342 8.6 Majority Rule Games Without Cores 344 8.7 Parliamentary Coalitions 354 8.8 Problems and Some Incomplete Ideas 358 8.9 The Balance of Power Versus Collective Security 361 8.10 Key Ideas and Concepts 371 Exercises for Chapter 8 371 Appendix 377 Index 413 This page intentionally left blank 1 Politics as a Game 1.1 Decision Versus Game Theoretic Decision Making Over twenty five hundred years ago, the Chinese scholar Sun Tzu, in The Art of War, proposed a codification of the general strategic character of armed conflict and, in the process, offered practical advice for securing military vic- tory. His advice is credited, for example, with having greatly influenced Mao Zedong’s approach to conflict and the subtle tactics of revolution and the ways in which North Vietnam and the Viet Cong thwarted America’s military advan- tages. The formulation of general strategic principles—whether applied to war, parlor games such as Go, or politics—has long fascinated scholars. And regard- less of context, the study of strategic principles is of interest because it grapples with fundamental facts of human existence—first, people’s fates are interde- pendent; second, this interdependence is characterized generally by conflicting goals; and, finally, as a consequence of the first two facts, conflicts such as war are not accidental but are the purposeful extension of a state’s or an individual’s motives and actions and must be studied in a rational way. The Art of War is, insofar as we know, our first written record of the attempt to understand strategy and conflict in a coherent and general way. It is impor- tant, moreover, to recall that it was written at a time of prolonged conflict within an emerging China whereby the leaders of competing kingdoms pos- sessed considerable experience not only in the explicit conduct of war, but also in diplomacy and strategic maneuver. As such, then, we should presume that it codifies the insights of an era skilled at strategy and tactics, including those of planning, deception and maneuver. This assumption, though, occasions a ques- tion: Although The Art of War was ostensibly written for the leader of a specific kingdom, what if all sides to a conflict have a copy of the book (or, equivalently, an advisor no less insightful than Sun Tzu)? How might our reading of Sun Tzu change if it is common knowledge that everyone studied The Art of War or its equivalent—where by “common knowledge” we mean that everyone knows that everyone has a copy of the book, everyone knows that everyone knows that everyone . . . and so on, ad infinitum. The assumption of common knowledge presumes that not only is each decision maker aware of the situation, but each is aware that the other is aware, each knows that the other knows, and so on and so forth, and after being told by Sun Tzu himself that the great trap to be