GAMES and DECISION MAKING Charalambos D. Aliprantis Subir K

GAMES AND DECISION MAKING Charalambos D. Aliprantis Departments of Economics & Mathematics Purdue University Subir K. Chakrabarti Department of Economics Indiana University Purdue University Indianapolis c 1998 All Rights Reserved ! To our wives and to our children: Bernadette and Tuhina; and Claire, Dionissi, Anisha, Devika, and Sharmitha Foreword In the fall of 1995 the National Science Foundation awarded a grant to Indiana University System for the project Mathematics Through- out the Curriculum whose objective was to develop inderdisciplinary undergraduate courses with substantial mathematical content. The primary focus was to expose students in a wide variety of subjects to the usefulness of mathematical techniques. We were asked to design a course that would include topics from economic theory. Our response was a course that put together material from decision theory and game theory in a novel way. This book is the culmination of that effort. The book evolved from the need to offer a course in decision theory that would include the classical material on optimization theory as well as from the newer and more recent literature in game theory. Since no existing text covered the topics in the manner required for the project, we put together our own material that included classical optimization theory, game theory, auctions and bargaining. The book has been written with the objective of introducing the rather sophisticated concepts of modern decision theory to a readership that is familiar with only elementary calculus and elementary probability theory. It is a self-contained treatment of almost everything that can be called decision theory—from classical optimization theory to modern game theory. We have included applications from economics, political science, finance, and management. Examples are used to show both the need for a theory and to demonstrate the framework within which the theory can be applied. Thus, the examples and applications are used as a major pedagogic device. In writing the book our objective was to introduce some of the cen- tral ideas of decision theory and game theory without trying to treat either subject exhaustively; which we think cannot be done adequately in a single text. Therefore, we tried to be eclectic. Whether we suc- ceeded in this effort is for the reader to judge. There is a pattern we follow, which the careful reader may disern. We start with the most iv Foreword elementary decision making problem, that of the single decision maker and move on to progressively more complex decision problems which culminate in the discussion of sequential rationality. We then move on to see how the intellectual capital that has been acquired can be used to study topics of more immediate interest like auctions and bargaining. A special feature of the book is that it treats decision theory and game theory as part of the same body of knowledge. This feature sep- arates it from the texts which are devoted to only game theory or deal solely with decision theory. In the book, single-person decision theory is used as the building block of game theory. This is the essential theme underlying the book. It highlights the interplay that exists between single-person decision problems and multi-person decision problems. The text leaves out important elements from both decision theory as well as game theory. We never discuss dynamic programming, though hints of the theory are present when we discuss backward in- duction. Our treatment of multivariate decision problems is at best cursory. Large chunks of game theory are also not discussed. There is a huge literature on the refinements of Nash equilibrium that we haven’t even touched, and we totally bypassed the extensive work on repeated games. We are well aware of all these omissions and many more that we have not mentioned. Our excuse is that we did this to avoid writing an incredibly formitable manuscript that would intimi- date the student. We wanted to write a text that would serve as a solid introduction to some of the basic issues and techniques of modern decision making and games. The reader with a year of college level mathematics that includes some elementary calculus and finite mathematics would be comfortable with the material in the book. We have made every effort to make the book as self-contained as possible. Thus, although some of the material in the later chapters requires some extra knowledge of mathematics, the necessary material is developed as an integral part of the text. The book is written with some thought to the sequence in which topics are introduced. Each chapter builds on the material presented in the earlier chapters. For instance, the chapters on sequential games build on the chapter of sequential decision-making. We paid special attention to this because much of game theory is technical and, as a result, it becomes unduly difficult if the ideas are not presented in their proper order. Foreword v This text can be used for a course at the undergraduate level as well as at the introductory graduate level. It is aimed at the undergraduate student who wants to take a serious look at decision theory; for instance, the economics major or the business major or even the mathematics major. The book is also appropriate for use at the graduate level to teach introductory courses in optimization and game theory and MBA courses in decision theory. Because of its content, the book would appeal to anyone who thinks seriously about making good decisions. Thus, the material in this book is useful to the business manager, the policy maker, the budding entrepreneur as well as to the serious undergraduate student in economics or business who aspires to acquire a graduate degree. We strongly believe that the book inter- weaves the different strands of decision theory and game theory well, and can be used for a fairly sophisticated course in game theory—in many contexts—even for students who have only a moderate back- ground in mathematics. In the first chapter we develop classical optimization theory using elementary calculus. The material in this chapter is similar to that usu- ally found in texts written for courses in mathematics for economists. The first chapter thus is a condensed version of the chapters on optimization in a book like Mathematics for Economic Analysis [23] by Sydsaeter and Hammond or a book like Fundamental Methods of Math- ematical Economics [2] by Chiang. The novelty in the first chapter is in the way we introduce the material. Using problems from economics, business, operations research and so forth, we demonstrate how to use the first and second order conditions of calculus to find optimal solutions to optimization problems. In this chapter our aim is to show how to use the mathematics in a variety of useful and interesting ways. While Chapter 1 deals with classical optimization theory with a single decision-maker, Chapter 2 introduces the fundamentals of modern game theory. Chapter 2 builds on the techniques developed in Chap- ter 1, and focuses on game theory, while at the same time drawing extensively from the material in Chapter 1. In the second chapter, we develop the theory of strategic form games and their solutions. We also present applications from markets, voting, auctions, resource extrac- tion and so forth. In Chapter 3, we study sequential decision-making and introduce the language of trees and graphs to define sequential games. Thus, we avoid the usual questions about graphs and trees vi Foreword that inevitably arise when one starts to define sequential games. In Chapters 4 and 5 we discuss sequential games using the techniques of Chapter 3 as building blocks. We deal with the concepts of subgame perfection and sequential rationality in some detail. Chapter 6 is devoted to a discussion of auctions. We study various types of auctions and the possible outcomes. Our discussion of auctions is done within the framework of game theory. Thus, we view an auction as a game and examine the equilibrium outcome of the game. Chapter 7 is concerned with bargaining. We treat both the axiomatic approach to bargaining as well as the analysis of bargaining problems using sequential games. Thus, Chapter 7 investigates the Nash bargaining solution, the Shapley value and the core. It also deals with sequential bargaining. Chapters 6 and 7 are, in fact, applications of the principles developed in the earlier chapters. We take this opportunity to thank the principal investigators of the project, Professors Bart Ng of IUPUI and Dan Maki of Indiana Uni- versity, for providing us with summer support to develop the course. We also thank Robert Sandy, the Chairman of the Department of Eco- nomics, IUPUI, for his support. We would like to thank our students Eric Broadus, Lisa Whitecotton, and David Parent for their critical impact while the text was being developed. Finally, we would like to acknowledge the patient understanding of our wives, Bernadette and Tuhina, who allowed us to spend many hours away from our families. C. D. Aliprantis & S. K. Chakrabarti Indianapolis, May, 1998 Contents Foreword iii 1 Choices and Opportunities 1 1.1 Functions . 2 1.2 The optimization problem . 4 1.3 First order conditions . 9 1.4 Optimizing using the Lagrange method . 16 1.5 Uncertainty and chance . 22 1.6 Decision making under uncertainty . 32 2 Decisions and Games 45 2.1 Two-person matrix games . 48 2.2 Strategic form games . 55 2.3 Applications . 62 2.4 Solving matrix games with mixed strategies . 78 3 Sequential Decisions 87 3.1 Graphs and trees . 88 3.2 Single-person decisions . 93 3.3 Uncertainty and single-person decisions .

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