with Economic Applications

Editors: Robert J. Aumann and Sergiu Hart

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Robert J. Aumann and Sergiu Hart

This is the second of three volumes planned for the

We follow the rough division into "noncooperative,"
"cooperative," and "general"
adopted in the first volume. Chapters 20 through 31 are mainly
noncooperative; 32
through 37, cooperative; 38 through 40, general.
This division should not be taken
too seriously; chapters may well contain aspects
of both approaches. Indeed, we
hope that the *Handbook* will help demonstrate that
noncooperative and cooperative
game theory are two sides of the same coin,
which complement each other well.

In game theory's early period, from the twenties through the early fifties, two-person zero-sum games -- where one player's gain is the other's loss -- were the main object of study; many thought that that is all there is to the discipline. Those days are long gone; but two-person zero-sum games still play a fundamental role, and it is fitting that we start the second volume with them. Chapter 20 is devoted to the basic minimax theory and its ramifications, with an appendix on duels. Chapter 21 covers the branch of statistical decision theory that deals with conservative or "worst case" analysis, i.e., in which nature is treated as a malevolent player who seeks to minimize the payoff to the decision maker. This is one of the earliest applications of two-person zero-sum games. The next two chapters treat differential games, i.e., games played over continuous time, like pursuit games. The mathematical theory -- which both benefitted from and contributed to control theory and differential equations -- is in Chapter 22; economic applications, zero-sum as well as non-zero-sum, in Chapter 23.

This brings us to the non-zero-sum world. Here, there is room for correlation and communication between the players. Chapter 24 treats correlated equilibria, communication equilibria and "mechanisms" in general; these concepts are extensions of the classical Nash strategic equilibrium.

"Actions speak louder than words": in addition to the explicit communication considered in the previous chapter, players may also communicate implicitly, bymeans of their actions -- as in Spence's seminal 1972 work, in which education serves as a signal of ability. Signalling is the subject of Chapter 25, the first of three chapters in this volume on economic applications of noncooperative game theory (in addition to Chapters 8 - 11 on such applications in the first volume).

To serve as signals, actions must be observed by the other players. Chapter 26 discusses the opposite case, in which actions cannot be directly monitored. The principal-agent problem concerns relationships between owner and manager, patient and physician, insurer and insured, and so on. Characteristic of these relationships is that the principal must devise incentives to motivate the agent to act in the principal's interests. A particular aspect of this is the moral hazard problem, of which a typical instance is insurance: Insuring one's home for more than its value would create an incentive to burn it down.

Chapter 27 concerns game theoretic aspects of search models in economics. Not classical zero-sum search such as destroyer-submarine, but non-zero-sum problems like shopping and marketing, job hunting and recruiting, and so on.

We come next to one of the most promising newer applications of game theory: biology. Rather surprisingly, the inequalities defining a Nash strategic equilibrium are, when properly reinterpreted, identical to those that characterize an equilibrium between populations (or within a population), in the sense of evolutionary ecology. This field, initiated in 1972 by John Maynard Smith, has spawned a large literature, which is surveyed in Chapter 28.

The following five chapters deal with applications to political science and related topics; these chapters bridge the noncooperative and the cooperative parts of the volume. Chapter 29 surveys models of international conflict -- an area to which game theory was applied already in the fifties, at the height of the cold war. Chapter 30 presents a game theoretic analysis of voting systems, such as proportional representation, plurality voting, approval voting, ranking methods, single transferable vote, and others. Much of the analysis deals with strategic considerations of voters facing the various systems. These questions may also be studied in the more general framework of "social choice" -- group decision problems. Social choice constitutes a large and much studied area; Chapter 31 deals with its game theoretic aspects, i.e., how to devise schemes that implement certain outcomes when the participants act strategically in their own best interest, and whether this is at all possible. This concludes the noncooperative part of the volume.

The cooperative part starts with three chapters on applications, to political science and economics. All use the concepts of core and value. Chapter 32 deals with measures of power and notions of stability in various political applications. Chapter 33 is devoted to a subject with both economic and political content -- public economics; this concerns taxation, provision of public goods, and so on.

In most applications of game theory, the "players" are either human individuals or collectives of humans, like companies, unions, political parties or nations. To some extent this is so even in the applications to statistics (Chapter 21) and computer science (Chapter 38); though there the "players" are not necessarily human, we ascribe to them human motives, which in one sense or another correspond to the goals of the architect or programmer. There is, however, another kind of application, where the mathematical formalism of game theory -- the "equations defining the game" -- is interpreted in a way that is quite different from standard. One such application is to evolutionary biology (Chapter 28). Another, surveyed in Chapter 34, is to the problem of allocating joint costs. For example, airport landing fees, overhead costs billed by universities, phone charges within an organization, and so on. Here, the players are individual aircraft landings; activities like specific research projects or student-hours taught in a specific course; single minutes of long-distance phone calls. The worth of a "coalition" of such activities is defined as the hypothetical cost of carrying out the activities in that coalition only.

Three chapters on theory end the cooperative part of this volume. The first is on bargaining problems, which were studied from the noncooperative viewpoint in Chapter 7 (Volume 1). Chapter 35 presents the axiomatic approach to these problems: solutions are sought that satisfy certain desirable properties. The axiomatic method studies various solution concepts from the viewpoint of their properties rather than their definitions, and helps us to compare them. Those that appear again and again in different setups, like the classic 1951 bargaining solution of Nash, gain credibility.

In bargaining problems, only the individual players and the "grand coalition" -- that of all players -- matter. In general cooperative games, also the intermediate coalitions play a role. Such "coalitional games" are presented and classified in Chapter 36. An important question arising in this connection is which coalitions actually form. Chapter 37 surveys some of the approaches to this problem, which have both cooperative and noncooperative aspects.

We have already said that game theory has significant ties to various disciplines, some of which may, at first glance, seem quite unrelated. In the case of computer science, the influence goes in both directions. For example, to model bounded rationality, game theory has borrowed from the theory of finite automata, Turing machines, and so on (some discussion of this is included in Chapter 4 of Volume 1). Chapter 38 surveys the other direction -- game theoretic ideas in computer science. Two examples are the application of iterated knowledge in the study of distributed systems, and the use of power indices -- from cooperative game theory -- to estimate the vulnerability of a system to "crashes" because of possible failures by relatively small sets of parts.

The last two chapters in this volume pertain to the foundations of game theory. Chapter 39 deals with how the players evaluate the possible outcomes; the subjective measurement of utilities and probabilities is at the basis of decision theory, both interactive and one-person. Finally, information drastically affects the way in which games are played. A coherent model of knowledge, knowledge about others' knowledge, and so on, has become essential equipment in game theory. The theory of these levels of knowledge, culminating in "common knowledge," is the subject of Chapter 40.

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