1. Interpersonal comparability of utility is generally regarded as an unsound basis on which to erect theories of multipersonal behavior. Nevertheless, it enters naturally—and, I believe, properly—as a nonbasic, derivative concept playing an important if sometimes hidden role in the theories of bargaining, group decisionmaking, and social welfare. The formal and conceptual framework of game theory is well adapted for a broad and unified approach to this group of theories, though it tends to slight the psychological aspects of group interaction in favor of the structural aspects—e.g., complementary physical resources, the channels of information and control, the threats and other strategic options open to the participants, etc. In this note I shall discuss two related topics in which game theory becomes creatively involved with questions of interpersonal utility comparison.

The first topic concerns the nature of the utility functions that are admissible in a bargaining theory that satisfies certain minimal requirements. I shall show, by a simple argument, that while cardinal utilities are admissible, purely ordinal utilities are not. Some intriguing intermediate systems are not excluded. The argument does not depend on the injection of probabilities or uncertainty into the theory.

The second topic concerns a method of solving general n-person games by making use of the interpersonal comparisons of utility that are implicit in the solution.

Introduction

The competitive equilibrium, core, and value are solution notions widely used in economics and are based on disparate ideas. The competitive equilibrium is a notion of noncooperative equilibrium based on individual optimization. The core is a notion of cooperative equilibrium based on what groups of individuals can extract from society. The value can be interpreted as a notion of fair division based on what individuals contribute to society. It is a remarkable fact that, under appropriate assumptions, these solution notions (nearly) coincide in large economies. The (near) coincidence of the competitive equilibrium and the core for large exchange economies was first suggested by Edgeworth (1881) and rigorously established by Debreu and Scarf (1963) in the context of replica economies and by Aumann (1964) in the context of continuum economies. This pioneering work has since been extended to much wider contexts; see Hildenbrand (1974) and Anderson (1986) for surveys. The (near) coincidence of the value and the competitive equilibrium (and hence the core) for large exchange economies was first suggested by Shubik and rigorously established by Shapley (1964) in the context of replica economies with money. This pioneering work, too, has since been extended to much wider contexts; see, for example, Shapley and Shubik (1969), Aumann and Shapley (1974), Aumann (1975), Champsaur (1975), Hart (1977), Mas-Colell (1977), and Cheng (1981).

Abstract

Shapley's combinatorial representation of the Shapley value is embodied in a formula that gives each player his expected marginal contribution to the set of players that precede him, where the expectation is taken with respect to the uniform distribution over the set of all orders of the players. We obtain alternative combinatorial representations that are based on allocating to each player the average relative payoff of coalitions that contain him, where one averages first over the sets of fixed cardinality that contain the player and then averages over the different cardinalities. Different base levels in comparison to which relative payoffs are evaluated yield different combinatorial formulas.

Introduction

The familiar representation of the Shapley value gives each player his “average marginal contribution to the players that precede him,” where averages are taken with respect to all potential orders of the players; see Shapley (1953). This chapter looks at three alternative representations of the Shapley value, each expressing the idea that a player gets the “average relative payoff to coalitions that contain him.” The common feature of the three representations we obtain is the way averages are taken, whereas the distinctive feature is the base level in comparison to which relative payoffs are evaluated.

Among the obligations facing a community of scholars is to make accessible to a wider community the ideas it finds useful and important. A related obligation is to recognize lasting contributions to ideas and to honor their progenitors. In this volume we undertake to fill part of both obligations.

The papers in this volume review and continue research that has grown out of a remarkable 1953 paper by Lloyd Shapley. There he proposed that it might be possible to evaluate, in a numerical way, the “value” of playing a game. The particular function he derived for this purpose, which has come to be called the Shapley value, has been the focus of sustained interest among students of cooperative game theory ever since. In the intervening years, the Shapley value has been interpreted and reinterpreted. Its domain has been extended and made more specialized. The same value function has been (re)derived from apparently quite different assumptions. And whole families of related value functions have been found to arise from relaxing various of the assumptions.

The reason the Shapley value has been the focus of so much interest is that it represents a distinct approach to the problems of complex strategic interaction that game theory seeks to illuminate.

Introduction

At the foundation of the theory of games is the assumption that the players of a game can evaluate, in their utility scales, every “prospect” that might arise as a result of a play. In attempting to apply the theory to any field, one would normally expect to be permitted to include, in the class of “prospects,” the prospect of having to play a game. The possibility of evaluating games is therefore of critical importance. So long as the theory is unable to assign values to the games typically found in application, only relatively simple situations—where games do not depend on other games—will be susceptible to analysis and solution.

In the finite theory of von Neumann and Morgenstern difficulty in evaluation persists for the “essential” games, and for only those. In this note we deduce a value for the “essential” case and examine a number of its elementary properties. We proceed from a set of three axioms, having simple intuitive interpretations, which suffice to determine the value uniquely.

Our present work, though mathematically self-contained, is founded conceptually on the von Neumann—Morgenstern theory up to their introduction of characteristic functions. We thereby inherit certain important underlying assumptions: (a) that utility is objective and transferable; (b) that games are cooperative affairs; (c) that games, granting (a) and (b), are adequately represented by their characteristic functions.

Introduction

The study of methods for measuring the “value” of playing a particular role in an n-person game is motivated by several considerations. One is to determine an equitable distribution of the wealth available to the players through their participation in the game. Another is to help an individual assess his prospects from participation in the game.

When a method of valuation is used to determine equitable distributions, a natural defining property is “efficiency”: The sum of the individual values should equal the total payoff achieved through the cooperation of all the players. However, when the players of a game individually assess their positions in the game, there is no reason to suppose that these assessments (which may depend on subjective or private information) will be jointly efficient.

This chapter presents an axiomatic development of values for games involving a fixed finite set of players. We primarily seek methods for evaluating the prospects of individual players, and our results center around the class of “probabilistic” values (defined in the next section). In the process of obtaining our results, we examine the role played by each of the Shapley axioms in restricting the set of value functions under consideration, and we trace in detail (with occasional excursions) the logical path leading to the Shapley value.

THE ENGINEERING OF CHOICE AND ORDINARY CHOICE BEHAVIOR

Recently I gave a lecture on elementary decision theory, an introduction to rational theories of choice. After the lecture, a student asked whether it was conceivable that the practical procedures for decision making implicit in theories of choice might make actual human decisions worse rather than better. What is the empirical evidence, he asked, that human choice is improved by knowledge of decision theory or by application of the various engineering forms of rational choice? I answered, I think correctly, that the case for the usefulness of decision engineering rested primarily not on the kind of direct empirical confirmation that he sought, but on two other things: on a set of theorems proving the superiority of particular procedures in particular situations if the situations are correctly specified and the procedures correctly applied, and on the willingness of clients to purchase the services of experts with skills in decision sciences.

The answer may not have been reasonable, but the question clearly was. It articulated a classical challenge to the practice of rational choice, the possibility that processes of rationality might combine with properties of human beings to produce decisions that are less sensible than the unsystematized actions of an intelligent person, or at least that the way in which we might use rational procedures intelligently is not self-evident.

This chapter concerns the proper role of values and the formation of values in decision-making processes. Such values, as used in this chapter, refer to preferences for states or things. We suggest that values should play a more central role in formalizing decision-making processes than is currently the case. By using value-focused thinking, a style of thinking that concentrates more and earlier on values, it may be reasonable to expect more appealing decision problems than those that currently face us. In other words, value-focused thinking should lead to better alternatives than those generated by existing “conventional” procedures.

There are four topics in this chapter. The first concerns identification of the proper role for values in the decision-making environment. The next discusses structuring and quantifying values to state unambiguously what the decision maker, decision makers or individuals concerned about the problem wish to achieve. The third indicates some approaches to facilitate the creation of alternatives based on stated values. The fourth suggests that the study of values has sufficient breadth and depth as well as sufficient potential rewards for researchers and students to be a legitimate discipline for serious study.

IDENTIFYING DECISION OPPORTUNITIES

Much of the focus of decision making is on the choice among alternatives. Indeed, it is common to characterize a decision problem by the alternatives available. Often it seems as if the alternatives present themselves with little background investigation and the decision problem begins when at least two alternatives have appeared.

Background and summary

One of the main axioms that characterizes the Shapley value is the axiom of symmetry. However, in many applications the assumption that, except for the parameters of the games the players are completely symmetric, seems unrealistic. Thus, the use of nonsymmetric generalizations of the Shapley value was proposed in such cases.

Weighted Shapley values were discussed in the original Shapley (1953a) Ph.D. dissertation. Owen (1968, 1972) studied weighted Shapley values through probabilistic approaches. Axiomatizations of nonsymmetric values were done by Weber (Chapter 7 this volume), Shapley (1981), Kalai and Samet (1987), and Hart and Mas-Colell (1987).

Consider, for example, a situation involving two players. If the two players cooperate in a joint project, they can generate a unit profit that is to be divided between them. On their own they can generate no profit. The Shapley value views this situation as being symmetric and would allocate the profit from cooperation equally between the two players. However, in some applications lack of symmetry may be present. It may be, for example, that for the project to succeed, a greater effort is needed on the part of player 1 than on the part of player 2. Another example arises in situations where player 1 represents a large constituency with many individuals and player 2's constituency is small (see, for example, Kalai 1977 and Thomson 1986).

INTRODUCTION

The technical feasibility of Bayesian statistics is rapidly improving as research and development activity exploits opportunities created by the computing revolution. This chapter offers an analysis and critique of Bayesian statistics, not from an internal aspect of flourishing technical development, but rather from an external aspect of the real world analyses and problem-solving tasks which the technology is designed to aid. I wish to raise for discussion the question: are the current norms and prescriptions of Bayesian statistics adequate to the tasks?

From an external standpoint, the central function of Bayesian statistics is the provision of probabilities to quantify prospective uncertainties given a current state of knowledge. The uncertainties refer to questions of fact about natural and social phenomena and about the effect of human decisions on these phenomena. The external motivation can be purely scientific, but in statistical practice there are usually decision- or policy-analytic components.

How broadly should Bayesian statistics be defined? I believe that the statistics profession has been hindered by the orthodoxy of academic mathematical statistics over the past 50 years which has largely removed evaluation of prospective uncertainties from the domain of statistical science. Thus, although statistics is the dominant source of useful probabilistic technologies, statisticians are often perceived as narrowly focused, and new professions such as “decision analysis” or “risk analysis” are created to fill the void.

The Shapley — Shubik and Banzhaf indices

In 1954 Lloyd Shapley and Martin Shubik published a short paper [12] in the American Political Science Review, proposing that the specialization of the Shapley value to simple games could serve as an index of voting power. That paper has been one of the most frequently cited articles in social science literature of the past thirty years, and its “Shapley—Shubik power index” has become widely known. Shapley and Shubik explained the index as follows:

“When English-speaking philosophers think of economics, they usually have a particular kind of pure theory in mind. This is the class of theories predominantly taught in western universities and often called neo-Classical. Purity here is a matter of conceiving homo economicus in abstraction from his social setting and, more excusably, of forswearing the attempt to make economics part (or all) of a general theory of society. By contrast, political economy, as the term is now used, is just such an attempt and its champions insist that no economic theory can be as pure as neo-Classicals pretend.” This view of the links between philosophy and economic theory, espoused by Frank Hahn and Martin Hollis, is one that the contributors to this volume embrace, even if none of them is a practitioner of “political economy” precisely as it is defined here. This book had its origins in an effort to place neoclassical economic theory (especially conventional textbook microeconomic theory) in the broader context of modern economics with special concern for the boundaries between economics and the other social sciences. The widespread use of textbook theory in business, economic, and political analysis is a clear testament to its power. Yet the restrictions and artificialities of neoclassical economic assumptions also give cause for worry to some of the finest minds in the discipline. These chapters examine two related themes that complicate the conventional “economist's” view of conduct and thereby provide a more complex (and humane) subject of study than the traditional Homo economicus.

Time has usually been treated in two very different ways in economic and social analysis. An analytical high road ponders the Nature of Time and the metaphysical conundrums of St. Augustine and Einstein, whereas a low road considers the time context of human behavior and its analysis to be self-evident, uncomplicated, and not very interesting. Both of these are appealing but both, I will argue, are inadequate: the high road because of its irrelevance to social analysis, the low road because its inattention to temporality creates important obstacles to understanding social behavior. A middle road needs building – or at least widening and surfacing – a road that acknowledges the importance of the time structure of social behavior and our analysis of it while avoiding both mystification, on the one hand, and carelessness, on the other. I hope this chapter will make a contribution to that end.

The high road, the low road, and a middle road

Consideration of the Ultimate Nature of Time is mind boggling. Sympathy with Augustine's (1955:354) famous complaint has certainly increased in the past fifteen hundred years: “What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know.” Philosophers have asked whether time has reality, whether the passage of time has reality, whether time exists apart from human observation of it, and whether statements about time can be made independent of time itself.

In his own account of his intellectual formation, F. A. Hayek has always acknowledged his indebtedness to the thinkers of the Scottish school and, above all, to Ferguson, Smith, and Hume. Indeed, in his contributions to the intellectual history of classical liberal political economy and social philosophy, Hayek has gone so far as to distinguish two divergent and opposed intellectual traditions – that of the French Enlightenment, which he sees as inspired ultimately by a variation of Cartesian rationalism, and that of the Scottish Enlightenment, with its roots in a Christian and skeptical recognition of the limits of human understanding – and has identified himself explicitly with the Scottish tradition. That Hayek's thought converges with that of the leading Scottish political economists on many fundamental questions is not in serious doubt and can easily be demonstrated. At the same time, the thought of the Scottish school is only one of the influences that have shaped Hayek's complex intellectual makeup, and these other influences, especially that of his teachers in the Austrian school, are responsible for many of the points of sharp and real divergence between Hayek and the Scottish philosophers. It is by virtue of these other influences that we may say that Hayek's thought diverges from that of the Scottish philosophers as often as it converges with it – and, often enough, in ways Hayek has not himself perceived.