Introduction

John Nash has contributed to game theory and economics two solution concepts for nonconstant sum games. One, the non-cooperative solution [9], is a generalization of the minimax theorem for two person zero sum games and of the Cournot solution; and the other, the cooperative solution [10], is completely new. It is the purpose of this paper to present a non-cooperative equilibrium concept, applicable to supergames, which fits the Nash (non-cooperative) definition and also has some features resembling the Nash cooperative solution. “Supergame” describes the playing of an infinite sequence of “ordinary games” over time. Oligopoly may profitably be viewed as a supergame. In each time period the players are in a game, and they know they will be in similar games with the same other players in future periods.

The most novel element of the present paper is in the introduction of a completely new concept of solution for non-cooperative supergames. In addition to proposing this solution, a proof of its existence is given. It is also argued that the usual notions of “threat” which are found in the literature of game theory make no sense in non-cooperative supergames. There is something analogous to threat, called “temptation,” which does have an intuitive appeal and is related to the solution which is proposed.

In section II the ordinary game will be described, the non-cooperative equilibrium defined and its existence established. Section III contains a description of supergames and supergame strategies.

Overview

Three papers comprise this section. First, Chapter Seven of Cournot's book, “Of the Competition of Producers”, is reproduced. In this chapter Cournot provides the first model of independent multiagent choice, initially analyzing two duopolists under constant marginal cost production; this is later extended to n firms and nonconstant marginal cost. As Fisher [1898] points out (and as is still true today), much of the main body of Cournot's book contains contributions to economic theory (see also the discussion in the Introduction) that few economists credit Cournot for. Cournot was the first to introduce aggregate demand and supply curves, and to provide the notion of diminishing returns (see Fisher [1898]). His original work on monopoly provides the basic model still in use today (Hicks [1935]).

Nathanial T. Bacon, brother-in-law of Irving Fisher's wife, translated Cournot's 1838 work into English in 1897. Fisher [1898] initially believed that Cournot had been grossly careless in the preparation of his manuscript, but it later transpired that Cournot was partially blind during this period (Nichol [1938], Fisher [1938]). Further reflections on Cournot's contributions and life may be found in the papers of Fisher and Nichol.

Cournot was a mathematician and, given the level of mathematics of most 19th-century economists, it is probably not surprising that at some point his work would be reviewed by another mathematician, in this case Joseph Bertrand, a name identified in mathematics with areas such as probability theory and differential geometry.

Overview

Utilitarianism is a philosophical thesis two centuries old. It judges collective action on the basis of the utility levels enjoyed by the individual agents and of those levels only. This is literally justice by the ends rather than by the means. Welfarism is the name, coined by Amartya Sen, of the theoretical formulation of utilitarianism, especially useful in economic theory and other social sciences. Its axiomatic presentation, developed in the last three decades, is the subject of Chapters 2 and 3.

For the utilitarianist, social cooperation is good only inasmuch as it improves upon the welfare of individual members of society. The means of cooperation (social and legal institutions, such as private contracts and public firms) do not carry any ethical value; they are merely technical devices - some admittedly more efficient than others - to promote individual welfares. For instance, protecting certain rights - say, freedom of speech - is not a moral imperative; it should be enforced only if the agents derive enough utility from it.

This very dry social model rests entirely upon the concept of individual preferences determined by the agents “libre-arbitre” while deliberately ignoring all its mitigating factors (among them education and the shaping of individual opinion by the social environment, as well as kinship, friendship, or any specific interpersonal relation). Ever since Bentham, utilitarianists have been aware of these limitations. Yet, oversimplified as it is, the utilitarian model is easily applicable, and its "liberal" ideology has the force of simplicity. The philosophical debate on utilitarianism is still quite active (see, e.g., Sen and Williams [1982]).

Overview

The economic behavior of a public firm is among the oldest problems of welfare economics. The two main instances where a public firm is called for are the provision of a public good and that of a private good of which the production technology has increasing returns to scale (IRS). Those are cases of natural monopoly. On the economic theory of natural monopolies see Baumol, Panzar, and Willig [1982]; on the optimal provision of a public good see the historical review by Musgrave and Peacock [1958] and the theoretical survey of Milleron [1972]. In both cases, the only efficient organization of production utilizes a single production unit, yet it would be socially wasteful to let this unit be operated by a profit-maximizing monopolist. Thus, there exist the need for a regulated public firm and the compelling normative question of its pricing policy.

Even under decreasing returns to scale (DRS), the case for joint production frequently arises. The workers of a cooperative jointly own its machines, and the partners of a law firm share its patrons. They must choose an equitable scheme to share the fruits of their cooperation. Thus, the general discussion of a public firm involves an arbitrary technology; if it has decreasing returns, we mean that it is operated jointly by the agents with no possibility of duplication (there is a large implicit fixed cost to open a new production unit).

Overview

Nash [1950] proposed to generalize SWOs to more complex choice rules that we call social choice functions (SCFs). The idea is to take into account the whole feasible set of utility vectors to guide the choice of the most equitable one. Accordingly, any two utility vectors may no longer be compared independently of the context (namely, the actual set of feasible vectors), as was the case with SWOs. To understand how this widens the range of conceivable choice methods, think of relative egalitarianism as opposed to plain egalitarianism. Say that two agents must divide some commodity bundle. The egalitarian program simply chooses the highest feasible equal utility allocation (assuming away the equality-efficiency dilemma) independently of the feasible unequal utility vectors.Relative egalitarianism, on the other hand, computes first the utility w, that each agent would derive from consuming alone the whole bundle; then it chooses thisefficient utility vector where the ratio of the actual utility to the highest conceivable utility w, is the same for each agent. In other words, relative egalitarianism equalizes individual ratios of satisfaction (or of frustration) by defining full satisfaction (i.e., zero frustration) in a context-dependent manner. We give a numerical example (Example 1.1), stressing the difference between egalitarianism and relative egalitarianism.

Overview

In Part III we apply the tools forged in Parts I and II to specific microeconomic surplus-sharing problems. The two problems discussed in this chapter are the simplest models of surplus and cost sharing, in which the distributive issue has no obvious answer.

The cost-sharing problem is best thought of as the provision of an indivisible public good, say, a bridge or any such public facility. The issue is how to share the cost of the facility. The only available data are the total cost of building the bridge and the benefits that each agent derives from the facility (a single dollar figure per agent since we assume quasilinear utilities). Another interpretation of the same formal model is the settlement of a bankruptcy. Here the agents are creditors of the bankrupt firm; each agent holds a monetary claim over its estate, but the total value of the estate falls short of the sum of the claims. The problem is to share the estate among its creditors.

The surplus-sharing problem is that of dividing the proceeds of an indivisible cooperative venture among several partners. Think of an orchestra giving a single performance. The receipts must be shared among the musicians; the sharing rule must be based upon the estimated “market value” of each musician, again, a single number per agent representing his opportunity cost of joining the orchestra. Assuming that the receipts exceed the sum of the opportunity costs, how should it be divided among the agents?

Overview

A cooperative game in society N consists of a feasible utility set for the grand coalition N as well as a utility set for each and every coalition (non-empty subset) of N, including the coalitions containing one agent only. Each of those 2n —1 utility sets is viewed, as in the welfarist models of Part I, as a feasible set of cooperative opportunities: If the agents in a given coalition all agree on it, they can enforce any utility distribution in this set. The game model does not describe the course of action they must take to achieve this utility distribution. This must be made clear by each particular microeconomic model generating a cooperative game.

We view the cooperative game model as an extension of the axiomatic bargaining model (Chapter 3). The latter specifies the feasible utility set for the grand coalition TV and for each coalition containing a single agent. Indeed, the disagreement utility of an agent corresponds to his opportunity cost for joining the grand coalition. Thus, the only new ingredients in a cooperative game are the opportunity sets of intermediate coalitions (containing at least two, but not more than n — 1, agents).

Many decisions of public concern cannot be left to the market because cooperative opportunities will not be efficiently utilized by decentralized actions of the agents. The most prominent examples include the provision of public goods, pricing of a natural monopoly, as well as all decisions taken by vote. To remedy these market failures, welfare economists have come up with a variety of normative solutions and tried to convince the decision makers of their relevance.

The theoretical foundation of these normative arguments is axiomatic. This point was clearly made in Amartya Sen’s landmark book (Collective Choice and Social Welfare, first published in 1970). Since then the axiomatic literature has considerably expanded its scope and refined its methods: The whole theory of cooperative games has played a central role in the analysis of cost sharing when there are increasing returns to scales; our understanding of voting rules now encompasses the impact of strategic manipulations; several refined measurements of inequality have been constructed and abundantly tested; and so on.

This book describes the recent successes of the axiomatic method in four areas: welfarism (the construction of collective utility functions and inequality measures as well as the axiomatic bargaining approach); cooperative games (the core and the two most popular value operators - Shapley value and nucleolus); public decision making (cost sharing of a public good and pricing of a regulated monopoly, in both the first-best and strategic-second-best perspectives); and voting and social choice (majority voting a la Condorcet and scoring methods a la Borda; the impossibility of aggregating individual preferences into a social preference).

Overview

“Democracy uses, as a method of governing, social summaries of citizens' decisions in elections and legislators' decisions in representative bodies” (Riker [1982], p. 21). Indeed, most public allocative decisions (such as taxes and public expenses) are made by voting. Elections are also used to fill many public offices. These are important examples of pure public goods (all citizens of a given town consume their mayor, with no possibility of exclusion) chosen by voting and precluding side payments.

Ever since the political philosophy of the Enlightenment, the choice of a voting rule has been a major ethical question with far-reaching implications on the behavior of most political institutions. The debate about fairness of various voting methods has been with us since the contributions of de Borda [1781] and Condorcet [1785]. In 1952, Arrow proposed the formal model that framed for three decades a voluminous mathematically oriented literature known as social choice (see Arrow [1963]). It studies the properties of various voting rules from an axiomatic angle. The object of Part IV is to discuss the most important contributions of the social choice approach.

Formally, a voting rule solves the collective decision problem where several individual agents (voters) must jointly choose one among several outcomes (also called candidates), about which their opinions conflict. In this chapter we assume that a finite set N of voters must pick one candidate within a finite set A (we discuss other options in Chapter 10). For simplicity we assume that individual opinions (or preferences) display no ties; they are arbitrary linear orders of A (i.e., complete, transitive, and asymmetric binary relations). This assumption entails no serious loss of generality.

Overview

The egalitarian and classical utilitarian programs, in spite of all their differences, have one common functional feature. Both utilize a collective utility function (CUF) aggregating individual utilities into a single utility index representing the social welfare. Within the feasible utility set, they then select the socially optimum utility vector by maximizing the CUF. This function is the sum of individual utilities for classical utilitarianism and their minimum for egalitarianism (see Chapter 1).

The welfarist axioms considered in this chapter develop this anthropomorphic idea. Society's welfare is described by a collective utility index computed mechanically from individual utilities. Thus, collective choice follows the same rationale as individual choice: “Il faut que les méthodes d'une assemblée délibérante se rapprochent autant qu'il est possible de celles des individus qui la composent” (Condorcet [1785]; the methods of a deliberating assembly must be as close as possible to those of its individual members). In particular, any two vectors of individual utilities can be compared, and those comparisons are transitive.

The primary economic application of CUFs is to the measurement of inequality. Estimating the welfare consequences of the distribution of incomes (or of that of any variable related to individual welfare) is an important task of public economics. Doing this systematically means that we must be able to compare any two income distributions and tell which one yields the highest social welfare. In other words, we must choose a social welfare ordering (SWO). This choice will be guided by additional ethical postulates. The consequences of those postulates on the mathematical form of the SWO is the subject of this chapter.

Overview

The most ambitious task of cooperative game theory is to build a universal solution concept based on widely acceptable equity axioms, picking out of every cooperative game a unique utility distribution, just as the social choice function of Chapter 3 does. Such an object is called a value, or value operator. For more than 30 years, this viewpoint has been tested for TU games. By and large, it proves to be successful. Surely, no single solution concept has emerged that would satisfy everyone's sense of equity for all TU games. All the same, no single social choice function is the universal panacea of axiomatic bargaining (see Part I). However, two prominent values have been discovered and prove useful in a wide range of economic models. These are the Shapley value and the nucleolus, to which most of the discussion of this chapter is devoted.

In a nutshell, the nucleolus applies egalitarianism to TU games, whereas the Shapley value follows from a utilitarian principle. Indeed, the nucleolus minimizes the leximin SWO over long utility vectors in which every coordinate corresponds to a different coalition (see Definition 5.4). The Shapley value, on the other hand, renumerates each agent by averaging his marginal contributions to all coalitions containing him; it is utilitarian inasmuch as classical utilitarianism (Chapter 1) is likewise based upon average utility. Accordingly, the nucleolus tends to be harder to compute than the Shapley value (just as maximizing the leximin SWO is a more involved program than maximizing the utilitarian one).

Overview

In plurality voting it is sometimes rational to cast one's vote for another candidate than one's first-best choice: If I know that my most preferred candidate a will not pass because b and c are both going to get more votes, then I had better help whomever of b, c I like more. In any voting method when a voter realizes that his own vote may influence the final outcome, he thinks twice before casting it. Maybe the naive ballot suggested by his true preferences does not serve his interest best. Rather than passively reporting his opinion about candidates, he acts as a player in the game of election, trying to maximize the returns from his vote.

In real-world elections it is impossible to distinguish a strategically biased report of a voters' preferences from a truthful one. One's opinion is private information de jure, and hence an openly untruthful report is a perfectly legal move.

Several early analysts of voting methods were aware of their strategic aspects, thought of as the nuisance of a dishonest vote; see the quotation by Borda in Straffin [1980] or by Dodgson in Farquharson [1969]. A mathematically rigorous attack of the problem has come only in the last 15 years. The seminal question is this: Can we design a strategyproof voting rule, namely, such that each individual voter would always want to report his opinion truthfully while isolated in the voting booth?