Peter J. Hammond

The Implementation of Social Choice Rules: Some General Results on Incentive Compatibility

We shall assume that the objectives of a society are embodied in a certain social choice rule. A social choice rule (SCR) selects a set of feasible social states for each possible configuration of individual preferences and other characteristics. One interprets the choice set as the set of welfare optima. For example, given an Arrow social welfare function which embodies individual preferences in a social ordering, then a natural social choice rule is derived by maximizing this social ordering over the feasible set. Alternatively, the Pareto rule is the social choice rule which selects all Pareto efficient states, given individual preferences and the feasible set. These are two particular social choice rules which have received much attention, but our discussion will cover social choice rules in general. If the relevant characteristics of individual agents, such as preferences, happen to be publicly known, then the social choice rule can be implemented trivially because the choice set itself is known. The problem of incentive compatibility arises precisely because these characteristics are not known by the planner a priori. The planner may attempt to learn characteristics directly by asking agents to reveal them. In general, however, if the agents realize how the information they reveal is to be used, they will have an incentive to misrepresent. Then the task of the planner in implementing the social choice rule is more difficult. Obviously, he must use a planning mechanism of some kind, whose outcomes are possible social states. We shall assume that, when he devises the mechanism, the planner knows what social states are feasible, so that he can ensure that the final outcome is feasible. (See, however, Hurwicz, Maskin and Postlewaite (1978), which considers the more general problem where feasibility itself depends on unknown characteristics.) The planner, however, relies on signals from the individual agents to help him implement the social choice rule. It is assumed that each individual agent sends his own signal. The planners mechanism is then a rule which specifies a social state for each list of signals sent by the individual agents. It is assumed that each agent knows the precise form of the mechanism the planner is using. Then each agent realizes that he is involved in a game, because the outcome of the mechanism depends on the signals which he and all the other

Handbook of utility theory

Preface. 14. Alternatives to Expected Utility: Foundations R. Sugden. 15. Alternatives to Expected Utility: Formal Theories U. Schmidt. 16. State-Dependent Utility and Decision Theory J.H. Dreze, A. Rustichini. 17. Ranking Sets of Objects S. Barbera, W. Bossert, P.K. Pattanaik. 18. Expected Utility in Non-Cooperative Game Theory P.J. Hammond. 19. Utility Theories in Cooperative Games M. Kaneko, M.H. Wooders. 20. Utility in Social Choice W. Bossert, J.A. Weymark. 21. Interpersonally Comparable Utility M. Fleurbaey, P.J. Hammond. Subject Index. Name Index.

Changing Tastes and Coherent Dynamic Choice

Consider any economic agent who is choosing a plan which is going to take some time to implement. Examples are a firm undertaking a long-term investment project, a consumer contemplating the choice of a superannuation scheme, and a department of government making any kind of longor even medium-term plan for the future. Before such a plan has been fully implemented, there is plenty of time in which the agent has the chance to change his mind and, perhaps, the plan he originally chose. If his plan does change, there is, prima facie, an instance of inconsistent choice. It is a convenient shorthand to describe this as a problem of changing tastes, even though this suggests an unintended limitation of the phenomenon to consumer choice. Changing tastes have been separated into two categories: exogenously changing tastes, and endogenously changing tastes. Most previous work has considered one aspect or the other, yet the distinction is unhelpful and one that will be abandoned in this paper. The first work on exogenously changing tastes was by Allais [2]. He considered the welfare implications of a consumer regretting his earlier choices; however, he specifically restricted his discussion to tastes which never give rise to changing or inconsistent choices (as explained in [7], Appendix 2) (see also [15], [16]). Strotzs [26] paper is a discussion of the consistency problem for a particular type of consumer planning his savings over a finite time-interval; it has been followed by Pollak [19], and by Blackorby, Nissen, Primont and Russell [3] (see also Heal [11], Section 10.2). The latter considered the general problem of a consumer with exogenously changing tastes making choices from budget sets. Finally, Peleg and Yaari [17] discuss certain existence problems. On the other hand, the literature on endogenous tastes has steered clear of the consistency problem. Instead, it has concentrated on the following two questions:

Interpersonal Comparisons of Well-being: Interpersonal comparisons of utility: Why and how they are and should be made

A satisfactory complete normative criterion for individualistic ethical decisionmaking under uncertainty such as Harsanyi’s (Journal of Political Economy , 1955) requires a single fundamental utility function for all individuals which is fully interpersonally comparable. The paper discusses reasons why interpersonal comparisons of utility (ICU’s) have been eschewed in the past and argues that most existing approaches, both empirical and ethical, to ICU’s are flawed. Either they confound facts with values, or they are based on unrealistic hypothetical decisions in an “original position”. Instead ICU’s need to be recognized for what they really are — preferences for different kinds of people. INTERPERSONAL COMPARISONS OF UTILITY . . . I still believe that it is helpful to speak as if inter-personal comparisons of utility rest upon scientiAEc foundations – that is, upon observation or introspection. . . . I still think, when I make interpersonal comparisons . . . that my judgments are more like judgments of value than judgments of veriAEable fact. Nevertheless, to those of my friends who think differently, I would urge that, in practice, our difference is not very important. They think that propositions based upon the assumption of equality are essentially part of economic science. I think that the assumption of equality comes from outside, and that its justiAEcation is more ethical than scientiAEc. But we all agree that it is AEtting that such assumptions should be made and their implications explored with the aid of the economist’s technique. — Robbins (1938, pp. 640–641)

Continuum economies with finite coalitions: Core, equilibria, and widespread externalities

Abstract We develop a new model of a continuum economy with coalitions consisting of only finite numbers of agents. The core, called the f -core, is the set of allocations that are stable against improvement by finite coalitions and feasible by trade within finite coalitions. Even with widespread externalities—preferences depend on own consumptions and also on the entire allocation up to the null set—we obtain the result that the f -core coincides with the Walrasian allocations. Without widespread externalities, the f -core, the Aumann core, and the Walrasian allocations all coincide; however, with widespread externalities there is no obvious natural definition of the Aumann core.

Interpersonally Comparable Utility

Over many years, interpersonal comparisons of utility have had a significant role to play in economics. Utility began as a basic concept on which Prances Hutcheson, Cesare Beccarla, Jeremy Bentham, John Stuart Mill, and Henry Sidgwick sought to build a general ethical theory that is simple yet profound. The resulting classical utilitarian theory relied on interpersonal comparisons because it required a common unit with which to measure each person’s pleasure or happiness, before adding to arrive at a measure of total happiness. According to the standard reading of Bentham, one should then proceed to subtract each person’s pain or misery, also measured in the same common unit, in order to arrive at a measure of total utility.1

Elementary Non-Archimedean Representations of Probability for Decision Theory and Games

In an extensive form game, whether a player has a better strategy than in a presumed equilibrium depends on the other players’ equilibrium reactions to a counterfactual deviation. To allow conditioning on counterfactual events with prior probability zero, extended probabilities are proposed and given the four equivalent characterizations: (i) complete conditional probability systems; (ii) lexicographic hierarchies of probabilities; (iii) extended logarithmic likelihood ratios; and (iv) certain ‘canonical rational probability functions’ representing ‘trembles’ directly as infinitesimal probabilities. However, having joint probability distributions be uniquely determined by independent marginal probability distributions requires general probabilities taking values in a space no smaller than the non-Archimedean ordered field whose members are rational functions of a particular infinitesimal.

Overlapping expectations and Hart's conditions for equilibrium in a securities model

Hart (J. Econ. Theory 9 (1974), 293–311) gave conditions for equilibrium to exist in a securities model where each agent undertakes asset transactions to maximize expected utility of wealth. These conditions rule out agents wanting to undertake unbounded balanced transactions to reach a Pareto superior allocation given their expectations. With mild extra assumptions to make agents unwilling to risk incurring unbounded losses on their portfolios, Harts conditions become equivalent to an assumption of “overlapping expectations,” which is comparable to a much weaker form of Greens “common expectations” (Econometrica 41 (1973), 1103–1124).

Endogenous tastes and stable long-run choice

Abstract Suppose that short-run preferences depend upon consumption one period earlier. Then there is an acyclic long-run strict preference relation iff, for every finite set, every conservative choice sequence converges. If long-run preferences are acyclic, then a unique long-run choice from a compact set is globally stable. If the long-run choice set includes multiple choices, there is a weaker stability property. Under special assumptions these results are extended to cases when the short-run consumption set is endogenous, and when more previous periods affect the present.

On Reconciling Arrow’s Theory of Social Choice with Harsanyi’s Fundamental Utilitarianism

The last paragraph of Kenneth Arrow’s Nobel Lecture is characteristic of the man: The philosophical and distributive implications of the paradox of social choice are still not clear. Certainly there is no simple way out. I hope that others will take this paradox as a challenge rather than as a discouraging barrier (Arrow 1973a; 1974).