Lloyd S. Shapley

COLLEGE ADMISSIONS AND THE STABILITY OF MARRIAGE

A procedure for assigning applicants to colleges which removes all uncertainties and, assuming there are enough applicants, assigns to each college precisely its quota.

A Method for Evaluating the Distribution of Power in a Committee System

In the following paper we offer a method for the a priori evaluation of the division of power among the various bodies and members of a legislature or committee system. The method is based on a technique of the mathematical theory of games, applied to what are known there as “simple games†and “weighted majority games.†We apply it here to a number of illustrative cases, including the United States Congress, and discuss some of its formal properties.The designing of the size and type of a legislative body is a process that may continue for many years, with frequent revisions and modifications aimed at reflecting changes in the social structure of the country; we may cite the role of the House of Lords in England as an example. The effect of a revision usually cannot be gauged in advance except in the roughest terms; it can easily happen that the mathematical structure of a voting system conceals a bias in power distribution unsuspected and unintended by the authors of the revision. How, for example, is one to predict the degree of protection which a proposed system affords to minority interests? Can a consistent criterion for “fair representation†be found? It is difficult even to describe the net effect of a double representation system such as is found in the U. S. Congress (i.e., by states and by population), without attempting to deduce it a priori. The method of measuring “power†which we present in this paper is intended as a first step in the attack on these problems.

Cores of convex games

The core of ann-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one that is based on a convex set function. In this paper it is shown that the core of a convex game is not empty and that it has an especially regular structure. It is further shown that certain other cooperative solution concepts are related in a simple way to the core: The value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core.

The assignment game I: The core

The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.

On cores and indivisibility

Abstract An economic model of trading in commodities that are inherently indivisible, like houses, is investigated from a game-theoretic point of view. The concepts of balanced game and core are developed, and a general theorem of Scarfs is applied to prove that the market in question has a nonempty core, that is, at least one outcome that no subset of traders can improve upon. A number of examples are discussed, and the final section reviews a series of other models involving indivisible commodities, with references to the literature.

Mathematical Properties of the Banzhaf Power Index

The Banzhaf index of power in a voting situation depends on the number of ways in which each voter can effect a “swing” in the outcome. It is comparable---but not actually equivalent---to the better-known Shapley-Shubik index, which depends on the number of alignments or “orders of support” in which each voter is pivotal. This paper investigates some properties of the Banzhaf index, the main topics being its derivation from axioms and its behavior in weighted-voting models when the number of small voters tends to infinity. These matters have previously been studied from the Shapley-Shubik viewpoint, but the present work reveals some striking differences between the two indices. The paper also attempts to promote better communication and less duplication of mathematical effort by identifying and describing several other theories, formally equivalent to Banzhaf’s, that are found in fields ranging from sociology to electrical engineering. An extensive bibliography is provided.

Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts

Two solution concepts for cooperative games in characteristic-function form, the kernel and the nucleolus, are studied in their relationship to a number of other concepts, most notably the core. The unifying technical idea is to analyze the behavior of the strong e-core as e varies. One of the central results is that the portion of the prekernel that falls within the core, or any other strong e-core, depends only on the latters geometrical shape. The prekernel is closely related to the kernel and often coincides with it, but has a simpler definition and simpler analytic properties. A notion of “quasi-zero-monotonicity” is developed to aid in enlarging the class of games in which kernel considerations can be replaced by prekernel considerations. The nucleolus is approached through a new, geometrical definition, equivalent to Schmeidlers original definition but providing very elementary proofs of existence, unicity, and other properties. Finally, the intuitive interpretations of the two solution concepts are clarified: the kernel as a kind of multi-bilateral bargaining equilibrium without interpersonal utility comparisons, in which each pair of players bisects an interval which is either the battleground over which they can push each other aided by their best allies if they are strong or the no-mans-land that lies between them if they are weak; the nucleolus as the result of an arbitrators desire to minimize the dissatisfaction of the most dissatisfied coalition.

QUASI-CORES IN A MONETARY ECONOMY WITH NONCONVEX PREFERENCES

Abstract : A model of a pure exchange economy is investigated without the usual assumption of convex preference sets for the participating traders. The concept of core, taken from the theory of games, is applied to show that if there are sufficiently many participants, the economy as a whole will possess a solution that is sociologically stable--i.e., that cannot profitably be upset by any coalition of traders.

THE KERNEL AND BARGAINING SET FOR CONVEX GAMES

It is shown that for convex games the bargaining setℳ1(i) (for the grand coalition) coincides with the core. Moreover, it is proved that the kernel (for the grand coalition) of convex games consists of a unique point which coincides with the nucleolus of the game.

Long Term Competition -- A Game-Theoretic Analysis

There have been continuing expressions of interest from a variety of quarters in the development of techniques for modelling national behavior in a long-term context of continuing international rivalry — for short, “long term competition”. The most characteristic feature of these models is that they extend over time in a fairly regular or repetitive manner. The underlying structure of possible actions and consequences remains the same, though parameters may vary and balances shift, and the decisions and policies of the national decision makers are by no means constrained to be constant or smoothly-varying, or even “rational” in any precisely identifiable sense. The use of game theory or an extension thereof is obviously indicated, and considerable theoretical progress has been made in this area. But the ability of the theory to handle real applications is still far from satisfactory. The trouble lies less with the descriptive modelling, — i.e., formulating the “rules of the game” in a dynamic setting, than with the choice of a solu- tion concept that will do dynamic justice to the interplay of motivations of the actors. (Game theoreticians, like mathematical economists, have always been more comfortable with static than dynamic models.) Since any predictions, recommendations, etc. that a mathematical analysis can produce will likely be very sensitive to the rationale of the solution that is used, and since the big difficulties are conceptual rather than technical, it seems both possible and worthwhile to discuss salient features of the theory without recourse to heavy mathematical apparatus or overly formal arguments, and thereby perhaps make the issues involved accessible to at least some of the potential customers for the practical analyses that we wish we could carry out in a more satisfactory and convincing manner.