Bezalel Peleg

Coalition-Proof Nash Equilibria I. Concepts

Abstract In an important class of “noncooperative” environments, it is natural to assume that players can freely discuss their strategies, but cannot make binding commitments. In such cases, any meaningful agreement between the players must be self-enforcing. Although the Nash best-response property is a necessary condition for self-enforceability, it is not sufficient—it is in general possible for coalitions arrange plausible, mutually beneficial deviations from Nash agreements. We provide a stronger definition of self-enforceability, and label the class of efficient self-enforcing agreements “coalition-proof.”

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.

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.

Strategy-Proof Voting Schemes With Continuous Preferences

We prove that all nondictatorial voting schemes whose range has more than two alternatives will be manipulable, when their domain is restricted to the set of all continuous preferences over alternatives. Our result neither implies nor is implied by the original Gibbard-Satterthwaite theorem, except if the number of alternatives is finite, when they coincide. A new, direct line of reasoning is used in the proof. It is presented in an introductory section, which may be useful in classroom situations.

An axiomatization of the core of cooperative games without side payments

Abstract Let M be a set of m players, m ≧3, and let Γ be the set of all (finite) games (without side payments) that have a non-empty core. When M is finite, the following four (independent) axioms fully characterize the core on Γ: (i) non-emptiness, (ii) individual rationality, (iii) the reduced game property, and (iv) the converse reduced game property. If M is infinite, then the converse reduced game property is redundant.

Utility Functions for Partially Ordered Topological Spaces

Abstract : Sufficient conditions are given for the existence of continuous numerical representations (utilities) for partially ordered topological spaces. (Author)

Coalition formation in simple games with dominant players

A player, in a proper and monotonic simple game, is dominant if he holds a “strict majority” within a winning coalition. A (non-dictatorial) simple game is dominated if it contains exactly one dominant player. We investigate several possibilities of coalition formation in dominated simple games, under the assumption that the dominant player is given a mandate to form a coalition. The relationship between the various hypotheses on coalition formation in dominated games is investigated in the first seven sections. In the last section we classify real-life data on European parliaments and town councils in Israel.

An Axiomatic Characterization of the Lexicographic Maximin Extension of an Ordering Over a Set to the Power Set

The lexicographic maximin extension of an ordering is an important and widely used tool in social choice theory. We provide an axiomatization of it by means of five axioms. When the basic ordering is linear the following four (independent) axioms are sufficient: (1) Gärdenfors principle; (2) Neutrality; (3) Strong Fishburn monotonicity; and (4) Extension. Our result may also have applications in the theory of individual choice under uncertainty.

ON WEIGHTS OF CONSTANT-SUM MAJORITY GAMES.

Abstract : It is shown that the nucleolus of a constant-sum weighted majority game G is a system of weights for G; moreover, if G is homogeneous then the nucleolus of G supplies homogeneous weights. Minimum and minimal integer weights are discussed with respect to their relations to the kernel and the nucleolus.

Co-evolution of preferences and information in simple games of trust

Abstract In standard rational choice modelling decisions are made according to given information and preferences. In the model presented here the ‘information technology’ of individual decision-makers as well as their preferences evolve in a dynamic process. In this process decisions are made rationally by players who differ in their informational as well as in their preference type. Relative success of alternative decisions feeds back on the type composition of the population which in turn influences rational decision-making. An indirect evolutionary analysis of an elementary yet important basic game of trust shows that under certain parameter constellations the population dynamics of the evolutionary process specify a unique completely mixed rest point. However, as opposed to previous studies of preference formation in the game of trust there is no convergence to but only circumventing the rest point if the informational status of individuals evolves rather than being chosen strategically.