Tadeusz Radzik

A Solidarity Value for n-Person Transferable Utility Games

In this paper, we introduce axiomatically a new value for cooperative TU games satisfying the efficiency, additivity, and symmetry axioms of Shapley (1953) and some new postulate connected with the average marginal contributions of the members of coalitions which can form. Our solution is referred to as the solidarity value. The reason is that its interpretation can be based on the assumption that if a coalition, sayS, forms, then the players who contribute toS more than the average marginal contribution of a member ofS support in some sense their “weaker” partners inS. Sometimes, it happens that the solidarity value belongs to the core of a game while the Shapley value does not.

On axiomatizations of the weighted Shapley values

The family of weighted Shapley values for cooperative n-person transferable utility games is studied. We assume first that the weights of the players are given exogenously and provide two axiomatic characterizations of the corresponding weighted Shapley value. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced by a new postulate called the [omega]-mutual dependence. In our second axiomatization we use among other things the strong monotonicity property of Young (1985, Int. J. Game Theory 14, 65-72). Finally, we give a new axiomatic characterization of the family of all weighted Shapley values. Journal of Economic Literature Classification Number: C71, D46.

Saddle point theorems

This paper considers three classes of matrices in terms of the existence ofsaddle points. The classes are described by conditions which are very closely related to the property ofquasi-concavity-convexity of functions of two variables. For the matrices in those classes, necessary and sufficient conditions for the existence of saddle points have been found.

A Weighted Pseudo-potential Approach to Values for TU-games

This paper provides a twofold generalization of the well-known characterization of the Shapley value for TU-games as the discrete gradient of a so-called potential function. On the one hand the potential approach is extended to the so-called weighted pseudo-potential approach in the sense that the extended representation may incorporate, besides a fraction of the discrete gradient, a fraction of the underlying pseudo-potential function itself, as well as a fraction of the average of all the components of the gradient. On the other hand the paper fully characterizes the class of values for TU-games that admit a weighted pseudo-potential representation. Besides two individual constraints, these values have to be efficient, symmetric, and linear. The theory developed is illustrated by several examples of such values and their weighted pseudo-potential representations are discussed.

On a family of values for TU-games generalizing the Shapley value

In this paper we study a family of efficient, symmetric and linear values for TU-games, described by some formula generalizing the Shapley value. These values appear to have surprising properties described in terms of the axioms: Fair treatment, monotonicity and two types of acceptability. The results obtained are discussed in the context of the Shapley value, the solidarity value, the least square prenucleolus and the consensus value.

Pure-strategy ε-Nash equilibrium in two-person non-zero-sum games

Abstract This paper considers two-person non-zero-sum games where the strategy spaces of players are compact intervals of real numbers and the payoff functions are upper semicontinuous . Under certain quasi-concavity assumptions, it is proved that such games possess e- Nash equilibria in pure strategies for every e > 0. AMS 1980 subject classification: 90D05, 90D13.

Nash equilibria of discontinuous non-zero-sum two-person games

The paper investigates two classes of non-zero-sum two-person games on the unit square, where the payoff function of Player 1 is convex or concave in the first variable. It is shown that this assumption together with the boundedness of payoff functions imply the existence of ɛ-Nash equilibria consisting of two probability measures concentrated at most at two points each.

Extensions of Hart and Mas-Colell’s Consistency to Efficient, Linear, and Symmetric Values for TU-Games

By Hart and Mas-Colell’s axiomatization, it is known that the Shapley value for TU-games is fully characterized by its 1-standardness for two-person games and its consistency property with respect to a particular reduced game. In the framework of TU-games, this paper establishes a similar axiomatization (with reference to some kind of consistency and standardness for two-person games) for values that verify efficiency, linearity, and symmetry. The fundamental idea in this unified consistency approach involves the introduction of a new type of reduced game. The construction of this game takes into account, besides the value itself, the probabilities of two events that a removed player joins or does not join a proposed coalition. Although the reduced game varies whenever the efficient, linear, and symmetric value varies, an operational criterion is presented to determine the appropriate reduced game by solving an associated system of linear equations recursively. Finally, the impact of the unified consistency approach is illustrated in the context of several known values, in particular the least square values and the Shapley value.

Characterization of optimal strategies in matrix games with convexity properties

Abstract. This paper gives a full characterization of matrices with rows and columns having properties closely related to the (quasi-) convexity-concavity of functions. The matrix games described by such payoff matrices well approximate continuous games on the unit square with payoff functions F (x, y) concave in x for each y, and convex in y for each x. It is shown that the optimal strategies in such matrix games have a very simple structure and a search-procedure is given. The results have a very close relationship with the known theorem of Debreu and Glicksberg about the existence of a pure Nash equilibrium in n-person games.

An alternative characterization of the weighted Banzhaf value

Abstract. We provide a new characterization of the weighted Banzhaf value derived from some postulates in a recent paper by Radzik, Nowak and Driessen [7]. Our approach owes much to the work by Lehrer [4] on the classical Banzhaf value based on the idea of amalgamation of pairs of players and an induction construction of the value. Compared with the approach in [7] we consider two new postulates: a weighted version of Lehrer’s “2-efficiency axiom” [4] and a generalized “null player out” property studied in terms of symmetric games by Derks and Haller [2].