Andrzej S. Nowak

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.

Existence of stationary correlated equilibria with symmetric information for discounted stochastic games

The following theorem is proved: Every nonzero-sum discounted stochastic game in countably generated measurable state space with compact metric action spaces admits a stationary correlated equilibrium point with symmetric public information whenever the immediate rewards and transition densities are measurable with respect to the state variable and continuous with respect to joint actions.

On a new class of nonzero-sum discounted stochastic games having stationary Nash equilibrium points

AbstractA new class of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria is presented. Some applications to economic theory are also included.

Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space

This paper considers discounted noncooperative stochastic games with uncountable state space and compact metric action spaces. We assume that the transition law is absolutely continuous with respect to some probability measure defined on the state space. We prove, under certain additional continuity and integrability conditions, that such games have ε-equilibrium stationary strategies for each ε>0. To prove this fact, we provide a method for approximating the original game by a sequence of finite or countable state games. The main result of this paper answers partially a question raised by Parthasarathy in Ref. 1.

Measurable Selection Theorems for Minimax Stochastic Optimization Problems

This paper resolves the measurability questions which arise in the analysis of various minimax stochastic optimization models posed in Borel spaces. For a large class of such models, we provide a set of sufficient conditions which ensure that the questions have positive answers. These conditions are essentially weaker than those described in the existing literature. Moreover, they allow us to conclude a fundamental theorem yielding some new results on the existence of universally measurable selections of extrema for both zero-sum stochastic games and minimax stochastic optimal control problems. In particular, a random version of the well-known minimax theorem of Fan is in this way established. The fundamental theorem yields also a descriptive set theoretic fact concerning the projections of Borel sets. The paper indicates also some counterexamples to possibly more general problems.

On stochastic games in economics

In this paper we show that many results on equilibria in stochastic games arising from economic theory can be deduced from the theorem on the existence of a correlated equilibrium due to Nowak and Raghavan. Some new classes of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria are also presented. Three nontrivial examples of dynamic stochastic games arising from economic theory are given closed form solutions.

Zero-Sum Ergodic Stochastic Games with Feller Transition Probabilities

We consider zero-sum stochastic games with Borel state spaces satisfying a generalized geometric ergodicity condition. The main objective of this paper is to establish a minimax theorem for a class of ergodic stochastic games with the Feller transition probability function. All previous results on ergodic stochastic games are based on a strong continuity of the transition probabilities in the actions of the players.

Stochastic Games with Unbounded Payoffs: Applications to Robust Control in Economics

We study a discounted maxmin control problem with general state space. The controller is unsure about his model in the sense that he also considers a class of approximate models as possibly true. The objective is to choose a maxmin strategy that will work under a range of different model specifications. This is done by dynamic programming techniques. Under relatively weak conditions, we show that there is a solution to the optimality equation for the maxmin control problem as well as an optimal strategy for the controller. These results are applied to the theory of optimal growth and the Hansen–Sargent robust control model in macroeconomics. We also study a class of zero-sum discounted stochastic games with unbounded payoffs and simultaneous moves and give a brief overview of recent results on stochastic games with weakly continuous transitions and the limiting average payoffs.

Existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation

In this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game. A suitable change in the terminology used in this paper provides (in the case of perfect altruism between generations) a new Nash equilibrium theorem for standard stochastic games with uncountable state space.

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.