Normalized equilibrium in Tullock rent seeking game

International audienceGames with Common Coupled Constraints represent manyreal life situations. In these games, if one player fails tosatisfy its constraints common to other players, then theother players are also penalised. Therefore these games canbe viewed as being cooperative in goals related to meetingthe common constraints, and non cooperative in terms ofthe utilities. We study in this paper the Tullock rent seekinggame with additional common coupled constraints. We havesucceded in showing that the utilities satisfy the property ofdiagonal strict concavity (DSC), which can be viewed asan extention of concavity to a game setting. It not onlyguarantees the uniqueness of the Nash equilibrium but also of the normalized equilibrium

Finite-Step Algorithms for Single-Controller and Perfect Information Stochastic Games

Abstract. After a brief survey of iterative algorithms for general stochas-tic games, we concentrate on finite-step algorithms for two special classes of stochastic games. They are Single-Controller Stochastic Games and Per-fect Information Stochastic Games. In the case of single-controller games, the transition probabilities depend on the actions of the same player in all states. In perfect information stochastic games, one of the players has exactly one action in each state. Single-controller zero-sum games are effi-ciently solved by linear programming. Non-zero-sum single-controller stochastic games are reducible to linear complementary problems (LCP). In the discounted case they can be modified to fit into the so-called LCPs of Eave’s class L. In the undiscounted case the LCP’s are reducible to Lemke’s copositive plus class. In either case Lemke’s algorithm can be used to find a Nash equilibrium. In the case of discounted zero-sum perfect informa-tion stochastic games, a policy improvement algorithm is presented. Many other classes of stochastic games with orderfield property still await efficient finite-step algorithms. 1

Monotonicity of optimal policies in a zero sum game: a flow control model

The purpose of this paper is to illustrate how value iteration can be used in a zero-sum game to obtain structural results on the optimal (equilibrium) value and policy. This is done through the following example. We consider the problem of dynamic flow control of arriving customers into a finite buffer. The service rate may depend on the state of the system, may change in time and is unknown to the controller. The goal of the controller is to design a policy that guarantees the best performance under the worst case service conditions. The cost is composed of a holding cost, a cost for rejecting customers and a cost that depends on the quality of the service. We consider both discounted and expected average cost. The problem is studied in the framework of zero-sum Markov games where the server, called player 1, is assumed to play against the flow controller, called player 2. Each player is assumed to have the information of all previous actions of both players as well as the current and past states of the system. We show that there exists an optimal policy for both players which is stationary (that does not depend on the time). A value iteration algorithm is used to obtain monotonicity properties of the optimal policies. For the case that only two actions are available to one of the players, we show that his optimal policy is of a threshold type, and optimal policies exist for both players that may need randomization in at most one state