János Flesch

Cyclic Markov equilibria in stochastic games

We examine a three-person stochastic game where the only existing equilibria consist of cyclic Markov strategies. Unlike in two-person games of a similar type, stationary ε-equilibria (ε > 0) do not exist for this game. Besides we characterize the set of feasible equilibrium rewards.

Recursive repeated games with absorbing states

We show the existence of stationary limiting average e-equilibria e > 0 for two-person recursive repeated games with absorbing states. These are stochastic games where all states but one are absorbing, and in the nonabsorbing state all payoffs are equal to zero. A state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions. For the purpose of our proof, we introduce properness for stationary strategy pairs. Our result is sharp since it extends neither to the case with more nonabsorbing states, nor to the n-person case with n > 2. Moreover, it is well known that the result cannot be strengthened to the existence of 0-equilibria and that repeated games with absorbing states generally do not admit stationary e-equilibria.

Perfect-Information Games with Lower-Semicontinuous Payoffs

We prove that every multiplayer perfect-information game with bounded and lower-semicontinuous payoffs admits a subgame-perfect e-equilibrium in pure strategies. This result complements Example 3 in Solan and Vieille [Solan, E., N. Vieille. 2003. Deterministic multi-player Dynkin games. J. Math. Econom.39 911--929], which shows that a subgame-perfect e-equilibrium in pure strategies need not exist when the payoffs are not lower-semicontinuous. In addition, if the range of payoffs is finite, we characterize in the form of a Folk Theorem the set of all plays and payoffs that are induced by subgame-perfect 0-equilibria in pure strategies.

Subgame Perfection in Positive Recursive Games with Perfect Information

We consider a class of n-player stochastic games with the following properties: (1) in every state, the transitions are controlled by one player; (2) the payoffs are equal to zero in every nonabsorbing state; (3) the payoffs are nonnegative in every absorbing state. We propose a new iterative method to analyze these games. With respect to the expected average reward, we prove the existence of a subgame-perfect e-equilibrium in pure strategies for every e > 0. Moreover, if all transitions are deterministic, we obtain a subgame-perfect 0-equilibrium in pure strategies.

Pure subgame-perfect equilibria in free transition games

We consider a class of stochastic games, where each state is identified with a player. At any moment during play, one of the players is called active. The active player can terminate the game, or he can announce any player, who then becomes the active player. There is a non-negative payoff for each player upon termination of the game, which depends only on the player who decided to terminate. We give a combinatorial proof of the existence of subgame-perfect equilibria in pure strategies for the games in our class.

Stochastic Games on a Product State Space

We examine product-games, which are n-player stochastic games satisfying: (1) the state space is a product S(1)A—…A—S(n); (2) the action space of any player i only depends of the i-th coordinate of the state; (3) the transition probability of moving from s(i) ∈ S(i) to t(i) ∈S(i), on the i-th coordinate S(i) of the state space, only depends on the action chosen by player i. So, as far as the actions and the transitions are concerned, every player i can play on the i-th coordinate of the product-game without interference of the other players. No condition is imposed on the payoff structure of the game. We focus on product-games with an aperiodic transition structure, for which we present an approach based on so-called communicating states. For the general n-player case, we establish the existence of 0-equilibria, which makes product-games one of the first classes within n-player stochastic games with such a result. In addition, for the special case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies. Both proofs are constructive by nature.

Non-existence of subgame-perfect ε-equilibrium in perfect information games with infinite horizon

Every finite extensive-form game with perfect information has a subgame-perfect equilibrium. In this note we settle to the negative an open problem regarding the existence of a subgame-perfect

On refinements of subgame-perfect epsilon-equilibrium

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