Recursive Games

Abstract

Recursive games are stochastic games with the property that any nonzero-payoff is absorbing, i.e., play immediately moves to an absorbing state where each player has only one action available and these actions give this particular non-zero payoff at all further stages. By its structure, it is natural to examine such games using limiting average rewards, or total rewards on the assumption of stopping play as soon as a non-zero payoff occurs. Everett [1] introduced the recursive game model and immediately solved it for the zero-sum case. We shall briefly discuss his approach. Later Thuijsman and Vrieze [7] presented an asymptotic algebraic proof for the existence of stationary ε-optimal strategies for recursive¹ games, which can be derived from any arbitrary sequence of stationary λ-discounted optimal strategies, converging for λ going to O. Because of their simple structure, one might hope that recursive games always allow for stationary ε-equilibria as well, but that such is not true can clearly be seen from an example in Flesch et al. [2]. However, if we wish to solve the general existence problem for ε-equilibria in stochastic games, then one should certainly be able to tackle the problem for recursive games. Such has indeed been done by Vieille [8], [9]. Vieille [8] shows that if one can exhibit the existence of ε-equilibria in recursive games, then it follows that ε-equilibria exist in any stochastic game.² Vieille [9] then shows that equilibria exist in recursive games. Hence the two papers together comprise a proof for the existence of ε-equilibria in any arbitrary stochastic game. These results we shall leave to him for discussion. Instead, based on the paper by Flesch et al. [2], we shall exhibit