Improving strategies in stochastic games

In a zero-sum limiting average stochastic game, we evaluate a\nstrategy π for the maximizing player, player 1, by the reward φ\ns(π) that π guarantees to him when starting in state s.\nA strategy π is called non-improving if\nφs(π)⩾φs(π[h]) for any state s\nand for any finite history h, where π[h] is the strategy π\nconditional on the history h; otherwise the strategy is called\nimproving. We investigate the use of improving and non-improving\nstrategies, and explore the relation between (non-)improvingness and\n(ε-) optimality. Improving strategies appear to play a very\nimportant role for obtaining ε optimality, while 0-optimal\nstrategies are always non-improving. Several examples are given to\nclarify all these issue