Improving strategies in stochastic games

In a zero-sum limiting average stochastic game, we evaluate a strategy π for the maximizing player, player 1, by the reward φ _s(π) that π guarantees to him when starting in state s. A strategy π is called non-improving if φ_s(π)⩾φ_s(π[h]) for any state s and for any finite history h, where π[h] is the strategy π conditional on the history h; otherwise the strategy is called improving. We investigate the use of improving and non-improving strategies, and explore the relation between (non-)improvingness and (ε-) optimality. Improving strategies appear to play a very important role for obtaining ε optimality, while 0-optimal strategies are always non-improving. Several examples are given to clarify all these issues