Orderfield property of mixtures of stochastic games

Abstract

We consider certain mixtures, Γ, of classes of stochastic games and provide sufficient conditions for these mixtures to possess the orderfield property. For 2-player zero-sum and non-zero sum stochastic games, we prove that if we mix a set of states S₁ where the transitions are controlled by one player with a set of states S₂ constituting a sub-game having the orderfield property (where S₁∩S₂=∅), the resulting mixture Γ with states S=S₁∪S₂ has the orderfield property if there are no transitions from S₂ to S₁. This is true for discounted as well as undiscounted games. This condition on the transitions is sufficient when S₁ is perfect information or SC (Switching Control) or ARAT (Additive Reward Additive Transition). In the zero-sum case, S₁ can be a mixture of SC and ARAT as well. On the other hand,when S₁ is SER-SIT (Separable Reward - State Independent Transition), we provide a counter example to show that this condition is not sufficient for the mixture Γ to possess the orderfield property. In addition to the condition that there are no transitions from S₂ to S₁, if the sum of all transition probabilities from S₁ to S₂ is independent of the actions of the players, then Γ has the orderfield property even when S₁ is SER-SIT. When S₁ and S₂ are both SERSIT, their mixture Γ has the orderfield property even if we allow transitions from S₂ to S₁. We also extend these results to some multi-player games namely, mixtures with one player control Polystochastic games. In all the above cases, we can inductively mix many such games and continue to retain the orderfield property.