Stochastic Games, Practical Motivation and the Orderfield Property for Special Classes

Stochastic games concentrate on decision situations where at different time moments the players have to make a choice. The joint choices of all the players together have two implications. First, each player receives some reward, or loses some amount when this reward is negative. Second, the underlying system moves on along its trajectory. However, it is assumed that nature here plays a role in the sense that the transition might be the outcome of a random experiment, which might be dependent on the choices the players made. We only consider games where the decision moments are discrete points on a time axis and just for convenience we shall let these decision moments coincide with the set of positive natural numbers {1, 2,…}. In stochastic games, perfect recall is assumed as well as complete information. That means that all the players know all the data of the game and at any future time moment all the players perfectly remember what has happened in the past.