#### Introducing more players

Involving more players poses no great difficulty, other than complicating the notation. For example, suppose that a set of players, , , , , takes turns playing a game. Consider using a game tree representation. A stage is now stretched into substages, in which each player acts individually. Suppose that always starts, followed by , and so on, until . After acts, then the next stage is started, and acts. The circular sequence of player alternations continues until the game ends. Again, many different information models are possible. For example, in the stage-by-stage model, each player does not know the action chosen by the other players in the current stage. The bottom-up computation method can be used to compute Nash equilibria; however, the problems with nonuniqueness must once again be confronted.

A state-space formulation that generalizes Formulation 10.4 can be made by introducing action sets for each player and state . Let denote the action chosen by at stage . The state transition becomes

 (10.120)

There is also a cost function, , for each . Value iteration, adapted to maintain multiple equilibria and cost vectors can be used to compute Nash equilibria.

Steven M LaValle 2012-04-20