Realization Plans for Extensive Form Games without Perfect Recall

Given a game in extensive form and a player p in the game, we want to find a small set of parameters describing a set M of mixed strategies with the property that every mixed strategy for p has an equivalent mixed strategy in M. In the case that the player has perfect recall, behavioral strategies describe such a set. [2] For computational purposes, it is more useful to work with corresponding path probabilities because the relationships among these probabilities are linear [3, 4, 5, 7]. A tree-like description of these linear relationships is often called a “realization plan.” Here we generalize the idea of a realization plan so that, in some cases, a player without perfect recall may also limit consideration to mixed strategies described by a small set of linearly related probabilities. We then describe techniques whereby such descriptions might be found and classes of games where these techniques might be effective. In the worst case, the generalized plans are too large to be useful. However, individual games with good enough recall will have small generalized realization plans. The point is that, whenever results are obtained using path probabilities, the results may immediately extend to certain more general situations merely by replacing traditional realization plans with the more general plans. To demonstrate this point, we define a class of near perfect recall games where the number of parameters is linear in the size of the game tree.