The Complexity of Interior Point Methods for Solving Discounted Turn-Based Stochastic Games

We study the problem of solving discounted, two player, turn based, stochastic games (2TBSGs). Jurdziński and Savani showed that in the case of deterministic games the problem can be reduced to solving P -matrix linear complementarity problems (LCPs). We show that the same reduction also works for general 2TBSGs. This implies that a number of interior point methods can be used to solve 2TBSGs. We consider two such algorithms: the unified interior point method of Kojima, Megiddo, Noma, and Yoshise, and the interior point potential reduction algorithm of Kojima, Megiddo, and Ye. The algorithms run in time O((1 + κ)nL) and O(−δ θ n 4 log −1), respectively, when applied to an LCP defined by an n× n matrix M that can be described with L bits, and where the potential reduction algorithm returns an -optimal solution. The parameters κ, δ, and θ depend on the matrix M . We show that for 2TBSGs with n states and discount factor γ we get κ = Θ( n (1−γ)2 ), −δ = Θ( √ n 1−γ ), and 1/θ = Θ( n (1−γ)2 ) in the worst case. The lower bounds for κ, −δ, and 1/θ are all obtained using the same family of deterministic games.