Digital Object Identifier (DOI) 10.1007/s101079900100 Math. Program., Ser. A 86: 637–648 (1999)  Springer-Verlag 1999 S.R. Mohan · S.K. Neogy · T. Parthasarathy · S. Sinha Vertical linear complementarity and discounted zero-sum stochastic games with ARAT structure Received July 8, 1998 / Revised version received April 16, 1999 Published online September 15, 1999 Abstract. In this paper we consider a two-person zero-sum discounted stochastic game with ARAT structure and formulate the problem of computing a pair of pure optimal stationary strategies and the corresponding value vector of such a game as a vertical linear complementarity problem. We show that Cottle-Dantzig’s algorithm (a generalization of Lemke’s algorithm) can solve this problem under a mild assumption. Key words. ARAT – Cottle-Dantzig’s algorithm – VLCP 1. Introduction In this paper we consider a two-person discounted zero-sum stochastic game in which for each state s, Player I and Player II have a finite set of actions A s and B s respectively. Let S be the set of states and let k be its cardinality. When the game is played in state s, Player I chooses an action i ∈ A s and Player II chooses an action j ∈ B s , the payoff to Player I is r(s, i , j); the payoff to Player II is −r(s, i , j). The game makes a transition to state t with probability p(t |s, i , j) on the next day. The stream of resulting payoffs to Player I over an infinite number of days, i.e., the time horizon of the game, is evaluated by the total discounted sum ∑ ∞ N=1 β N−1 r(s, i , j) assuming that on day N the game is played in state s, and the actions chosen by players are i and j respectively. The transition probability p(t |s, i , j) and the reward function r(s, i , j) satisfy the following additive property: p(t |s, i , j) = p 1 (t |s, i ) + p 2 (t |s, j) r(s, i , j) = r 1 (s, i ) + r 2 (s, j) Due to this additive property assumed on the transition and reward functions, the game is called β-discounted zero-sum ARAT(Additive Reward & Additive Transition) Game. As is usual in game theory, players are allowed to choose a probability distribution over the set of actions available to them in each state and then choose an action with the probability specified by the chosen distribution. The space of probability distributions over A s is called the space of mixed strategies for player I in state s. A mixed strategy that assigns probability mass 1 to a particular action is called a pure strategy. In a stochastic game the players are required to choose a mixed strategy each day and such a sequence S.R. Mohan, S.K. Neogy, T. Parthasarathy: Indian Statistical Institute, New Delhi-110016, India e-mail: {srm,skn,tps}@isid.ac.in S. Sinha: Jadavpur University, Calcutta-700 032, India