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We discuss the application of indirect learning methods in zero-sum and identical payoff learning automata games. We propose a novel decentralized version of the well-known pursuit learning algorithm. Such a decentralized algorithm has significant computational advantages over its centralized counterpart. The theoretical study of such a decentralized algorithm requires the analysis to be carried out in a nonstationary environment. We use a novel bootstrapping argument to prove the convergence of the algorithm. To our knowledge, this is the first time that such analysis has been carried out for zero-sum and identical payoff games. Extensive simulation studies are reported, which demonstrate the proposed algorithm's fast and accurate convergence in a variety of game scenarios. We also introduce the framework of partial communication in the context of identical payoff games of learning automata. In such games, the automata may not communicate with each other or may communicate selectively. This comprehensive framework has the capability to model both centralized and decentralized games discussed in this paper.