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We consider Markov Decision Processes (MDPs) in which every stationary policyinduces the same graph structure for the underlying Markov chain and further,the graph has the following property: if we replace each recurrent class by anode, then the resulting graph is acyclic. For such MDPs, we prove theconvergence of the stochastic dynamics associated with a version of optimisticpolicy iteration (OPI), suggested in Tsitsiklis (2002), in which the valuesassociated with all the nodes visited during each iteration of the OPI areupdated.

Optimistic Policy Iteration for MDPs with Acyclic Transient State Structure