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Markov decision chains; second order optimality; optimality conditions for transient; discounted and average models; policy iterations; value iterations
The article is devoted to Markov reward chains in discrete-time setting with finite state spaces. Unfortunately, the usual optimization criteria examined in the literature on Markov decision chains, such as a total discounted, total reward up to reaching some specific state (called the first passage models) or mean (average) reward optimality, may be quite insufficient to characterize the problem from the point of a decision maker. To this end it seems that it may be preferable if not necessary to select more sophisticated criteria that also reflect variability-risk features of the problem. Perhaps the best known approaches stem from the classical work of Markowitz on mean variance selection rules, i. e. we optimize the weighted sum of average or total reward and its variance. The article presents explicit formulae for calculating the variances for transient and discounted models (where the value of the discount factor depends on the current state and action taken) for finite and infinite time horizon. The same result is presented for the long run average nondiscounted models where finding stationary policies minimizing the average variance in the class of policies with a given long run average reward is discussed.
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