Effect of Reward Function Choices in MDPs with Value-at-Risk
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This paper studies Value-at-Risk (VaR) problems in short- and long-horizon Markov decision processes (MDPs) with finite state space and two different reward functions. Firstly we examine the effects of two reward functions under two criteria in a short-horizon MDP. We show that under the VaR criterion, when the original reward function is on both current and next states, the reward simplification will change the VaR. Secondly, for long-horizon MDPs, we estimate the Pareto front of the total reward distribution set with the aid of spectral theory and the central limit theorem. Since the estimation is for a Markov process with the simplified reward function only, we present a transformation algorithm for the Markov process with the original reward function, in order to estimate the Pareto front with an intact total reward distribution.
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