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Policies of Markov Decision Processes (MDPs) determine the next action toexecute from the current state and, possibly, the history (the past states).When the number of states is large, succinct representations are often used tocompactly represent both the MDPs and the policies in a reduced amount ofspace. In this paper, some problems related to the size of succinctlyrepresented policies are analyzed. Namely, it is shown that some MDPs havepolicies that can only be represented in space super-polynomial in the size ofthe MDP, unless the polynomial hierarchy collapses. This fact motivates thestudy of the problem of deciding whether a given MDP has a policy of a givensize and reward. Since some algorithms for MDPs work by finding a succinctrepresentation of the value function, the problem of deciding the existence ofa succinct representation of a value function of a given size and reward isalso considered.

On Polynomial Sized MDP Succinct Policies