Strategy Complexity of Limsup and Liminf Threshold Objectives in Countable MDPs, with Applications to Optimal Expected Payoffs

We study Markov decision processes (MDPs) with a countably infinite number of states. The $\mathrm{lim\hspace{0.17em}sup}$ (resp. $\mathrm{lim\hspace{0.17em}inf}$) threshold objective is to maximize the probability that the $\mathrm{lim\hspace{0.17em}sup}$ (resp. $\mathrm{lim\hspace{0.17em}inf}$) of the infinite sequence of directly seen rewards is non-negative. We establish the complete picture of the strategy complexity of these objectives, i.e., the upper and lower bounds on the memory required by $\epsilon$-optimal (resp. optimal) strategies. We then apply these results to solve two open problems from (Sudderth, Decisions in Economics and Finance, 2020) about the strategy complexity of optimal strategies for the expected $\mathrm{lim\hspace{0.17em}sup}$ (resp. $\mathrm{lim\hspace{0.17em}inf}$) payoff.