General limit value in Dynamic Programming

We consider a dynamic programming problem with arbitrary state space and bounded rewards. Is it possible to define in an unique way a limit value for the problem, where the "patience" of the decision-maker tends to infinity ? We consider, for each evaluation $θ$ (a probability distribution over positive integers) the value function $v_θ$ of the problem where the weight of any stage $t$ is given by $θ_t$, and we investigate the uniform convergence of a sequence $(v_{θ^k})_k$ when the "impatience" of the evaluations vanishes, in the sense that $\sum_{t} |θ^k_{t}-θ^k_{t+1}| \rightarrow_{k \to \infty} 0$. We prove that this uniform convergence happens if and only if the metric space ${v_{θ^k}, k\geq 1}$ is totally bounded. Moreover there exists a particular function $v^*$, independent of the particular chosen sequence $({θ^k})_k$, such that any limit point of such sequence of value functions is precisely $v^*$. Consequently, while speaking of uniform convergence of the value functions, $v^*$ may be considered as the unique possible limit when the patience of the decision-maker tends to infinity. The result applies in particular to discounted payoffs when the discount factor vanishes, as well as to average payoffs where the number of stages goes to infinity, and also to models with stochastic transitions. We present tractable corollaries, and we discuss counterexamples and a conjecture.

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