International Journal of Game Theory (1993) 22:1-11 Game Theory Asymptotic Properties in Dynamic Programming Dov MONDERER Department of Economics, Queen's University, Kingston, Canada, K7L 3N6, Canada and Faculty of Industrial Engineering and Management, the Technion, Haifa, Israel SYLVAIN SORIN DMJ (URA CNRS 762), Ecole Normale Sup6rieure, 45 rue d'Ulm, 75230 Paris, France Abstract: In the framework of dynamic programming we provide two results: - An example where uniform convergence of the T-stage value does not imply equality of the limit and the lower infinite value. - Generalized Tauberian theorems, that relate uniform convergence of the T-stage value to uniform convergence of values associated with a general distribution on stages. 1 Introduction Let S be a state space. For each sES let OCF(s)C_S, and let f be a real bounded function on S. Consider the dynamic programming problem where the decision maker on day t, at stage st, has to choose a new state st+lEF(st), and receives a payoff f(st). A play at ssS is a sequence (st)~~ with So=S and st+~ ~F(st) for all t ~ 0. One traditionally considers the X-discounted value Vx (s): oo V~(s)= sup (1-2) ~. Xtf(st), (st) 7~o t = 0 or the T-stage value Vr(s): 1 T VT(S) = sup - - ~, f(s,), (s,)Y=o T+ 1 t=o where in both cases the supremum ranges over all plays at s. One can also consider other evaluations: Let 0= (0(t))F=o be a probability on the set of non-negative integers and define: oo Vo(s)= sup ~, O(t) f(st). (st) t= o t=O Lehrer and Sorin (1992) proved that if either one of the limits limz~l Vz(s), or limr~= Vr(s) exists uniformly in s~S, then the other limit also exists uniformly, and the limit functions coincide. In Section 3 we give sufficient conditions on linearly ordered families (| <) of probabilities on the integers to get analogous results for (Vo)o~o and (Vr)r>_o- This research was supported by the fund for the promotion of research in the technion. 0020-7276/93 / 1 / 1-11 $ 2.50 9 1993 Physica-Verlag, Heidelberg