Tauberian Theorem for Value Functions

作     者：Khlopin, Dmitry 

作者机构：Russian Acad Sci Krasovskii Inst Math & Mech Ural Branch 16 S Kovalevskaja St Ekaterinburg 620990 Russia Ural Fed Univ Inst Math & Comp Sci Chair Appl Math 4 Turgeneva St Ekaterinburg 620083 Russia 

出 版 物：《DYNAMIC GAMES AND APPLICATIONS》 (动态博弈与应用)

年 卷 期：2018年第8卷第2期

页      面：401-422页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Russian Foundation for Basic Research [16-01-00505] 

主　　题：Dynamic Programming Principle Zero-sum games Abel mean Cesaro mean Differential games 

摘      要：For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of -discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian theorem-that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point;a very simple stochastic game model is also considered.