Borel状态空间中平均零和随机博弈的新条件  

New conditions for zero-sum stochastic games with average criteria in Borel space

作　　者：郭先平[1] 廖景浩 谭梓祺 温馨 Xianping Guo;Jinghao Liao;Ziqi Tan;Xin Wen

机构地区：[1]中山大学数学学院,广州510275 [2]中山大学管理学院,广州510275

出　　处：《中国科学：数学》2024年第12期1963-1978,共16页Scientia Sinica：Mathematica

基　　金：国家重点研发计划(批准号:2022YFA1004600);国家自然科学基金(批准号:11931018,72342006和72301304)资助项目。

摘　　要：本文研究Borel状态空间的离散时间Markov平均博弈.对报酬函数可以无界的一般情形,本文用平均最优双不等式取代相应的Shapley方程,提出比现有的几何遍历性条件更弱的新条件.在此新的条件下,本文建立上述平均最优双不等式的可解性,并由此证明平均博弈的值和Nash均衡策略的存在性.进而,在较强的几何遍历性条件下,用本文的最优双不等式,证明Shapley方程的可解性.最后,用电力系统与金融保险中的例子验证本文的条件,阐明本文的结果.In this paper,we study the expected average criterion in discrete-time Markov games with Borel spaces.For the general case of unbounded reward functions,we first replace the corresponding Shapley equation for the average criterion with average-optimality two-inequalities.Then,by using the relative difference of the values of the discounted games,we give a new set of optimality conditions,which are weaker than the geometric ergodicity condition in the existing literature.Under these new conditions,we not only establish the solvability of the average-optimality two-inequalities but also show the existence of both the value and a Nash equilibrium of the game.Moreover,under the stronger geometric ergodicity condition,by the average-optimality two-inequalities,we also establish the solvability of the Shapley equation.Finally,we present two examples of renewable resources and financial insurance to verify the conditions and illustrate the results in this paper.

关 键 词：零和平均随机博弈 最优性条件 平均最优双不等式 Shapley方程 Nash均衡策略 

分 类 号：O225[理学—运筹学与控制论]