Stochastic fictitious play with continuous action sets
将随机虚拟博弈学习模型从有限行动集扩展到连续行动集，利用Banach空间上的随机逼近方法分析其极限行为，证明在两人零和博弈和负定单种群博弈中收敛到均衡。
Continuous action space games are ubiquitous in economics. However, whilst learning dynamics in normal form games with finite action sets are now well studied, it is not until recently that their continuous action space counterparts have been examined. We extend stochastic fictitious play to the continuous action space framework. In normal form games with finite action sets the limiting behaviour of a discrete time learning process is often studied using its continuous time counterpart via stochastic approximation. In this paper we study stochastic fictitious play in games with continuous action spaces using the same method. This requires the asymptotic pseudo-trajectory approach to stochastic approximation to be extended to Banach spaces. In particular the limiting behaviour of stochastic fictitious play is studied using the associated smooth best response dynamics on the space of finite signed measures. Using this approach, stochastic fictitious play is shown to converge to an equilibrium point in two-player zero-sum games and a stochastic fictitious play-like process is shown to converge to an equilibrium in negative definite single population games.