在$3\times 3$游戏中玩虚拟游戏：混乱和抖动行为
Fictitious Play in $3\times 3$ games: chaos and dithering behaviour
摘要：
在60年代沙普利提供了一个例子，两个球员虚构的游戏与周期性的行为。在这个游戏中，玩家$A$的目标是复制$B$的行为，玩家$B$的目标是在玩家$A$之前玩一个。本文继续研究一类博弈，通过引入外部参数推广了Shapley的例子，证明了博弈中存在大量的周期性和混沌行为，参与者在不同的策略之间摇摆不定。所有这一切的原因，是存在一个周期性轨道（包括播放混合策略），这是{ \em“抖动类型”}：这样的轨道既不是吸引，排斥或鞍型，因为附近的轨道抖动接近和远离它的方式，这是令人想起一个随机游动运动。我们证明了这种行为的开集的游戏。
英文摘要：
In the 60's Shapley provided an example of a two player fictitious game with periodic behaviour. In this game, player $A$ aims to copy $B$'s behaviour and player $B$ aims to play one ahead of player $A$. In this paper we continue to study a family of games which generalize Shapley's example by introducing an external parameter, and prove that there exists an abundance of periodic and chaotic behavior with players dithering between different strategies. The reason for all this, is that there exists a periodic orbit (consisting of playing mixed strategies) which is of {\em `jitter type'}: such an orbit is neither attracting, repelling or of saddle type as nearby orbits jitter closer and further away from it in a manner which is reminiscent of a random walk motion. We prove that this behaviour holds for an open set of games.
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