3x3游戏中的虚构游戏：周期性和混沌行为之间的过渡
Fictitious Play in 3x3 Games: the transition between periodic and chaotic 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。本文通过引入一个外参数推广了Shapley的例子。我们表明，在沙普利的例子中，在一些关键参数值的周期性行为分解成不可预测的（混乱）的行为，与球员之间的不同策略抖动了大量的时间。在另一个临界参数下，动态再次变成周期性的，并且两个参与者的目标是在另一个之前玩一个。我们研究了两个玩家连续时间双矩阵虚拟游戏的动态与动态的一个参数家庭的$3 \times 3$游戏，其中包括一个著名的例子Shapley的特殊情况。本文采用几何（动力系统）方法研究简单周期轨道的分岔。在这里，我们集中在周期性的行为，而在续集中，我们将描述的混沌行为。
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 generalize Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again and both players aim to play one ahead of the other. We study the dynamics of a two player continuous time bimatrix fictitious play with the dynamics of a one-parameter family of $3 \times 3$ games that includes a well-known example of Shapley's as a special case. In this paper we adopt a geometric (dynamical systems) approach and study the bifurcations of simple periodic orbits. Here we concentrate on the periodic behaviour, while in a sequel we shall describe the chaotic behaviour.
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