简化的三人库恩扑克
Simplified three player Kuhn poker
摘要：
我们研究了一个非常小的三人扑克游戏（三分之一街库恩扑克），和一个简化版本的游戏是有趣的，因为它有三个不同的均衡解。对于三分之一街库恩扑克，我们能够找到所有的均衡解分析。对于足够大的底池大小，$P$，在解决方案中有一定的自由度，允许一个玩家在其他两个玩家之间转移利润而不改变他们自己的利润。这在重复玩游戏时具有潜在的有趣后果。我们还表明，在一个简化版本的游戏与$P>5$，有一个平衡的解决方案，如果$5 < P < P^* \equiv (5+\sqrt{73})/2$，和三个不同的平衡解决方案，如果$P > P^*$。这可能是最简单的非平凡的多人扑克游戏与一个以上的不同的均衡解决方案，并为我们提供了一个测试案例的动态策略调整理论在多个实现的游戏。然后，我们研究了三阶系统的常微分方程模型的动态三个球员谁试图最大限度地提高他们的期望，不断改变他们的投注频率。我们发现这个系统的动力学是振荡的，有两种不同类型的解决方案。然后，我们研究了一个差分方程模型，基于重复玩的游戏，其中每个球员不断更新他们的估计其他球员的投注频率。我们发现，动态是嘈杂的，但基本上振荡足够短的估计周期和足够慢的频率调整，但其他参数值的动态可以是非常不同的。
英文摘要：
We study a very small three player poker game (one-third street Kuhn poker), and a simplified version of the game that is interesting because it has three distinct equilibrium solutions. For one-third street Kuhn poker, we are able to find all of the equilibrium solutions analytically. For large enough pot size, $P$, there is a degree of freedom in the solution that allows one player to transfer profit between the other two players without changing their own profit. This has potentially interesting consequences in repeated play of the game. We also show that in a simplified version of the game with $P>5$, there is one equilibrium solution if $5 < P < P^* \equiv (5+\sqrt{73})/2$, and three distinct equilibrium solutions if $P > P^*$. This may be the simplest non-trivial multiplayer poker game with more than one distinct equilibrium solution and provides us with a test case for theories of dynamic strategy adjustment over multiple realisations of the game. We then study a third order system of ordinary differential equations that models the dynamics of three players who try to maximise their expectation by continuously varying their betting frequencies. We find that the dynamics of this system are oscillatory, with two distinct types of solution. We then study a difference equation model, based on repeated play of the game, in which each player continually updates their estimates of the other players' betting frequencies. We find that the dynamics are noisy, but basically oscillatory for short enough estimation periods and slow enough frequency adjustments, but that the dynamics can be very different for other parameter values.
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