《罗宾汉》的温度
Temperatures of Robin Hood
AI总结：
该研究聚焦累积游戏实例《罗宾汉》的温度特性，发现其温度随玩家财富比率呈二分变化，与黄金比例及斐波那契序列相关，且高温下走最热组件未必是最优策略。
摘要：
累积游戏（Cumulative Games）由Larsson、Meir和Zick于2020年提出，旨在弥合组合博弈论（CGT）与经济博弈论之间的部分概念与技术鸿沟。分方规则集{\textsc Robin Hood}（《罗宾汉》）是累积游戏的一个实例，即{\textsc Wealth Nim}（财富尼姆游戏）。该游戏在多堆棋子上进行，每堆对应一对累积值，在此处被解释为财富。每名玩家选择其中一堆，从该堆移除不超过自身财富数量的棋子，同时将对手的财富减少相同数量。在组合博弈论中，析取和游戏组件的温度是对在该组件中先行走子紧迫性的估计。研究发现，《罗宾汉》的大多数局面都是“热”的。单个大堆上《罗宾汉》的温度表现出二分行为，这取决于玩家的财富比率。有趣的是，这种分岔与平加拉（斐波那契）序列以及黄金比例$ϕ$相关：当财富比率处于区间$(ϕ^{-1},ϕ)$内时，温度随堆的规模线性增长，否则温度保持恒定，且均值具有倒数性质。研究还发现，尽管《罗宾汉》呈现出高温特性，但在最热的组件中走子可能是一种次优策略。
英文摘要：
Cumulative Games were introduced by Larsson, Meir, and Zick (2020) to bridge some conceptual and technical gaps between Combinatorial Game Theory (CGT) and Economic Game Theory. The partizan ruleset {\sc Robin Hood} is an instance of a Cumulative Game, viz., {\sc Wealth Nim}. It is played on multiple heaps, each associated with a pair of cumulations, interpreted here as wealth. Each player chooses one of the heaps, removes tokens from that heap not exceeding their own wealth, while simultaneously diminishing the other player's wealth by the same amount. In CGT, the {\em temperature} of a {\em disjunctive sum} game component is an estimate of the urgency of moving first in that component. It turns out that most of the positions of {\sc Robin Hood} are {\em hot}. The temperature of {\sc Robin Hood} on a single large heap shows a dichotomy in behavior depending on the ratio of the wealths of the players. Interestingly, this bifurcation is related to Pingala (Fibonacci) sequences and the Golden Ratio $ϕ$: when the ratio of the wealths lies in the interval $(ϕ^{-1},ϕ)$, the temperature increases linearly with the heap size, and otherwise it remains constant, and the mean values has a reciprocal property. It turns out that despite {\sc Robin Hood} displaying high temperatures, playing in the hottest component might be a sub-optimal strategy.
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