Temperatures of Robin Hood

TL;DR

This paper investigates the temperature of Robin Hood, a Cumulative Game bridging combinatorial and economic game theory, revealing a dichotomy in behavior related to wealth ratios and Fibonacci sequences.

Contribution

It introduces the concept of temperature in Robin Hood, analyzes its behavior depending on wealth ratios, and uncovers a connection to Fibonacci sequences and the Golden Ratio.

Findings

Most Robin Hood positions are hot.

Temperature depends on wealth ratio, showing bifurcation.

High temperature does not necessarily imply optimal play.

Abstract

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 $\phi$: when the ratio of the wealths lies in the interval $(\phi^{-1},\phi)$, 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.