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Temperatures of Robin Hood

Ankita Dargad, Urban Larsson, Niranjan Balachandran

TL;DR

This work analyzes the Robin Hood (Wealth Nim) game as a disjunctive sum of single-heap components, focusing on the temperature and mean value that quantify urgency and desirability of moving first. By bridging Robin Hood with a simplified Little John game and leveraging Pingala/MП-sequence structure, it derives a complete description of the temperature and mean value on large heaps, revealing a sharp dependence on the wealth ratio relative to the golden ratio φ. The main result yields explicit piecewise formulas for t(G_n) and m(G_n) and shows that Robin Hood and Little John share identical thermographs in the large-heap regime, enabling a unified analysis of hot positions. The findings illuminate when starting is advantageous, how the ratio a/b governs hotness, and open avenues for middle-region behavior, canonical forms, and broader Wealth Nim variants.

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 $φ$: 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.

Temperatures of Robin Hood

TL;DR

This work analyzes the Robin Hood (Wealth Nim) game as a disjunctive sum of single-heap components, focusing on the temperature and mean value that quantify urgency and desirability of moving first. By bridging Robin Hood with a simplified Little John game and leveraging Pingala/MП-sequence structure, it derives a complete description of the temperature and mean value on large heaps, revealing a sharp dependence on the wealth ratio relative to the golden ratio φ. The main result yields explicit piecewise formulas for t(G_n) and m(G_n) and shows that Robin Hood and Little John share identical thermographs in the large-heap regime, enabling a unified analysis of hot positions. The findings illuminate when starting is advantageous, how the ratio a/b governs hotness, and open avenues for middle-region behavior, canonical forms, and broader Wealth Nim variants.

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 : when the ratio of the wealths lies in the interval , 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.
Paper Structure (9 sections, 27 theorems, 27 equations, 6 figures, 2 tables)

This paper contains 9 sections, 27 theorems, 27 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Some basics
  4. Sherwood Organization
  5. Robin Hood meets Little John
  6. Pingala sequences and their properties
  7. Little John's Ruleset
  8. Little John guides Robin Hood
  9. A solution for the Pingala Era wetland tribes

Key Result

Theorem 1.1

Let $G = (n;a,b)$ be an instance of Robin Hood, where $n$, $a$ and $b$ are positive integers. Let $(U_k)_{k\geq 0}$ be the unique sequence of positive integers such that If $n\geq a+b$, then $G$ is a hot game and for all sufficiently large $n$, the temperature of the game $G$, denoted by $t(G)$, is and the mean value of the game $G$, denoted by $m(G)$, is

Figures (6)

  • Figure 1: A three days' war started by the red tribe. The war ended on the 3rd day as only one tribe is left on each island. Ruined land pieces and beaten soldiers are colored gray. The colored arrows specify the attacking tribe.
  • Figure 2: The thermograph of $G=\Bigl\{6, \bigl\{10\mid\{5\mid3\}\bigr\}\mid-5\Bigr\}$.
  • Figure 3: Mast and Tents.
  • Figure 4: The Little John Paths of $(n;a,b)$, for $a/b = 1.4$ and sufficiently large $n$.
  • Figure 5: Thermographs of $(n;a,b)^*$ for small $a$ and $b$.
  • ...and 1 more figures

Theorems & Definitions (70)

  • Theorem 1.1: Main Theorem
  • Theorem 1.2: No Hotstrat
  • Theorem 2.1: cole1969game
  • Definition 3.1: Numbers
  • Definition 3.2: Mean Value
  • Theorem 3.3
  • proof
  • Definition 3.4: Stops
  • Remark 3.4.1
  • Proposition 3.5: S2013*Proposition 3.17
  • ...and 60 more