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Strategies in a misère two-player tree searching game

Ben Andrews

TL;DR

The paper analyzes a misère two-player tree searching game where guesses remove vertices and reveal the component containing a hidden poisoned vertex. For path graphs, it derives exact winning probabilities for two optimal players, depending on $n$ modulo $4$, and characterizes optimal first moves; it also studies a random vs exploitative pair, proving an asymptotic exploitative win rate of about $0.598889$, with the optimal exploitative moves being $2$ or $n-1$. It extends the analysis to star graphs, showing star structures maximize exploitative advantage, and discusses extensions to more general trees, including spider and caterpillar graphs, via Nim-style couplings and related combinatorial methods. The work combines strong induction, recurrence relations, and asymptotic analysis to illuminate strategic behavior in misère tree search and provides computational tools and directions for future exploration. The results contribute to a deeper understanding of competitive search under misère objectives and offer a framework for extending to broader tree families and related combinatorial games.

Abstract

In this paper, we analyse a misere tree searching game, where players take turns to guess vertices in a tree with a secret `poisoned' vertex. After each turn, the guessed vertex is removed from the tree and the game continues on the component containing the poisoned vertex, and as soon as a player guesses the poisoned vertex, they lose. We describe and prove the solution when the game is played on a path graph, both between two optimal players and between a player who makes their decisions uniformly at random and an opponent who plays to exploit this. We show that, with two perfect players, the solution involves different guessing strategies depending on the value of n modulo 4. We then show that, with a random and an exploitative player, the probability that the exploitative player wins approaches a constant (approximately 0.599) as n increases, and that the vertices one away from the leaves of the path are always optimal guesses for them. We also solve the game played on a star graph, and briefly discuss the possibility for extending the analysis to more general trees.

Strategies in a misère two-player tree searching game

TL;DR

The paper analyzes a misère two-player tree searching game where guesses remove vertices and reveal the component containing a hidden poisoned vertex. For path graphs, it derives exact winning probabilities for two optimal players, depending on modulo , and characterizes optimal first moves; it also studies a random vs exploitative pair, proving an asymptotic exploitative win rate of about , with the optimal exploitative moves being or . It extends the analysis to star graphs, showing star structures maximize exploitative advantage, and discusses extensions to more general trees, including spider and caterpillar graphs, via Nim-style couplings and related combinatorial methods. The work combines strong induction, recurrence relations, and asymptotic analysis to illuminate strategic behavior in misère tree search and provides computational tools and directions for future exploration. The results contribute to a deeper understanding of competitive search under misère objectives and offer a framework for extending to broader tree families and related combinatorial games.

Abstract

In this paper, we analyse a misere tree searching game, where players take turns to guess vertices in a tree with a secret `poisoned' vertex. After each turn, the guessed vertex is removed from the tree and the game continues on the component containing the poisoned vertex, and as soon as a player guesses the poisoned vertex, they lose. We describe and prove the solution when the game is played on a path graph, both between two optimal players and between a player who makes their decisions uniformly at random and an opponent who plays to exploit this. We show that, with two perfect players, the solution involves different guessing strategies depending on the value of n modulo 4. We then show that, with a random and an exploitative player, the probability that the exploitative player wins approaches a constant (approximately 0.599) as n increases, and that the vertices one away from the leaves of the path are always optimal guesses for them. We also solve the game played on a star graph, and briefly discuss the possibility for extending the analysis to more general trees.

Paper Structure

This paper contains 14 sections, 10 theorems, 52 equations.

Table of Contents

  1. Overview and main results
  2. Introduction and definitions
  3. Main results
  4. Two optimal players on a path graph
  5. Random player versus exploitative player on a path graph
  6. Results
  7. Proofs of Propositions 3.1 to 3.6, and Theorem 1.2
  8. Nature of the other solutions of the $s_n$ recurrence
  9. Stars
  10. Possible directions for future research
  11. Extensions to spider graphs, caterpillar graphs and multi-pile Nim analogues
  12. Nim-style coupling for the spider graph
  13. Other extensions
  14. Appendix

Key Result

Theorem 1.1

Assuming that both players play optimally, and know this about each other, the probability that player $1$ wins in $MS(P_n)$ is given by Further, the guesses that achieve these maximum winning probabilities are described as follows.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem 1.1.
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 9 more