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Generalized Three and Four Person Hat Game

Theo van Uem

TL;DR

This work tackles generalized Ebert’s hat problem for 3 and 4 players with two colors, allowing per-player color probabilities and simultaneous guesses with optional passes. It introduces the adequate-set framework and two accompanying algorithms, ASG and DMG, to derive winning strategies independent of the underlying probabilities and then tailor them to asymmetric and symmetric subcases. The authors derive explicit optimal win probabilities and corresponding decision matrices across multiple regimes, showing when adding a fourth player improves performance and when it does not, and provide a computational complexity analysis demonstrating substantial efficiency gains over brute-force search. Overall, the paper advances a codified, probability-agnostic method for designing optimal hat-game strategies and clarifies the impact of player asymmetry on performance, with practical implications for combinatorial game design and coding-theoretic connections.

Abstract

This paper studies Ebert's hat problem for three and four players and two colors, where the probabilities of the colors may be different for each player. Our goal is to maximize the probability of winning the game and to describe winning strategies We use the concept of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of our general case.

Generalized Three and Four Person Hat Game

TL;DR

This work tackles generalized Ebert’s hat problem for 3 and 4 players with two colors, allowing per-player color probabilities and simultaneous guesses with optional passes. It introduces the adequate-set framework and two accompanying algorithms, ASG and DMG, to derive winning strategies independent of the underlying probabilities and then tailor them to asymmetric and symmetric subcases. The authors derive explicit optimal win probabilities and corresponding decision matrices across multiple regimes, showing when adding a fourth player improves performance and when it does not, and provide a computational complexity analysis demonstrating substantial efficiency gains over brute-force search. Overall, the paper advances a codified, probability-agnostic method for designing optimal hat-game strategies and clarifies the impact of player asymmetry on performance, with practical implications for combinatorial game design and coding-theoretic connections.

Abstract

This paper studies Ebert's hat problem for three and four players and two colors, where the probabilities of the colors may be different for each player. Our goal is to maximize the probability of winning the game and to describe winning strategies We use the concept of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of our general case.

Paper Structure

This paper contains 16 sections, 18 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Adequate set
  3. Generalized three person two color Hat Game
  4. Asymmetric three person two color Hat Game
  5. Symmetric two color three person Hat Game
  6. Generalized four person two color Hat Game
  7. Asymmetric four person Two Color Hat Game
  8. Symmetric four person two color Hat Game
  9. Comparison between three and four person Hat Game
  10. Computational complexity
  11. ASG three persons
  12. DMG three persons
  13. Adequate sets; three players; das=3
  14. Dominated adequate sets; General case (three persons, two colors, das=4)
  15. four persons, das=4,
  16. ...and 1 more sections

Key Result

Theorem 1

BAD CASES are adequate to GOOD CASES.

Theorems & Definitions (37)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • ...and 27 more