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Combinatorial games played randomly: Chomp and nim

Pat Devlin, Paulina Trifonova

TL;DR

This model provides closed-form expressions for the expected number of turns in a game of Chomp with any starting condition and derive and prove formulas for the win probabilities for any game of Chomp with at most two rows.

Abstract

In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position.

Combinatorial games played randomly: Chomp and nim

TL;DR

This model provides closed-form expressions for the expected number of turns in a game of Chomp with any starting condition and derive and prove formulas for the win probabilities for any game of Chomp with at most two rows.

Abstract

In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position.
Paper Structure (6 sections, 6 theorems, 43 equations, 1 table)

This paper contains 6 sections, 6 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. General notation
  3. Expected game length in Chomp
  4. Two-rowed Chomp: win probability
  5. Analyzing nim: proof of Theorem \ref{['theorem:nim']}
  6. Concluding remarks

Key Result

Theorem 1

A game of random Chomp starting starting from $B$ has expected length where the summation is taken over all cells $(x,y)$ in the board $B$. In particular, if $B$ is an $m \times n$ rectangle, then the expected length is $\sum_{j=1}^{m} \frac{1}{j} \times \sum_{j=1}^{n} \frac{1}{j}$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 1
  • Conjecture 2
  • Theorem 5
  • Lemma 1
  • proof