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The generalized Alice HH vs Bob HT problem

Svante Janson, Mihai Nica, Simon Segert

Abstract

In 2024, Daniel Litt posed a simple coinflip game pitting Alice's "Heads-Heads" vs Bob's "Heads-Tails": who is more likely to win if they score 1 point per occurrence of their substring in a sequence of n fair coinflips? This attracted over 1 million views on X and quickly spawned several articles explaining the counterintuitive solution. We study the generalized game, where the set of coin outcomes, {Heads, Tails}, is generalized to an arbitrary finite alphabet A, and where Alice's and Bob's substrings are any finite A-strings of the same length. We find that the winner of Litt's game can be determined by a single quantity which measures the amount of prefix/suffix self-overlaps in each string; whoever's string has more overlaps loses. For example, "Heads-Tails" beats "Heads-Heads" in the original problem because "Heads-Heads" has a prefix/suffix overlap of length 1 while "Heads-Tails" has none. The method of proof is to develop a precise Edgeworth expansion for discreteMarkov chains, and apply this to calculate Alice's and Bob's probability to win the game correct to order O(1/n).

The generalized Alice HH vs Bob HT problem

Abstract

In 2024, Daniel Litt posed a simple coinflip game pitting Alice's "Heads-Heads" vs Bob's "Heads-Tails": who is more likely to win if they score 1 point per occurrence of their substring in a sequence of n fair coinflips? This attracted over 1 million views on X and quickly spawned several articles explaining the counterintuitive solution. We study the generalized game, where the set of coin outcomes, {Heads, Tails}, is generalized to an arbitrary finite alphabet A, and where Alice's and Bob's substrings are any finite A-strings of the same length. We find that the winner of Litt's game can be determined by a single quantity which measures the amount of prefix/suffix self-overlaps in each string; whoever's string has more overlaps loses. For example, "Heads-Tails" beats "Heads-Heads" in the original problem because "Heads-Heads" has a prefix/suffix overlap of length 1 while "Heads-Tails" has none. The method of proof is to develop a precise Edgeworth expansion for discreteMarkov chains, and apply this to calculate Alice's and Bob's probability to win the game correct to order O(1/n).

Paper Structure

This paper contains 29 sections, 26 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Cumulants
  5. Background on Edgeworth expansions
  6. Group Inverse
  7. An Edgeworth expansion for finite-state Markov chains
  8. Markov chain preliminaries
  9. The main result
  10. Coin tossing
  11. Proof of Theorems \ref{['T1']}--\ref{['T2']}
  12. A lemma
  13. Proof of Theorem \ref{['T1']}
  14. Proof of Theorem \ref{['T2']}
  15. The aperiodicity condition and non-degeneracy
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let Alice and Bob play Litt's game with distinct words $A$ and $B$ of the same length $\ell$ in an alphabet $\mathcal{A}$ with $q$ letters, and assume that $n$ letters are chosen at random, uniformly and independently. Exclude the two cases, both with $q=2$: and their variants obtained by interchanging Alice and Bob or $\mathsf{H}$ and $\mathsf{T}$ (or both). Then, with $\theta_{UU}$ given by the

Theorems & Definitions (65)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Proposition 4.1
  • ...and 55 more