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arXiv · 2503.19035

The generalized Alice HH vs Bob HT problem

Svante Janson·Mihai Nica·Simon Segert
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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).

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Svante Janson, Mihai Nica, Simon Segert. 2025-03-24. The generalized Alice HH vs Bob HT problem. https://doi.org/10.1007/s10959-025-01452-7

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