On cases where Litt's game is fair

TL;DR

This paper proves that Litt's game, involving scoring based on the appearance of two words in coin flips, is fair when the words share the same auto-correlation structure, regardless of sequence length.

Contribution

It establishes the fairness of Litt's game for words with identical auto-correlation, using a bijection that swaps Alice and Bob's scores, revealing a surprising invariance.

Findings

Litt's game is fair for words with the same auto-correlation.

A bijection exchanges Alice and Bob's scores, proving fairness.

Auto-correlation structure determines game fairness.

Abstract

A fair coin is flipped $n$ times, and two finite sequences of heads and tails (words) $A$ and $B$ of the same length are given. Each time the word $A$ appears in the sequence of coin flips, Alice gets a point, and each time the word $B$ appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt's game that recently set Twitter abuzz. We show that Litt's game is fair for any value of $n$ and any two words that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores; the fact that the inter-correlation does not come into play in this case may come up as a surprise.