On cases where Litt’s game is fair

Anne-Laure Basdevant; Olivier Hénard; Édouard Maurel-Segala; Arvind Singh

doi:10.5802/crmath.783

Research article - Probability theory

On cases where Litt’s game is fair

Anne-Laure Basdevant ¹; Olivier Hénard ²; Édouard Maurel-Segala ²; Arvind Singh ²

¹Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, 75205 Paris, France
²Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 977-984

Abstract (Original language)
Résumé

A fair coin is flipped $n$ times, and two finite sequences of heads and tails with the same length are given, say $A$ and $B$. Each time $A$ appears in the sequence of fair coin flips, Alice gets a point, and each time $B$ appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt’s game [10]. In this note, we show that the game is fair for any value of $n$ and any two words $A$, $B$ that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores. It is remarkable that the inter-correlations between $A$ and $B$ do not play any role in this case. Additionally, we propose a conjecture for cases where the game is unfair, providing insights into the underlying structure of the game for a fixed $n$.

Une pièce équilibrée est lancée $n$ fois, et deux suites finies $A$ et $B$ de piles et faces de même longueur sont données. Chaque fois que $A$ apparaît dans la suite de lancers de la pièce, Alice gagne un point, et chaque fois que $B$ apparaît, Bob gagne un point. Qui a le plus de chances de gagner ? Ce problème est une légère extension du jeu de Litt [10]. Nous montrons que le jeu est équitable pour toute valeur de $n$ et pour toute paire de mots $A$, $B$ ayant la même structure d’auto-corrélation, en construisant une bijection qui échange les scores de Bob et Alice. Le fait que les inter-corrélations entre $A$ et $B$ ne jouent aucun rôle ici est remarquable. De plus, nous proposons une conjecture pour les cas où le jeu est inéquitable, qui témoigne de la structure riche du jeu de taille finie.

Received: 2025-03-26
Revised: 2025-06-19
Accepted: 2025-07-14
Published online: 2025-09-05

DOI: 10.5802/crmath.783

Classification: 60C05
Keywords: Coin flips, pattern counts, stochastic games
Mots-clés : Pile ou face, motifs, jeu aléatoire

Author's affiliations:

Anne-Laure Basdevant ¹ ; Olivier Hénard ² ; Édouard Maurel-Segala ² ; Arvind Singh ²

¹ Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, 75205 Paris, France
² Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Anne-Laure Basdevant and Olivier H\'enard and \'Edouard Maurel-Segala and Arvind Singh},
     title = {On cases where {Litt{\textquoteright}s} game is fair},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {977--984},
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Anne-Laure Basdevant; Olivier Hénard; Édouard Maurel-Segala; Arvind Singh. On cases where Litt’s game is fair. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 977-984. doi: 10.5802/crmath.783

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