The Number of Optimal Strategies in the Penney-Ante Game

TL;DR

This paper investigates the number of optimal strategies for Player I in the Penney-Ante game, deriving a recurrence relation and asymptotic estimate showing about 4% of length-$n$ strings are optimal.

Contribution

It introduces a recurrence relation for counting Player I's optimal strategies and provides the first sharp asymptotic estimate for their proportion among all strings.

Findings

Approximately 4.06% of length-$n$ strings are optimal for Player I as $n$ grows large.

A recurrence relation for the number of optimal strategies $c_n$ is established.

Asymptotic analysis reveals the fixed proportion of optimal strategies among all possible strings.

Abstract

In the Penney-Ante game, Player I chooses a head/tail string of a predetermined length $n\ge3$. Player II, upon seeing Player I's choice, chooses another head/tail string of the same length. A coin is then tossed repeatedly and the player whose string appears first in the resulting head/tail sequence wins the game. The Penney-Ante game has gained notoriety as a source of counterintuitive probabilities and nontransitivity phenomena. For example, Player II can always choose a string that beats the choice of Player I in the sense of being more likely to appear first in a random head/tail sequence. It is known that Player II has a unique optimal strategy that maximizes her winning chances in this game. On the other hand, for Player I there exist multiple equivalent optimal strategies. In this paper we investigate the number, $c_n$, of optimal strategies for Player I, i.e., the number of head/tail strings of length $n$ that maximize the winning probability for Player I assuming optimal play by Player II. We derive a recurrence relation for $c_n$ and use this to obtain a sharp asymptotic estimate for $c_n$. In particular, we show that, as $n\to\infty$, a fixed proportion $\alpha \approx 0.04062\dots$ of the $2^n$ head/tail strings of length $n$ are optimal from Player I's perspective.