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In the Penney-Ante game, Player I chooses a head/tail string of apredetermined length $n\ge3$. Player II, upon seeing Player I's choice, choosesanother head/tail string of the same length. A coin is then tossed repeatedlyand the player whose string appears first in the resulting head/tail sequencewins the game. The Penney-Ante game has gained notoriety as a source ofcounterintuitive probabilities and nontransitivity phenomena. For example,Player II can always choose a string that beats the choice of Player I in thesense 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 herwinning chances in this game. On the other hand, for Player I there existmultiple equivalent optimal strategies. In this paper we investigate thenumber, $c_n$, of optimal strategies for Player I, i.e., the number ofhead/tail strings of length $n$ that maximize the winning probability forPlayer I assuming optimal play by Player II. We derive a recurrence relationfor $c_n$ and use this to obtain a sharp asymptotic estimate for $c_n$. Inparticular, we show that, as $n\to\infty$, a fixed proportion $\alpha \approx0.04062\dots$ of the $2^n$ head/tail strings of length $n$ are optimal fromPlayer I's perspective.

The Number of Optimal Strategies in the Penney-Ante Game