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Optimal Strategy for the First Player in the Penney Ante Game

Published online by Cambridge University Press:  12 September 2008

János A. Csirik
János A. Csirik
Affiliation:
Trinity College, Cambridge, England
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Abstract

In the Penney Ante game two players choose one binary string of length k each in turn, and toss a coin repeatedly. If at some stage the last k outcomes match one of their strings, the player with that string wins. The case k ≤ 4 is somewhat exceptional and in any case easily done. For k ≥ 5, Guibas and Odlyzko proved that the second player's optimal strategy is to choose the first k − 1 digits of the first player's string prefixed by 0 or 1. They conjectured that these two choices are never equally good. We prove that this conjecture is correct. Then we prove that 01…100 (with k − 1 ones) is an optimal strategy for the first player, and find all the strategies that are equally good.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 4 , December 1992 , pp. 311 - 321
Copyright
Copyright © Cambridge University Press 1992

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References

[1]Coxeter, H. S. M. (1969) Introduction to Geometry, John Wiley & Sons 209.Google Scholar
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