Advertisement

First-Past-the-Post Games

  • Roland Backhouse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7342)

Abstract

Informally, a first-past-the-post game is a (probabilistic) game where the winner is the person who predicts the event that occurs first among a set of events. Examples of first-past-the-post games include so-called block and hidden patterns and the Penney-Ante game invented by Walter Penney. We formalise the abstract notion of a first-past-the-post game, and the process of extending a probability distribution on symbols of an alphabet to the plays of a game.

Analysis of first-past-the-post games depends on a collection of simultaneous (non-linear) equations in languages. Essentially, the equations are due to Guibas and Odlyzko but they did not formulate them as equations in languages but as equations in generating functions detailing lengths of words.

Penney-Ante games are two-player games characterised by a collection of regular, prefix-free languages. For such two-player games, we show how to use the equations in languages to calculate the probability of winning. The formula generalises a formula due to John H. Conway for the original Penney-Ante game. At no point in our analysis do we use generating functions. Even so, we are able to calculate probabilities and expected values. Generating functions do appear to become necessary when higher-order cumulatives (for example, the standard deviation) are also required.

Keywords

algorithmic problem solving regular language generating function probabilistic game Penney-Ante block pattern hidden pattern 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bac86]
    Backhouse, R.C.: Program Construction and Verification. Prentice-Hall International (1986)Google Scholar
  2. [Bac03]
    Backhouse, R.: Program Construction. Calculating Implementations From Specifications. John Wiley & Sons, Ltd. (2003)Google Scholar
  3. [FS09]
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009)Google Scholar
  4. [GKP94]
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: a Foundation for Computer Science, 2nd edn. Addison-Wesley Publishing Company (1994)Google Scholar
  5. [GO81]
    Guibas, L.J., Odlyzko, A.M.: String overlaps, pattern matching and nontransitive games. Journal of Combinatorial Theory, Series A30, 183–208 (1981)Google Scholar
  6. [GS93]
    Gries, D., Schneider, F.B.: A Logical Approach to Discrete Math. Springer (1993)Google Scholar
  7. [Pen74]
    Penney, W.: Problem 95: Penney-Ante. Journal of Recreational Mathematics, 321 (1974)Google Scholar
  8. [Sol66]
    Solov’ev, A.D.: A combinatorial identity and its application to the problem concerning the first occurrence of a rare event. Theory of Probability and its Applications 11, 276–282 (1966)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Roland Backhouse
    • 1
  1. 1.School of Computer ScienceUniversity of NottinghamNottinghamEngland

Personalised recommendations