First-Past-the-Post Games

Informally, a rst-past-the-post game is a (probabilistic) game where the winner is the person who predicts the event that occurs rst among a set of events. Examples of rst-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 rst-past-the-post game, and the process of extending a probability distribution on symbols of an alphabet to the plays of a game. We establish a number of properties of such games, for example , the property that an incomplete rst-past-the-post game is also a rst-past-the-post game. Penney-Ante games are multi-player games characterised by a collection of regular, pre x-free languages. Analysis of such 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. For such games, we show how to use the equations in languages to calculate the probability of winning and how to calculate the expected length of a game for a given outcome. We also exploit the properties of rst-past-the-post games to show how to calculate the probability of winning in the course of a play of the game. Our analysis does not depend on the construction of a deterministic nite-state machine or use generating functions, the two methods traditionally used for the task. We observe that Aho and Corasick's generalisation of the Knuth-Morris-Pratt pattern-matching algorithm can be used to construct the deterministic nite-state machine that recognises the language underlying a Penney-Ante game. The two methods of calculating the probabilities and expected values, one based on the nitestate machine and the other based on the non-linear equations in languages, have been implemented and veri ed to yield the same results. Penney-Ante is the name of a game with pennies invented by Walter Penney [Pen74]. The two-player game has attracted much interest because it is non-transitive [Gar74]; ∗School of Computer Science, University of Nottingham, Nottingham NG8 1BB, England