First-past-the-post and Penney-Ante Games
نویسندگان
چکیده
Penney-Ante is the name of a (probabilistic) game with pennies invented by Walter Penney. Analysis of the game depends on a collection of simultaneous (nonlinear) equations in languages. In all the publications I am aware of, the equations are complicated by the introduction of generating functions detailing lengths of words. This document records a proof in which the unnecessary naming of word length is avoided. We introduce the abstract notion of a rst-past-the-post game, and we formalise the process of extending a probability distribution on symbols of an alphabet to the plays of a game. Penney-Ante-type games are shown to be instances of rstpast-the-post games. Such games are characterised by a collection of regular, pre xfree languages. We derive a collection of simultaneous non-linear equations in these languages and use these to show how to calculate the probability of winning. The formula we derive generalises a well-known formula due to John H. Conway for the original Penney-Ante game. Penney-Ante is the name of a game with pennies invented by Walter Penney [Pen74]. Analysis of the game is substantially facilitated by a collection of simultaneous (nonlinear) equations between languages. In the publications I am aware of either the equations are stated without proof [GKP89] or the equations are complicated by the introduction of generating functions detailing lengths of words [GO81]. This document records a statement and proof of the equations in which the unnecessary naming of word length is avoided. Our proof has several novel features. We introduce the abstract notion of a rstpast-the-post game, and we formalise the process of extending a probability distribution on symbols of an alphabet to the plays of a game (section 2). A simple instance of a rst-past-the-post game is a game in which some xed number of dice is thrown and the winner is the one who chose the outcome. Penney-Ante-type games are also shown to be ∗School of Computer Science, University of Nottingham, Nottingham NG8 1BB, England
منابع مشابه
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 dist...
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