Firstpastthepost gamesTools Backhouse, Roland (2012) Firstpastthepost games. In: Mathematics of program construction: 11th International Conference, MPC 2012, Madrid, Spain, June 2527, 2012: proceedings. Lecture notes in computer science (7342). Springer, Berlin, pp. 157176. ISBN 9783642311130 (electronic); 9783642311123 (print)
AbstractInformally, a firstpastthepost 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 firstpastthepost games include socalled block and hidden patterns and the PenneyAnte game invented by Walter Penney. We formalise the abstract notion of a firstpastthepost game, and the process of extending a probability distribution on symbols of an alphabet to the plays of a game. Analysis of firstpastthepost games depends on a collection of simultaneous (nonlinear) 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. PenneyAnte games are twoplayer games characterised by a collection of regular, prefixfree languages. For such twoplayer 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 PenneyAnte 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 higherorder cumulatives (for example, the standard deviation) are also required.
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