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Register a new userFrom Unequal Chance to a Coin Game Dance: Variants of Penneys Game
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We introduce and analyze several variations of Penneys game aimed to find a more equitable game.
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Penneys game is a two player zero-sum game in which each player chooses a three-flip pattern of heads and tails and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occur first. This paper provides a novel no-arbitrage argument that generates the winning odds corresponding to any pair of distinct patterns. The resulting odds formula is equivalent to that generated by Conways leading number algorithm. The accompanying betting odds intuition adds insight into why Conways algorithm works. The proof is simple and easy to generalize to games involving more than two outcomes, unequal probabilities, and competing patterns of various length. Additional results on the expected duration of Penneys game are presented. Code implementing and cross-validating the algorithms is included.
Consider equipping an alphabet $mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penneys game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.
We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers
In this note, we present a compatibility test based on John Nashs game-theoretic notion of equilibrium strategy. The test must be taken separately by both partners, making it difficult for either partner alone to control the outcome. The mathematics behind the test including Nashs celebrated theorem and an example from the film, A Beautiful Mind, are discussed as well as how to customize the test for more accurate results and how to modify the test to evaluate interpersonal relationships in other settings, not only romantic. To investigate the long-term dynamics of give and take in a relationship we introduce the iterated dating dilemma and apply the notion of zero-determinant payoff strategy introduced by Dyson and Press in 2012 for the iterated prisoners dilemma.
We detail the rules and mathematical structure of Al-Jabar, a game invented by the authors based on intuitive concepts of color-mixing and ideas from abstract algebra. Game-play consists of manipulating colored game pieces; we discuss how these colored pieces form a group structure and how this structure, along with an operation used to combine the pieces, is used to create a game of strategy. We also consider extensions of the game rules to other group structures. Note: While this is an article for general readership originally published online by Gathering for Gardner in honor of Martin Gardners birthday (Oct. 2011), Al-Jabar has been played in university abstract algebra courses as a teaching tool, as well as by game enthusiasts, since its release. Moreover, the algebraic game structure described has sparked further work by other mathematicians and game designers. Thus, we submit this article to the ArXiV as a resource for educators as well as those interested in mathematical games.
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