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Rock, Paper, Scissors, Etc -- Topics in the Theory of Regular Tournaments

The classic Rock-Paper-Scissors game of size 3 and its extension, Rock-Paper-Scissors-Lizard-Spock, are modeled by directed graphs called tournaments. They can be further extended to any odd size. The extended games are regular tournaments where each strategy beats and is beaten by exactly half of the alternatives. We survey the properties of regular tournaments, which we will call games. In the process we describe a number of constructions for such games. These include games on groups of odd order and the associated games on coset spaces. We obtain a new lower bound for the number of games of size 2n+1.