How to Generalize Janken - Rock-Paper-Scissors-King-Flea

摘要

Janken, which is a very simple game and it is usually used as a coin-toss in Japan, originated in China, and many variants are seen throughout the world. A variant of janken can be represented by a tournament, where a vertex corresponds a sign and an arc (x, y) means sign x defeats sign y. However, not all tournaments define useful janken variants, i.e., some janken variants may include a useless sign, which is strictly inferior than another sign in any case. We first shows that for any positive integer n except 2 and 4, we can construct a janken variant with n signs without useless signs. Next we introduces a measure of amusement of janken variants by using the variation of the difference of out-degree and in-degree. Under this measure, we show that a janken variant has the best amusement among ones with n signs if and only if it corresponds to one of the tournaments defined by J. W. Moon in 1993. Following these results, we present a janken variant “King-fles-janken,” which is the best amusing janken variant among ones with five signs.