Number of Ties and Undefeated Signs in a Generalized Janken

Abstract

Janken, which is a very simple game and 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 (a complete asymmetric digraph), where a vertex corresponds to a sign and an arc (x, y) indicates that 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 to any other sign in any case. In a previous paper by one of the authors, a variant of janken (or simply janken) was said to be efficient if it contains no such useless signs, and some properties of efficient jankens were presented. The jankens considered in the above research had no tie between different signs. However, some actual jankens do include such ties. In the present paper, we investigate jankens that are allowed to have a tie between different signs. That is, a janken can be represented as an asymmetric digraph, where no edge between two vertices x and y indicates a tie between x and y. We first show the tight upper and lower bounds of the number of ties in an efficient janken with n-vertices. Moreover, it is shown that for any integer t between the upper and lower bounds, there is an efficient janken having just t ties. We next consider undefeated vertices, which are vertices that are not defeated by any sign. We show that there is an efficient janken with n vertices such that the number of vertices that are not undefeated is o(n), i.e., almost all vertices are undefeated.