Nash Equilibria of Rock Paper Scissors Variants

Abstract

We generalize Rock Paper Scissors to complete directed graphs, or tournaments, on $n$ vertices. Properties of the mixed-strategy Nash equilibria of these tournaments are discussed, particularly those with Nash equilibria where all of the strategies have a nonzero probability. We find graph-theoretic properties of such games and tabulate them for $n \leq 7$.

Figures (7)

• Figure 1: A directed graph $G$ demonstrating which player wins in Rock Paper Scissors. If there exists an edge pointing from Player 1's choice to Player 2's choice, then Player 1 wins.
• Figure 2: Rock Paper Scissors Lizard Spock, an Eulerian variant with $n=5$.
• Figure 3: A Rock Paper Scissors variant with $n=5$. This game may be seen as Rock Paper Scissors with $n=3$, where the strategies are to pick strategy 4, 5, or any of $\{1,2,3\}$. 1, 2, and 3 form another Rock Paper Scissors game, which substitutes for a strategy in the first. We call this game Rock Paper Scissors Turtle Balloon.
• Figure 4: An example game that functions the same as Rock Paper Scissors, but with two Scissors options. Scissors 1 beats Scissors 2.
• Figure 5: Maurer's example of a royal flock with $n=6$. It is not all-positive.

Theorems & Definitions (19)

• Theorem 2.1: Nash Equilibria, Nash 1950
• Lemma 2.2
• Lemma 2.3
• Corollary 2.3.1
• Lemma 2.4
• Lemma 2.5
• Corollary 2.5.1
• Corollary 2.5.2
• Theorem 2.6: Uniqueness of Nash Equilibria
• Theorem 2.7: Reduction of Strategies with 0 Probability