Cycles with Undistinguished Actions and Extended Rock-Paper-Scissors Games

The present paper examines zero-sum games that are based on a cyclic preference relation defined over anonymous actions. For each of these games, the set of Nash equilibria is characterized. When the number of actions is odd, a unique Nash equilibrium is always obtained. On the other hand, in the case of an even number of actions, every such game exhibits an infinite number of Nash equilibria. As a special case, a proof of the uniqueness of the Nash equilibrium for the Rock-Paper-Scissors game obtains.