I am looking for a $3$-player game that meets the following requirements:
- It should be a zero-sum-game
- It should be simple in the sense that the payoff table is not too large (ideally memorisable by an average human)
- Every Nash Equilibrium involves only mixed strategies
One instance is the $3$-player version of "rock-paper-scissors", see Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?. Do you have other nice examples?
One could generalized the $3$-player version of "rock-paper-scissors": To any symmetric $2$-player zero-sum-game one can construct a $3$-player in the following way. The options of each player are the same as in the $2$-player game. If the players all play the same or all a different option, then declare a stalemate. If only two different options are played, then we can consider those players that play the same option as one player in the $2$-player game and split the outcome. I call such a $3$-player game induced by a $2$-player game.
Are there nice examples for $3$-player games that meet the above requirements but are not induced by a $2$-player game?
