Games with no optimal strategy

Question

I am looking for some examples of (mathematical) games for more than 2 players (the more the better) that do not have an optimal strategy for any player. The games must be symmetric and players cannot "collude".

Answer 1

The easiest is to set up a game with a non-compact action space, so that no maximum exists.

For example, we have $N$ players who each choose an action $a_i\in \mathbb{R}$, and get payoff $u_i(a_i)=a_i$. Then, since there exists no maximal $a_i$, there is also no Nash equilibrium in this game.