Example of infinite game without any Nash equiblibria I have to find an example of a game that does not admit (mixed strategy) Nash equilibria.

Consider a game in normal form. Let $N=\{1,2\}$ be set of players and $S_i=\mathbb{R}$ a set of possible strategies for each player (uncountably many), $i=1,2$.

The rules of this game are as follows: each player chooses one real number (his strategy). Player who chose a bigger number than the other player wins $1$, player who chose smaller number gets $0$. If both players choose the same number, they both get $0$.

Formally, payoff functions of each player one looks like this:
$$v_1(s_1,s_2)=\left\{\begin{matrix} 1 \ \mathrm{if}\ s_1&gt;s_2 \\ 0 \ \mathrm{if}\ s_1\leq s_2 \end{matrix}\right. $$
$$v_2(s_1,s_2)=\left\{\begin{matrix} 1 \ \mathrm{if}\ s_2&gt;s_1 \\ 0 \ \mathrm{if}\ s_2\leq s_1 \end{matrix}\right. $$

Do you think it's a good example?
There is none, unless the problem is wrongly planned. By definition, every game has at least one equilibrium in mixed strategies.
There is none, unless the problem is wrongly planned. By definition, every game has at least one equilibrium in mixed strategies.