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: $$v_1(s_1,s_2)=\left\{\begin{matrix} 1 \ \mathrm{if}\ s_1>s_2 \\ 0 \ \mathrm{if}\ s_1\leq s_2 \end{matrix}\right. $$ $$v_2(s_1,s_2)=\left\{\begin{matrix} 1 \ \mathrm{if}\ s_2>s_1 \\ 0 \ \mathrm{if}\ s_2\leq s_1 \end{matrix}\right. $$

Do you think it's a good example?

  • $\begingroup$ It looks ok, but – just my two cents – I would choose $(0,1)$ instead of $\mathbb{R}$ to emphasize that the set of strategies of a player can indeed be uncountably infinite, but it has to be compact. $\endgroup$ – Kolmin Jul 20 '15 at 10:18
  • $\begingroup$ But $(0,1)$ is not compact as far as I know... $\endgroup$ – luka5z Jul 20 '15 at 10:19
  • 2
    $\begingroup$ Indeed, that's the point. If you choose $[0,1]$ (that is compact), you have uncountably many strategies for every player and a NE (think about Cournot's duopoly). $\endgroup$ – Kolmin Jul 20 '15 at 10:20
  • $\begingroup$ In my game also - if set of strategies was $[0,1]$, then $(1,1)$ would be pure strategy Nash equilibrium. Ok, thanks. $\endgroup$ – luka5z Jul 20 '15 at 10:25
  • $\begingroup$ You just wrote the example. You did not prove that it has no mixed strategy Nash equilibrium. $\endgroup$ – Sergio Parreiras Jul 23 '15 at 20:55

There is none, unless the problem is wrongly planned. By definition, every game has at least one equilibrium in mixed strategies.

  • $\begingroup$ Please post this as a comment. $\endgroup$ – Max0815 Apr 29 at 22:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.