Pure Nash equilibrium in Zero sum

Question

Do zero sum games have a pure Nash equilibrium and if so how do I find the pure Nash equilibrium

Answer 1

Unfortunately not every zero sum game needs to have a pure Nash equilibrium; you can see this quite easily from the example of matching pennies.

Computing Nash equilibria is a hard problem in general, but for pure equilibria it turns out to be quite easy. From the definition, a pure Nash equilibrium is a strategy profile in which no player can increase their utility by unilaterally changing their strategy (i.e. no one can deviate by themselves in order to get a better outcome). If we only consider pure strategies (and two player games in bimatrix form with finite strategy spaces), then a particular strategy profile is a pure equilibrium if the row players payout is the largest in that column while simultaneously the column players payout is the largest in that row. You can generalize that idea to more than two players readily enough.

Finding all pure equilibria then becomes polynomial in the size of the game, which is about as much as you can hope for.