# Can you have a pure strategy Nash Equilibrium if there's tie in payoffs?

**URL:** <https://boards.straightdope.com/t/can-you-have-a-pure-strategy-nash-equilibrium-if-theres-tie-in-payoffs/400361>\
**Category:** Factual Questions\
**Created:** [April 16, 2007, 5:28pm UTC](https://boards.straightdope.com/t/can-you-have-a-pure-strategy-nash-equilibrium-if-theres-tie-in-payoffs/400361 "2007-04-16T17:28:55Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![Rysto](https://avatars.discourse-cdn.com/v4/letter/r/ecccb3/32.png) [@Rysto](https://boards.straightdope.com/u/Rysto)\
**Post date:** [April 16, 2007, 5:28pm UTC](https://boards.straightdope.com/t/can-you-have-a-pure-strategy-nash-equilibrium-if-theres-tie-in-payoffs/400361/1 "2007-04-16T17:28:55Z")

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Consider the following game: Player 1 can choose A or B. Player 2 can choose C or D. The payoff matrix is as follows:

(A, C): (2, 2)  
(A, D): (4, 2)  
(B, C): (1, 1)  
(B, D): (3, 5)

Is (A, D) a Nash Equilibrium? Player 1’s best choice if 2 chooses D is A. However, if 1 chooses A, C and D are equally good to 2, so my intuition is that (A, D) is not a Nash Equilibrium.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [April 16, 2007, 5:34pm UTC](https://boards.straightdope.com/t/can-you-have-a-pure-strategy-nash-equilibrium-if-theres-tie-in-payoffs/400361/2 "2007-04-16T17:34:39Z")

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Yes, it is still considered a Nash equilibrium, as long as no one has a strict incentive to unilaterally deviate.

First cite: [Wikipedia](http://en.wikipedia.org/wiki/Nash_equilibrium#Formal_definition)

Second cite:  
Recall that every n-player game of finitely many possible strategies has a Nash equilibrium in mixed strategies (due to Nash), or the weaker result that every two-player zero-sum game of finitely many possible strategies has a Nash equilibrium in mixed strategies (due to von Neumann and Morgenstern). Consider a game where there’s more than one possible strategy profile, but the payout is 0 for all players no matter what, anyway. By the referenced results, the game has a Nash equilibrium; by the design of the game, we know that this must mean Nash equilibria are allowed to have “ties”.