example of Nash equilibrium

Consider the first two games given as examples of normal form games.

In Prisoner’s Dilemma the only Nash equilibriumMathworldPlanetmath is for both players to play D: it’s apparent that, no matter what player 1 plays, player 2 does better playing D, and vice-versa for 1.

Battle of the Sexes has three Nash equilibria. Both (O,O) and (F,F) are Nash equilibria, since it should be clear that if player 2 expects player 1 to play O, player 2 does best by playing O, and vice-versa, while the same situation holds if player 2 expects player 1 to play F. The third is a mixed equilibrium; player 1 plays O with 23 probability and player 2 plays O with 13 probability. We confirm that these are equilibria by testing the first derivativesMathworldPlanetmath (if 0 then the strategy is either maximal or minimal). Technically we also need to check the second derivative to make sure that it is a maximum, but with simple games this is not really necessary.

Let player 1 play O with probability p and player 2 plays O with probability q.


And indeed the derivatives are 0 at p=23 and q=13.

Title example of Nash equilibrium
Canonical name ExampleOfNashEquilibrium
Date of creation 2013-03-22 12:52:48
Last modified on 2013-03-22 12:52:48
Owner Henry (455)
Last modified by Henry (455)
Numerical id 6
Author Henry (455)
Entry type Example
Classification msc 91A99