This lecture looks at the battle of the sexes from a generalized perspective, again replacing specific numerical payoffs with “exogenous variables” to cover any case with the specific preference ordering. We now see how to prove a mixed strategy Nash equilibrium is a valid probability distribution.

**Takeaway Points**

- Even without specific numbers, we can use the same methods as before to solve these games.
- When solving for a mixed strategy Nash equilibrium, remember that all strategies in the support of the equilibrium must be played with a probability no less than 0 and no greater than 1.
- To check whether a fraction is no less than 0 and no greater than 1, the absolute value of the numerator must be less than the absolute value of the denominator, and the numerator and denominator must be both positive or both negative.