Let’s revisit the battle of the sexes. We have seen the pure strategy Nash equilibria last time, viz. both meeting at the football (Football, Football) or both at Dance (Dance: Dance). Here is the payoff matrix for reference.
| B | |||
| Football | Dance | ||
| A | Football | A:10, B:5 | A:0, B:0 |
| Dance | A:0, B:0 | A:5, B:10 |
But we know, unfortunately, they might fail to reach the equilibria due to communication failure. We continue from there to explore if there is any other equilibrium exists. To find these (mixed) equilibria, let’s assume that A is going to mix at p and (1-p) and B is going to mix at q and (1-q). This means A go to football at probability p and B at probability q. Needless to say, A go to dance at probability (1-p) and B at (1-q).
| B | ||||
| Football | Dance | |||
| A | Football | A:10, B:5 | A:0, B:0 | p |
| Dance | A:0, B:0 | A:5, B:10 | 1-p | |
| q | 1-q |
To estimate B’s equilibrium mix ( = q), we need to get A’s payoff. We know how to do that, i.e., calculate A’s expected payoff (pure payoff x probability of finding (q) B at the spot) for football and equate it to its payoff for dance.
10 x q + 0 x (1-q) = 0 x q + 5 x (1-q)
q = 5/15 = 1/3
On the other hand, to estimate A’s equilibrium mix ( = p), we get B’s payoff. B’s payoff for football and equate it to its payoff for dance.
5 x p + 0 x (1-p) = 0 x p + 10 x (1-p)
p = 10/15 = 2/3
So A will go 2/3 of the time for football and 1/3 for dance, and B to go for football 1/3 of the time and dance 2/3 of the time. In other words, even if A mixes the strategy, a better payoff comes by leaning towards football with a higher (2/3) probability. Similar is the case for B with dance.
