Mixed strategies need to be analysed in game theory when there are many possible equilibria, which is especially the case for coordination games. The battle of the sexes is a common example of a coordination game where two Nash equilibria appear (underlined in red), meaning that no real equilibrium can be reached.
In the battle of the sexes, a couple argues over what to do over the weekend. Both know that they want to spend the weekend together, but they cannot agree over what to do. The woman prefers to go shopping for a new pair of shoes, whereas the man wants to go a boxing match. The game matrix is therefore as follows:
In this case, knowing your opponent’s strategy will not help you decide on your own course of action, and there is a chance an equilibrium may not be reached. The way to solve this dilemma is through the use of mixed strategies, in which we look at the probability of our opponent choosing one or the other strategy and balance our pay off against it.
Let’s suppose that the woman is likely to choose boxing with probability q and shopping with probability (1-q). Likewise, the man is likely to choose boxing with a probability of r and shopping with a probability of (1-r). In that case, our outcomes are as follows:
- Boxing-Boxing: qr
- Shopping-boxing: (1-r) q
- Boxing-shopping: r (1-q)
- Shopping-shopping: (1-q) (1-r)
Analogously, for the woman, q= 2/3. Now she must balance what q (the man’s chances of valuing his own happiness over hers) really is. If r>1/3, they’ll go to a boxing match. If r=1/3, either could happen, and if r<1/3, the woman will get her own way and they’ll go shopping. She must balance this carefully, because if she makes a mistake in valuing his probability (likewise for the man) then, as this is still a simultaneous game and there are no second chances, they could end up spending the weekend apart, which would mean less utility for both.