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 man prefers to go watch a boxing match, whereas the woman wants to go shopping. This is a classical example of a coordination game, analysed in game theory for its applications in many fields, such as business management or military operations.
Since the couple wants to spend time together, if they go separate ways, they will receive no utility (set of payoffs will be 0,0). If they go either shopping or to a boxing match, both will receive some utility from the fact that they’re together, but one of them will actually enjoy the activity. The description of this game in strategic form 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. This can be easily seen by looking for a dominant strategy, eliminating all dominated strategies. However, there will be two dominant strategies, two Nash equilibria (underlined in red). 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.