# Battle of the sexes: Barack and Michelle

In ‘The Battle of the Sexes’ game a man and a woman are trying to co-ordinate where they meet up one evening.

In the usual version of the game the woman would rather go to the opera and the man to the football, but they don’t know where the other is going, so they have to guess. This is obviously nonsense as they could just communicate beforehand. Let’s try a new version:

Barack and Michelle have agreed to meet at the cinema at 7:45pm. They know that there are two movies starting at 8pm but they haven’t had time to agree which movie they are going to see. They will do that once they meet up.

Michelle arrives at 7:45pm but, because of a crisis at work, Barack is late. He isn’t answering his phone and so by 8pm Michelle has to decide which of the two movies to watch. She would rather watch Bridesmaids but she knows that Barack would rather see Horrible Bosses (it might give him some ideas about how to sort out his problems at the office!).

She’s pretty sure that Barack will turn up soon and knows that she would rather watch a movie with him than on her own. Should she choose Bridesmaids and hope that he also chooses that when he arrives, or should she choose Horrible Bosses in the expectation that he will pick that?

Barack runs into the cinema at 8:05pm and can still buy a ticket for either movie. He would rather be in whichever movie Michelle has chosen than on his own, so he has to guess which movie she has picked. Money is a bit tight for Barack at the moment so he can only afford to buy one ticket.

This game has two situations where both Michelle and Barack will be happy with their choices. The first is if they both pick Bridesmaids and the second is if they both pick Horrible Bosses. Each of these is known as a Nash equilibrium: given the choice of the other person they made the best possible choice. If they pick different movies then they will both wish they had made a different choice.

If this situation is repeated over a number of weeks then we can start to see a problem with these equilibrium points. The problem is that they are not fair. If they settle on always going to Michelle’s favorite movie then she will always be happier than Barack. If they settle on always seeing his preferred movie then he will always be happier.

There is another equilibrium which leaves Barack and Michelle seeing their preferred movie the same amount of times so neither one does better than the other. This happens if they randomly choose which film they will see each week. If they do that then sometimes they will both see Michelle’s preferred movie and sometimes they will both see Barack’s preferred movie. Unfortunately they will also not co-ordinate some of the time and they will both be unhappily watching different movies. Although this is a ‘fair’ equilibrium neither person ends up as happy as they would be if they were always together.

What is interesting about this game is that it gives us three possible equilibrium points. Two of them aren’t fair and the other one is inefficient.

Today’s takeaway: Sensible strategies can still give unfair or inefficient results.

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