Dating Game

QUESTION: A man and a woman are trying to decide where to go on a date.  The woman prefers ballet, but the man prefers going to a football game.  There is some gain to going together, but each would rather go to their preferred activity alone, than together to their less preferred activity.  Diagram this game, and show how best to play.

 ANSWER:

                                                 Man
                                              Football             Ballet
                              Football   (1,4)                  (0,0)
Woman        
                                  Ballet   (2,2)                  (4,1)


The man does better by going to the football game, regardless what the woman does, and the woman does better by going to the ballet, regardless what the man does.  These are called "dominant strategies."  The equilibrium of the game is for each to go to their preferred activity.

Notice, however, that the two players could make themselves better off by cooperating.  Self interest is taking them to a place (2,2) with a lower group payoff than the cells on the main diagonal. 

There are two ways to change the game to increase group payouts.

1. Alternate.  If the couples take turns, their group payout goes up.

2. Have the player that receives the higher payoff, compensate the other player for going to their less preferred activity.

In this case, the man could give 1.5 units to the woman if they go to the football game, which would change the payoff in the upper left to (2.5, 2.5).  This would change the equilibrium of the game.

Alternatively, the woman could give 1.5 units to the man if they go to the ballet.  This is the premise of an off-color South Park episode.