How Game Theory informs Valentine's Day

Written by Barry Hughes Monday, 13 February 2012 22:12
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How long should you keep looking through your options before you settle on one?

The marriage problem in game theory can help us with the answer.

In the problem a man is seeking a wife. He has a number of potential girlfriends (we assume that they will all accept him should he propose). He can date each one in turn but once he has turned one down and moved on to the next girlfriend he can’t go back to an earlier girlfriend.

When should he stop searching and propose to his current girlfriend?

The answer is surprisingly simple. If there are a number of potential girlfriends, using the letter ‘n’ to represent the number, then he should reject the first n divided by ‘e’ (e is a key number in mathematics and has a value of about 2.72). He should then accept the next girlfriend after that who is better than all the preceding ones. If none of them are better than all the earlier ones then he ends up with the last girlfriend.

This is his optimal strategy and he ends up with the best girlfriend as his wife about 37% of the time (actually with a probability of 1 divided by e).

For a bit of fun let’s think about how soon a man (or woman) should get married using this rule. Let’s assume that he is 16 and wants to get married sometime before he is 46. This gives him 30 years to find his optimum partner. According to the theory he should not settle down until at least 30 divided by 2.72 years have gone by. This works out to be 11 years, so he shouldn’t get married before he is 27 otherwise he has reduced his chance of finding his best wife!

Of course, plenty of people do marry before 27 and when the similar games are played in experiments people also tend to stop earlier than they should.

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