Battle of the Sexes - game theory

“Battle of the sexes” has always been one of my favorite topics for FB, gtalk status messages….and when I actually came across the scientific proof for this…Well you cannot imagine how overjoyed I was…so here goes the formal description…

The Battle of the Sexes is a two-player coordination game used in game theory. Imagine a couple that agreed to meet this evening, but cannot recall if they will be attending the opera or a football match. The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go?

Pointers to the above that are in sync with the real time battle of the sexes –
1. They agreed to meet but cannot recall the location…very typical of the husband I’d say
2. The husband would like to go to the football game….typical
3. The wife would like to go to the opera….typical
4. Both would prefer to go to the same place rather than different ones…of course! Don’t we all agree that we should all agree!
5. They cannot communicate…..of course they can’t we wouldn’t be here if they could!

Game theory solves this problem using the pay off matrix…which I am still trying to understand! ….Watch out this space for more interpretation once I understand that…

There can be four cases –
1. Both go to the Opera
2. Wife goes to the opera and husband to the football game
3. Husband goes to the opera and wife to the football game
4. Both go to the football game.

But in event 3, additional harm may occur!! He goes to the opera while she goes to the football game, satisfying neither…and both of them come back and have a nice fight!

Another aspect that can occur in this game is called “burning money”, sounds very realistic doesn’t it? When the husband agrees to go to the opera, he is “burning money” when the wife goes to the football game she is burning money….So now there are additional conditions to the game, the player can choose to burn or not burn money!

The scenario where the wife agrees to go to the football game, she is achieving the objective of going with the husband, but she is choosing to “burn money”! How very foolish of her to choose what Game theory calls the “weakly dominated strategy” where she dominates at the cost of choosing to burn money! Had the wife been smart, she would have known that Game theory suggests that if one iteratively deletes weakly dominated strategies then one arrives at a unique solution where the wife does not burn the money and chooses Opera and where the husband has to choose Opera. The odd thing about this result is that by simply having the opportunity to burn money (but not actually using it), the wife is able to secure her favored equilibrium. Smart, ain’t it?

Another way of attaining equilibrium according to game theory involves the use of a correlated equilibrium. Using a randomized event we can reach a decision. In simple words, this is like flipping a coin…before choosing their strategies, they might agree to correlate their strategies based on the coin flip by, say, choosing football in the event of heads and opera in the event of tails. Notice that once the results of the coin flip are revealed neither the husband nor wife have any incentives to alter their proposed actions – that would result in miscoordination and a lower payoff than simply adhering to the agreed upon strategies. So nothing’s fair when everything’s fair!!

If you’re still confused, don’t worry so am I! But the only point of writing so much about it is to tell you that this is interesting ain’t it?