Wednesday, November 30, 2011

Game Theory: Part VIII: Metagames: The Battle of the Sexes

Now, on to something a bit more optimistic. In game theory, it is a basic assumption that you are basing your actions on what your opponent can do, and your opponent is basing their actions on what you can do.
It can be to your advantage to change your own options, since this changes your opponent's decision. Does this extend to limiting yourself?
The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.
Let's look at a classic game. We have a husband and a wife deciding on what that night's entertainment is going to be. Since this was developed by someone living in a culture based heavily in gender stereotypes, the husband wants to go to the fights, and the wife wants to go to the opera. However, each would rather go to their less-preferred entertainment with the other than to their preferred entertainment alone.
So, the game looks like this, with the husband across the top and the wife down the side:



Fights

Opera
Fights (B,A) (D,D)
Opera(C,C)(A,B)
where A>B>C>D.
So, it is in the best interest of each of them to convince the other to go to their preferred entertainment.
But, there is a solution to this. Suppose we offer one player the ability to make themselves unable to go with one of the options. Or, alternately, let's compare the outcome for a player who is unable to that for one who is not.
The classic example in this case (again, heavily based in Victorian gender stereotypes) is that the wife faints at the sight of blood, so let's apply that.
Now then, we will need to set up a metagame: a game that determines the rules of the next game. This, by the way, is something else that gets absolutely everywhere, from law to history to kids' games.
Our metagame looks a bit like this:



Disable


Fights

Opera
Opera(C,C)(A,B)
Do not disable


Fights

Opera
Fights (B,A) (D,D)
Opera(C,C)(A,B)
And now, let's calculate the value of these games. On the Disable side, the husband is the only one left with a choice. He would rather go to the opera with his wife than to the fights alone, so he will pick "Opera". The value of the first option for the wife is A, and the value for the husband is B.
And now, let's look at the "Do not disable" option. We will, for simplicity's sake, assume that the two are able to discuss and coordinate ahead of time (a subject for later discussion), and that they pick each option half of the time. That is, they alternate between going to the fights together or to the opera together, or they flip a coin to decide. The value for each player is therefore (A+B)/2.
So, we conclude that each player stands to benefit from the ability to disable certain options, as long as the other player does not share that ability. (If that were the case, the value for each player would obviously be C.)
For extra credit, go back to the "Chicken" game, and try thinking about it in terms of metagames and disabled options. What happens if one player locks their wheel into a particular position?

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