## Monday, April 18, 2011

### A Beautiful Mind

"A society is a group that shares conventions. There are many societies, and I dare say every convention in every society, has an opposite convention in an opposite society. The point is not that one convention is better than another, but that within one society one convention is correct. Many societies have the convention that they drive on the left side of the road, which is no different from the convention of driving on the right side of the road, but stick with the convention or risk a head on collision. A set of laws is a convention. A set of laws is a convention of a society that keeps individuals from running into one another. A set of laws provides a level of awareness to the individuals who can maintain the convention of the set of laws of their society. The point is that the law will increase the awareness of the individuals. Once one had become aware through the maintenance of the convention that they find themselves in, the awareness continues beyond the limits of the law or society's convention. The law has served its' purpose."

Nash Equilibrium

A Nash Equilibrium (NE) is a vector of strategies (pure or mixed), one per player, such that no player can improve her own payoff by unilaterally changing strategies. For an N player game $G = (N,A = \times_{i=1}^N A_i, u: A \rightarrow \reals^N)$, a NE is $\sigma \in \Delta = \Delta_1 \times \ldots \times \Delta_N$, such that,

$u_i(\sigma)\geq u_i(\sigma_i',\sigma_{-i}),$ for all $\sigma \in \Delta, \sigma_i'\in \Delta_i$.

Note that it is equivalent to the following definition:

$u_i(\sigma)\geq u_i(a_i',\sigma_{-i}),$ for all $\sigma \in \Delta, a_i'\in A_i$.

Some nice examples of Nash Eq. are:

• Coordination game: it is a NE to all drive on the left side of roads, or all drive on right side of roads (there is one more... :)
• Prisoner's dilemma: it is a NE to not cooperate in prisoner's dilemma.

A game is finite if it has a finite number of players and each player has a finite number of actions (or pure strategies).

I watched the movie "A Beautiful Mind" last night where Nash's theory was explained as the best payoff for all participants is to pursue the brunettes who make up 4 out of the 5 available women, rather than disappoint and offend all the brunettes by all going for the blonde when only, theoretically, just one could win the blonde. It is also stated in the movie that the best payoff is to consider BOTH what is best for oneself AND what is best for the group. This would be the same as balancing Whole and Part, Female and Male, and Many and One.