Tuesday, October 11, 2005

Nash Equilibrium

I might not have learnt to spell equilibrium correctly but I learnt Nash's definition of it for oligopoly/duopoly markets (markets which have 2 competing firms who want to maximize profits by competing/colluding at the same time). I have to say the primary advantage of being in B school is you learn Nash's equilibrium :-) (obviously I am kidding). No this is not Ogden Nash from the Xth Std Gulmohar textbook, the poet who wrote "Ode to Clothes" and talked about Kister Monductor. This is the person, the movie Beautiful Mind, was based out of. Nash got Nobel prize in 1994 for this.

Think about this: An interesting example given to illustrate two competitors who mistrust each other but still want to collude/compete at the same time. Criminals A and B are caught. Police tell A that if he confesses he will go free and B will get 5 years in the slammer. B is told the same thing. So the matrix is now: if A & B are silent both get 1 year. A confesses & B is silent then A goes free and B gets 5 years. A is silent and B confesses means A gets 5 years and B goes free. If both confess both get 3 years. If you are A what is your dominant strategy?

The traditional dominant strategy says that in order to reach maximizing equilibrium each party should take a position regardless of what the other party is doing or can do. So if A confesses he either goes free (if B remains silent) or gets 3 years(if B talks). So either way, whatever B does, A is better off or in the same situation as B.

But think about the situation where a dominant strategy is not possible. Or two "best situations" exist. Nash's equilibrium was illustrated with the example of the classic "battle of the sexes". Let us say a man and woman wanted to go out. The woman wants to go for a ballet and the man wants to go for prize-fighting show. They can only go to one show. The woman gets 1 unit of happiness if she goes to the ballet. The man gets 1 unit of happiness if he goes to the prize fight. They both get 1 unit of happiness if their partner accompanies them. They get -1 unit of happiness if their partner does not accompany them. Think about this situation as a 2 X 2 matrix with man as Y axis and woman as X axis. 0,0 represents man and woman together in prize-fight competition(man =2, woman =1). 0,1 represents man in prize fight and woman in ballet (man=-1, woman =-1). 1,0 represent man in ballet and woman in prizefight(both are so unhappy that man and woman each are -5 units unhappy). 1,1 is where man and woman are in ballet (man =1, woman =2). When man and woman sit and decide where they want to go, how do they decide? Their decision should result in a large amount of happiness units. Now clearly a more sophisticated dominant strategy is required.

Nash changed this to say that in order to acheive happiness/profit maximizing equilibrium A should do the best he/she can given what B does. B does his/her best given what A does. If you think A and B are like Coke and Pepsi fighting for market space, Nash's equilibrium changes the way firms thought about dominant strategy.


Golden Face said...

Your blog provoked me to visualize the relationships with in-laws as a series of linear algebraic equations, and I end up solving them by complex simplex method with lots of iterations.

tilotamma said...

I don't think Ogden Nash went around translating poems from Russian.The Muddlehead poem you are referring too - someone has blogged that one for us


Artful Badger said...

complex simplex method
This method is not guaranteed to terminate. It could go on forever.
Try the Karmarkar Algorithm.

Well, I guess it's trust. In the prisoner's dilemna, an element of trust and co-operation between the two lead to higher payoffs for both eventually. Trying to maximize just your own payoff lead to both doing the same, and lower payoffs for both.
International politics is a good example of that.