Monday, February 9, 2009

A Beautiful Equilibrium?

The movie A Beautiful Mind details the life of mathematician John Nash, winner of the 1994 Nobel Prize in economics for his contribution to game theory known as the Nash Equilibrium. Consider the following cola pricing game as described by the matrix below.


Cut Price Maintain Price
Coca-Cola Cut Price 50, 50 100, 40
Maintain Price 40, 100 80, 80

Each firm has two strategies: either maintain the current price of their cola, or to cut price. The numbers inside the matrix represent the profits that each firm earns depending on the strategies chosen by each firm. Thus, if Pepsi maintains its price and Coke cuts price, then Coke will earn $100 and Pepsi will earn $40.

What is the Nash Equilibrium for this game? Explain carefully.

Congratulations to Jake Verdoorn for correctly identifying and describing the only Nash Equilibrium in the game above.


Anthony Sayre said...

There are two Nash equilibria in this game. Each firm is making a choice to maximize their profit. Either they will both maintain their price, or they both will cut their price. Neither will let the other cut their price without cutting their own. This way both companies will maximize profit.

Greg Delemeester said...

Not quite, Anthony. Give it another try.

Jake Verdoorn said...

The Nash Equilibrium of this problem would be both Coca-Cola and Pepsi cutting prices. While this seems absurd, it is actually the only logical thing to expect from a competitive firm.
Setting lower prices is a dominant strategy for a competitive firm. Both Pepsi and Coca-cola will cut their price, in hopes of a lucrative $100 profit if the other company maintains original price and in the end they will both simply make less than before. This is called competitive outcome. While we may see equilibrium as keeping the status-quo, Coca-Cola and Pepsi will look to under-cut each other and drive the price down.
This was shown in A Beautiful Mind by John Nash and his friends strategically going after lesser attractive women at a bar. If they all went for the obviously more attractive girl they would most likely get shot down. While they are all settling for less they still “get some”. Yet this phenomenon of cooperative outcome is almost never seen.
In the end, both companies will make less than they would have because neither can afford to try to maintain price.