The Cola Wars
Multiple Nash Equilibria and Coordination Problems
There are two cola companies, Pepsi and Coke, that each own a vending machine in the dormitory. Each must decide how to stock its machine. They can fill the machine with diet soda, regular soda, or a combination of the two. The payoffs are in the strategic form of the game presented below.
Coca-Cola | ||||
Diet | Both | Classic | ||
Pepsi | Diet | 25, 25 | 50, 30 | 50, 20 |
Both | 30, 50 | 15, 15 | 30, 20 | |
Classic | 20, 50 | 20, 30 | 10, 10 |
There is no dominant strategy for either Pepsi or Coke. Both of them are able to eliminate 'classic' as a dominated strategy. In the smaller game you can use 'best response' to find that there are two Nash equilibria.
Coca-Cola | ||||
Diet | Both | |||
Pepsi | Diet | 25, 25 | 50, 30 | |
Both | 30, 50 | 15, 15 | ||
There are two features of the two equilibria:
1. The total payoff is no greater for any other strategy pairs.
2. The total payoff is the same for both equilibria.
Without some coordination between the players, this game could end in disaster! Suppose Coke decides that they will leave it to Pepsi to play 'both' and chooses 'diet' for itself. It is reasonable for us to conjecture that Pepsi plays the same way. In which case they both end up playing 'diet' and the outcome is worse for both of them than either of the Nash equilibria.
Alternatively, suppose that Coke is in a magnanimous mood and decides they will play 'both', believing that greedy Pepsi will surely play 'diet' and the result will be a Nash equilibrium rather than one of the other less desirable outcomes. But, again, it is reasonable for us to think that Pepsi is going through the same reasoning process and chooses to play 'both'. Now they have earned the worst payoffs of the game.
What mechanism can be used to coordinate the strategies of the two players? Keep in mind that it is illegal for them to collude.