# An Example of Duopoly Nash Equilibrium

Companies A and B are the only competitors in the market. Each has to decide what price to set for its product. Once prices are set, they cannot be changed for the year. Both firms set prices at the same time.

The options for setting price are the same for both firms: $8,000 or $4,000.

If Firm A and B both set price at $4,000: A's payoff is $10 million and B's payoff is $10 million.

If Firm A and B both set price at $8,000: A's payoff is $4 million and B's payoff is $5 million.

If Firm A sets price at $4,000 and B sets price at $8,000: A's payoff is -$4 million and B's payoff is $8 million.

If Firm B sets price at $4,000 and A sets price at $8,000: B's payoff is -$4 million and A's payoff is $8 million.

What is the Nash Equilibrium?

The answer is set both A and B at $8,000. How do you get to this answer?

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#### Solution Preview

There are two players in the game: Company A and Company B. Both have two possible actions: set price at $4000, or set price at $8000. The best way to find the Nash equilibrium of the game is to first form the game matrix. Assuming the first element in each cell is Company A's payoff and that in the second is Company B's payoff, the game matrix can be drawn as

Company B

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#### Solution Summary

A simple solution mechanism to find the Nash Equilibrium in a two player game.