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I'm reviewing game theory and came across these slides: http://www.econ.ohio-state.edu/jpeck/Econ601/Econ601L10.pdf.

On the last few pages it talks about Stackelberg Competition.

Two firms set quantities just like in Cournot. Marginal production cost is equal to 100, and market inverse demand is given by p = 1000 − q1 − q2. Firm 1 moves first.

It goes on to derive the best response of Firm 1 and Firm 2. We get that the subgame perfect Nash equilibrium is (450, 225).

What I'm confused about is the last slide:

Although the game has one SPNE, there are many NE that violate sequential rationality, for example:

q1 = 64

q2(q1) = 418 if q1 = 64

q2(q1) = 1000 otherwise.

How is the above a NE? I understand that a set of strategies is NE if each player's action is the best response given the other player's action. But in this case why is Firm 1's quantity (64) a best response to Firm 2?

Thank you so much!

1 Answers 1

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A way to understand the difference between Nash equilibrium and subgame perfect equilibrium is the following. In a Nash equilibrium every party is playing a best reply (to others' actions) on the equilibrium path. In a subgame perfect equilibrium, every party is also (planning to) play a best reply off the equilibrium path.

In your example, consider Player 1. The equilibrium path has 1 playing 64 and 2 playing 418. As shown in the slides you mention, each party is playing a best reply. So it is a Nash equilibrium.

We get off the equilibrium path when 1 deviates. Suppose 1 plays $q_1 \ne 64$; f.i., he plays $q_1 = 50$. Then 2's best reply should be $q^*_2(50) = 425$ whereas her plan is to play $q_2 = 1000$. So she is not best replying to 1's possible deviations.