What is the mixed strategy Nash equilibrium in the game below? Part III: Cournot

Question

What is the mixed strategy Nash equilibrium in the game below? Part III: Cournot Competitors Suppose two firms ure Cournot competitors with TC(q) = 4q (where q is a given firm's output) and demand curve p = 100 - 4Q, where Q is total industry output (summing all firm's quantities, as usual). Compute the Cournot equilibrium price, profits for each firm, and the overall consumer surplus. Suppose a third firm enters (with the same cost function). Recalculate the equilibrium. Are total industry profits (summing across all firms) higher in (a) or (b)? What about consumer surplus in (a) vs. (b)? What about CS+ profits in (a) vs. (b)?

Explanation / Answer

Multiple questions asked.

Q1 is answered below.

Mary plays Left and Right with probabilities q and 1 – q, Sarah’s expected payoff from Up is

q (2) + (1-q) (0) = 2q

and Sarah’s expected payoff from Down is

q (1) + (1-q) (1) = 1

Equating these expected payoffs and solving for q yields

2q = 1

or

q = 1/2.

Similarly, Sarah plays Up with probability p and Down with probability 1-p, Mary’s expected payoff from Left is

p (2) + (1-p) (4) = 4 – 2p

And Mary’s expected payoff from Right is

p (5) + (1-p)(3) = 3+2p.

Equating these and solving yields

3+2p = 4-2p,

4p = 1

p = 1/4.

The mixed-strategy profile for this game is: (1/4; 3/4), (1/2; 1/2).

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