Question

5. Three firms are considering entering a new market. The payoff for each firm that enters is 150, where n is the number of firms that enter. The cost of entering is 62. Find a symmetric mixed-strategy Nash equilibrium in which all three players enter with the same probability. (15 points)

Answer 1

Suppose, each of the three players enter the market with probability p,

Since, the response of a firm is mixed i.e. 0<p<1, the firm must be indifferent between joining and not joining.

Thus can only happen if the payoff from joining the market is 0

Now, for that firm's perspective, two other firms join the market with probability p*p=p^2 and one of the firm join with probability 2p*(1-p) and none of the other two join with probability (1-p)^2

Thus, he gets a pay-off of (150/3)*p^2+(150/2)*2p*(1-p)+150*(1-p)^2-62=50p^2+150p-150p^2+150+150p^2-300p-62

=50p^2-150p+88

Thus equals 0

50p^2-150p+88=0

p=0.8

Thus, the mixed Nash Equilibrium would be 0.8E+0.2N for all the players where E is enter into the market and N is not entering the market.