Final answer:
The pure-strategy Nash equilibrium is when all three firms enter the market. The symmetric mixed-strategy equilibrium in which all three players enter with the same probability is not possible.
Step-by-step explanation:
In this scenario, three firms are considering entering a new market. The payoff for each firm that enters is 150/n, where n is the number of firms that enter. The cost of entering is 62.
a. To find all the pure-strategy Nash equilibria, we need to determine when no firm has an incentive to deviate from their chosen strategy. In this case, if one firm decides to enter, the payoff for that firm would be 150/1 = 150. However, if two firms enter, the payoff for each firm would be 150/2 = 75. Therefore, the only pure-strategy Nash equilibrium is when all three firms enter the market.
b. To find the symmetric mixed-strategy equilibrium in which all three players enter with the same probability, we need to determine the potential probabilities that each firm enters. Let's assume each firm enters with probability p. The expected payoff for a firm entering is 150/n = 150/3 = 50, and the expected payoff for a firm not entering is 0. Therefore, in equilibrium, the expected payoffs for both entering and not entering should be equal. So we have the equation: p ×50 + (1-p) × 0 = 62. Solving this equation, we find that p = 62/50 = 1.24. Therefore, in the symmetric mixed-strategy equilibrium, all three firms enter the market with a probability of 1.