Question

3 firms are thinking about entering a new market. If a firm decides not to enter, the payoff is 0. If a firm enters, its payoff depends on whether other firms also enter. Assume the total market profit is 100. If a single firm enters, it gets the whole profit. If multiple firms enter, they'll share the total profit equally. Furthermore, the cost of entering for each firm is 60.

a. Formulate the game in the normal-form representation. (Note: you only need to specify the strategy set and the payoff function for one firm, because they are all the same.)
b. Find all the pure-strategy Nash equilibria.
c. What will the Nash equilibrium be if the cost of entering is 40 instead of 60?

Answer 1

The pure-strategy Nash equilibria for firms considering entry into a market is determined by their entry costs and potential profits from the market. When the cost of entry is 60, either one firm will enter while others don't or none will enter. If the entry cost is lowered to 40, it becomes profitable for two firms to enter and share the market, though three firms entering would still lead to losses.

Step-by-step explanation:

Pure-Strategy Nash Equilibria and Market Entry

A student has posed a question about the entry decisions of firms into a new market within an economic context. We can address this by using game theory and the concept of Nash equilibrium. With three firms considering market entry, we can define their strategies as 'Enter' or 'Not Enter'. The payoffs depend on the shared profit if multiple firms enter and the cost of entry.

If we denote 'E' as Enter and 'N' as Not Enter, and considering the cost to enter is 60, the payoff function for each firm could be represented as:

• If one firm enters: Payoff = 100 (profit) - 60 (cost) = 40.
• If two firms enter: Payoff = 100/2 (shared profit) - 60 (cost) = -10 (loss).
• If three firms enter: Payoff = 100/3 (shared profit) - 60 (cost) = approximately -20 (loss).
• If no firms enter or a firm doesn't enter: Payoff = 0.

The pure-strategy Nash equilibria occurs when each firm's strategy is the best response to the other firms' strategies. In this setup, there can be multiple equilibria:

• One firm enters, and two do not, which occurs in three variations depending on which firm enters.
• All firms decide not to enter.

If the cost of entering is reduced to 40, the payoff changes:

• If one firm enters: Payoff = 100 - 40 = 60.
• If two firms enter: Payoff = 100/2 - 40 = 10.
• If three firms enter: Payoff = 100/3 - 40 < 0 (The exact number has to be computed, but it is a loss since dividing the profit by three nets less than the cost).

With the lower cost, the Nash equilibrium changes. Firms have an incentive to enter as they do not incur losses when one or two firms enter the market. Therefore, the Nash equilibria can now include:

• Two firms enter, and one does not, which occurs in three variations.
• One firm enters, and two do not, still viable in three variations.

The incentive to enter the market becomes stronger with a lower entry cost, and hence, the likelihood that multiple firms will share the market increases. However, when three firms enter at the cost of 40, they would still incur losses, indicating that some other equilibrium would be slightly preferable.