Market Structure and Game Theory Application

Question

Two firms, Alpha Corp and Beta Inc, operate in an oligopolistic market and produce identical products. They each have two strategies: Set a high price or Set a low price. If both firms set a high price, they each earn a profit of $1,000,000. If one firm sets a low price while the other sets a high price, the low-pricing firm captures the market, earning $1,500,000, while the high-pricing firm suffers a loss, earning only $300,000. If both firms set a low price, they engage in a price war, each earning $600,000.

1. Construct the payoff matrix for this pricing game.
2. Determine the Nash equilibrium of the game.
3. If the firms could collude, what would be the optimal strategy?
4. Assume the game is repeated indefinitely. Under what conditions can the firms sustain the collusive outcome using a trigger strategy?
5. If the government imposes an antitrust policy that fines firms $500,000 for price-fixing agreements, how does this affect the firms’ strategic decisions?

Answer 1

This question requires an understanding of oligopoly, game theory, Nash equilibrium, collusion, and antitrust policy. Let's go step by step.

1. Construct the Payoff Matrix

We create a payoff matrix where each firm's strategies (high price or low price) result in different payoffs (profits). The first number in each cell represents Alpha Corp’s profit, and the second represents Beta Inc’s profit.

Beta IncHigh Price (H)Low Price (L)
Alpha Corp
High Price (H)	(1,000,000, 1,000,000)	(300,000, 1,500,000)
Low Price (L)	(1,500,000, 300,000)	(600,000, 600,000)

1. If both set a high price, they each earn $1,000,000.
2. If Alpha sets a low price while Beta sets a high price, Alpha captures the market and earns $1,500,000, while Beta earns only $300,000.
3. If Beta sets a low price while Alpha sets a high price, Beta captures the market and earns $1,500,000, while Alpha earns $300,000.
4. If both set a low price, they engage in a price war and each earn $600,000.

2. Determine the Nash Equilibrium

A Nash equilibrium occurs when no firm wants to change its strategy given the strategy of the other firm.

1. Suppose Alpha sets a high price (H).
2. If Beta also sets H, Alpha earns $1,000,000.
3. But Beta can do better by switching to L, earning $1,500,000 instead of $1,000,000.
4. Suppose Alpha sets a low price (L).
5. If Beta sets H, Alpha earns $1,500,000, but Beta earns only $300,000.
6. Beta would prefer to switch to L because it would earn $600,000 instead of $300,000.
7. The same reasoning applies to Alpha.

Thus, (Low Price, Low Price) (L, L) is the Nash equilibrium, because if one firm is setting a low price, the other cannot improve its position by changing its strategy.

Conclusion: The dominant strategy for both firms is to set a low price, leading to a price war and lower profits ($600,000 each).

3. If the Firms Could Collude, What Would Be the Optimal Strategy?

If the firms could collude, they would both set a high price (H, H) to maximize profits.

1. This outcome (1,000,000, 1,000,000) is better than (600,000, 600,000) from (L, L).
2. However, collusion is illegal in many countries because it leads to higher prices for consumers.

4. Can the Firms Sustain the Collusive Outcome in a Repeated Game?

If this game is played indefinitely, the firms may sustain the collusive outcome using a trigger strategy:

1. Each firm starts by setting a high price (H).
2. If the other firm cheats (switches to L), the firm that was cheated will punish it by setting L forever (leading to price war).
3. If the firms value future profits highly, they will cooperate rather than defect.

Mathematically, if the firms’ discount factor (δ) is high enough, meaning they care more about long-term profits than short-term gains, collusion can be sustained.

5. Effect of an Antitrust Fine ($500,000) on Collusion

If the government imposes a $500,000 fine on firms caught colluding:

1. The firms now risk losing money if they continue to set high prices (H, H).
2. This increases the incentive to defect and set a low price (L) to avoid being caught.
3. As a result, the Nash equilibrium (L, L) becomes even stronger, making price wars more likely.

Conclusion: Antitrust laws discourage price-fixing, forcing firms to compete rather than collude, leading to lower consumer prices.