Duopoly Strategies and Payoff Matrix

Duopoly Strategies and Payoff Matrix

Understanding the Payoff Matrix

The payoff matrix represents a duopoly, which is an oligopoly consisting of two firms. The matrix illustrates the profits of the two firms based on their pricing strategies:

• Strategies: Each firm can choose to charge either a High Price or a Low Price.
• Payoffs: The matrix shows the profit outcomes for each firm depending on the strategy chosen by both.

Payoff Matrix Diagram

(The following represents the provided payoff matrix. Profits are in dollars.)

Note: The values in each cell represent (Firm 1 Profit, Firm 2 Profit).

Dominant Strategy

Definition

A dominant strategy exists for a firm if, regardless of the other firm's choice, one strategy always yields a better payoff for that firm.

Identifying Dominant Strategy for Firm 1

• If Firm 2 chooses High Price, Firm 1's payoffs are:
  • Low Price: $20,000$
  • High Price: $4,000$
  • Therefore, if Firm 2 chooses High Price, Firm 1 is better off choosing Low Price
• If Firm 2 chooses Low Price, Firm 1's payoffs are:
  • Low Price: $40,000$
  • High Price: $25,000$
  • Therefore, if Firm 2 chooses Low Price, Firm 1 is better off choosing Low Price

Since Low Price is the better option for Firm 1 regardless of Firm 2's choice, Low Price is Firm 1's dominant strategy.

Is there a dominant strategy for both firms?

To determine if Firm 2 also has a dominant strategy, a similar analysis is needed:

• If Firm 1 chooses High Price, Firm 2's payoffs are:
  • High Price: $45,000$
  • Low Price: $30,000$
• If Firm 1 chooses Low Price, Firm 2's payoffs are:
  • High Price: $25,000$
  • Low Price: $5,000$

In both scenarios charging high price yields a higher payoff than charging a low price. Thus, both firms have a dominant strategy.

Nash Equilibrium

Definition

A Nash Equilibrium is a state in which no player can benefit from unilaterally changing their strategy if the other players maintain their strategies. It's a stable state of mutual best responses.

Identifying Nash Equilibrium

1. Both Firms Low Price:
   • Firm 1 gets $40,000$, Firm 2 gets $5,000$. If Firm 1 switches to High Price, they get $25,000$. If Firm 2 switches to High Price they get $25,000$.
2. Both Firms High Price:
   • Firm 1 gets $4,000$, Firm 2 gets $45,000$. If Firm 1 switches to Low Price, they get $20,000$. If Firm 2 switches to Low Price they get $30,000$.

Thus, the Nash Equilibrium is where both firms charge a Low Price.

Cooperative Equilibrium

Definition

A cooperative equilibrium occurs when firms collude to maximize their joint profits. This typically involves both firms charging a high price.

Analysis

• If both firms charge a High Price each firms profits would be: Firm 1 gets $4,000$, Firm 2 gets $45,000$.

This equilibrium leads to greater profits for both firms as a whole.

Sustainability

A cooperative equilibrium is often unsustainable due to:

• Incentive to Cheat: Each firm has an incentive to lower its price to capture a larger market share, thereby increasing its own profits at the expense of the other firm.
• Lack of Trust: Maintaining cooperation requires trust, which can be difficult in a competitive environment.
• Anti-trust Laws: Laws designed to prevent monopolies and collusion can deter firms from explicitly coordinating prices.

Factors Preventing Cooperative Equilibrium

• Competition: The competitive nature of the market pushes firms to undercut each other.
• New Entrants: The threat of new firms entering the market with lower prices can destabilize cooperation.
• Anti-trust Laws: These laws discourage explicit agreements to fix prices.