Analyzing the Duopoly Game Payoff Matrix

This question asks us to analyze a duopoly game between Firm A and Firm B based on the given payoff matrix. We need to determine if either firm has a dominant strategy and if a Nash equilibrium exists.

Understanding the Payoff Matrix

The matrix shows the profits (payoffs) for Firm A and Firm B for each combination of strategies (Advertise or Don't Advertise). The first number in each cell is Firm A's payoff, and the second number is Firm B's payoff.

Determining Dominant Strategy for Firm A

A dominant strategy for Firm A is a strategy that is always better for Firm A, regardless of what Firm B does. Let's examine Firm A's options:

• Suppose Firm B chooses to Advertise:
  • If Firm A Advertises, A gets 11.
  • If Firm A Doesn't Advertise, A gets 7.
  • Comparing 11 and 7, Firm A prefers to Advertise ($11 > 7$).
• Suppose Firm B chooses not to Advertise:
  • If Firm A Advertises, A gets 16.
  • If Firm A Doesn't Advertise, A gets 20.
  • Comparing 16 and 20, Firm A prefers not to Advertise ($20 > 16$).

Since Firm A's best strategy (Advertise when B Advertises, Don't Advertise when B Doesn't Advertise) changes depending on Firm B's action, Firm A does not have a single strategy that is always best. Therefore, Firm A has no dominant strategy.

Statement (a) "Firm A has no dominant strategy" is true.

Determining Dominant Strategy for Firm B

A dominant strategy for Firm B is a strategy that is always better for Firm B, regardless of what Firm A does. Let's examine Firm B's options:

• Suppose Firm A chooses to Advertise:
  • If Firm B Advertises, B gets 6.
  • If Firm B Doesn't Advertise, B gets 0.
  • Comparing 6 and 0, Firm B prefers to Advertise ($6 > 0$).
• Suppose Firm A chooses not to Advertise:
  • If Firm B Advertises, B gets 9.
  • If Firm B Doesn't Advertise, B gets 2.
  • Comparing 9 and 2, Firm B prefers to Advertise ($9 > 2$).

Since Firm B always prefers to Advertise, regardless of Firm A's strategy, Firm B has a dominant strategy, which is to Advertise.

Statement (b) "Firm B has a dominant strategy" is true.

Determining Nash Equilibrium

A Nash equilibrium is a state where neither firm can improve its payoff by unilaterally changing its strategy, assuming the other firm's strategy remains unchanged. We can find Nash equilibria by checking each outcome in the matrix:

• Consider (Advertise, Advertise) with payoffs (11, 6):
  • If Firm A unilaterally changes to Don't Advertise (while B stays Advertise), A's payoff changes from 11 to 7. A would not switch ($11 > 7$).
  • If Firm B unilaterally changes to Don't Advertise (while A stays Advertise), B's payoff changes from 6 to 0. B would not switch ($6 > 0$).
  • Since neither firm wants to switch, (Advertise, Advertise) is a Nash equilibrium.
• Consider (Advertise, Don't Advertise) with payoffs (16, 0):
  • If Firm A unilaterally changes to Don't Advertise (while B stays Don't Advertise), A's payoff changes from 16 to 20. A would switch ($20 > 16$).
  • This is not a Nash equilibrium.
• Consider (Don't Advertise, Advertise) with payoffs (7, 9):
  • If Firm A unilaterally changes to Advertise (while B stays Advertise), A's payoff changes from 7 to 11. A would switch ($11 > 7$).
  • This is not a Nash equilibrium.
• Consider (Don't Advertise, Don't Advertise) with payoffs (20, 2):
  • If Firm B unilaterally changes to Advertise (while A stays Don't Advertise), B's payoff changes from 2 to 9. B would switch ($9 > 2$).
  • This is not a Nash equilibrium.

The game has one Nash equilibrium: (Advertise, Advertise).

Statement (c) "The game has a Nash equilibrium" is true.

Evaluating Statement (d)

Statement (d) says "Neither Firm A nor Firm B has a dominant strategy". Based on our analysis, Firm B *does* have a dominant strategy (Advertise). Therefore, statement (d) is false.

Conclusion

Based on the analysis of the duopoly game payoff matrix:

• Statement (a): Firm A has no dominant strategy - True.
• Statement (b): Firm B has a dominant strategy - True.
• Statement (c): The game has a Nash equilibrium - True.
• Statement (d): Neither Firm A nor Firm B has a dominant strategy - False.

The statements that are true are (a), (b), and (c).

Revision Table: Key Game Theory Concepts

Additional Information: Duopoly and Strategic Interaction

A duopoly is a market structure where only two firms operate. In such markets, firms often engage in strategic interaction, meaning that the outcome for one firm depends not only on its own actions but also on the actions of the other firm. Game theory provides a framework to analyze these strategic decisions.

In this specific duopoly game:

• Firm B having a dominant strategy simplifies its decision-making process; it will always choose to Advertise.
• Firm A, knowing that Firm B will Advertise (because it's B's dominant strategy), can anticipate B's move. Firm A then chooses its best response to B choosing Advertise, which is Advertise (since 11 > 7).
• This leads both firms to choose Advertise, resulting in the (Advertise, Advertise) outcome. This outcome is also the Nash equilibrium, reinforcing its stability.
• The concept of dominant strategy equilibrium (if one exists) is a stronger prediction than Nash equilibrium, but not all games have dominant strategy equilibria. This game has a dominant strategy for one player and a Nash equilibrium.