For Each Of The Following Games, Indicate A Or Do

For Each Of The Following Games Indicate A If There Is A Do

123 For Each Of The Following Games Indicate A If There Is A Do

For each of the following games, indicate (a) if there is a dominant solution— if there is one, state what it is and explain how you got it (make your explanation VERY clear if finding a dominant solution involves iterative dominance), and (b) all Nash equilibriums. I Player 2 II Player 2 Player 1 Left Middle Right Player 1 Left Middle Right Up 1,,,1 Up 1,,,0 Center 2,,,1 Center 0,,,0 Down 3,,,9 Down 0,,,) There are two firms that are considering entering a new market and must make their decision without knowing what the other firm is going to do. Once a decision is made, it becomes a commitment that the firm cannot back out of or change after learning what the other firm will do. Unfortunately, the market is only big enough to support one of the two firms. If both firms enter the market, then they will each lose $10,000. If only one firm enters the market, that firm will earn a profit of $50,000 with the other firm breaking even (i.e., earning $0 profit). Obviously, if neither firm enters the market, neither will earn any profits in that market. Draw the normal form game layout for the above problem and identify any dominant and pure Nash equilibrium solutions as well as any mixed-form Nash equilibrium strategies. If there is a mixed-form Nash equilibrium strategy, state it precisely. Show all work.

Paper For Above instruction

The strategic decision of firms contemplating market entry is a classic problem in game theory, illustrating key concepts such as dominant strategies, Nash equilibria, and mixed strategies. This analysis explores a specific scenario involving two firms considering whether to enter a new market, with outcomes contingent on the combined decisions of both firms. Throughout this examination, we will construct the normal form game, identify dominant solutions, and determine all Nash equilibrium strategies, including any mixed strategies, supported by detailed reasoning and calculations.

Constructing the Normal Form Game

The game involves two firms: Firm A and Firm B. Each firm has two strategies: to Enter (E) or Not Enter (N) the market. The payoff matrix can be summarized as follows:

Firm B: Enter Firm B: Not Enter
Firm A: Enter (-10, -10) (50, 0)
Firm A: Not Enter (0, 50) (0, 0)

This matrix summarizes the payoffs: when both enter, both lose $10,000; when one enters and the other doesn't, the entrant profits $50,000 while the other breaks even; neither enters results in zero profit for both.

Analysis of Dominant Strategies

A dominant strategy is one that yields a higher payoff for a player regardless of the other player's choice. Checking for dominate strategies:

  • Firm A's perspective:
  • If Firm B enters:
    • Enter: -$10,000
    • Not Enter: $0
  • Preferably, Firm A chooses to not enter (since $0 > -$10,000).
  • If Firm B does not enter:
    • Enter: $50,000
    • Not Enter: $0
  • Therefore, Firm A prefers to enter if Firm B does not enter but prefers not to enter if Firm B enters. No single strategy strictly dominates for Firm A.
  • Firm B's perspective:
    • If Firm A enters:
      • Enter: -$10,000
      • Not Enter: $0
    • If Firm A does not enter:
      • Enter: $50,000
      • Not Enter: $0
    • Similarly, Firm B prefers to not enter if Firm A enters but prefers to enter if Firm A does not enter. No dominant strategies for Firm B either.

    Identifying Nash Equilibria

    A Nash equilibrium occurs where neither player can improve their payoff by unilaterally changing their strategy.

    1. Both enter: (E, E): Each gets -$10,000.

      - If Firm A switches to N: payoff becomes 0, which is better.

      - If Firm B switches to N: payoff becomes 0, which is better.

      - Thus, (E, E) is not a Nash equilibrium.

    2. Both do not enter: (N, N): Each gets $0.

      - If Firm A switches to E: payoff becomes $50,000 (better).

      - If Firm B switches to E: payoff becomes $50,000 (better).

      - So, (N, N) is not a Nash equilibrium.

    3. (E, N): Firm A: $50,000; Firm B: $0.

      - Firm A cannot gain by switching to N: payoff remains $0.

      - Firm B cannot gain by switching to E: payoff would be -$10,000.

      - Therefore, (E, N) is a Nash equilibrium.

    4. (N, E): Firm A: $0; Firm B: $50,000.

      - Firm A cannot gain by switching to E: payoff would be -$10,000.

      - Firm B cannot gain by switching to N: payoff would be -$10,000.

      - Therefore, (N, E) is a Nash equilibrium.

    Summary of Pure-Strategy Nash Equilibria

    There are two pure-strategy Nash equilibria:

    • (E, N): Firm A enters, Firm B does not.
    • (N, E): Firm B enters, Firm A does not.

    Mixed Strategy Nash Equilibrium Analysis

    In cases where no pure strategy Nash equilibrium exists or where players are indifferent, mixed strategies can be optimal. Here, the pure equilibria (E, N) and (N, E) are clear and stable; however, to analyze the possibility of a mixed-strategy equilibrium, we denote:

    • Let p be the probability that Firm A enters.
    • Let q be the probability that Firm B enters.

    In an equilibrium, each firm must be indifferent between entering and not entering, given the other firm's mixed strategy. The expected payoffs are computed as follows:

    Firm A's expected payoff from entering:

    EU_A(E) = q (-10,000) + (1 - q) 50,000

    Firm A's expected payoff from not entering:

    EU_A(N) = q 0 + (1 - q) 0 = 0

    Indifference condition for Firm A:

    EU_A(E) = EU_A(N) => q (-10,000) + (1 - q) 50,000 = 0

    Expanding:

    -10,000q + 50,000 - 50,000q = 0

    Combine like terms:

    -60,000q + 50,000 = 0

    Solve for q:

    q = 50,000 / 60,000 = 5/6 ≈ 0.8333

    Firm B's expected payoff from entering:

    EU_B(E) = p (-10,000) + (1 - p) 50,000

    Firm B's expected payoff from not entering:

    EU_B(N) = p 0 + (1 - p) 0 = 0

    Indifference condition for Firm B:

    p (-10,000) + (1 - p) 50,000 = 0

    Expanding:

    -10,000p + 50,000 - 50,000p = 0

    -60,000p + 50,000 = 0

    p = 50,000 / 60,000 = 5/6 ≈ 0.8333

    Therefore, the mixed-strategy equilibrium involves each firm entering with probability approximately 0.8333 and not entering with probability 0.1667. At these probabilities, each firm is indifferent between entering and not, making neither profitable to deviate unilaterally.

    Conclusion

    This analysis demonstrates key concepts in game theory through the classic entry game. The two pure-strategy Nash equilibria—where one firm enters and the other stays out—highlight the strategic asymmetry and potential for coordination. The mixed-strategy equilibrium reflects the uncertainty and strategic mixing when neither pure strategy dominates. Recognizing the stability of these equilibria can inform real-world decision-making, especially in competitive markets where entry costs and payoffs are significant considerations.

    References

    • Fudenberg, D., & Tirole, J. (1991). Game Theory. MIT Press.
    • Myerson, R. B. (1997). Game Theory: Analysis of Conflict. Harvard University Press.
    • Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. MIT Press.
    • Binmore, K. (2007). Playing for Real: A Text on Game Theory. Oxford University Press.
    • Gibbons, R. (1992). Game Theory for Applied Economists. Princeton University Press.
    • Friedman, J. (1996). Game Theory with Applications to Economics. Oxford University Press.
    • Lucking-Reiley, D., & Williams, J. (1999). The Economics of Information and Negotiation Strategies. Journal of Business & Economic Statistics, 17(1), 86-99.
    • Vives, X. (1999). Oligopoly Pricing: Old Ideas and New Tools. MIT Press.
    • Kuhn, H. W. (1953). Extensive Games and Their Applications. Annals of Mathematics Studies, No. 28. Princeton University Press.
    • Luce, R. D., & Raiffa, H. (1957). Games and Decisions. Wiley.

    Related Assignments