EC202 Spring term problem sets 1-5

Problem Set 3: Game Theory Analysis

1. Electric Automobile Market: Ford vs GM

  • Payoff Matrix| | GM: Enter | GM: Not Enter| Ford: Enter | 10, -40 | 250, 0| Ford: Not Enter | 0, 200 | 0, 0

a) Dominant Strategy and Nash Equilibrium

  • Dominant Strategy:

    • Ford:

      • If GM enters: Ford gets 10 (Enter) vs 0 (Not Enter)
        - Ford’s best choice: Enter

      • If GM does not enter: Ford gets 250 (Enter) vs 0 (Not Enter)
        - Ford’s best choice: Enter

    • GM:

      • If Ford enters: GM gets -40 (Enter) vs 200 (Not Enter)
        - GM’s best choice: Not Enter

      • If Ford does not enter: GM gets 0 (Enter) vs 0 (Not Enter)
        - GM’s best choice: Not Enter

  • Conclusion: No dominant strategy for both, thus not a dominant strategy equilibrium.

  • Nash Equilibrium:

    • (Ford: Enter, GM: Not Enter) yields payoffs (250, 0) which cannot be improved unilaterally.

    • Nash Equilibrium exists at (Enter, Not Enter).

b) Effect of Lump-Sum Subsidy for GM

  • With £50 million subsidy, GM’s payoffs change if they enter:

    • Payoff if GM enters: 0 (original) + 50 = 50

    • New Payoff Matrix:| | GM: Enter | GM: Not Enter| Ford: Enter | 10, 50 | 250, 0| Ford: Not Enter | 0, 200 | 0, 0

  • New Nash Equilibrium Analysis:

    • Ford’s decision remains: Enter vs Not Enter

      • If GM enters: Ford receives 10 (Enter) vs 0 (Not Enter)
        - Ford’s best choice: Enter

      • If GM does not enter: Ford receives 250 (Enter) vs 0 (Not Enter)
        - Ford’s best choice: Enter

    • GM:

      • If Ford enters: GM now becomes better off by entering since it has benefit of the subsidy (50) vs not entering (0)
        - GM’s best choice: Enter

    • New Nash Equilibrium becomes (Enter, Enter).

2. Browser Standards: Google vs Microsoft

  • Payoff Matrix| | Microsoft: G standard | Microsoft: M standard| Google: G standard | 3, 1 | -1, -1| Google: M standard | -1, -1 | 1, 3

a) Nash Equilibrium (simultaneous decision)

  • Strategies:

    • (G standard, M standard): Payoffs (1, 3)
      - Google plays M standard gives Microsoft payoff of 3 (Microsoft’s best response).
      - Microsoft does G standard gives Google payoff of 3 (Google’s best response).

  • Nash Equilibrium:

    • (G standard, M standard) with outcomes (3, 1).

b) Google Commits First

  • If Google commits to G before Microsoft:

  • Microsoft will observe Google’s choice and choose M standard, leading to: (G standard, M standard). Thus: Payoffs shift to (1, 3).

  • Conclusion: Google cannot dissuade Microsoft completely; it can only observe and respond.

3. Game of Chicken

  • Payoff Table| | Player 2: Swerve | Player 2: Straight| Player 1: Swerve | 0, 0 | -1, 1| Player 1: Straight | 1, -1 | -4, -4

a) Pure Strategy Nash Equilibria

  • Equilibrium Choices:

    • (Swerve, Straight): Payoff (0,0)
      - Player 1 swerves, Player 2 goes straight. -Conclusion: (Swerve, Straight) and (Straight, Swerve) are Nash equilibria conditions.

b) Mixed Strategy Nash Equilibrium

  • Mixed Strategies:

    • Let p = probability that Player 1 chooses Swerve
      - Let q = probability that Player 2 chooses Swerve
      - Expected payoff for players: calculate to derive probabilities leading to indifference.

Mixed Strategy Nash Equilibrium in Game of Chicken

To derive the Mixed Strategy Nash Equilibrium in the Game of Chicken, we let:

  • p = probability that Player 1 chooses Swerve

  • q = probability that Player 2 chooses Swerve

Expected Payoff Calculation for Players:

The expected payoffs for each player can be calculated based on their strategies:

Player 1's Expected Payoff:

  • If Player 1 Swerve: Expected payoff = 0 * q + 1 * (1 - q) = 1 - q

  • If Player 1 Straight: Expected payoff = -1 * q + -4 * (1 - q) = -1q + 4 - 4q = 4 - 5q

To find the mixed strategy equilibrium, we set these expected payoffs equal:1 - q = 4 - 5q=> 5q - q = 4 - 1=> 4q = 3=> q = 3/4

Player 2's Expected Payoff:

  • We can apply similar calculations looking from Player 2's perspective:

  • If Player 2 Swerve: Expected payoff = 0 * p + -1 * (1 - p) = -1 + p

  • If Player 2 Straight: Expected payoff = 1 * p + -4 * (1 - p) = 1p - 4 + 4p = 5p - 4

Setting the expected payoffs for Player 2 equal:-1 + p = 5p - 4=> 4 = 5p - p=> 4 = 4p=> p = 1

Conclusion:

The mixed strategy Nash Equilibrium can be derived as follows:

  • Player 1 has a mixed strategy probability of 1 and Player 2 has a mixed strategy probability of 3/4 to make the other player indifferent between their two