ECO 4341: Notes on Nash Equilibria, Mixed Strategies, and Stackelberg

Nash Equilibria: Pure Strategy NE

  • a. First 2-player 3x3 game (L, C, R) vs (T, M, B)

    - Payoff matrix (row, column):

    \begin{array}{c|ccc}\n & L & C & R\\hline T& (0,0)& (0,44)& (0,31)\\n M& (44,0)& (14,14)& (2,16)\\n B& (31,0)& (16,2)& (1,1)\\end{array}

    • Pure strategy NE: (B,C) and (M,R)
    • At (B,C) the payoff is (16,2). Row cannot improve by deviating: T,C gives 0, M,C gives 14, both < 16; column cannot improve by deviating: B,L gives 0, B,R gives 1, both < 2.
    • At (M,R) the payoff is (2,16). Row cannot improve by deviating: T,R gives 0, B,R gives 1, both < 2; column cannot improve by deviating: M,L gives 0, M,C gives 14, both < 16.

  • b. 2-player 2×3 game with Up/Down vs L/C/R

    • Payoff matrix:
      \begin{array}{c|ccc}\n & L & C & R\\hline U& (10,0)& (7,9)& (15,8)\\n D& (10,15)& (5,11)& (12,12)\\end{array}
    • Pure strategy NE: (Up, C) and (Down, L)
    • (Up,C): column best response to Up is C (payoff 9, compared to 0 at L and 8 at R); row best response to C is Up (7 vs 5 if Down).
    • (Down,L): column best response to Down is L (15 vs 11 at C and 12 at R); row best response to L is Down (10 vs 10 for Up, but Down is not worse and supports the equilibrium under the given best-response structure).

  • c. Subsidies and International Trade (Airbus vs Boeing)

    • Baseline payoffs (both decide simultaneously):
    • If both produce: each loses $10B. If exactly one produces: the producer earns $50B. If neither produces: $0.
    • Normal-form representation (two pure strategies per firm):
      \begin{array}{c|cc}\n & \text{Produce} & \text{Don\'t Produce}\\hline \n\text{Airbus} & (-10,-10) & (50,0) \\n\text{Boeing} & (0,50) & (0,0) \\end{array}
    • Pure strategy NE: (Produce, Don’t Produce) and (Don’t Produce, Produce)
    • Each is a coordination-friendly outcome: if Airbus produces, Boeing’s best response is not to produce (to avoid dual production losses), and vice versa. Both are NE in pure strategies.

  • d. Subsidies: EC subsidy of $15B to Airbus if Airbus produces

    • Modified payoff matrix (Airbus EC subsidy on Airbus producing):
      \begin{array}{c|cc}\n& \text{Produce} & \text{Don\'t Produce}\\hline \n\text{Airbus} & (-10,5) & (0,0) \\n\text{Boeing} & (50,0) & (0,0) \\end{array}
    • Result: The subsidy eliminates one NE and leaves only (Don\'t Produce, Produce) in which Airbus earns 0? Wait: with the subsidy, the NE identified is (Don\'t Produce, Produce). The EC subsidy effectively removes the NE where both would threaten to produce, eliminating the coordination problem and concentrating profit in Airbus’s favor.
    • Discussion: Subsidies can reallocate payoffs and shift NE, potentially solving coordination failures but creating winners/losers. They change efficiency and welfare depending on perspective (aircraft producers, workers, taxpayers). The broader question of whether subsidies are "good" depends on definitions of social vs private welfare and the vantage point. The takeaway is that policy can alter NE and market outcomes in predictable, strategic ways, even if it does not always improve overall efficiency.

  • e. Connections to real-world trade policy: subsidies can change strategic incentives between international producers and can affect market structure, price, and welfare. Considerations include coordination failures, welfare distribution, and political economy (lobbying, retaliation).