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Game Theory

Game Theory. Section 2: Externalities. Agenda. Key terms and definitions Complementarity and cross-partial derivatives) Partnership game Cournot game Mixed strategy equilibria Conceptualization How to solve Application: Iran and IAEA. Key Terms and Definitions.

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Game Theory

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  1. Game Theory Section 2: Externalities

  2. Agenda • Key terms and definitions • Complementarity and cross-partial derivatives) • Partnership game • Cournot game • Mixed strategy equilibria • Conceptualization • How to solve • Application: Iran and IAEA

  3. Key Terms and Definitions • Set of rationalizable strategies • Those that survive iterated dominance • Motivated by common knowledge assumptions • Congruity • Weakly congruous strategies • Each strategy is a best response to others’ strategy • Best response complete • Set includes i’s best response to all others’ strategies

  4. Quick Congruity Quiz Pareto Optima Nash Eqm (TR & DC) strategy profiles Set of Rationalizable Strategies: {T,D} x {C,R} Set of Weakly Congruent Strategies: {T,D} x {C,R} Set of Best Response Complete: {T,D} x {C, R} or {T,D} x {L,C,R} Set of Congruent Strategies: {T,D} x {C, R}

  5. Classic Examples Pollution Lightening rods Security Grades and curving LaTeX and presentations Cooperation and competition Contextual Strategic Issues Complementarity Agents share the benefits of each other’s actions Substitutability Agents share the costs of each other’s actions Pareto efficiency Externalities: The Dirty Little Secret When One Person’s Actions Affect Others’ Welfare

  6. Partnership Game: Setting • 2 Partners in business • Complete information about the situation • Simultaneous moves (no verifiable contract) • Two players: i, j • Each expends effort: i, j • Payoffs are: πi = 2(i + j + cij) - i2 πj = 2(i + j + cij) - j2 • Interpret these equations

  7. Partnership Game: Analysis • Payoffs are: • πi = 2(i + j + cij) - i2 • πj = 2(i + j + cij) - j2 • Best response functions: • d (πi)/di = 2+2cj - 2i • set equal to zero: 2i = 2 + 2cj • i*=1+cj … This is our BR function for agent i • symmetry requires thatj*=1+ci • If each is responding to the BR of the other • j*=1+c(1+cj*), j*=1/(1-c) = i* also by symmetry • πi,j= 2[1/(1-c) + 1/(1-c) + c(1/(1-c))2] - (1/(1-c))2 • πi,j= (3-2c)/(1-c)2

  8. Partnership Game: Pareto-Efficiency In Strategic Setting, Equilibrium was i*,j*=1/(1-c), πi,j= (3-2c)/(1-c)2 • Now we leave the strategic setting (S.S.) and maximize total profits (pareto efficiency) πF = 4(i + j + cij) - i2- j2 • d (πF)/di = 4 +4cj - 2i • d (πF)/dj = 4 +4ci - 2j Set equal to zero: 2i = 4 + 4cj • i* = 2 + 2cj, j* = 2 + 2ci, i* = 2 + 2c(2 + 2ci*) • i* = 2/(1-2c) = j*, πF = 8/(1-2c), • Profit per partner, πi.j= 4/(1-2c) > (3-2c)/(1-c)2in S.S. • Note that output is also lower in the strategic setting

  9. Partnership Game: Explanation Positive Cross-Partial Derivatives = Complementarity πi = 2(i + j + cij) - i2- j2 • d (πi)/di = 2 +2cj - 2i • d ((πi)/di)/dj = 2c > 0 • Why? What does this mean? • Agent j’s efforthas positive effect on agent i’s productivity • They complement each other • All benefits are shared, but costs are private • In the strategic setting, contributing more than i* would contribute to total output (through an indirect effect via j) • But would not contribute to i’s welfare • Therefore this contribution is not made • Outcome is not Pareto Efficient (there is deadweight loss)

  10. Cournot Oligopoly: The Setting • 2 Companies in competition: Firm 1 and Firm 2 • Each sets quantity (q1 , q2), which in turn effects price p = 1000 - q1 - q2 • When nothing is produced, p = 1000 • Price = 0 when 1000 units are produced • Every additional unit produced lowers price by $1 • Quantity and price are inversely related • Firms are profit-maximizing • Cost (to a company) of each item sold: $100 π1 = (1000 - q1 - q2)q1 - 100 q1 π2 = (1000 - q1 - q2)q2 - 100 q2

  11. Cournot Oligopoly: Analysis • Profit functions for firms 1 and 2 π1 = (1000 - q1 - q2)q1 - 100q1 π2 = (1000 - q1 - q2)q2 - 100q2 • d (π1)/dq1 = 1000 - 2q1 - q2 - 100 • Set equal to zero, so... 2q1 = 1000 - q2 - 100 • q1* = 450 - (q2)/2 q2* = 450 - (q1)/2 • q1* = 450 - (450 - (q1)/2)/2 = 225 + q1/4 • (3/4)q1* = 225 • 225(4/3) = q1* = $300 = q2* by symmetry • Price is 1000 - q1 - q2 = $400, so π1,2 = $120,000

  12. Cournot Oligopoly:Pareto Problemo Recall: in strategic setting, q1,2* = $300, π1,2* = $120,000 • Now we leave the strategic setting and maximize total welfare (pareto efficiency) πF = (1000 - Q)Q - 100Q • d (πF)/dQ= 1000 - 2Q- 100 • Set equal to zero… • 2Q* = 1000 -100soQ* = 450 • Firms split 450 equally, so q1,2* = $225 • Price = 1000 - 450 = $550, so π1,2 = $123,750 • Profits in pareto world > Profits in strategic world • Why?

  13. Cournot Oligopoly: Explanation Positive Cross-Partial Derivatives = Complementarity π1 = (1000 - q1 - q2)q1 - 100q1 d (π1)/dq1 = 1000 - 2q1 - q2 - 100 d ((π1)/dq1)/dq2 = -1 < 0 • Why? What does this mean? • Firm 2’s productionnegatively effects firm 1’s production • They substitute for each other goods • All benefits are private, but costs are shared • In the strategic setting, contributing less than i* would add to total profit (by raising prices across all goods sold) • But individual firms produce until marginal revenue = $100 • Therefore firms overproduce, outcome is not efficient

  14. Mixed Strategy Equilibria Mixed Strategy Nash Equilibrium: mixed-strategy profile such that no player can increase his payoff by switching strategies, given the other players’ strategies Mixed Strategy Nash Equilibrium put positive probability (support) only on pure strategies that are themselves best responses Assume Player 2 follows a mixed strategy supporting Left and Right.Assume Player 1 plays a mixed strategy, and the two are in a MSNE. What are Player 1’s payoffs to playing Up vs. Down in any game? What are Player 2’s payoffs to playing Left vs. Right in any game?

  15. Mixed Strategies: Conceptualization • Many situations: more than one pure strategy equilibrium • Agents try to influence others’ actions • This can be done by keeping others’ off balance by selecting one’s own actions probabilistically • Mixed strategies force rivals to thin out capabilities by not being able to concentrate them in response to a single strategic action • Another way: proportions in a population

  16. Mixed Strategies: Logic The Questions that Motivate the Analysis • What mixed strategy would make my rival indifferent between some mix of his pure strategies? • If he’s indifferent, he may find he also have an incentive to mix. If so, what is the equilibrium? Prob (q) Prob (1-q) Prob (p) Prob (1-p) Player 1: (p, 1-p) to make Player 2 indifferent between Left and Right (p)(b)+(f)(1-p) = (p)(d) + (h)(1-p) p* = (h - f)/[(b+h)-(f+d)] Player 2: (q, 1-q) to make Player 1 indifferent between Up and Down (q)(a)+(1-q)(c) = (q)(e)+(1-q)(g) q* = (g - c)/[(a+g)-(c+e)]

  17. Mixed Strategies: Application Iran and the International Atomic Energy Agency • Story • Iran is eager to build nuclear weapons • IAEA wants to prohibit nuclear proliferation • Players have rival preferences • Iran wants to eliminate IAEA’s incentive to inspect • IAEA wants to eliminate Iran’s incentive to make nukes • (p)(5) + (1-p)(-1) = (p)(-10) + 0 • (q)(-10) + (1-q)(8) = (q)(3) + 0 • Mixed Strategy N.E. {(1/16),(8/21)} IAEA (q) (1-q) (p) (1-p) Consider effect of IAEA’s power

  18. Mixed Strategy: Dénouement For make nukes, Iran gets [(8/21)*-10 + (13/21)*8] = 24/21 For no nukes, Iran gets [(8/21)*3 + (13/21)*0] = 24/21 For inspect, IAEA gets [(1/16)*5 + (15/16)*-1] = -10/16 For don’t inspect, IAEA gets [(1/16)*-10 + (15/16)*0] = -10/16 IAEA (8/21) (13/21) (1/16) (15/16)

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