Game Theory “I Used to Think I Was Indecisive - But Now I’m Not So Sure” - Anonymous Mike Shor Lecture 5

Game Theory - Mike Shor2 Review Predicting likely outcome of a game Sequential games Look forward and reason back Simultaneous games Look for best replies What if there are multiple equilibria? What if there are no equilibria?

Game Theory - Mike Shor3 Employee Monitoring Employees can work hard or shirk Salary: $100K unless caught shirking Cost of effort: $50K Managers can monitor or not Value of employee output: $200K Profit if employee doesn’t work: $0 Cost of monitoring: $10K

Game Theory - Mike Shor4 Best replies do not correspond No equilibrium in pure strategies What do the players do? Employee Monitoring Manager MonitorNo Monitor Employee Work 50, 9050, 100 Shirk 0, , -100

Game Theory - Mike Shor5 Mixed Strategies Randomize – surprise the rival Mixed Strategy: Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. Nash Equilibrium in Mixed Strategies: A probability distribution for each player The distributions are mutual best responses to one another in the sense of expectations

Game Theory - Mike Shor6 Finding Mixed Strategies Suppose: Employee chooses (shirk, work) with probabilities (p,1-p) Manager chooses (monitor, no monitor) with probabilities (q,1-q) Find expected payoffs for each player Use these to calculate best responses

Game Theory - Mike Shor7 Employee’s Payoff First, find employee’s expected payoff from each pure strategy If employee works: receives 50 E e (work)= 50q+ 50(1-q) = 50 If employee shirks: receives 0 or 100 E e (shirk)= 0q+ 100(1-q) = 100 – 100q

Game Theory - Mike Shor8 Employee’s Best Response Next, calculate the best strategy for possible strategies of the opponent For q<1/2: SHIRK E e (shirk) = q > 50 = E e (work) For q>1/2: WORK E e (shirk) = q < 50 = E e (work) For q=1/2: INDIFFERENT E e (shirk) = q = 50 = E e (work)

Game Theory - Mike Shor9 Manager’s Best Response E m (mntr)= 90(1-p)- 10p E m (no m)= 100(1-p)-100p For p<1/10: NO MONITOR E m (mntr) = p < p = E m (no m) For p>1/10: MONITOR E m (mntr) = p > p = E m (no m) For p=1/10: INDIFFERENT E m (mntr) = p = p = E m (no m)

Game Theory - Mike Shor10 Cycles q 01 1/2 p 0 1/10 1 shirk work monitorno monitor

Game Theory - Mike Shor11 Mutual Best Replies q 01 1/2 p 0 1/10 1 shirk work monitorno monitor

Game Theory - Mike Shor12 Mixed Strategy Equilibrium Employees shirk with probability 1/10 Managers monitor with probability ½ Expected payoff to employee: Expected payoff to manager:

Game Theory - Mike Shor13 Properties of Equilibrium Both players are indifferent between any mixture over their strategies E.g. employee: If shirk: If work: Regardless of what employee does, expected payoff is the same

Game Theory - Mike Shor14 Indifference 1/2 MonitorNo Monitor 9/10Work 50, 90 50, 100= 50 1/10Shirk 0, , -100= 50 = 80

Game Theory - Mike Shor15 Why Do We Mix? Since a player does not care what mixture she uses, she picks the mixture that will make her opponent indifferent! COMMANDMENT Use the mixed strategy that keeps your opponent guessing.

Game Theory - Mike Shor16 Examples Standards and Compatibility Microsoft’s market dominance means that compatibility is very important Microsoft doesn’t want compatibility Competitors do Policy Enforcement Random drug testing Government compliance policies Coincidence vs. divergence

Game Theory - Mike Shor17 Multiple Equilibria Natural Monopoly Two firms are considering entry A market generates $300K of profit, divided by all entering firms Fixed cost of entry is $200K Firm 2 InOut Firm 1 In -50, , 0 Out 0, 100 0, 0

Game Theory - Mike Shor18 Mixed Strategies in Natural Monopoly Firm 1 enters with probability p Firm 2 enters with probability q Firm 1: E 1 (in) = -50q + 100(1-q) = q E 1 (out)= 0q + 0(1-q) = 0 For q 0 For q>2/3 out q<0 For q=2/3 indifferent q=0

Game Theory - Mike Shor19 Mutual Best Replies q 01 2/3 p 0 1

Game Theory - Mike Shor20 Multiple Equilibria Three equilibria exist: ( p, q ) = ( 1, 0 ) pure strategy: (In,Out) ( p, q ) = ( 0, 1 ) pure strategy: (Out,In) ( p, q ) = ( 2/3, 2/3 ) each randomizes Expected Payoff from mixed strategy equilibrium:

Game Theory - Mike Shor21 Interpretation Coordination failure: The probability that both firms enter is (2/3)  (2/3) = (4/9) Loss of opportunity: The probability that neither firm enters is (1/3)  (1/3) = (1/9)

Game Theory - Mike Shor22 Coordination and Mixing Move fast Commit yourself first to guarantee your preferred outcome. Use mixed strategies as a threat force opponent to bargaining table. “Mix jointly” If you each rely on the SAME coin, expected profits rise from 0 to 50!!!