Principles of Game Theory

1. Principles of Game Theory Lecture 8: Mixed Strategies

2. Administrative • I’ll post homework that will be due by Sunday at the beginning of class • We’ll be going over problems Thursday and Sunday. To prepare for the exam. • Do the problems! I’ll be calling on you during class both days…. • Quiz results • Not good... We’ll go over the quiz now.

3. Quiz 1 • Common problems: Proper Trees; Payoff order 3,2,2 3,2,1 A A Cassie Payoffs of the form: <Bart, Cassie, Dad> Dad S 2,3,1 S A M 1,1,3 Bart 3,2,1 A Dad S S A 2,3,1 M Cassie 1,1,3 S 2,3,2

4. Quiz 2 • No one got the first question right • <U,L,B>, <D,R,B>, <D,R,A>, <U,L,C>, <D,R,C>, <U,R,A> • Question 2: • Take the FOC; some of you didn’t – just wrote down the profit function. • Most common error (which I didn’t penalize much): FOC vs best response function

5. Penalty kicks Many of you are football (soccer) fans How would you play this game? What would you do if you knew that the keeper jumps left 75% of the time?

6. Penalty Kicks (Pee Wee league) Consider the simplified version: Allow the keeper to randomize • Suppose he jumps left p proportion of the time • What is the kicker’s best response? • p=1  kick right • P=0  kick left

7. Probabilistic Soccer • Kicker’s expected payoff: • Kick Left: -1 * p + 1 * (1-p) = 1-2p • Kick Right: 1*p + (-1) *(1-p) = 2p -1 If p < ½,  1 – 2p > 2p – 1  should kick left Similarly, if p > ½  should kick right. p 1-p

8. Probabilistic Soccer • Mixed Strategies: • If opponent knows what I’ll do, I’ll lose • Correct Randomization takes away any ability to be taken advantage of • Make opponent indifferent between her strategies. • Weird aspects of mixed strategies • A player chooses his strategy so as to make his opponent indifferent • If done correctly, the other player earns the same payoff from either of her strategies.

9. Employee Monitoring Recall the employee monitoring game: • Employees can work hard or shirk • Salary: $100K unless caught shirking, if shirking then 0 • Cost of effort: $50K

10. Employee Monitoring • Managers can monitor or not • Value of employee output: $200K • Profit if employee doesn’t work: $0 • Cost of monitoring: $10K 90 = 200 - salary - monitoring Manager

11. Employee Monitoring • No equilibrium in pure strategies • What do the players do? Manager

12. Back to Monitoring Suppose each player plays a mixed strategy: • Employee chooses (shirk, work) with probabilities (p,1-p) • Manager chooses (monitor, no monitor) with probabilities (q,1-q)

13. Mixed Monitoring Calculate each expected payoff: • Employee: • Working: • 50q + 50(1-q) = 50 • Shirking: • 0q + 100(1-q) = 100-100q These are easy to plot!

14. Employee’s Payoff What should the employee do? Employee’s payoff • Payoff from shirking • = 100-100q 100 Payoff from working: 50q + 50(1-q) = 50 50 1 0 q: Manager’s probability of monitoring

15. Employee’s decision Optimal payoff as a function of the probability of monitoring: • Payoff from shirking 100 Payoff from working: 50 Point of indifference: 100-100q = 50  q = ½ 1 0 q .5

16. Mixed Monitoring Best responses: • 50 = 100-100q  q = ½ and employee is indifferent. • If q < ½ ? • shirking has a higher expected payoff and he should shirk. • If q > ½ ? • working has a higher payoff. Therefore: if the manager wants to keep the employee from shirking, she should monitor at least ½ of the time.

17. Employee Best Response • Plot Employee’s best response: p q 0.5

18. Mixed Monitoring Now do the same for the Manager • Monitor: 90(1-p) – 10p • No Monitor: 100(1-p)-100p 100 90 Monitor p: prob of shirking 0 1 0.1 -10 No Monitor

19. Mixed Monitoring Now do the same for the Manager • Monitor: 90(1-p) – 10p • No Monitor: 100(1-p)-100p • Best responses: • For p=1/10: INDIFFERENT monitor = 90-100p = 100-200p = no monitor • For p<1/10: NO MONITOR monitor = 90-100p < 100-200p = no monitor • For p>1/10: MONITOR monitor = 90-100p > 100-200p = no monitor

20. Manager Best Response • Plot Manager’s best response: p 0.1 q

21. Mutual Best Responses • Equilibrium: Mutual Best Response p p 0.1 q q 0.5

22. Equilibrium Here we have no pure strategy equilibria but one mixed strategy equilibria: Note: Manager’s strategy of monitoring half of the time must mean that there is a 50% chance of being monitored every day. • She cannot just monitor every other day!

23. Real Life? Are mixed strategies just a mathematical construction? • Sports • Football (American and Soccer) • Tennis • Baseball • Law Enforcement • Traffic tickets • Policy compliance • Random drug testing

24. Using Mixed Strategies • How do you actually play a mixed strategy? • Study: • A fifteen percent chance of being stopped at an alcohol checkpoint will deter drinking and driving • Implementation • Set up checkpoints one day a week (1 / 7 ≈ 14%) • How about Fridays? • What does “Random” mean?

25. Change in Monitoring Costs • What happens if the cost of Monitoring was 50?