Game Theory

1. Game Theory Topic 4Mixed Strategies “I used to think I was indecisive – but now I’m not so sure.” - Anonymous

2. Review • Predicting likely outcome of a game • Sequential games • Look forward and reason back • Simultaneous games • Look for simultaneous best replies • What if (seemingly) there are no equilibria?

3. 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

4. Employee Monitoring Manager • Best replies do not correspond • No equilibrium in pure strategies • What do the players do?

5. Employee Monitoring • John Nash proved: • Every finite game has a Nash equilibrium • So, if there is no equilibrium in pure strategies, we have to allow for mixing or randomization

6. Mixed Strategies • Unreasonable predictors of one-time interaction • Reasonable predictors of long-term proportions

7. Game Winning Goal

8. G O A L I E L R K I C K E R L R Soccer Penalty Kicks (Six Year Olds Version)

9. Soccer Penalty Kicks • There are no mutual best responses • Seemingly, no equilibria • How would you play this game? • What would you do if you know that the goalie jumps left 75% of the time?

10. Probabilistic Soccer • Allow the goalie to randomize • Suppose that the goalie jumps left p proportion of the time • What is the kicker’s best response? • If p=1, goalie always jumps left • we should kick right • If p=0, goalie always jumps right • we should kick left

11. Probabilistic Soccer (continued) • The kicker’s expected payoff is: • Kick left: - 1 x p + 1 x (1-p) = 1 – 2p • Kick right: 1 x p - 1 x (1-p) = 2p – 1 • should kick left if: p < ½ (1 – 2p > 2p – 1) • should kick right if: p > ½ • is indifferent if: p = ½ • What value of p is best for the goalie?

12. Probabilistic Soccer (continued)

13. Probabilistic Soccer (continued) • Mixed strategies: • If opponent knows what I will do, I will always lose! • Randomizing just right takes away any ability for the opponent to take advantage • If opponent has a preference for a particular action, that would mean that they had chosen the worst course from your perspective. • Make opponent indifferent between her strategies

14. Mixed Strategies • Strange Implications • A player chooses his strategy so as to make her opponent indifferent • If done right, the other player earns the same payoff from either of her strategies

15. Mixed Strategies COMMANDMENT Use the mixed strategy that keeps your opponents guessing.

16. Employee Monitoring Manager • Suppose: • Employee chooses (shirk, work) with probabilities (p,1-p) • Manager chooses (monitor, no monitor) with probabilities (q,1-q)

17. Keeping Employees from Shirking • First, find employee’s expected payoff from each pure strategy • If employee works: receives 50 • Profit(work) = 50  q + 50  (1-q) = 50 • If employee shirks: receives 0 or 100 • Profit(shirk) = 0  q + 100 (1-q) = 100 – 100q

18. Employee’s Best Response • Next, calculate the best strategy for possible strategies of the opponent • For q<1/2: SHIRK Profit(shirk) = 100-100q > 50 = Profit(work) • For q>1/2: WORK Profit(shirk) = 100-100q < 50 = Profit(work) • For q=1/2: INDIFFERENT Profit(shirk) = 100-100q = 50 = Profit(work)

19. Manager’s Equilibrium Strategy • Employees will shirk if q<1/2 • To keep employees from shirking, must monitor at least half of the time • Note: Our monitoring strategy was obtained by using employees’ payoffs

20. Manager’s Best Response • Monitor: 90  (1-p) - 10  p • No monitor: 100  (1-p) -100  p • 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 • For p=1/10:INDIFFERENT monitor = 90-100p = 100-200p = no monitor

21. Cycles 1 shirk p 1/10 work 0 0 1 1/2 no monitor q monitor

22. Mutual Best Responses 1 shirk p 1/10 work 0 0 1 1/2 no monitor q monitor

23. Equilibrium Payoffs

24. Solving Mixed Strategies • Seeming random is too important to be left to chance! • Determine the probability mix for each player that makes the other player indifferent between her strategies • Assign a probability to one strategy (e.g., p) • Assign remaining probability to other strategy • Calculate opponent’s expected payoff from each strategy • Set them equal

25. New Scenario • What if cost of monitoring were 50? Manager

26. New Scenario • To make employee indifferent:

27. Real Life? • Sports • Football • Tennis • Baseball • Law Enforcement • Traffic tickets • Price Discrimination • Airline stand-by policies • Policy compliance • Random drug testing

28. IRS Audits • 1997 • Offshore evasion compliance study • Calibrated random audits • 2002 IRS Commissioner Charles Rossotti: • Audits more expensive now than in ’97 • Number of audits decreased slightly • Offshore evasion alone increased to $70 billion dollars! • Recommendation: As audits get more expensive, need to increase budget to keep number of audits constant!

29. Law Enforcement • Motivate compliance at lower monitoring cost • Audits • Drug Testing • Parking • Should punishment fit the crime?

30. Football • You have a balanced offense • Equilibrium: • run half of the time • defend the run half of the time Defense

31. Football • You have a balanced offense • The run now works better than before What is the equilibrium? Defense

32. Effects of Payoff Changes • Direct Effect: • The player benefitted should take the better action more often • Strategic Effect: • Opponent defends against my better strategy more often, so I should take the action less often

33. Mixed Strategy Examples • Market entry • Stopping to help • All-pay auctions

34. Market Entry • N potential entrants into market • Profit from staying out: 10 • Profit from entry: 40 – 10 m • m is the number that enter • Symmetric mixed strategy equilibrium: • Earn 10 if stay out. Must earn 10 if enter!

35. Stopping to Help • N people pass a stranded motorist • Cost of helping is 1 • Benefit of helping is B > 1 • i.e., if you are the only one who could help, you would, since net benefit is B-1 > 0 • Symmetric Equilibria • p is the probability of stopping • Help: B-1 • Don’t help: B x chance someone stops

36. Stopping to Help (continued) • Don’t help: • B x chance someone stops = B x ( 1 – chance no one stops ) = B x ( 1 – (1-p)N-1 ) • Set help = Don’t help • B x ( 1 – (1-p)N-1 ) = B – 1 • p = 1 - (1/B)1/(N-1)

37. Probability of Stopping B=2

38. Probability of Someone Stopping

39. All-Pay Auctions • Players decide how much to spend • Expenditures are sunk • Biggest spender wins a prize worth V • How much would you spend?

40. Pure Strategy Equilibria? • Suppose rival spends s < V • Then you should spend just a drop higher • Then rival will also spend a drop higher • Suppose rival spends s ≥ V • Then you should spend 0 • Then rival should spend a drop over 0 • No equilibrium in pure strategies

41. Mixed Strategies • We need a probability of each amount • Use a distribution function F • F(s) is the probability of spending up to s • Imagine I spend s • Profit: V x Pr{win} – s = V x F(s) – s • ε

42. Mixed Strategies • For an equilibrium, I must be indifferent between all of my strategies V x F(s) – s = V x F(s’) – s’ for any s, s’ • What about s=0? • Probability of winning = 0 • So V x 0 – 0 = 0 • V x F(s) – s = 0 • F(s) = s/V

43. Mixed Strategies • F(s) = s/V on [0,V] • This implies that every amount between 0 and V is equally likely • Expected bid = V/2 • Expected payment = V • There is no economic surplus to firms competing in this auction

44. All-Pay Auctions • Patent races • Political contests • Wars of attrition • Lesson: With equally-matched opponents, all economic surplus is competed away • If running the competition: all-pay auctions are very attractive

45. Mixed Strategies in Tennis • Study: • Ten grand slam tennis finals • Coded serves as left or right • Determined who won each point • Found: • All serves have equal probability of winning • But: serves are not temporally independent

46. What Random Means • 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?

47. Exploitable Patterns CAVEAT Use the mixed strategy that keeps your opponents guessing. BUT Your probability of each action must be the same period to period.

48. Exploitable Patterns • Manager’s strategy of monitoring half of the time must mean that there is a 50% chance of being monitored every day! • Cannot just monitor every other day. • Humans are very bad at this. • Exploit patterns!