1. Mixed Strategies For Managers

2. Overview
Dominant and dominated strategies Dominant strategy equilibrium Prisoners’ dilemma Nash equilibrium in pure strategies Games with multiple Nash equilibria Equilibrium selection Games with no pure strategy Nash equilibria Mixed strategy Nash equilibrium

3. Outline
Games with no pure strategy Nash equilibrium Mixed Strategies What is the idea? How do we compute them? Mixed strategies in practice Examples Evidence from football penalty kicks Minimax strategies in zero-sum games

4. costly auditing (waste)
Mixed strategies are strategies that involve randomization.
Example: filing taxes
Fiscal Authority
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
Cheat
Fiscal Authorithy
low tax revenue
costly auditing
Taxpayer
pays high taxes
Don’t cheat
pays high taxes
costly auditing (waste)
high tax revenue

5. Best response of the taxpayer
Fiscal Authority
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
Cheat
Fiscal Authority
low tax revenue
costly auditing
Taxpayer
pays high taxes
Don’t cheat
pays high taxes
costly auditing (waste)
high tax revenue

6. Best response of the Fiscal Authorithy
Fiscal Authority
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
Cheat
Fiscal Authority
low tax revenue
costly auditing
Taxpayer
pays high taxes
Don’t cheat
pays high taxes
costly auditing (waste)
high tax revenue

7. Best responses do not coincide:
No Nash equilibrium in pure strategies
Fiscal Authority
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
Cheat
Fiscal Authority
low tax revenue
costly auditing
Taxpayer
pays high taxes
Don’t cheat
pays high taxes
costly auditing (waste)
high tax revenue

8. Example: to work or not to work…
Players Employee Work Shirk Manager Monitor Do not monitor

9. Payoffs
The employee Salary: $100K unless caught shirking Cost of effort: $50K The manager Value of the employee output: $200K Profit if the employee doesn’t work: $0 Cost of monitoring: $10K

10. The payoff matrix Manager Employee Monitor No monitor Monitor
Work
Employee
Shirk

11. The payoff matrix Manager Employee Monitor No monitor 50 , 90 50 , 100
0 , -10
100 , -100
Manager
Monitor
No Monitor
Work
Employee
Shirk

12. Best response of the employee
Monitor
No monitor
50 , 90
50 , 100
0 , -10
100 , -100
Manager
Monitor
No Monitor
Work
Employee
Shirk

13. Best response of the manager
Monitor
No monitor
50 , 90
50 , 100
0 , -10
100 , -100
Manager
Monitor
No Monitor
Work
Employee
Shirk

14. No Nash equilibrium in pure strategies
Monitor
No monitor
50 , 90
50 , 100
0 , -10
100 , -100
Manager
Monitor
No Monitor
Work
Employee
Shirk

15. Mixed Strategies
(1) What is the idea? (2) How do we compute mixed strategies?

16. The Idea Mixed Strategies
The idea is to prevent the other player to anticipate my strategy. Randomizing “just right” takes away any ability to be taken advantage of. Just right: Making other player indifferent to her strategies.

17. Computing mixed strategies
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100 q
1q p
1p
Suppose that:
The employee chooses to work with probability p
(and shirk with 1p)
The manager chooses to monitor with probability q
(and no monitor with 1q)

18. The manager’s perspective: how can I avoid shirking?
Mixed Strategies
Calculate the employee’s expected payoff.
Find out his best response to each possible strategy of the manager.

19. 1. Expected payoff of the employee
Mixed Strategies
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100 q
1q
Expected payoff from working:
Expected payoff from shirking:
(50 x q) +
(50 x (1q))= 50
(0 x q) + (100 x (1q))= 100100q

20. 2. The employee’s best response
Mixed Strategies
What is the employee’s best response for all possible strategies of the manager? The manager’s possible strategies: q=0, q=0.1, …, q=0.5, ..., q=1 Technically, q[0,1]

21. 2. The employee’s best response
Expected payoff from working: 50 Expected payoff from shirking:100100q
Recap:
E. P. working > E.P. of shirking
50 > 100 – 100q
if q >1/2
E. P. working < E.P. of shirking
50 < 100 – 100q
if q <1/2
E. P. working = E.P. of shirking
if q =1/2

22. 2. The employee’s best response
Mixed Strategies
Best response to all q >1/2 : Work Best response to all q <1/2 : Shirk Best response to q=1/2 : Work or Shirk (i.e., the employee is indifferent) If you want to keep the employee from shirking, you should set q >1/2 (i.e., monitor more than half of the time).

23. Not done yet… Mixed Strategies
All this was from the Manager’s perspective; she wants to determine the best q to induce the Employee not to shirk. To do so, she tried to figure out how the employee would respond to different q. Now look at things from the Employee’s perspective. The employee will also try to determine the best p.

24. The employee’s perspective: follow the same steps
Mixed Strategies
Calculate the manager’s expected payoff.
Find out her best response to each possible strategy of the employee.

25. 1. Expected payoff of the manager
Mixed Strategies
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100 p
1p
Expected payoff from monitoring:
Expected payoff from not monitoring:
(90 x p) +
(-10 x (1p))=
100p 10
(100 x p) + (-100 x (1p))= 200p100

26. 2. The manager’s best response
Mixed Strategies
What is the manager’s best response for all possible strategies of the employee? The employee’s possible strategies: p=0, p=0.1, …, p=0.5, ..., p=1 Technically, p[0,1]

27. 2. The manager’s best response
Expected payoff from monitoring: 100p 10 Expected payoff from not monitoring:200p100
Recap:
E. P. of monitoring > E.P. of no monitoring
100p-10 > 200p – 100
if p <9/10
E. P. of monitoring < E.P. of no monitoring
if p >9/10
E. P. of monitoring = E.P. of no monitoring
if p =9/10

28. 2. The manager’s best response
Mixed Strategies
Best response to all p <9/10: Monitor Best response to all p >9/10: No monitor Best response to p=9/10 : Monitor or No Monitor (i.e., the manager is indifferent) If you want keep the manager from monitoring, you should set p > 9/10 (work “most of the time”).

29. Nash equilibrium in mixed strategies
The employer works with probability 9/10 and shirks with probability 1/10. The manager monitors with probability ½ and does not monitor with probability ½.

30. Nash equilibrium in mixed strategies1 p
Probability of working
Can this be an equilibrium?
1/3
1/4 1 q
Probability of monitoring

31. Nash equilibrium in mixed strategies1 p
What is the employee’s best response to q =1/4?
Probability of working
Shirk!
1/3
( Shirk if q <1/2 )
1/4 1 q
Probability of monitoring

32. Nash equilibrium in mixed strategies1 p
Probability of working
Can this be an equilibrium?
1/4 1 q
Probability of monitoring

33. Nash equilibrium in mixed strategies1 p
Probability of working
What is the manager’s best response to p =0 (shirk)?
Monitor!
( Monitor if p <9/10 )
1/4 1 q
Probability of monitoring

34. Nash equilibrium in mixed strategies1 p
Probability of working
Can this be an equilibrium? 1 q
Probability of monitoring

35. Nash equilibrium in mixed strategies1
shirk
work p
Probability of working
1/2 1 q
Probability of monitoring

36. Nash equilibrium in mixed strategies1
no monitor
9/10 p
Probability of working
monitor 1 q
Probability of monitoring

37. Nash equilibrium in mixed strategies
The employee is Indifferent between “work” and “shirk” 1
The manager is Indifferent between “monitor” and “no monitor”
no monitor
9/10
Unique N.E. in mixed strategies
shirk
work p
Probability of working
monitor
1/2 1 q
Probability of monitoring

38. Equilibrium Payoffs: the employee
Mixed Strategies
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
1/2
1/2
9/10
1/10
Expected payoff from working:
(50 x ½ ) + (50 x ½ ) = 50
Expected payoff from shirking:
(0 x ½ ) + (100 x ½ ) = 50
Gets (50 x 9/10) + (50 x 1/10) = 50

39. Equilibrium Payoffs: the manager
Mixed Strategies
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
1/2
1/2
9/10
1/10
Expected payoff from monitoring:
(90 x 9/10 ) + (-10 x 1/10) = 80
Expected payoff from no monitoring:
(100 x 9/10 ) + (-100 x 1/10 ) = 80
Gets (80 x 1/2) + (80 x 1/2) = 80

40. What if cost of monitoring was 50 (instead of 10)?
Mixed Strategies
Initial Payoff Matrix
Manager
Monitor
No monitor
Employee
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
New Payoff Matrix
Manager
Monitor
No monitor
Employee
Work
50 , . . .
50 , 100
Shirk
0 , . . .
100 , -100 50
-50

41. A change in the manager’s payoffs
Mixed Strategies
New Payoff Matrix
Manager
Monitor
No monitor
Employee
Work
50 , 50
50 , 100
Shirk
0 , -50
100 , -100
Which player’s equilibrium strategy will change?
The employee’s equilibrium strategy:
“Work with probability ½ and shirk with probability ½”
(As opposed to “work with probability 9/10 …” with a less expensive monitoring technology)

42. Properties of mixed strategy equilibria
Mixed Strategies
A player chooses his strategy so as to make his rival indifferent. As a player, you want to prevent others from exploiting any systematic behavior of yours. A player earns the same expected payoff for each pure strategy chosen with positive probability. When a player’s own payoff from a pure strategy changes (e.g., more costly monitoring), his mixture does not change but his opponent’s does.