Mixed Strategies For Managers. Dominant and dominated strategies Dominant strategy equilibrium...
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Transcript of Mixed Strategies For Managers. Dominant and dominated strategies Dominant strategy equilibrium...
Mixed Strategies For Managers
Dominant and dominated strategiesDominant strategy equilibriumPrisoners’ dilemma
Nash equilibrium in pure strategiesGames with multiple Nash equilibriaEquilibrium selection
Games with no pure strategy Nash equilibriaMixed strategy Nash equilibrium
Games with no pure strategy Nash equilibrium
Mixed StrategiesWhat is the idea?How do we compute them?
Mixed strategies in practiceExamplesEvidence from football penalty kicks
Minimax strategies in zero-sum games
Fiscal Authority
Taxpayer
Cheat
Don’t cheat
Audit Don’t audit
pays low taxes
gets punished pays low taxes
pays high taxes pays high taxes
costly auditing
costly auditing (waste)
low tax revenue
high tax revenue
Mixed strategies are strategies that involve randomization.
Fiscal Authority
Taxpayer
Cheat
Don’t cheat
Audit Don’t audit
pays low taxes
gets punished pays low taxes
pays high taxes pays high taxes
costly auditing
costly auditing (waste)
low tax revenue
high tax revenue
Fiscal Authority
Taxpayer
Cheat
Don’t cheat
Audit Don’t audit
pays low taxes
gets punished pays low taxes
pays high taxes pays high taxes
costly auditing
costly auditing (waste)
low tax revenue
high tax revenue
Fiscal Authority
Taxpayer
Cheat
Don’t cheat
Audit Don’t audit
pays low taxes
gets punished pays low taxes
pays high taxes pays high taxes
costly auditing
costly auditing (waste)
low tax revenue
high tax revenue
No Nash equilibrium in pure strategies
Players
Employee
Work
Shirk
Manager
Monitor
Do not monitor
The employeeSalary: $100K unless caught shirking Cost of effort: $50K
The managerValue of the employee output: $200KProfit if the employee doesn’t work: $0Cost of monitoring: $10K
Monitor No monitor
Employee
Manager
Work
Shirk
Monitor No Monitor
Monitor No monitor
50 , 90 50 , 100
0 , -10 100 , -100
Employee
Manager
Work
Shirk
Monitor No Monitor
Monitor No monitor
50 , 90 50 , 100
0 , -10 100 , -100
Employee
Manager
Work
Shirk
Monitor No Monitor
Monitor No monitor
50 , 90 50 , 100
0 , -10 100 , -100
Employee
Manager
Work
Shirk
Monitor No Monitor
Monitor No monitor
50 , 90 50 , 100
0 , -10 100 , -100
Employee
Manager
Work
Shirk
Monitor No Monitor
(1) What is the idea?
(2) How do we compute 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.
Mix
ed
Str
ate
gie
s
Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Suppose that:
The employee chooses to work with probability p
(and shirk with 1p)
The manager chooses to monitor with probability q
(and no monitor with 1q)
Mix
ed
Str
ate
gie
s
q 1q
p
1p
1. Calculate the employee’s expected payoff.
2. Find out his best response to each possible strategy of the manager.
Mix
ed
Str
ate
gie
s
Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Mix
ed
Str
ate
gie
s
Expected payoff from working:
Expected payoff from shirking:
q 1q
(50 x q) + (50 x (1q))= 50
(0 x q) + (100 x (1q))= 100100q
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]
Mix
ed
Str
ate
gie
s
Expected payoff from working: 50
Expected payoff from shirking:100100q
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 shirkingif q =1/2
Recap:
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).
Mix
ed
Str
ate
gie
s
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.Mix
ed
Str
ate
gie
s
1. Calculate the manager’s expected payoff.
2. Find out her best response to each possible strategy of the employee.
Mix
ed
Str
ate
gie
s
Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Mix
ed
Str
ate
gie
s
Expected payoff from monitoring:
Expected payoff from not monitoring:
p
1p
(90 x p) + (-10 x (1p))= 100p 10
(100 x p) + (-100 x (1p))= 200p100
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]
Mix
ed
Str
ate
gie
s
Expected payoff from monitoring: 100p 10
Expected payoff from not monitoring:200p100
E. P. of monitoring > E.P. of no monitoring100p-10 > 200p – 100
if p <9/10
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 monitoringif p =9/10
Recap:
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”).
Mix
ed
Str
ate
gie
s
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.
Mix
ed
Str
ate
gie
s
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
Can this be an equilibrium?
1/4
1/3
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
What is the employee’s best response to q =1/4?
1/4
1/3
Shirk!
( Shirk if q <1/2 )
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
1/4
Can this be an equilibrium?
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
1/4
What is the manager’s best response to p =0 (shirk)?
Monitor!
( Monitor if p <9/10 )
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
Can this be an equilibrium?
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
1/2
shirk work
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
monitor
no monitor9/1
0
0 1
1
q
p
Probability of monitoring
Pro
bab
ilit
y o
f w
ork
ing
1/2
shirk workmonitor
no monitor9/1
0
The employee is Indifferent between “work” and
“shirk”The
manager is Indifferent between
“monitor” and “no monitor”
Unique N.E. in mixed
strategies
Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Mix
ed
Str
ate
gie
s
Expected payoff from working: (50 x ½ ) + (50 x ½ ) = 50Expected payoff from shirking: (0 x ½ ) + (100 x ½ ) = 50
Gets (50 x 9/10) + (50 x 1/10) = 50
9/10
1/10
1/2 1/2
Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Mix
ed
Str
ate
gie
s
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
9/10
1/10
1/2 1/2
Initial Payoff Matrix Manager
Monitor No monitor
EmployeeWork 50 , 90 50 , 100
Shirk 0 , -10 100 , -100
Mix
ed
Str
ate
gie
s
New Payoff Matrix Manager
Monitor No monitor
EmployeeWork 50 , . . . 50 , 100
Shirk 0 , . . . 100 , -100
50
-50
Mix
ed
Str
ate
gie
s
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)
New Payoff Matrix Manager
Monitor No monitor
EmployeeWork 50 , 50 50 , 100
Shirk 0 , -50 100 , -100
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.
Mix
ed
Str
ate
gie
s