Mixed Strategies For Managers
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
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
Mixed strategies are strategies that involve randomization.
Fiscal Authority
Cheat
Taxpayer
Don’t
cheat
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
costly auditing
low tax revenue
pays high taxes
costly auditing
(waste)
pays high taxes
high tax revenue
Fiscal Authority
Cheat
Taxpayer
Don’t
cheat
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
costly auditing
low tax revenue
pays high taxes
costly auditing
(waste)
pays high taxes
high tax revenue
Fiscal Authority
Cheat
Taxpayer
Don’t
cheat
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
costly auditing
low tax revenue
pays high taxes
costly auditing
(waste)
pays high taxes
high tax revenue
No Nash equilibrium in pure strategies
Fiscal Authority
Cheat
Taxpayer
Don’t
cheat
Audit
Don’t audit
pays low taxes
gets punished
pays low taxes
costly auditing
low tax revenue
pays high taxes
costly auditing
(waste)
pays high taxes
high tax revenue
Players
Employee
Work
Shirk
Manager
Monitor
Do not monitor
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
Manager
Monitor
Monitor
Work
Employee
Shirk
No Monitor
No monitor
Manager
Monitor
Monitor
No Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
Employee
Manager
Monitor
Monitor
No Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
Employee
Manager
Monitor
Monitor
No Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
Employee
Manager
Monitor
Monitor
No Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
Employee
(1) What is the idea?
(2) How do we compute mixed strategies?
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.
Mixed Strategies
q
Manager1q
Monitor
No monitor
p
Work
50 , 90
50 , 100
1p
Shirk
0 , -10
100 , -100
Employee
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)
Mixed Strategies
1.
2.
Calculate the employee’s expected payoff.
Find out his best response to each possible strategy of
the manager.
Mixed Strategies
q
Employee
Manager1q
Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
Expected payoff from working:
(50 x q) + (50 x (1q))= 50
Expected payoff from shirking:
(0 x q) + (100 x (1q))= 100100q
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]
Expected payoff from working: 50
Recap:
Expected payoff from shirking:100100q
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
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).
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.
Mixed Strategies
1.
2.
Calculate the manager’s expected payoff.
Find out her best response to each possible strategy of
the employee.
Mixed Strategies
Manager
Monitor
No monitor
p
Work
50 , 90
50 , 100
1p
Shirk
0 , -10
100 , -100
Employee
Expected payoff from monitoring:
(90 x p) + (-10 x (1p))= 100p 10
Expected payoff from not monitoring:
(100 x p) + (-100 x (1p))= 200p100
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]
Expected payoff from monitoring: 100p 10
Recap:
Expected payoff from not monitoring:200p100
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
100p-10 > 200p – 100
if p >9/10
E. P. of monitoring = E.P. of no monitoring
if p =9/10
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”).
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.
Probability of working
1
p
Can this be an equilibrium?
1/3
0
1/4
q
Probability of monitoring
1
Probability of working
1
What is the employee’s
best response to q =1/4?
p
1/3
0
Shirk!
( Shirk if q <1/2 )
1/4
q
Probability of monitoring
1
Probability of working
1
p
Can this be an equilibrium?
0
1/4
q
Probability of monitoring
1
Probability of working
1
p
What is the manager’s best
response to p =0 (shirk)?
Monitor!
( Monitor if p <9/10 )
0
1/4
q
Probability of monitoring
1
Probability of working
1
p
Can this be an equilibrium?
0
q
Probability of monitoring
1
Probability of working
1
shirk
p
0
work
1/2
q
Probability of monitoring
1
Probability of working
1
9/10
p
no monitor
monitor
0
q
Probability of monitoring
1
Probability of working
1
9/10
The employee is Indifferent
between “work” and “shirk”
no monitor
Unique N.E.
in mixed
strategies
shirk
p
0
monitor
The manager
is Indifferent
between
“monitor” and
“no monitor”
work
1/2
q
Probability of monitoring
1
Mixed Strategies
1/2
Manager 1/2
Monitor
No monitor
9/10
Work
50 , 90
50 , 100
1/10
Shirk
0 , -10
100 , -100
Employee
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
Mixed Strategies
1/2
Manager 1/2
Monitor
No monitor
9/10
Work
50 , 90
50 , 100
1/10
Shirk
0 , -10
100 , -100
Employee
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
Mixed Strategies
Initial Payoff Matrix
Employee
Manager
Monitor
No monitor
Work
50 , 90
50 , 100
Shirk
0 , -10
100 , -100
New Payoff Matrix
Employee
Manager
Monitor
No monitor
Work
50 , 50
...
50 , 100
Shirk
0 , .-50
..
100 , -100
Mixed Strategies
New Payoff Matrix
Employee
Manager
Monitor
No monitor
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)
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.