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.

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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

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q 1q

p

1p

1. Calculate the employee’s expected payoff.

2. Find out his best response to each possible strategy of the manager.

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Manager

Monitor No monitor

EmployeeWork 50 , 90 50 , 100

Shirk 0 , -10 100 , -100

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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]

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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).

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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

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1. Calculate the manager’s expected payoff.

2. Find out her best response to each possible strategy of the employee.

Mix

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Manager

Monitor No monitor

EmployeeWork 50 , 90 50 , 100

Shirk 0 , -10 100 , -100

Mix

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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

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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”).

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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.

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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

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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

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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

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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

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New Payoff Matrix Manager

Monitor No monitor

EmployeeWork 50 , . . . 50 , 100

Shirk 0 , . . . 100 , -100

50

-50

Mix

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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.

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