Game Theory

“I Used to Think I Was Indecisive- But Now I’m Not So Sure”

-Anonymous

Mike ShorLecture 5

Game Theory - Mike Shor 2

Review Predicting likely outcome of a game Sequential games

• Look forward and reason back

Simultaneous games• Look for best replies

What if there are multiple equilibria? What if there are no equilibria?

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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

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Best replies do not correspond No equilibrium in pure strategies What do the players do?

Employee MonitoringManager

Monitor No Monitor

EmployeeWork 50 , 90 50 , 100

Shirk 0 , -10 100 , -100

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Mixed Strategies Randomize – surprise the rival

Mixed Strategy:• Specifies that an actual move be chosen

randomly from the set of pure strategies with some specific probabilities.

Nash Equilibrium in Mixed Strategies:• A probability distribution for each player• The distributions are mutual best responses

to one another in the sense of expectations

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Finding Mixed Strategies Suppose:

• Employee chooses (shirk, work) with probabilities (p,1-p)

• Manager chooses (monitor, no monitor) with probabilities (q,1-q)

Find expected payoffs for each player Use these to calculate best responses

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Employee’s Payoff First, find employee’s expected

payoff from each pure strategy

If employee works: receives 50 Ee(work) = 50 q + 50 (1-q)

= 50

If employee shirks: receives 0 or 100 Ee(shirk) = 0 q + 100(1-q)

= 100 – 100q

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Employee’s Best Response Next, calculate the best strategy for

possible strategies of the opponent

For q<1/2: SHIRK

Ee(shirk) = 100-100q > 50 = Ee(work)

For q>1/2: WORK

Ee(shirk) = 100-100q < 50 = Ee(work) For q=1/2: INDIFFERENT

Ee(shirk) = 100-100q = 50 = Ee(work)

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Manager’s Best Response Em(mntr) = 90 (1-p) - 10 p

Em(no m) = 100 (1-p) -100p For p<1/10: NO MONITOR

Em(mntr) = 90-100p < 100-200p = Em(no m)

For p>1/10: MONITOR

Em(mntr) = 90-100p > 100-200p = Em(no m)

For p=1/10: INDIFFERENT

Em(mntr) = 90-100p = 100-200p = Em(no m)

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Cycles

q0 11/2

p

0

1/10

1

shirk

work

monitorno monitor

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Mutual Best Replies

q0 11/2

p

0

1/10

1

shirk

work

monitorno monitor

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Mixed Strategy Equilibrium Employees shirk with probability 1/10 Managers monitor with probability ½ Expected payoff to employee:

Expected payoff to manager:

50 ]5050[]1000[21

21

109

21

21

101

80 ]100100[]1090[101

109

21

101

109

21

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Properties of Equilibrium Both players are indifferent between any

mixture over their strategies E.g. employee:

If shirk:

If work:

Regardless of what employee does, expected payoff is the same

50 ]1000[21

21

50 ]5050[21

21

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Indifference

1/2 1/2

Monitor No Monitor

9/10 Work 50 , 90 50 , 100 = 50

1/10 Shirk 0 , -10 100 , -100 = 50

= 80 = 80

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Why Do We Mix? Since a player does not care what

mixture she uses, she picks the mixture that will make her opponent indifferent!

COMMANDMENT

Use the mixed strategy that keeps your opponent guessing.

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Examples Standards and Compatibility

• Microsoft’s market dominance means that compatibility is very important

• Microsoft doesn’t want compatibility• Competitors do

Policy Enforcement• Random drug testing• Government compliance policies

Coincidence vs. divergence

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Multiple Equilibria Natural Monopoly

• Two firms are considering entry• A market generates $300K of profit,

divided by all entering firms• Fixed cost of entry is $200K

Firm 2In Out

Firm 1In -50 , -50 100 , 0

Out 0 , 100 0 , 0

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Mixed Strategies in Natural Monopoly

• Firm 1 enters with probability p• Firm 2 enters with probability q

Firm 1:• E1(in) = -50q + 100(1-q) = 100 - 150q

• E1(out) = 0q + 0(1-q) = 0

For q<2/3 in 100 - 150q>0

For q>2/3 out 100 - 150q<0

For q=2/3 indifferent 100 - 150q=0

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Mutual Best Replies

q0 12/3

p

0

2/3

1

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Multiple Equilibria Three equilibria exist:

• ( p , q ) = ( 1 , 0 ) pure strategy: (In,Out) • ( p , q ) = ( 0 , 1 ) pure strategy: (Out,In) • ( p , q ) = ( 2/3 , 2/3 ) each randomizes

Expected Payoff from mixed strategy equilibrium:

0 ]50100[]00[32

31

32

32

31

31

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Interpretation Coordination failure:

• The probability that both firms enter is (2/3) (2/3) = (4/9)

Loss of opportunity:• The probability that neither firm enters is

(1/3) (1/3) = (1/9)

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Coordination and Mixing Move fast

•Commit yourself first to guarantee your preferred outcome.

Use mixed strategies as a threat•force opponent to bargaining table.

“Mix jointly”•If you each rely on the SAME coin,

expected profits rise from 0 to 50!!!