Game Theory
“I Used to Think I Was Indecisive
- But Now I’m Not So Sure”
-Anonymous
Mike Shor
Lecture 5
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?

Game Theory - Mike Shor
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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
Game Theory - Mike Shor
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Employee Monitoring
Work
Employee
Shirk



Manager
Monitor
No Monitor
50 , 90 50 , 100
0 , -10 100 , -100
Best replies do not correspond
No equilibrium in pure strategies
What do the players do?
Game Theory - Mike Shor
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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
Game Theory - Mike Shor
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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
Game Theory - Mike Shor
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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
Game Theory - Mike Shor
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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)
Game Theory - Mike Shor
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Manager’s Best Response



Em(mntr) = 90 (1-p) - 10 p
Em(no m) = 100 (1-p) -100p
For p<1/10:
NO MONITOR
Em(mntr) = 90-100p < 100-200p = Em(no m)

For p>1/10:

For p=1/10:
MONITOR
Em(mntr) = 90-100p > 100-200p = Em(no m)
INDIFFERENT
Em(mntr) = 90-100p = 100-200p = Em(no m)
Game Theory - Mike Shor
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Cycles
1
shirk
p
1/10
work
0
0
no monitor
1
1/2
q
monitor
Game Theory - Mike Shor
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Mutual Best Replies
1
shirk
p
1/10
work
0
0
no monitor
1
1/2
q
monitor
Game Theory - Mike Shor
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Mixed Strategy Equilibrium



Employees shirk with probability 1/10
Managers monitor with probability ½
Expected payoff to employee:
1 [ 1 0  1 100]  9 [ 1 50  1 50]  50
10 2
2
10 2
2

Expected payoff to manager:
1 [ 9 90  1 10]  1 [ 9 100  1 100]  80
2 10
10
2 10
10
Game Theory - Mike Shor
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Properties of Equilibrium


Both players are indifferent between any
mixture over their strategies
E.g. employee:
[ 1 0  1 100]  50

If shirk:

If work:

Regardless of what employee does,
expected payoff is the same
2
2
[ 1 50  1 50]  50
2
2
Game Theory - Mike Shor
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Indifference
9/10 Work
1/10 Shirk
1/2
1/2
Monitor
No Monitor
50 ,
90 50 , 100 = 50
0 , -10 100 , -100 = 50
= 80
= 80
Game Theory - Mike Shor
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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.
Game Theory - Mike Shor
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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
Game Theory - Mike Shor
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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 2
Firm 1
In
Out
In
-50 , -50
0 , 100
Game Theory - Mike Shor
Out
100 , 0
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
 For q>2/3 out
 For q=2/3 indifferent

Game Theory - Mike Shor
100 - 150q>0
100 - 150q<0
100 - 150q=0
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Mutual Best Replies
1
2/3
p
0
0
2/3
1
q
Game Theory - Mike Shor
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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:
1 [1 0  2 0]  2 [1100  2 50]  0
3 3
3
3 3
3
Game Theory - Mike Shor
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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)
Game Theory - Mike Shor
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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!!!
Game Theory - Mike Shor
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