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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?
Game Theory - Mike Shor 3
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 4
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
Game Theory - Mike Shor 5
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 6
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 7
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 8
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 9
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)
Game Theory - Mike Shor 12
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
Game Theory - Mike Shor 13
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
Game Theory - Mike Shor 14
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
Game Theory - Mike Shor 15
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 16
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 17
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
Game Theory - Mike Shor 18
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
Game Theory - Mike Shor 20
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
Game Theory - Mike Shor 21
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)