Topic 4
Mixed Strategies
“I used to think I was indecisive
– but now I’m not so sure.”
- Anonymous

Review
 Predicting likely outcome of a game
 Sequential games
 Look forward and reason back
 Simultaneous games
 Look for simultaneous best replies
Mike Shor
 What if (seemingly) there are no equilibria?
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Mike Shor
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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Mike Shor
Employee Monitoring
Employee
Manager
Monitor No Monitor
Work
50 , 90 50 , 100
Shirk
0 , -10 100 , -100
 Best replies do not correspond
 No equilibrium in pure strategies
 What do the players do?
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Mike Shor
Employee Monitoring
 John Nash proved:
 Every finite game has a Nash equilibrium
 So, if there is no equilibrium in pure strategies, we have to allow for mixing or randomization
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Mike Shor
Mixed Strategies
 Unreasonable predictors of one-time interaction
 Reasonable predictors of long-term proportions
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Mike Shor
Game Winning Goal
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Mike Shor
K
I
C
K
E
R
Soccer Penalty Kicks
(Six Year Olds Version)
G O A L I E
L R
L
R
-1
1 ,
, 1
-1
1
-1 ,
, -1
1
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Mike Shor
Soccer Penalty Kicks
 There are no mutual best responses
 Seemingly, no equilibria
 How would you play this game?
 What would you do if you know that the goalie jumps left 75% of the time?
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Mike Shor
Probabilistic Soccer
 Allow the goalie to randomize


Suppose that the goalie jumps left p proportion of the time
What is the kicker’s best response?
 If p=1, goalie always jumps left
 we should kick right
 If p=0, goalie always jumps right
 we should kick left
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Mike Shor
Probabilistic Soccer (continued)

The kicker’s expected payoff is:
 Kick left: -1 x p + 1 x (1-p) = 1 – 2p
 Kick right: 1 x p - 1 x (1-p) = 2p – 1


 should kick left if: p < ½
(1 – 2p > 2p – 1) should kick right if: p > ½ is indifferent if: p = ½
 What value of p is best for the goalie?
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Mike Shor
Probabilistic Soccer (continued)
Goalie’s p
L (p = 1)
R (p = 0) p = 0.75
p = 0.25
p = 0.55
p = 0.50
Kicker’s strategy
R
L
R
L
R
Either
Goalie’s
Payoff
-1.0
-1.0
-0.5
-0.5
-0.1
0
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Mike Shor
Probabilistic Soccer (continued)
 Mixed strategies:
 If opponent knows what I will do, I will always lose!
 Randomizing just right takes away any ability for the opponent to take advantage
 If opponent has a preference for a particular action, that would mean that they had chosen the worst course from your perspective.
 Make opponent indifferent between her strategies
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Mike Shor
Mixed Strategies
 Strange Implications
 A player chooses his strategy so as to make her opponent indifferent
 If done right, the other player earns the same payoff from either of her strategies
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Mike Shor
Mixed Strategies
COMMANDMENT
Use the mixed strategy that keeps your opponents guessing.
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Mike Shor
Employee Monitoring
Employee
Manager
Monitor No Monitor
Work 50 , 90 50 , 100
Shirk
0 , -10 100 , -100
 Suppose:
 Employee chooses (shirk, work) with probabilities (p,1-p)
 Manager chooses (monitor, no monitor) with probabilities (q,1-q)
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Mike Shor
Keeping Employees from Shirking

First, find employee’s expected payoff from each pure strategy


If employee works: receives 50
Profit(work) = 50
 q + 50

(1-q)
= 50


If employee shirks: receives 0 or 100
Profit(shirk) = 0
 q + 100

(1-q)
= 100 – 100q
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Mike Shor
Employee’s Best Response
 Next, calculate the best strategy for possible strategies of the opponent
 For q<1/2 : SHIRK
Profit (shirk) = 100-100q > 50 = Profit(work)
 For q>1/2 : WORK
Profit (shirk) = 100-100q < 50 = Profit(work)
 For q=1/2 : INDIFFERENT
Profit(shirk) = 100-100q = 50 = Profit(work)
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Mike Shor
Manager’s Equilibrium Strategy
 Employees will shirk if q<1/2
 To keep employees from shirking, must monitor at least half of the time
 Note: Our monitoring strategy was obtained by using employees’ payoffs
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Mike Shor
Manager’s Best Response


Monitor: 90

(1-p) - 10
 p
No monitor: 100

(1-p) -100
 p
 For p<1/10 : NO MONITOR monitor = 90-100p < 100-200p = no monitor
 For p>1/10 : MONITOR monitor = 90-100p > 100-200p = no monitor
 For p=1/10 : INDIFFERENT monitor = 90-100p = 100-200p = no monitor
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Cycles
1 shirk
Mike Shor p
1/10 work
0
0 no monitor
1/2 q
1 monitor
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Mutual Best Responses
1 shirk
Mike Shor p
1/10 work
0
0 no monitor
1/2 q
1 monitor
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Mike Shor
Equilibrium Payoffs
1/2
Monitor
1/2
No Monitor
9/10
Work
50 , 90 50 , 100
1/10
Shirk
0 , -10 100 , -100
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Solving Mixed Strategies
 Seeming random is too important to be left to chance!
 Determine the probability mix for each player that makes the other player indifferent between her strategies
Mike Shor
 Assign a probability to one strategy (e.g., p)


Assign remaining probability to other strategy
Calculate opponent’s expected payoff from each strategy
 Set them equal
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Mike Shor
New Scenario
 What if cost of monitoring were 50?
Employee
Manager
Monitor No Monitor
Work
50 , 50 50 , 100
Shirk 0 , -50 100 , -100
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Mike Shor
New Scenario
 To make employee indifferent:
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Mike Shor
Real Life?
 Sports
 Football
 Tennis
 Baseball
 Law Enforcement
 Traffic tickets
 Price Discrimination
 Airline stand-by policies
 Policy compliance
 Random drug testing
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Mike Shor
IRS Audits



1997


Offshore evasion compliance study
Calibrated random audits
2002
IRS Commissioner Charles Rossotti:

Audits more expensive now than in ’97


Number of audits decreased slightly
Offshore evasion alone increased to $70 billion dollars!
Recommendation:
As audits get more expensive, need to increase budget to keep number of audits constant!
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Mike Shor
Law Enforcement
 Motivate compliance at lower monitoring cost
 Audits
 Drug Testing
 Parking
 Should punishment fit the crime?
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Mike Shor
Football
 You have a balanced offense
Offense
Run
Pass
Defense
Run
0 , 0
5 , -5
Pass
5 , -5
0 , 0
 Equilibrium:
 run half of the time
 defend the run half of the time
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Football
 You have a balanced offense
Offense
Run
Pass
Defense
Run
1 , -1
5 , -5
Pass
8 , -8
0 , 0
Mike Shor
 The run now works better than before
What is the equilibrium?
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Mike Shor
Effects of Payoff Changes
 Direct Effect:
 The player benefitted should take the better action more often
 Strategic Effect:
 Opponent defends against my better strategy more often, so I should take the action less often
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Mike Shor
Mixed Strategy Examples
 Market entry
 Stopping to help
 All-pay auctions
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Mike Shor
Market Entry
 N potential entrants into market
 Profit from staying out: 10
 Profit from entry: 40 – 10 m
 m is the number that enter
 Symmetric mixed strategy equilibrium:
 Earn 10 if stay out. Must earn 10 if enter!
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Stopping to Help
Mike Shor
 N people pass a stranded motorist
 Cost of helping is 1
 Benefit of helping is B > 1
 i.e., if you are the only one who could help, you would, since net benefit is B-1 > 0
 Symmetric Equilibria
 p is the probability of stopping
 Help: B-1

Don’t help: B x chance someone stops
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Mike Shor
Stopping to Help (continued)

Don’t help:
 B x chance someone stops
= B x ( 1 – chance no one stops )
= B x ( 1 – (1-p) N-1 )

Set help = Don’t help
 B x ( 1 – (1-p) N-1 ) = B – 1
 p = 1 - (1/B) 1/(N-1)
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Mike Shor
Probability of Stopping
1
0.9
0.8
0.7
0.6
p 0.5
0.4
0.3
0.2
0.1
0
0 2 4 6
N
8 10 12
B=2
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Mike Shor
Probability of Someone Stopping
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Mike Shor
All-Pay Auctions
 Players decide how much to spend
 Expenditures are sunk
 Biggest spender wins a prize worth V
 How much would you spend?
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Mike Shor
Pure Strategy Equilibria?
 Suppose rival spends s < V
 Then you should spend just a drop higher
 Then rival will also spend a drop higher

Suppose rival spends s ≥ V
 Then you should spend 0
 Then rival should spend a drop over 0
 No equilibrium in pure strategies
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Mike Shor
Mixed Strategies
 We need a probability of each amount
 Use a distribution function F
 F(s) is the probability of spending up to s
 ε
 Imagine I spend s
 Profit: V x Pr{win} – s
= V x F(s) – s
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Mixed Strategies
 For an equilibrium, I must be indifferent between all of my strategies
V x F(s) – s = V x F(s’) – s’ for any s, s’
 What about s=0?
 Probability of winning = 0
 So V x 0 – 0 = 0
Mike Shor
 V x F(s) – s = 0
 F(s) = s/V
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Mike Shor
Mixed Strategies
 F(s) = s/V on [0,V]
 This implies that every amount between
0 and V is equally likely
 Expected bid = V/2
 Expected payment = V
 There is no economic surplus to firms competing in this auction
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All-Pay Auctions
 Patent races
 Political contests
 Wars of attrition
Mike Shor
 Lesson: With equally-matched opponents, all economic surplus is competed away
 If running the competition: all-pay auctions are very attractive
44

Mike Shor
Mixed Strategies in Tennis
 Study:
 Ten grand slam tennis finals
 Coded serves as left or right
 Determined who won each point
 Found:
 All serves have equal probability of winning
 But: serves are not temporally independent
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Mike Shor
What Random Means
 Study:
 A fifteen percent chance of being stopped at an alcohol checkpoint will deter drinking and driving
 Implementation
 Set up checkpoints one day a week
(1 / 7 ≈ 14%)
 How about Fridays?
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Mike Shor
Exploitable Patterns
CAVEAT
Use the mixed strategy that keeps your opponents guessing.
BUT
Your probability of each action must be the same period to period.
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Exploitable Patterns

Manager’s strategy of monitoring half of the time must mean that there is a 50% chance of being monitored every day !
Mike Shor
 Cannot just monitor every other day.
 Humans are very bad at this.
 Exploit patterns!
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