實驗經濟學一：行為賽局論
Experimental Economics I: Behavioral Game Theory
補充材料：於現實驗證賽局理論：Swedish LUPI Lottery 賽局
Additional Material: “Testing Game Theory in the Field: Swedish
Östling, Joseph Tao-yi Wang,
LUPI Lottery Games” by Robert
Eileen Chou & Colin Camerer
授課教師：國立臺灣大學 經濟學系 王道一教授（Joseph Tao-yi Wang ）
本課程指定教材: Colin E. Camerer, Behavioral Game Theory: Experiments in
Strategic Interaction. New York: Russell Sage Foundation; New Jersey:
Princeton UP, 2003.
【本著作除另有註明外，採取創用CC「姓名標示
－非商業性－相同方式分享」臺灣3.0版授權釋
出】
1
Population Uncertainty
• Game theory often assumes fixed-N players
• Not realistic in entry situations:
– Voter turn-outs,
– (Travel) congestion games,
– Online auctions, etc.
• Games with population uncertainty (Myerson, IJGT
1998, GEB 2000, etc.)
2
Poisson Games
• Poisson Games: Assume N ~ Poisson(n)
– Environmental Equivalence (EE)
– Independence of Actions (IA)
• Applied to voting games by Myerson (1998)
• Contests: Myerson and Warneryd (2006)
• Other applications?
3
Research Questions
1. Where is a Poisson game relevant?
2. How good does Poisson equilibrium fit the data
(if there is such application)?
3. How did we get to equilibrium? Or, if it doesn’t,
why don’t we get to equilibrium?
4
Join the Swedish LUPI Game
• 49 games played daily: Jan. 29 – Mar. 18, 07’
• Each choose an integer from 1 to K=99999
• The person that chooses the lowest number that no one else
does wins
– LUPI: Lowest Unique Positive Integer
• Fixed Prize: Earn 10,000 Euros if win, 0 if not.
• Play against approximately 53,783 players
– Assume “approximately 54k” is Poisson(53783)
5
Why Care?
• LUPI is a part of the economy
• The Swedish Limbo game
• Lowest unique bid auctions (ongoing research by
Eichberger & Vinogradov, Raviv & Virag and Rapoport
et al)
• Unique opportunity to test the theory
• Close field-laboratory parallel
• Full vs bounded rationality
6
Solving the LUPI Game
• To win by picking k = I uniquely picked number k
and nobody uniquely picked numbers 1~(k-1)
• The mixed equilibrium is characterized by
Nobody chose 1
Nobody uniquely chose 1
7
Nobody
chose 2
The Unique Poisson Equilibrium
0.00025
1. Decreasing
probabilities
2. Full support
3. Concave/convex
4. Convergence to
uniform
Probability
0.00020
0.00015
0.00010
0.00005
0.00000
1
10000
Number chosen (truncated at 10000)
8
Average Daily Frequencies (Wk 1)
150
100
50
0
2000
6000
4000
Number guessed
9
8000
0
10000
Average Daily Frequencies (Wk 3)
150
100
50
0
2000
4000
6000
Number guessed
10
8000
0
10000
Average Daily Frequencies (Wk 5)
100
80
60
40
20
0
2000
4000
6000
Number guessed
11
8000
0
10000
Average Daily Frequencies (Wk 7)
60
40
20
0
2000
4000
6000
Number guessed
12
8000
0
10000
Details about the Swedish Game
• Players can bet (1 Euro each) up to 6 numbers from (1,
2, 3,…, 99999)
• The (first) prize fluctuates slightly (guaranteed >10,000
Euro until 3/18/07)
• Share prize if there is a tie
• Smaller second and third prizes offered
• Do people really think it’s Poisson?
13
Birth/Current Year Effects
3000
Total # guesses on number all days
1950
1940
2001
1930
2000
1990
1000
1960
1850
2007
1900
1970
1980
1950
2000
Number guessed
14
0
2050
Lab Experimental Design
•
•
•
•
CASSEL at UCLA
Choose between 1 and K=99
49 rounds, w/ winning number announced
Scale down prize and population by 2,000:
– Winning prize = USD $7.00
– n=26.9 (=53,783 / 2,000)
– Variance is smaller than Poisson (due to a technical
error; could have made it Poisson)
15
Aggregate Data in the Lab
0
20
500
Total # plays on number all rounds
Total # plays on number all rounds
500
400
300
200
100
40
60
Number played
80
0
100
0
16
400
300
200
100
5
10
15
Number played
20
0
0
100
200
300
400
500
Aggregate Data in the Lab
0
5
10
Number chosen
17
15
20
Week-by-Week data in the Lab
15
15
10
10
5
5
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
Week 1
15
10
10
5
5
2
4
6
8
10
12
12
14
16
18
20
Week 3
15
0
10
14
16
18
0
20
Week 5
2
4
6
8
10
12
14
Week 7
18
16
18
20
Week-by-Week data in the Lab
90
• Not quite in
equilibrium
• 95 percent
confidence
intervals for last
week in the lab
80
70
60
50
40
30
20
10
0
2
4
6
8
10
12
14
16
18
19
20
Learning in the Field
• Winning numbers are the only feedback
• Nobody except the winner is reinforced
• Can update beliefs about other’s strategy
since they don’t see the frequencies
• But, people do respond to winning numbers!
20
Learning in the Field
2500
2000
Median
of all
past
Median
winner
winning
numbers
until t-1
Median
guess
Median
number
today
in t
1500
1000
500
0
1
7
13 19 25 31 37 43 49
21
Imitation Learning
• Start with initial attractions A(1)
– Backed out by empirically using initial data
• Update attractions for a window (size W) close to
the previous winning number
• Why would this work at all?
– The winning number indicates undershooting!
• MLE estimates W=344 for field data
22
Learning in the Field (Week 1)
-3
x 10
2.5
Swedish Field Data
2
Imitation Learning
1.5
1
Poisson Equilibrium
0.5
0
500
1000 150039 2000 2500 3000
23
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 2)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
24
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 3)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
25
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 4)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
26
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 5)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
27
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 6)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
28
3500 4000 4500 5000 5500
6000
Learning in the Field (Week 7)
-3
x 10
2.5
2
1.5
1
0.5
0
500
1000 150039 2000 2500 3000
29
3500 4000 4500 5000 5500
6000
How did this START?
• Cognitive hierarchy (Camerer et al, 2004): Players have
incorrect & heterogeneous beliefs.
• Zero-step thinkers randomize uniformly
• Higher-step thinkers best respond given the belief
that other players are a mixture of lower-step
thinkers
• The type distribution is Poisson; players’ beliefs are a
truncated Poisson distribution
30
How did this START?
• We extend the standard model in two respects:
• Number of players is random (Poisson); allows
computation of expected payoffs
• Players best-respond noisily using a power function
• τ: Average number of thinking steps
• λ: Degree of precision in best responses
31
Cognitive Hierarchy
0.25
0.20
0.15
0.10
0.05
0.00
8
5
4
3
2
k=0
k=1
k=2
k=4
k=3
6
7
9
10
1
32
11
12
13
20
1819
1617
15
14
τ = 1.5
λ=2
Initial Response in the Field (Week 1)
Swedish Field Data
Cognitive Hierarchy
(τ = 1.80, λ=0.0034)
Poisson Equilibrium
39
33
Quantal Respone Equilibrium
• We maintain the assumption that N ~ Poisson (n).
• Replace best responses with noisy (quantal)
responses.
• QRE: Players know both are doing quantal response
(correct beliefs)
• Can’t explain overshooting
– Converges “UP” to equilibrium
34
Logit QRE
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
2
3
4
5
6
7
8
9 10 11 12 13
14 15 16 17 18 19
35
Poisson eq.
QRE(250)
QRE(50)
QRE(1)
Logit QRE Approx. from Below
0.12
0.1
Density
0.08
0.06
0.04
Equilibrium
0.02
200
150
0
100
2
4
6
8
10
50
12
14
36
16
18
1
Conclusion
•
•
•
•
Observe a well-defined game (LUPI) played in the field
Poisson equilibrium explains the data surprisingly well
Imitation learning explains convergence
CH (τ = 1.80) accounts for initial overshooting of low
numbers
• Shouldn’t we apply population uncertainty to other
games?
• LUBA (Least Unique Bid Auctions)
37
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作品
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本作品轉載自 Microsoft Office 2011 多媒體藝廊，
依據 Microsoft 服務合約及著作權法第 46、52、65 條合理使
用。
1-37
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.7
依據著作權法第46、52、65條合理使用
7
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.8
依據著作權法第46、52、65條合理使用
0.00025
Probability
0.00020
8
0.00015
0.00010
0.00005
0.00000
1
10000
Number chosen (truncated at 10000)
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.19
依據著作權法第46、52、65條合理使用
150
100
50
9
來源/作者
0
2000
4000
6000
Number guessed
8000
0
10000
38
版權聲明版權聲明
頁碼
作品
版權標示
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.19
依據著作權法第46、52、65條合理使用
150
100
10
50
0
2000
4000
6000
Number guessed
8000
0
10000
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
100
11
80
60
40
20
0
2000
4000
6000
Number guessed
8000
12
0
10000
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
60
40
20
0
2000
4000
6000
Number guessed
8000
0
10000
Total # guesses on number all days
3000
14
1850
1900
2000
1000
1950
2000
Number guessed
來源/作者
0
2050
39
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.12
依據著作權法第46、52、65條合理使用
版權聲明版權聲明
頁碼
作品
版權標示
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), ,pp.24
依據著作權法第46、52、65條合理使用
Total # plays on number all rounds
500
16
0
400
300
200
100
20
40
60
Number played
80
0
100
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.24
依據著作權法第46、52、65條合理使用
Total # plays on number all rounds
500
16
0
400
300
200
100
5
10
Number played
15
20
0
200
300
400
500
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.24
依據著作權法第46、52、65條合理使用
0
100
Total/expected frequency
17
來源/作者
0
5
10
Number chosen
15
20
15
18
10
5
0
2
4
6
8
10
12
14
16
18
20
40
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), ,pp.26
依據著作權法第46、52、65條合理使用
版權聲明版權聲明
頁碼
作品
版權標示
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011) ,pp.26
依據著作權法第46、52、65條合理使用
15
10
18
5
0
2
4
6
8
10
12
14
16
18
20
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.26
依據著作權法第46、52、65條合理使用
15
10
18
5
0
2
4
6
8
10
12
14
16
18
20
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.26
依據著作權法第46、52、65條合理使用
15
18
10
5
0
2
4
6
8
10
12
14
16
18
來源/作者
20
19
41
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011),pp.26
依據著作權法第46、52、65條合理使用
版權聲明版權聲明
頁碼
作品
版權標示
Erik Mohlin, Robert Östling, and Joseph Tao-yi Wang, “Learning by
Imitation in Games: Theory, Field, and Laboratory,” Oxford
University, Department of Economics, Discussion Paper Series,
pp.53. 依據著作權法第46、52、65條合理使用
2500
2000
21
Median winner
until t-1
Median number
in t
1500
1000
500
0
1
7
來源/作者
13 19 25 31 37 43 49
23
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.19
依據著作權法第46、52、65條合理使用
24
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.19
依據著作權法第46、52、65條合理使用
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.19
依據著作權法第46、52、65條合理使用
25
42
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來源/作者
26
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
27
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
28
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
29
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.20
依據著作權法第46、52、65條合理使用
43
版權聲明版權聲明
頁碼
作品
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來源/作者
32
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.18
依據著作權法第46、52、65條合理使用
33
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.18
依據著作權法第46、52、65條合理使用
35
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.16
依據著作權法第46、52、65條合理使用
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
2
3
4
5
6
7
8
9 10 11 12 13
14 15 16 17 18 19
Poisson eq.
QRE(250)
QRE(50)
QRE(1)
Östling, Robert, Joseph Tao-yi Wang, Eileen Y. Chou, and Colin F.
Camerer,"Testing Game Theory in the Field: Swedish LUPI Lottery
Games," American Economic Journal: Microeconomics,Vol.3, No.3,
(2011), pp.16
依據著作權法第46、52、65條合理使用
36
44