1. 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 LUPI Lottery Games”
by Robert Östling, Joseph Tao-yi Wang,
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】

2. 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.)

3. 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?

4. Research Questions Where is a Poisson game relevant?
How good does Poisson equilibrium fit the data (if there is such application)?
How did we get to equilibrium? Or, if it doesn’t, why don’t we get to equilibrium?

5. 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)

6. 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

7. Solving the LUPI Game The mixed equilibrium is characterized by
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 2
Nobody chose 1
Nobody uniquely chose 1

8. The Unique Poisson Equilibrium
Decreasing probabilities
Full support
Concave/convex
Convergence to uniform

9. Average Daily Frequencies (Wk 1)

10. Average Daily Frequencies (Wk 3)

11. Average Daily Frequencies (Wk 5)

12. Average Daily Frequencies (Wk 7)

13. 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?

14. Birth/Current Year Effects
1950
1940
1930
1960
1970
1980
1990
2007
2001

15. 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)

16. Aggregate Data in the Lab

17. Aggregate Data in the Lab

18. Week-by-Week data in the Lab

19. Week-by-Week data in the Lab
Not quite in equilibrium
95 percent confidence intervals for last week in the lab

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!

21. Learning in the Field Median of all past winning numbers
Median guess today

22. 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

23. Learning in the Field (Week 1)
Swedish Field Data
Imitation Learning
Poisson Equilibrium 39

24. Learning in the Field (Week 2)39

25. Learning in the Field (Week 3)39

26. Learning in the Field (Week 4)39

27. Learning in the Field (Week 5)39

28. Learning in the Field (Week 6)39

29. Learning in the Field (Week 7)39

30. 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

31. 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

32. Cognitive Hierarchy
τ = 1.5
λ = 2

33. Initial Response in the Field (Week 1)
Swedish Field Data
Cognitive Hierarchy
(τ = 1.80, λ=0.0034)
Poisson Equilibrium 39

34. 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

35. Logit QRE

36. Logit QRE Approx. from Below

37. 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)

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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。 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.7
依據著作權法第46、52、65條合理使用 8
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 9
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 38

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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 11
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 12 14
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 39

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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
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 17 18
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 40

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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
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 19
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 41

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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 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 25 42

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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
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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 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 36 44