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】

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.)

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?

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?

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)

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

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

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

Average Daily Frequencies (Wk 1)

Average Daily Frequencies (Wk 3)

Average Daily Frequencies (Wk 5)

Average Daily Frequencies (Wk 7)

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?

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

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)

Aggregate Data in the Lab

Aggregate Data in the Lab

Week-by-Week data in the Lab

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

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!

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

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

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

Learning in the Field (Week 2) 39

Learning in the Field (Week 3) 39

Learning in the Field (Week 4) 39

Learning in the Field (Week 5) 39

Learning in the Field (Week 6) 39

Learning in the Field (Week 7) 39

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

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

Cognitive Hierarchy 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

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

Logit QRE

Logit QRE Approx. from Below

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)

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 1-37 7 8 9 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

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 10 11 12 14 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.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

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 16 17 18 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.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

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 18 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 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

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 21 23 24 25 42 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

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 26 27 28 29 43 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 28 29 43

版權聲明版權聲明 頁碼 作品 版權標示 來源/作者 32 33 35 36 44 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