Learning Dynamics for Mechanism Design

Transcription

1 Learnng Dynamcs for Mechansm Desgn An Expermental Comparson of Publc Goods Mechansms Paul J. Healy Calforna Insttute of Technology

2 Overvew Insttuton (mechansm) desgn Publc goods Experments Equlbrum, ratonalty, convergence (How) Can experments mprove nsttuton/mechansm desgn?

3 Plan of the Talk Introducton The framework Mechansm desgn, exstng experments New experments Desgn, data, analyss A (better) model of behavor n mechansms Comparng the model to the data

4 A Smple Example Envronment Condo owners Preferences Income, exstng park Outcomes Gardenng budget / Qualty of the park Mechansm Proposals, votes, majorty rule Repeated Game, Incomplete Info

5 Mechansm Desgn Implementaton: g (e) F(e)

6 The Role of Experments Feld: e unknown => F(e) unknown Experment: everythng fxed/nduced except

7 The Publc Goods Envronment n agents 1 prvate good x, 1 publc good y Endowed wth prvate good only ( ) Preferences: u (x,y)=v (y)+x Lnear technology ( ) Mechansms: m M y( m) = y( m, m2, K, m t m) = t ( m, Km ) x = ω t 1 n ( 1 n ( m, y) )

8 Fve Mechansms Effcent => g (e) PO(e) Ineffcent Mechansms Voluntary Contrbuton Mech. (VCM) Proportonal Tax Mech. (Outcome-) Effcent Mechansms Domnant Strategy Equlbrum Vckrey, Clarke, Groves (VCG) (1961, 71, 73) Nash Equlbrum Groves-Ledyard (1977) Walker (1981)

9 The Expermental Envronment n = 5 Four sessons of each mech. 50 perods (repettons) Quadratc, quaslnear utlty Preferences are prvate nfo Payoff $25 for 1.5 hours Computerzed, anonymous Caltech undergrads Inexperenced subjects Hstory wndow What-If Scenaro Analyzer

10 What-If Scenaro Analyzer An nteractve payoff table Subjects understand how strateges outcomes Used extensvely by all subjects

11 Envronment Parameters Loosely based on Chen & Plott 96 u 2 ( x, y) = ( a y + b y + α ) + x a b Player Player Player Player Player = 100 Pareto optmum: y o =( b - )/( 2a )=4.8095

12 Voluntary Contrbuton Mechansm M = [0,6] y(m) = m t (m)= m Prevous experments: All players have domnant strategy: m * = 0 Contrbutons declne n tme Current experment: Players 1, 3, 4, 5 have dom. strat.: m * = 0 Player 2 s best response: m 2* = 1-2 m Nash equlbrum: (0,1,0,0,0)

13 VCM Results Average Message (4 sessons) Nash Equlbrum: (0,1,0,0,0) Domnant Strateges Player 2 PLR1 PLR2 PLR3 PLR4 PLR Perod

14 Proportonal Tax Mechansm M = [0,6] y(m) = m t (m)=( /n)y(m) No prevous experments (?) Foundaton of many effcent mechansms Current experment: No domnant strateges Best response: m * = y * k m k (y 1*,,y 5* ) = (7, 6, 5, 4, 3) Nash equlbrum: (6,0,0,0,0)

15 Prop. Tax Results Average Message PLR1 PLR2 PLR3 PLR4 PLR5 Player 1 Player Perod

16 Groves-Ledyard Mechansm Theory: Pareto optmal equlbrum, not Lndahl Supermodular f /n > 2a for every Prevous experments: Chen & Plott 96 hgher => converges better Current experment: =100 => Supermodular Nash equlbrum: (1.00, 1.15, 0.97, 0.86, 0.82) ( ) + = = ) ( 1 2 ) ( ) ( ) ( 2 2 m m m n n n m y m t m m y σ γ κ

17 Groves-Ledyard Results PLR1 PLR2 PLR3 PLR4 PLR5 Average Message Perod

18 Theory: Walker s Mechansm κ y( m) = m t ( m) = + m 1)modn m + mod n Implements Lndahl Allocatons Indvdually ratonal (nce!) Prevous experments: Chen & Tang 98 unstable Current experment: ( 1) n y( ) ( m Nash equlbrum: (12.28, -1.44, -6.78, -2.2, 2.94)

19 Walker Mechansm Results NE: (12.28, -1.44, -6.78, -2.2, 2.94) Average Message PLR1 PLR2 PLR3 PLR4 PLR Perod

20 VCG Mechansm: Theory Truth-tellng s a domnant strategy Pareto optmal publc good level Not budget balanced Not always ndvdually ratonal = + = = = = = Θ y n n y v z z n n z v y n n y v n y t y y v y b a m M j j j y j j j j j j y κ θ θ θ κ θ θ θ κ θ θ θ κ θ κ θ θ θ 1 ) ˆ ( arg max ) ˆ ( ) ˆ ( 1 ) ˆ ) ˆ ( ( ˆ) ( 1 ) ˆ ˆ) ( ( ˆ) ( ˆ) ( ) ˆ ( arg max ˆ) ( ) ˆ, ˆ ( ˆ 0 0

21 VCG Mechansm: Best Responses Truth-tellng ( ˆ θ = θ ) s a weak domnant strategy There s always a contnuum of best responses: BR ( ˆ θ ) = ˆ θ : y ˆ θ, ˆ θ y θ, ˆ θ { ( ) ( )} =

22 VCG Mechansm: Prevous Experments Attyeh, Francos & Isaac 00 Bnary publc good: weak domnant strategy Value revelaton around 15%, no convergence Cason, Sajo, Sjostrom & Yamato 03 Bnary publc good: 50% revelaton Many play non-domnant Nash equlbra Contnuous publc good wth sngle-peaked preferences: 81% revelaton Subjects play the unque equlbrum

23 VCG Experment Results Demand revelaton: 50 60% NEVER observe the domnant strategy equlbrum 10/20 subjects fully reveal n 9/10 fnal perods Fully reveal = both parameters 6/20 subjects fully reveal < 10% of tme Outcomes very close to Pareto optmal Announcements may be near non-revealng best responses

24 Summary of Expermental Results VCM: convergence to domnant strateges Prop Tax: non-equl., but near best response Groves-Ledyard: convergence to stable equl. Walker: no convergence to unstable equlbrum VCG: low revelaton, but hgh effcency Goal: A smple model of behavor to explan/predct whch mechansms converge to equlbrum Observaton: Results are qualtatvely smlar to best response predctons

25 A Class of Best Response Models A general best response framework: Predctons map hstores nto strateges ψ j ( 1 t 1 m ) j,k, m j M j Agents best respond to ther predctons m t BR ( ψ,, ψ ) A k-perod best response model: ψ ( m K j 1 K 1 t 1 1 j,, m j ) = Pure strateges only Convex strategy space Ratonal behavor, nconsstent predctons k n k s= 1 m t s j

26 Testable Predctons of the k-perod Model 1. No strctly domnated strateges after perod k 2. Same strategy k+1 tmes => Nash equlbrum 3. U.H.C. + Convergence to m * => m * s a N.E Asymptotcally stable ponts are N.E. 4. Not always stable 4.1. Global stablty n supermodular games 4.2. Global stablty n games wth domnant dagonal Note: Stablty propertes are not monotonc n k

27 Choosng the best k Whch k mnmzes t m t obs m t pred? Model k= k= k= k= k= k= k= k= k= k= k=5 s the best ft

28 15 Walker Sesson 2 Player 1 10 Message Perod Walker Sesson 2 Player 2 10 Message Perod

29 15 Walker Sesson 2 Player 3 10 Message Perod Walker Sesson 2 Player 4 10 Message Perod

30 15 Walker Sesson 2 Player 5 10 Message Perod Groves-Ledyard Sesson 1 Player 1 4 Message Perod

31 5-Perod Best Response vs. Equlbrum: Walker

32 5-Perod Best Response vs. Equlbrum: Groves-Ledyard

33 5-Perod Best Response vs. Equlbrum: VCM

34 5-Perod Best Response vs. Equlbrum: PropTax

35 Statstcal Tests: 5-B.R. vs. Equlbrum Null Hypothess: E[ m t BR t ] E[ m t EQ t ] Non-statonarty => perod-by-perod tests Non-normalty of errors => non-parametrc tests Permutaton test wth 2,000 sample permutatons t t Problem: If EQ then the test has lttle power BR Soluton: t t Estmate test power as a functon of ( EQ BR ) /σ Perform the test on the data only where power s suffcently large.

36 Prob. H 0 False Gven Reject H Smulated Test Power ( 1 2 ) Frequency of Rejectng H0 (Power)

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41 5-perod B.R. vs. Nash Equlbrum Voluntary Contrbuton (strct dom. strats): EQ t BR t Groves-Ledyard (stable Nash equl): EQ t BR t Walker (unstable Nash equl): 73/81 tests reject H 0 No apparent pattern of results across tme Proportonal Tax: 16/19 tests reject H 0 5-perod model beats any statc predcton

42 Best Response n the VCG Mechansm Convert data to polar coordnates:

43 Best Response n the cvcg Mechansm Orgn = Truth-tellng domnant strategy 0-degree Lne = Best response to 5-perod average

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45 The Testable Predctons 1. Weakly domnated ε-nash equlbra are observed (67%) The domnant strategy equlbrum s not (0%) Convergence to strct domnant strateges 2,3. 6 repettons of a strategy mples ε-equlbrum (75%) 4. Convergence wth supermodularty & dom. dagonal (G-L)

46 Conclusons Experments reveal the mportance of dynamcs & stablty Dynamc models outperform statc models New drectons for theoretcal work Applcatons for real world mplementaton Open questons: Stable mechansms mplementng Lndahl * Effcency/equlbrum tenson n VCG Effect of the What-If Scenaro Analyzer Better learnng models

47 An Almost-Trval Game Cyclng (ncludng equlbrum!) for k=3 Global convergence for k=1,2,4,5,

48 Effcency Effcency Confdence Intervals - All 50 Perods 1 Effcency No Pub Good 0.5 Walker VC PT GL VCG Mechansm

49 Publc Good Level Pareto Optmal Av e rage Publc Good Le v e ls Pers 1-50 Pers VC PT GL W K VC G VC G* Mechansm Standard Devaton Standard Devaton of PG Levels Perods 1-50 Perods VC PT GL W K VCG VCG* Mechansm

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51 Voluntary Contrbuton Mechansm Results