Learning Dynamics for Mechanism Design

Learning Dynamics for

Mechanism Design

An Experimental Comparison of

Public Goods Mechanisms





Paul J. Healy

California Institute of Technology

Overview

• Institution (mechanism) design

– Public goods





• Experiments

– Equilibrium, rationality, convergence





• (How) Can experiments improve

institution/mechanism design?

Plan of the Talk

• Introduction



• The framework

– Mechanism design, existing experiments





• New experiments

– Design, data, analysis





• A (better) model of behavior in mechanisms



• Comparing the model to the data

A Simple Example

• Environment

– Condo owners

– Preferences

– Income, existing park

• Outcomes

– Gardening budget / Quality of the park

• Mechanism

– Proposals, votes, majority rule

• Repeated Game, Incomplete Info

Mechanism Design









Implementation: g(e)F(e)

The Role of Experiments









Field: e unknown => F(e) unknown

Experiment: everything fixed/induced except 

The Public Goods Environment

• n agents

• 1 private good x, 1 public good y

• Endowed with private good only (gi)

• Preferences: ui(xi,y)=vi(y)+xi

• Linear technology ()

• Mechanisms: mi  M i

y (m)  y (m1 , m2 ,, mn )

ti (m)  ti (m1 , mn )

xi  i  ti (m, y )

Five Mechanisms

• “Efficient” => g(e)  PO(e)

• Inefficient Mechanisms

• Voluntary Contribution Mech. (VCM)

• Proportional Tax Mech.

• (Outcome-) Efficient Mechanisms

– Dominant Strategy Equilibrium

• Vickrey, Clarke, Groves (VCG) (1961, 71, 73)

– Nash Equilibrium

• Groves-Ledyard (1977)

• Walker (1981)

The Experimental Environment

• n=5

• Four sessions of each mech.

• 50 periods (repetitions)

• Quadratic, quasilinear utility

• Preferences are private info

• Payoff ≈ $25 for 1.5 hours

•Computerized, anonymous

•Caltech undergrads

•Inexperienced subjects



•History window

•“What-If Scenario Analyzer”

What-If Scenario Analyzer









• An interactive payoff table

• Subjects understand how strategies → outcomes

• Used extensively by all subjects

Environment Parameters

• Loosely based on Chen & Plott ’96



u i ( xi , y )  (ai y  bi y   i )  xi

2





ai bi i

Player 1 1 34 260

Player 2 8 116 140

Player 3 2 40 260

Player 4 6 68 250

Player 5 4 44 290



•  = 100

• Pareto optimum: yo =(bi - )/(2ai)=4.8095

Voluntary Contribution Mechanism

Mi = [0,6] y(m) = imi ti(m)= mi



• Previous experiments:

– All players have dominant strategy: m* = 0

– Contributions decline in time



• Current experiment:

– Players 1, 3, 4, 5 have dom. strat.: m* = 0

– Player 2’s best response: m2* = 1 - i2mi

– Nash equilibrium: (0,1,0,0,0)

VCM Results

6

PLR1

PLR2

5

Nash Equilibrium: (0,1,0,0,0) PLR3

PLR4

Average Message (4 sessions)









Dominant Strategies PLR5

4







3







2



Player 2

1







0

0 10 20 30 40 50

Period

Proportional Tax Mechanism

Mi = [0,6] y(m) = imi ti(m)=(/n)y(m)



• No previous experiments (?)

• Foundation of many efficient mechanisms

• Current experiment:

– No dominant strategies

– Best response: mi* = yi*  ki mk

– (y1*,…,y5*) = (7, 6, 5, 4, 3)

– Nash equilibrium: (6,0,0,0,0)

Prop. Tax Results

6

PLR1

PLR2

5 PLR3

PLR4

PLR5

4

Average Message









Player 1





3





2

Player 2



1





0

0 10 20 30 40 50

Period

Groves-Ledyard Mechanism

 y ( m)   n  1

y (m)   mi t i ( m)    mi  mi 2   2 (mi ) 



i n 2 n 

• Theory:

– Pareto optimal equilibrium, not Lindahl

– Supermodular if /n > 2ai for every i

• Previous experiments:

– Chen & Plott ’96 – higher  => converges better

• Current experiment:

–  =100 => Supermodular

– Nash equilibrium: (1.00, 1.15, 0.97, 0.86, 0.82)

Groves-Ledyard Results

6

PLR1

5 PLR2

PLR3

4 PLR4

PLR5

3

Average Message









2



1



0



-1



-2



-3



-4

0 10 20 30 40 50

Period

Walker’s Mechanism

 

y (m)   mi ti (m)    m(i 1) m odn  mi 1m odn  y (m)

i n 



• Theory:

– Implements Lindahl Allocations

– Individually rational (nice!)

• Previous experiments:

– Chen & Tang ’98 – unstable

• Current experiment:

– Nash equilibrium: (12.28, -1.44, -6.78, -2.2, 2.94)

Walker Mechanism Results

NE: (12.28, -1.44, -6.78, -2.2, 2.94)



12

PLR1

10 PLR2

PLR3

8 PLR4

PLR5

6

Average Message









4



2



0



-2



-4



-6



-8

0 10 20 30 40 50

Period

VCG Mechanism: Theory

M i  i ˆ ˆ ˆ

mi   i  (ai , bi )

 

y ( )  arg max vi ( y |  i )  y 

ˆ ˆ

y 0

 i 

ˆ

ˆ)  y ( )   v ( y ( ) |  )  n  1 y ( )    v ( z ( ) |  )  n  1 z ( ) 

 j

ti ( 

ˆ ˆ

j

ˆ  

  i j i ˆi ˆj n i ˆi  

n  j i n   j 

ˆ )  arg max v ( y |  )  n  1 y 

zi ( i  j ˆ 

j

y 0

 j i n 



• Truth-telling is a dominant strategy

• Pareto optimal public good level

• Not budget balanced

• Not always individually rational

VCG Mechanism: Best Responses

• Truth-telling (ˆi  i ) is a weak dominant strategy

• There is always a continuum of best responses:

ˆ  ˆ  ˆ ˆ

BR ( )   : y  ,  y  ,

i i i i i  

ˆ

i i 

VCG Mechanism: Previous Experiments



• Attiyeh, Franciosi & Isaac ’00

– Binary public good: weak dominant strategy

– Value revelation around 15%, no convergence

• Cason, Saijo, Sjostrom & Yamato ’03

– Binary public good:

• 50% revelation

• Many play non-dominant Nash equilibria

– Continuous public good with single-peaked

preferences:

• 81% revelation

• Subjects play the unique equilibrium

VCG Experiment Results

• Demand revelation: 50 – 60%

– NEVER observe the dominant strategy equilibrium





• 10/20 subjects fully reveal in 9/10 final periods

– “Fully reveal” = both parameters





• 6/20 subjects fully reveal Nash equilibrium



3. U.H.C. + Convergence to m* => m* is a N.E.

3.1. Asymptotically stable points are N.E.



4. Not always stable

4.1. Global stability in supermodular games

4.2. Global stability in games with dominant diagonal

Note: Stability properties are not monotonic in k

Choosing the best k

• Which k minimizest |mtobs  mtpred| ?

Model 2-50 3-50 4-50 5-50 6-50 7-50 8-50 9-50 10-50 11-50

k=1 1.407 1.394 1.284 1.151 1.104 1.088 1.072 1.054 1.054 1.049

k=2 - 1.240 1.135 0.991 0.967 0.949 0.932 0.922 0.913 0.910

k=3 - - 1.097 0.963 0.940 0.925 0.904 0.888 0.883 0.875

k=4 - - - 0.952 0.932 0.915 0.898 0.877 0.866 0.861

k=5 - - - - 0.924 0.9114 0.895 0.876 0.860 0.853

k=6 - - - - - 0.9106 0.897 0.881 0.868 0.854

k=7 - - - - - - 0.899 0.884 0.873 0.863

k=8 - - - - - - - 0.884 0.874 0.864

k=9 - - - - - - - - 0.879 0.870

k=10 - - - - - - - - - 0.875



• k=5 is the best fit

15 Walker Session 2 Player 1





10





Message 5





0





-5





-10

0 10 20 Period 30 40 50

15 Walker Session 2 Player 2



10

Message









5





0





-5





-10

0 10 20 Period 30 40 50

15 Walker Session 2 Player 3



10





Message 5





0





-5





-10

0 10 20 Period 30 40 50

15

Walker Session 2 Player 4



10

Message









5





0





-5





-10

0 10 20 Period 30 40 50

15 Walker Session 2 Player 5



10





Message

5





0





-5





-10

0 10 20 Period 30 40 50

6 Groves-Ledyard Session 1 Player 1



4

Message









2





0





-2





-4

0 10 20 Period 30 40 50

5-Period Best Response vs. Equilibrium: Walker

5-Period Best Response vs. Equilibrium: Groves-Ledyard

5-Period Best Response vs. Equilibrium: VCM

5-Period Best Response vs. Equilibrium: PropTax

Statistical Tests: 5-B.R. vs. Equilibrium



• Null Hypothesis: E[| mit  BRit |]  E[| mit  EQit |]



• Non-stationarity => period-by-period tests

• Non-normality of errors => non-parametric tests

– Permutation test with 2,000 sample permutations





• Problem: If EQit  BRit then the test has little power

• Solution:

– Estimate test power as a function of ( EQi  BRi ) / 

t t





– Perform the test on the data only where power is sufficiently large.

Simulated Test Power

1 0.95





0.9 0.95

Frequency of Rejecting H0 (Power)









Prob. H0 False Given Reject H0

0.8 0.94





0.7 0.93





0.6 0.92





0.5 0.91





0.4 0.89





0.3 0.86





0.2 0.8





0.1 0.67





0 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5



(1-2)/a

5-period B.R. vs. Nash Equilibrium

• Voluntary Contribution (strict dom. strats): EQit  BRit



• Groves-Ledyard (stable Nash equil): EQit  BRit



• Walker (unstable Nash equil): 73/81 tests reject H0

– No apparent pattern of results across time





• Proportional Tax: 16/19 tests reject H0



• 5-period model beats any static prediction

Best Response in the VCG Mechanism

• Convert data to polar coordinates:

Best Response in the cVCG Mechanism

Origin = Truth-telling dominant strategy

0-degree Line = Best response to 5-period average

The Testable Predictions

1. Weakly dominated ε-Nash equilibria are observed (67%)

– The dominant strategy equilibrium is not (0%)

– Convergence to strict dominant strategies

6





5

Avg. Contribution









4





3





2





1





0

0 5 10 15 20 25 30 35 40 45 50

Period





2,3. 6 repetitions of a strategy implies ε-equilibrium (75%)

4. Convergence with supermodularity & dom. diagonal (G-L)

Conclusions

• Experiments reveal the importance of

dynamics & stability

• Dynamic models outperform static models

• New directions for theoretical work

• Applications for “real world” implementation

• Open questions:

– Stable mechanisms implementing Lindahl*

– Efficiency/equilibrium tension in VCG

– Effect of the “What-If Scenario Analyzer”

– Better learning models

An Almost-Trivial Game

• Cycling (including equilibrium!) for k=3

• Global convergence for k=1,2,4,5,…

Efficiency

Efficiency Confidence Intervals - All 50 Periods

1

Efficiency









No Pub Good





0.5

Walker VC PT GL VCG

Mechanism

Av e rage Public Good Le v e ls

9

Pers 1-50

8

Pareto Optimal Pers 41-50

7



Public Good Level 6

5

4

3

2

1

0

VC PT GL WK VCG VCG*

Mechanism



Standard Deviation of PG Levels

7

Periods 1-50

6 Periods 41-50

Standard Deviation









5



4



3



2



1



0

VC PT GL WK VCG VCG*

Mechanism

Voluntary Contribution Mechanism

Results