Learning Dynamics for Mechanism Design

Paul J. HealyCalifornia Institute of Technology

An Experimental Comparison of Public Goods Mechanisms

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 i

• Preferences: ui(xi,y)=vi(y)+xi

• Linear technology ()• Mechanisms:

),,,()( 21 nmmmymy

),( ymtx iii ),()( 1 nii mmtmt

ii Mm

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

= 100

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

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

iiiiii xybyayx )(),(u 2

Voluntary Contribution Mechanism

• 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 - i2mi

– Nash equilibrium: (0,1,0,0,0)

Mi = [0,6] y(m) = imi ti(m)= mi

VCM Results

Player 2

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

Dominant Strategies

0

1

2

3

4

5

6

0 10 20 30 40 50

Period

Ave

rag

e M

essa

ge

(4 s

essi

on

s)

PLR1

PLR2

PLR3

PLR4

PLR5

Proportional Tax Mechanism

• No previous experiments (?)

• Foundation of many efficient mechanisms

• Current experiment:– No dominant strategies

– Best response: mi* = yi

* ki mk

– (y1*,…,y5

*) = (7, 6, 5, 4, 3)

– Nash equilibrium: (6,0,0,0,0)

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

Prop. Tax Results

Player 2

Player 1

0

1

2

3

4

5

6

0 10 20 30 40 50

Period

Ave

rag

e M

essa

ge

PLR1

PLR2

PLR3

PLR4

PLR5

Groves-Ledyard Mechanism

• 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)

)(

1

2

)()()( 22

iiiii

i mmmn

n

n

mymtmmy

Groves-Ledyard Results

-4

-3

-2

-1

0

1

2

3

4

5

6

0 10 20 30 40 50Period

Ave

rag

e M

essa

ge

PLR1

PLR2

PLR3

PLR4

PLR5

Walker’s Mechanism

• 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)

)()()( mod1mod)1( mymmn

mtmmy niniii

i

Walker Mechanism ResultsNE: (12.28, -1.44, -6.78, -2.2, 2.94)

-8

-6

-4

-2

0

2

4

6

8

10

12

0 10 20 30 40 50

Period

Av

era

ge

Me

ss

ag

e

PLR1

PLR2

PLR3

PLR4

PLR5

VCG Mechanism: Theory

• Truth-telling is a dominant strategy• Pareto optimal public good level• Not budget balanced• Not always individually rational

yn

nyvz

zn

nzvy

n

nyv

n

yt

yyvy

bamM

ijjj

yii

ijiijiij

ijjji

iii

y

iiiiii

1)ˆ|(maxarg)ˆ(

)ˆ(1

)ˆ|)ˆ(()ˆ(1

)ˆ|)ˆ(()ˆ(

)ˆ(

)ˆ|(maxarg)ˆ(

)ˆ,ˆ(ˆ

0

0

VCG Mechanism: Best Responses

• Truth-telling ( ) is a weak dominant strategy• There is always a continuum of best responses:

ii ˆ

iiiiiii yyBR ˆ,ˆ,ˆ:ˆ)ˆ(

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 < 10% of time

• Outcomes very close to Pareto optimal– Announcements may be near non-revealing best

responses

Summary of Experimental Results

• VCM: convergence to dominant strategies• Prop Tax: non-equil., but near best response• Groves-Ledyard: convergence to stable equil. • Walker: no convergence to unstable equilibrium• VCG: low revelation, but high efficiency

Goal: A simple model of behavior to explain/predict which mechanisms converge to equilibrium

Observation: Results are qualitatively similar to best response predictions

A Class of Best Response Models• A general best response framework:

– Predictions map histories into strategies

– Agents best respond to their predictions

• A k-period best response model:

– Pure strategies only– Convex strategy space– Rational behavior, inconsistent predictions

jtjj

ij Mmm 11 ,,

in

ii

ti BRm ,,1

k

s

stj

tjj

ij m

kmm

1

11 1),,(

Testable Predictions of the k-Period Model

1. No strictly dominated strategies after period k

2. Same strategy k+1 times => 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 minimizest |mtobs mt

pred| ?

• k=5 is the best fit

Model 2-50 3-50 4-50 5-50 6-50 7-50 8-50 9-50 10-50 11-50k=1 1.407 1.394 1.284 1.151 1.104 1.088 1.072 1.054 1.054 1.049k=2 - 1.240 1.135 0.991 0.967 0.949 0.932 0.922 0.913 0.910k=3 - - 1.097 0.963 0.940 0.925 0.904 0.888 0.883 0.875k=4 - - - 0.952 0.932 0.915 0.898 0.877 0.866 0.861k=5 - - - - 0.924 0.9114 0.895 0.876 0.860 0.853k=6 - - - - - 0.9106 0.897 0.881 0.868 0.854k=7 - - - - - - 0.899 0.884 0.873 0.863k=8 - - - - - - - 0.884 0.874 0.864k=9 - - - - - - - - 0.879 0.870

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

Walker Session 2 Player 1

-10

-5

0

5

10

15

0 10 20 30 40 50Period

Me

ss

ag

e

Walker Session 2 Player 2

-10

-5

0

5

10

15

0 10 20 30 40 50Period

Me

ss

ag

e

Walker Session 2 Player 3

-10

-5

0

5

10

15

0 10 20 30 40 50Period

Mes

sag

e

Walker Session 2 Player 4

-10

-5

0

5

10

15

0 10 20 30 40 50Period

Mes

sag

e

Walker Session 2 Player 5

-10

-5

0

5

10

15

0 10 20 30 40 50Period

Me

ss

ag

e

Groves-Ledyard Session 1 Player 1

-4

-2

0

2

4

6

0 10 20 30 40 50Period

Me

ss

ag

e

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:

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

• Non-normality of errors => non-parametric tests– Permutation test with 2,000 sample permutations

• Problem: If then the test has little power

• Solution: – Estimate test power as a function of

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

|][||][| ti

ti

ti

ti EQmEBRmE

ti

ti BREQ

/)( ti

ti BREQ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.67

0.8

0.86

0.89

0.91

0.92

0.93

0.94

0.95

0.95

Simulated Test PowerF

req

uen

cy o

f R

eje

ctin

g H

0

(Pow

er)

12

Pro

b. H

0 False

Give

n

Reje

ct H0

5-period B.R. vs. Nash Equilibrium• Voluntary Contribution (strict dom. strats):

• Groves-Ledyard (stable Nash equil):

• 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

ti

ti BREQ

ti

ti BREQ

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

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

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

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

Period

Avg

. C

on

trib

utio

n

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 Confidence Intervals - All 50 Periods

0.5

1

Mechanism

Eff

icie

ncy

Walker VC PT GL VCG

No Pub Good

Efficiency

Voluntary Contribution Mechanism

Results