The Public Goods Environmentn agents1 private good x, 1 public good yEndowed with private good only (gi)Preferences: ui(xi,y)=vi(y)+xiLinear technology ()Mechanisms:

Five Mechanisms Efficient => g(e) PO(e)Inefficient MechanismsVoluntary Contribution Mech. (VCM)Proportional Tax Mech.(Outcome-) Efficient MechanismsDominant Strategy EquilibriumVickrey, Clarke, Groves (VCG) (1961, 71, 73)Nash EquilibriumGroves-Ledyard (1977)Walker (1981)

The Experimental Environmentn = 5Four sessions of each mech.50 periods (repetitions)Quadratic, quasilinear utilityPreferences are private infoPayoff $25 for 1.5 hoursComputerized, anonymousCaltech undergradsInexperienced subjects

History windowWhat-If Scenario Analyzer

What-If Scenario AnalyzerAn interactive payoff tableSubjects understand how strategies outcomesUsed extensively by all subjects

Environment ParametersLoosely based on Chen & Plott 96

= 100Pareto optimum: yo =(bi - )/(2ai)=4.8095

aibiiPlayer 1134260Player 28116140Player 3240260Player 4668250Player 5444290

Voluntary Contribution MechanismPrevious experiments:All players have dominant strategy: m* = 0Contributions decline in time

Current experiment:Players 1, 3, 4, 5 have dom. strat.: m* = 0Player 2s best response: m2* = 1 - i2miNash equilibrium: (0,1,0,0,0)Mi = [0,6] y(m) = imi ti(m)= mi

VCM ResultsPlayer 2Nash Equilibrium: (0,1,0,0,0)Dominant Strategies

Proportional Tax MechanismNo previous experiments (?)Foundation of many efficient mechanismsCurrent experiment:No dominant strategiesBest 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 ResultsPlayer 2Player 1

Groves-Ledyard MechanismTheory:Pareto optimal equilibrium, not LindahlSupermodular if /n > 2ai for every iPrevious experiments:Chen & Plott 96 higher => converges betterCurrent experiment: =100 => SupermodularNash equilibrium: (1.00, 1.15, 0.97, 0.86, 0.82)

Groves-Ledyard Results

Walkers MechanismTheory:Implements Lindahl AllocationsIndividually rational (nice!)Previous experiments:Chen & Tang 98 unstableCurrent experiment:Nash equilibrium: (12.28, -1.44, -6.78, -2.2, 2.94)

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

VCG Mechanism: TheoryTruth-telling is a dominant strategyPareto optimal public good levelNot budget balancedNot always individually rational

VCG Mechanism: Best ResponsesTruth-telling ( ) is a weak dominant strategyThere is always a continuum of best responses:

VCG Mechanism: Previous ExperimentsAttiyeh, Franciosi & Isaac 00Binary public good: weak dominant strategyValue revelation around 15%, no convergence

Cason, Saijo, Sjostrom & Yamato 03Binary public good:50% revelationMany pairings play dominated Nash equilibriaContinuous public good with single-peaked preferences (strict dominant strategy):81% revelation

VCG Experiment ResultsDemand revelation: 50 60%NEVER observe the dominant strategy equilibrium

10/20 subjects fully reveal in 9/10 final periodsFully reveal = both parameters

6/20 subjects fully reveal < 10% of time

Outcomes very close to Pareto optimalAnnouncements may be near non-revealing best responses

Summary of Experimental ResultsVCM: convergence to dominant strategiesProp Tax: non-equil., but near best responseGroves-Ledyard: convergence to stable equil. Walker: no convergence to unstable equilibriumVCG: 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 ModelsA general best response framework:Predictions map histories into strategies

Agents best respond to their predictions

A k-period best response model:

Pure strategies onlyConvex strategy spaceRational behavior, inconsistent predictions

Testable Predictions of the k-Period ModelNo strictly dominated strategies after period k

Same strategy k+1 times => Nash equilibrium

U.H.C. + Convergence to m* => m* is a N.E.3.1. Asymptotically stable points are N.E.

Stability: 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 kWhich k minimizest |mtobs mtpred| ?

k=5 is the best fit

Sheet1

Model2-503-504-505-506-507-508-509-5010-5011-50

k=11.4071.3941.2841.1511.1041.0881.0721.0541.0541.049

k=2-1.2401.1350.9910.9670.9490.9320.9220.9130.910

k=3--1.0970.9630.9400.9250.9040.8880.8830.875

k=4---0.9520.9320.9150.8980.8770.8660.861

k=5----0.9240.91140.8950.8760.8600.853

k=6-----0.91060.8970.8810.8680.854

k=7------0.8990.8840.8730.863

k=8-------0.8840.8740.864

k=9--------0.8790.870

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

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

Null Hypothesis:

Non-stationarity => period-by-period testsNon-normality of errors => non-parametric testsPermutation test with 2,000 sample permutations

Problem: If then the test has little powerSolution: Estimate test power as a function ofPerform the test on the data only where power is sufficiently large.

5-period B.R. vs. Nash EquilibriumVoluntary Contribution (strict dom. strats):

Groves-Ledyard (stable Nash equil):

Walker (unstable Nash equil): 73/81 tests reject H0No 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 MechanismConvert data to polar coordinates:

Best Response in the cVCG MechanismOrigin = Truth-telling dominant strategy0-degree Line = Best response to 5-period average

Efficiency Confidence Intervals - All 50 Periods0.51MechanismEfficiency Walker VC PT GL VCGNo Pub GoodEfficiency

The Testable PredictionsWeakly dominated -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 -equilibrium (75%)Convergence with supermodularity & dom. diagonal (G-L)

ConclusionsImportance of dynamics & stabilityDynamic models outperform static modelsStrict vs. weak dominant strategiesApplications for real world implementationDirections for theoretical work:Developing stable mechanismsOpen experimental questions:Efficiency/equilibrium tension in VCGEffect of the What-If Scenario AnalyzerBetter learning models