P.J. Healypj@hss.caltech.edu

California Institute of Technology

Learning Dynamics for Mechanism Design

An Experimental Comparison of Public Goods Mechanisms

The Repeated Public Goods Implementation Problem

• Example: Condo Association “special assessment”– Fixed set of agents regularly choosing public good levels.– Goal is to maximize efficiency across all periods– What mechanism should be used?

• Questions: – Are the “one-shot” mechanisms the best solution to the

repeated problem?– Can one simple learning model approximate behavior in a

variety of games with different equilibrium properties?– Which existing mechanisms are most efficient in the

dynamic setting?

Previous Experiments on Public Goods Mechanisms I

• Dominant Strategy (VCG) mechanism experiments– Attiyeh, Franciosi and Isaac ’00– Kawagoe and Mori ’01 & ’99 pilot– Cason, Saijo, Sjostrom, & Yamato ’03

– Convergence to strict dominant strategies– Weakly dominated strategies are observed

Previous Experiments onPublic Goods Mechanisms II

• Nash Equilibrium mechanisms– Voluntary Contribution experiments– Chen & Plott ’96– Chen & Tang ’98

– Convergence iff supermodularity (stable equil.)

• Results consistent with best response behavior

• k-period Best Response model– Agents best respond to pure strat. beliefs– Belief = unweighted average of the others’

strategies in the previous k periods• Needs convex strategy space

– Rational behavior, inconsistent beliefs– Pure strategies only

A Simple Learning Model

– Strictly dominated strategies: never played– Weakly dominated strategies: possible

– Always converges in supermodular games– Stable/convergence => Nash equilibrium

– Can be very unstable (cycles w/ equilibrium)

A Simple Learning Model: Predictions

• New experiments over 5 public goods mechanisms– Voluntary Contribution– Proportional Tax– Groves-Ledyard– Walker– Continuous VCG (“cVCG”) with 2 parameters

• Identical environment (endow., prefs., tech.)• 4 sessions each with 5 players for 50 periods• Computer Interface

– History window & “What-If Scenario Analyzer”

A New Set of Experiments

• Agents:

• Private Good: Public Good: Endowments:

• Preferences:

• Technology:

• Mechanisms:

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• Proportional Tax

• Groves-Ledyard

• Walker

• VCG

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Experimental Results I: Choosing k

• Which value of k minimizes the M.A.D. across all mechanisms, sessions, players and periods?

• k=5 is the most accurate

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

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Walker Session 2 Player 2

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Walker Session 2 Player 3

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Walker Session 2 Player 4

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Walker Session 2 Player 5

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Groves-Ledyard Session 1 Player 1

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Experimental Results: 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.

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Simulated Test Power

5-period B.R. vs. 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

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Interesting properties of the2-parameter cVCG mechanism

• Best response line in 2-dimensional strategy space

Best Response in the cVCG mechanism

• Convert data to polar coordinates• Dom. Strat. = origin, B.R. line = 0-degree line

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Experimental Results III: Efficiency

• Outcomes are closest to Pareto optimal in cVCG– cVCG > GL ≥ PT > VC > WK (same for efficiency)– Sensitivity to parameter selection

• Variance of outcomes:– cVCG is lowest, followed by Groves-Ledyard – Walker has highest

• Walker mechanism performs very poorly– Efficiency below the endowment– Individual rationality violated 42% of last 10 periods

Discussion & Conclusions

• Data are consistent with the learning model.– Repercussions for theoretical research

• Should worry about dynamics– k-period best response studied here, but other learning

models may apply• Example: Instability of the Walker mechanism• cVCG mechanism can perform efficiently

• Open questions:– cVCG behavior with stronger conflict between incentives

and efficiency– Sensitivity of results to parameter changes– Effect of “What-If Scenario Analyzer” tool

Voluntary Contribution MechanismResults