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Learning Dynamics for Mechanism Design Paul J. Healy California Institute of Technology An Experimental Comparison of Public Goods Mechanisms

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Overview Institution (mechanism) design –Public goods Experiments –Equilibrium, rationality, convergence (How) Can experiments improve institution/mechanism design?

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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

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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

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Mechanism Design Implementation: g (e) F( e )

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The Role of Experiments Field: e unknown => F ( e ) unknown Experiment: everything fixed/induced except

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The Public Goods Environment n agents 1 private good x, 1 public good y Endowed with private good only i Preferences: u i (x i,y)=v i (y)+x i Linear technology ( ) Mechanisms:

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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)

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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”

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What-If Scenario Analyzer An interactive payoff table Subjects understand how strategies → outcomes Used extensively by all subjects

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Environment Parameters Loosely based on Chen & Plott ’96 = 100 Pareto optimum: y o =( b i - )/( 2a i )=4.8095 aiai bibi i Player 1134260 Player 28116140 Player 3240260 Player 4668250 Player 5444290

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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: m 2 * = 1 - i 2 m i –Nash equilibrium: (0,1,0,0,0) M i = [0,6] y(m) = i m i t i (m)= m i

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

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Proportional Tax Mechanism No previous experiments (?) Foundation of many efficient mechanisms Current experiment: –No dominant strategies –Best response: m i * = y i * k i m k –(y 1 *,…,y 5 * ) = (7, 6, 5, 4, 3) –Nash equilibrium: (6,0,0,0,0) M i = [0,6] y(m) = i m i t i (m)=( /n)y(m)

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Prop. Tax Results Player 2 Player 1

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Groves-Ledyard Mechanism Theory: –Pareto optimal equilibrium, not Lindahl –Supermodular if /n > 2a i 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)

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Groves-Ledyard Results

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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)

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

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VCG Mechanism: Theory Truth-telling is a dominant strategy Pareto optimal public good level Not budget balanced Not always individually rational

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VCG Mechanism: Best Responses Truth-telling ( ) is a weak dominant strategy There is always a continuum of best responses:

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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

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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

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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

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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

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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

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Choosing the best k Which k minimizes t |m t obs m t pred | ? k=5 is the best fit

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5-Period Best Response vs. Equilibrium: Walker

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

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5-Period Best Response vs. Equilibrium: VCM

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5-Period Best Response vs. Equilibrium: PropTax

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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.

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0 0.67 0.8 0.86 0.89 0.91 0.92 0.93 0.94 0.95 Simulated Test Power Frequency of Rejecting H 0 (Power) 1 2 Prob. H 0 False Given Reject H 0

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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 H 0 –No apparent pattern of results across time Proportional Tax: 16/19 tests reject H 0 5-period model beats any static prediction

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Best Response in the VCG Mechanism Convert data to polar coordinates:

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Best Response in the cVCG Mechanism Origin = Truth-telling dominant strategy 0-degree Line = Best response to 5-period average

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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)

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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

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An Almost-Trivial Game Cycling (including equilibrium!) for k=3 Global convergence for k=1,2,4,5,…

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Efficiency Confidence Intervals - All 50 Periods 0.5 1 Mechanism Efficiency Walker VC PT GL VCG No Pub Good Efficiency

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Voluntary Contribution Mechanism Results

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