Learning Dynamics for Mechanism Design Paul J. Healy California 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: u i (x i,y)=v i (y)+x i Linear technology (  ) Mechanisms:

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: y o =(  b i -  )/(  2a i )= aiai bibi  i Player Player Player Player Player

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

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

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)

Prop. Tax Results Player 2 Player 1

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)

Groves-Ledyard Results

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)

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

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

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

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

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 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 |m t obs  m t pred | ? k=5 is the best fit

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.

Simulated Test Power Frequency of Rejecting H 0 (Power)  1  2  Prob. H 0 False Given Reject H 0

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

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)

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 Mechanism Efficiency Walker VC PT GL VCG No Pub Good Efficiency

Voluntary Contribution Mechanism Results