The Public Goods Environment n agents 1 private good x, 1 public good y Endowed with private good only (gi) Preferences: ui(xi,y)=vi(y)+xi 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 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 Player 4 6 68 250 Player 5 4 44 290

Voluntary Contribution Mechanism Mi = [0,6] y(m) = imi ti(m)= mi 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 - i2mi Nash equilibrium: (0,1,0,0,0)

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

Proportional Tax Mechanism Mi = [0,6] y(m) = imi ti(m)=(/n)y(m) No previous experiments (?) Foundation of many efficient mechanisms Current experiment: No dominant strategies Best response: mi* = yi*  ki mk (y1*,…,y5*) = (7, 6, 5, 4, 3) Nash equilibrium: (6,0,0,0,0)

Prop. Tax Results Player 1 Player 2

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)

Groves-Ledyard Results

Walker’s Mechanism Theory: Previous experiments: Current experiment: 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 pairings play dominated Nash equilibria Continuous public good with single-peaked preferences (strict dominant strategy): 81% revelation

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 No 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 k Which k minimizest |mtobs  mtpred| ? k=5 is the best fit

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.

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

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

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

The Testable Predictions 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%) Convergence with supermodularity & dom. diagonal (G-L)

Conclusions Importance of dynamics & stability Dynamic models outperform static models Strict vs. weak dominant strategies Applications for “real world” implementation Directions for theoretical work: Developing stable mechanisms Open experimental questions: Efficiency/equilibrium tension in VCG Effect of the “What-If Scenario Analyzer” Better learning models