The Public Goods Environment

1
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

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Five Mechanisms

• Efficient gt 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)

3
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

5
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 40 260
Player 4 6 68 250
Player 5 4 44 290
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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 2s best response m2 1 - ?i?2mi
• Nash equilibrium (0,1,0,0,0)

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

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Prop. Tax Results
Player 1
Player 2
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Groves-Ledyard Mechanism

• Theory
• Pareto optimal equilibrium, not Lindahl
• Supermodular if ?/n gt 2ai for every i
• Previous experiments
• Chen Plott 96 higher?? gt converges better
• Current experiment
• ? 100 gt Supermodular
• Nash equilibrium (1.00, 1.15, 0.97, 0.86, 0.82)

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Groves-Ledyard Results
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Walkers 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 pairings play dominated Nash equilibria
• Continuous public good with single-peaked
  preferences (strict dominant strategy)
• 81 revelation

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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 lt 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

• No strictly dominated strategies after period k
• Same strategy k1 times gt Nash equilibrium
• U.H.C. Convergence to m gt 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

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Choosing the best k

• Which k minimizes??t mtobs ? mtpred ?
• k5 is the best fit

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Statistical Tests 5-B.R. vs. Equilibrium

• Null Hypothesis
• Non-stationarity gt period-by-period tests
• Non-normality of errors gt 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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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

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

• Weakly dominated e-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
  e-equilibrium (75)
• Convergence with supermodularity dom. diagonal
  (G-L)

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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