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:

2. 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)

3. 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”

4. 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
Player 4 6 68
250
Player 5 4 44
290

6. 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)

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

8. 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)

9. Prop. Tax Results
Player 1
Player 2

10. 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)

11. Groves-Ledyard Results

12. 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)

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

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

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

16. 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

17. 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

18. 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

19. 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

20. 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.
Asymptotically stable points are N.E.
Stability:
Global stability in supermodular games
Global stability in games with dominant diagonal
Note: Stability properties are not monotonic in k

21. Choosing the best k Which k minimizest |mtobs  mtpred| ?
k=5 is the best fit

22. 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.

23. 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

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

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

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

28. 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)

29. 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