1 / 29

The Public Goods Environment

The Public Goods Environment. n agents 1 private good x , 1 public good y Endowed with private good only ( g i ) Preferences: u i (x i ,y)=v i (y)+x i Linear technology (  ) Mechanisms:. Five Mechanisms . “Efficient” => g  ( e )  PO ( e ) Inefficient Mechanisms

nairi
Download Presentation

The Public Goods Environment

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 • 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”

  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

  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 > 2aifor 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: • 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. 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

  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

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

  27. 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)

  28. 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

More Related