Hierarchical Parametric Models for Social Dilemma Games

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Hierarchical Parametric Models for Social Dilemma
Games

• Klaus Moeltner
• James J. Murphy, U. Massachusetts
• John Stranlund, U. Massachusetts
• Maria Alejandra Velez, Columbia University

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Overview

• Experiments and Econometrics
• Social Dilemma Games
• Hierarchical Ordered Probit (HOP)
• Hierarchical Doubly-Truncated Poisson (HDTP)
• Model Selection and Prediction
• Empirical Application / Data
• Estimation Results
• Predicted Outcomes
• Conclusions

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Empirical Data
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Processing Experimental Data The Staple
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Parametric Motivations

• Recognition of different types of noise in data
• Largely unobserved heterogeneity
• Parametric model needed to control for it
• Example Quantal Response Equilibria
• Multiple treatments, heterogeneous subjects
• Isolation of marginal effects
• Summarize all effects in a concise way
• Hypothesis testing

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Parametric Simplifications
Parametric Regressions as an Afterthought
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Parametric Considerations

• Focus on theory testing
• Identify structural parameters, test for
  significance(warm-glow effects, altruism
  effects, etc)
• Limited attention given to
• Specification tests
• Limitations of dependent variable
• Consistency, Efficiency
• Predictive Performance, Fit with Data

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Social Dilemma Games

• Two varieties
• Public Goods Games
• Common Pool Resource (CPR) Games
• Interactive, Multiple Players
• Lots of room for unobserved heterogeneity
• Played with Integer Tokens
• Limited Dependent Variable!
• Often set in the field
• Potential to inform real-world policy decisions
• Predictive quality of econometric model becomes
  important

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Past Parametric Approaches

• Treat as binary data (e.g.contribute or not)
• Loss of information, efficiency
• Basic OLS
• Biased due to truncation
• Tobit
• Inconsistent due to measurement error(Stapleton
  Young, 1984)
• Ordered Probit (Palfrey Prisbrey, 1996)
• Reasonable, but needs built-in control for
  heterogeneity

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Hierarchical Truncated Count Data Model

• Captures Integer Nature of Data
• No lower truncation required if support starts at
  0
• Hierarchical layer(s) control for unobserved
  heterogeneity
• Allows for cardinal comparison of predictions
• Compare performance to Hierarchical Ordered Probit

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Bayesian Estimation

• Circumvents Classical Estimation Hurdles
• Evaluation of multi-fold integrals
• Sensitivity to starting values
• Notorious problems in hierarchical count data
  models
• Maximum flexibility in model comparison
• Via marginal likelihoods and Bayes Factors
• No nesting required
• Option to model-average results

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Hierarchical Ordered Probit (HOP)
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OP bins thresholds
Thresholds must be estimated -gt efficiency
losscompared to count data approach
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HOP Likelihood Function
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Re-parameterization
Nandram Chen, 1996 Li Tobias, 2007
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Priors and Structure of Gibbs Sampler
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Hierarchical Doubly-Truncated Poisson (HDTP)
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HDTP Likelihood Function
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Priors and Structure of Gibbs Sampler
Useful reference Chib et al., 1998
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Posterior Density and Marginal Likelihood
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Model Probabilities and Bayes Factors
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Posterior Predictive Distribution, HOP
For the probability of falling into a given tier
of extraction levels
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Posterior Predictive Distribution, HDTP
For a given extraction level
For expected extraction
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Application

• 3 Fishing villages in Columbia
• 2004 Common Pool Resource Experiment
• 12 groups _at_5 players / village
• 20 rounds
• 10 under Open Access, all groups
• 10 under 1 of 3 treatments
• Quota with low penalty
• Quota with medium penalty
• Open communication prior to each round
• Choose 1-9 extraction level each round

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Payoff Function

• Chosen by researcher based on
• Convenience for derivation of structural models
  of behavior
• Large gap between social optimum and Nash
  Equilibrium
• Payoffs are of reasonable magnitude(large enough
  to provide incentives, small enough to fit in
  budget...)

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Payoff Table
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Implementation
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Aggregate Sample Results
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Individual-Level Results
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Sub-Model Selection, HOP
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Sub-Model Selection, HDTP
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Features of Winning Model

• Unobserved heterogeneity varies systematically
  over treatments
• Warrants separate random effects for each
  treatment
• Period effects matter(first 5 rounds vs second 5
  rounds of each game)
• Period indicator should be included in model
• Period effects vary systematically over
  treatments
• Warrants interaction terms with each treatment

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Estimation Results RE Means
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Estimation Results RE VCOV
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Predictions (HDTP)
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PPD of Expected Effort
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PPD of Expected Effort
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Summary Econometrics

• Best Sub-model captures
• treatment time effects
• heterogeneity over treatments
• interactions between treatments time effects
• HOP estimates noisier than HDTP
• But captures essential patterns and trends
• HOP predictions imprecise
• But replicates general sample patterns

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Summary Policy

• Pronounced heterogeneity in
• How different individuals react to treatments
• How different communities react to treatments
• Experience with policy matters
• T2, T3 work well for Pacific only
• T4 has strongest effect for Magdalena, strong
  effect for Caribbean
• In absence of policy experience people use own
  heuristics (pos. COVs for CAR only)
• Learning / trust building matters
• CAR T4 effects take longer to kick in

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Conclusions Look Ahead

• HOP gets the basic job done, but HDTP is
  superior in all aspects
• re-run HOP with 3 tiers prob. more efficient
• HOP invariant to scaling
• Could be advantage for data combination
• Closer look warranted

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Policy relevance of Results?

• Community heterogeneity, effect of historical
  institutions prob. generalizable
• To say more about policy effect on resource use
• Must link payoff table to real HH production
• Must combine experiment with broader survey
• Use experiment as a fancy focus group
• Get a good Econometrician!!