Theses Doctoral

Essays on Bandit Games and Endogenously Missing Data

Grimme, William James

I characterize optimal strategies and equilibria in two bandit-type games of strategic information acquisition. The latter game admits a missing data problem, which I develop an estimator to address. I consider a third environment with an additional missing data problem, characterize estimation bias both statistically and empirically, and develop a pair of easily implementable unbiased estimators.

In the first chapter, I analyze a two-player, two-armed exponential bandit model in continuous time across several monitoring and disclosure environments, in which breakthroughs are fully informative of the state of the world. In particular, I consider an environment where player actions are observed but the results of experimentation are unobserved, and an environment where both actions and experimentation outcomes are unobserved. In the former environment, I construct a Perfect Bayesian Equilibrium that induces the efficient (cooperative) experimentation path.

This equilibrium relies on strategies that deter free-riding by moving to an undesirable, asymmetric experimentation path when the safe arm is played. In the latter environment, I provide conditions on starting beliefs for which it is possible to construct a Perfect Bayesian Equilibrium that induces the efficient experimentation path. This equilibrium requires that players privately, separately experiment and share the results of experimentation at a single point in time. I show how this hybrid Markov approach can be adapted to a general discrete-time setting and provide conditions for which strategies measurable with respect to both the Markov partition and a finite automaton are consistent.

In the second chapter, I consider the problem of a rideshare company which makes sequences of take it or leave it offers to drivers for individual trips. When drivers are privately and commonly informed of trip quality, rejected offers are informative about trip quality to the rideshare company. As such, the rideshare company faces a strategic information acquisition problem: offers influence both the current-period payoff and the expectation of future payoffs via posterior beliefs. When unobserved quality is Bernoulli, the value function increases in beliefs over quality, and I characterize the value function and optimal offer sequence using dynamic programming. Moreover, I show that a heuristic 𝑛-offer look-ahead sequence converges uniformly to the optimal revenue, and characterize bounds on the revenue gap. For a general bounded distribution of unobserved trip quality, I show that the value function is decreasing after rejections, and define analogous convergent bounds. Among heterogeneous drivers, driver and platform value can be collapsed into a single-dimensional match value parameter that governs the optimal order of offers. I also illustrate a source of potential bias when estimating driver preferences using a data set with repeated offers, and construct a consistent, easily-implementable estimator to address the issue.

In the third chapter, I consider a choice environment where an econometrician only observes certain covariates for goods that are chosen. This missing data may reflect literal omissions from a dataset or inherently counterfactual outcomes, such as in the Roy (1951) model. When partially observed covariates vary stochastically across consumers there is selection bias in the distribution of observed covariates: consumers are more likely to pick goods with preferential characteristic draws. I show that imputation methods that use the observed distribution of covariates to replace missing data bias discrete choice parameter estimates toward zero, regardless of whether missing data or all data are imputed. Moreover, this bias rapidly increases with the variance of the covariate distribution. Instead, I propose two full-information maximum likelihood estimation procedures that jointly estimate preferences and the underlying distribution of covariates. While these estimators necessarily involve Monte Carlo simulation when covariates are distributed continuously, I show that it is possible to avoid simulation when covariates take a discrete set of values.

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More About This Work

Academic Units
Economics
Thesis Advisors
Tebaldi, Pietro
Degree
Ph.D., Columbia University
Published Here
January 15, 2025