## Working Papers

**Mislearning from Censored Data: The Gambler’s Fallacy in Optimal-Stopping
Problems**

[show/hide abstract] [download pdf] [online
appendix] [arXiv]

I study endogenous learning dynamics for people expecting systematic reversals from random
sequences — the “gambler’s fallacy.” Biased agents face an
optimal-stopping problem, such as managers conducting sequential interviews. They are
uncertain about the underlying distribution (e.g. talent distribution in the labor pool) and
must learn its parameters from previous agents’ histories. Agents stop when early draws
are deemed “good enough,” so predecessors’ histories contain negative
streaks but not positive streaks. Since biased learners understate the likelihood of
consecutive below-average draws, histories induce pessimistic beliefs about the
distribution’s mean. When early agents decrease their acceptance thresholds due to
pessimism, later learners will become more surprised by the lack of positive reversals in
their predecessors’ histories, leading to even more pessimistic inferences and even
lower acceptance thresholds — a positive-feedback loop. Agents who are additionally
uncertain about the distribution’s variance believe in fictitious variation (exaggerated
variance) to an extent depending on the severity of data censoring. When payoffs are convex in
the draws (e.g. managers can hire previously rejected interviewees), variance uncertainty
provides another channel of positive feedback between past and future thresholds.

**Network Structure and Naive Sequential Learning** (with Krishna Dasaratha)

Revision requested at *Theoretical Economics*

[show/hide abstract] [download pdf] [slides]
[arXiv] [pre-registration]

We study a sequential learning model featuring naive agents on a network. Agents wrongly
believe their predecessors act solely on private information, so they neglect redundancies
among observed actions. We provide a simple linear formula expressing agents’ actions in
terms of network paths and use this formula to completely characterize the set of networks
guaranteeing eventual correct learning. This characterization shows that on almost all
networks, disproportionately influential early agents can cause herding on incorrect actions.
Going beyond existing social-learning results, we compute the probability of such mislearning
exactly. This lets us compare likelihoods of incorrect herding, and hence expected welfare
losses, across network structures. The probability of mislearning increases when link
densities are higher and when networks are more integrated. In partially segregated networks,
divergent early signals can lead to persistent disagreement between groups. We conduct an
experiment and find that the accuracy gain from social learning is twice as large on sparser
networks, which is consistent with naive inference but inconsistent with the rational-learning
model.

**Player-Compatible Equilibrium** (with Drew Fudenberg)

[show/hide abstract] [download
pdf] [arXiv]

*Player-Compatible Equilibrium* (PCE) imposes cross-player restrictions on the
magnitudes of the players’ “trembles” onto different strategies. These
restrictions capture the idea that trembles correspond to deliberate experiments by agents who
are unsure of the prevailing distribution of play. PCE selects intuitive equilibria in a
number of examples where trembling-hand perfect equilibrium (Selten, 1975) and proper
equilibrium (Myerson, 1978) have no bite. We show that rational learning and some near-optimal
heuristics imply our compatibility restrictions in a steady-state setting.

**Payoff Information and Learning in Signaling Games** (with Drew Fudenberg)

[show/hide abstract] [download
pdf] [arXiv]

We show how to add the assumption that players know their opponents' payoff functions to the theory of learning in games, and use it to derive restrictions on signaling-game play in the spirit of divine equilibrium. In our learning model, agents are born into player roles and play the game against a random opponent each period. Inexperienced agents are uncertain about the prevailing distribution of opponents' play, and update their beliefs based on their observations. Long-lived and patient senders experiment with every signal that they think might yield an improvement over their myopically best play. We show that divine equilibrium (Banks and Sobel, 1987) is nested between “rationality-compatible” equilibrium, which corresponds to an upper bound on the set of possible learning outcomes, and “uniform rationality-compatible” equilibrium, which provides a lower bound.

## Published Papers

**Learning and Type Compatibility in Signaling Games** (with Drew Fudenberg)

*Econometrica* 86(4):1215-1255, July 2018

[show/hide abstract] [download
pdf] [online appendix] [publisher’s DOI] [arXiv]

Which equilibria will arise in signaling games depends on how the receiver interprets
deviations from the path of play. We develop a micro-foundation for these off-path beliefs,
and an associated equilibrium refinement, in a model where equilibrium arises through
non-equilibrium learning by populations of patient and long-lived senders and receivers. In
our model, young senders are uncertain about the prevailing distribution of play, so they
rationally send out-of-equilibrium signals as experiments to learn about the behavior of the
population of receivers. Differences in the payoff functions of the types of senders generate
different incentives for these experiments. Using the Gittins index (Gittins, 1979), we
characterize which sender types use each signal more often, leading to a constraint on the
receiver’s off-path beliefs based on “type compatibility” and hence a
learning-based equilibrium selection.

**Bayesian Posteriors for Arbitrarily Rare Events** (with Drew Fudenberg and Lorens Imhof)

*Proceedings of the National Academy of Sciences* 114(19):4925-4929, May 2017

[show/hide abstract] [download pdf] [publisher’s DOI] [arXiv]

We study how much data a Bayesian observer needs to correctly infer the relative likelihoods
of two events when both events are arbitrarily rare. Each period, either a blue die or a red
die is tossed. The two dice land on side 1 with unknown probabilities \(p_1\) and \(q_1\),
which can be arbitrarily low. Given a data-generating process where \(p_1 \ge c q_1\), we are
interested in how much data is required to guarantee that with high probability the observer's
Bayesian posterior mean for \(p_1\) exceeds \((1-\delta)c\) times that for \(q_1\). If the
prior densities for the two dice are positive on the interior of the parameter space and
behave like power functions at the boundary, then for every \(\epsilon >0\), there exists a
finite \(N\) so that the observer obtains such an inference after \(n\) periods with
probability at least \(1-\epsilon\) whenever \(n p_1 \ge N\). The condition on \(n\) and
\(p_1\) is the best possible. The result can fail if one of the prior densities converges to
zero exponentially fast at the boundary.

**Differentially Private and Incentive Compatible Recommendation System for the
Adoption of Network Goods** (with Xiaosheng
Mu)

*Proceedings of the Fifteenth ACM Conference on Economics and Computation*
(EC’14):949-966, June 2014

[show/hide abstract] [download pdf] [slides]
[publisher’s DOI]

We study the problem of designing a recommendation system for network goods under the
constraint of differential privacy. Agents living on a graph face the introduction of a new
good and undergo two stages of adoption. The first stage consists of private, random
adoptions. In the second stage, remaining non-adopters decide whether to adopt with the help
of a recommendation system \(\mathcal{A}\). The good has network complimentarity, making it
socially desirable for \(\mathcal{A}\) to reveal the adoption status of neighboring agents.
The designer’s problem, however, is to find the socially optimal \(\mathcal{A}\) that
preserves privacy. We derive feasibility conditions for this problem and characterize the
optimal solution.

I was a Teaching Fellow for the following classes.

**Economics 1011A: Intermediate Microeconomics - Advanced**

Economics 1011A is similar to Economics 1010A, but more mathematical and covers more material.
The course teaches the basic tools of economics and their applications to a wide range of human
behavior.

[syllabus] [section notes] [teaching evaluations]

**Economics 2010A: Economic Theory**

Topics include extensive- and normal-form games, Nash equilibrium, rationalizability, Nash
implementation, auctions, bargaining, repeated games, signaling, and forward induction.

[syllabus] [section notes] [teaching evaluations]