**Information
Aggregation with Costly Reporting**,
with Martin
J. Osborne and
Jeffrey
S. Rosenthal

A group of individuals with common interests has to choose a binary option whose desirability depends on an unknown binary state of the world. The individuals independently and privately observe a signal of the state. Each individual chooses whether to reveal her signal, at a cost. We show that if for all revelation choices of the individuals the option chosen by the group is optimal given the signals revealed and the set of individuals who do not reveal signals, then in a large group few signals are revealed, and these signals are extreme. The correct decision is taken with high probability in one state but with probability bounded away from one in the other. No anonymous decision-making mechanism without transfers does better. However, the first-best average payoff can be attained using transfers among agents, and approximately attained with a non-anonymous mechanism without transfers.

**Optimal
Adaptive Testing: Informativeness and Incentives**,
with Rahul
Deb

We introduce a learning framework in which a principal seeks to determine the ability of a strategic agent. The principal assigns a test consisting of a finite sequence of questions or tasks. The test is adaptive: each question that is assigned can depend on the agent’s past performance. The probability of success on a question is jointly determined by the agent’s privately known ability and an unobserved action that he chooses to maximize the probability of passing the test. We identify a simple monotonicity condition under which the principal always employs the most (statistically) informative question in the optimal adaptive test. Conversely, whenever the condition is violated, we show that there are cases in which the principal strictly prefers to use less informative questions.

**Rational
Inattention Dynamics: Inertia and Delay in Decision-Making**,
with Jakub
Steiner and
Filip
Matějka,
forthcoming in *Econometrica*

We solve a general class of dynamic rational-inattention problems in which an agent repeatedly acquires costly information about an evolving state and selects actions. The solution resembles the choice rule in a dynamic logit model, but it is biased towards an optimal default rule that depends only on the history of actions, not on the realized state. We apply the general solution to the study of (i) the status quo bias; (ii) inertia in actions leading to lagged adjustments to shocks; and (iii) the tradeoff between accuracy and delay in decision-making.

**Identification
of Payoffs in Repeated Games**,
with Byung
Soo Lee,
*Games
and Economic Behavior*,
Vol. 99, September 2016, 82–88

In one-shot games, an analyst who knows the best response correspondence can only make limited inferences about the players’ payoffs. In repeated games, this is not true: we show that, under a weak condition, if the game is repeated sufficiently many times and players are sufficiently patient, the best response correspondence completely determines the payoffs (up to positive affine transformations).

**Perceiving
Prospects Properly**,
with Jakub
Steiner,
*American
Economic Review*,
Vol. 106 (7), July 2016, 1601–1631

When an agent chooses between prospects, noise in information processing generates an effect akin to the winner’s curse. Statistically unbiased perception systematically overvalues the chosen action because it fails to account for the possibility that noise is responsible for making the preferred action appear to be optimal. The optimal perception patterns share key features with prospect theory, namely, overweighting of small probability events (and corresponding underweighting of high probability events), status quo bias, and reference-dependent S-shaped valuations. These biases arise to correct for the winner’s curse effect.

**Price
Distortions under Coarse Reasoning with Frequent Trade**,
with Jakub
Steiner,
*Journal
of Economic Theory*,
Vol. 159 (Part A), September 2015, 574–595

We
study the effect of frequent trading opportunities and categorization
on pricing of a risky asset. Frequent opportunities to trade can lead
to large distortions in prices if some agents forecast future prices
using a simplified model of the world that fails to distinguish
between some states. In the limit as the period length vanishes,
these distortions take a particular form: the price must be the same
in any two states that a positive mass of agents categorize together.
Price distortions therefore tend to be large when different agents
categorize states in different ways, even if each individual’s
categorization is not very coarse.

Supplementary
Appendix

**Influential
Opinion Leaders**,
with Antoine
Loeper and
Jakub
Steiner,
*The
Economic Journal*,
Vol. 124, December 2014, 1147–1167

We
present a two-stage coordination game in which early choices of
experts with special interests are observed by followers who move in
the second stage. We show that the equilibrium outcome is biased
toward the experts’ interests even though followers know the
distribution of expert interests and account for it when evaluating
observed experts’ actions. Expert influence is fully decentralized
in the sense that each individual expert has a negligible impact. The
bias in favor of experts results from a social learning effect that
is multiplied through a coordination motive. We show that the total
effect can be large even if the direct social learning effect is
small. We apply our results to the diffusion of products with network
externalities and the onset of social movements.**Dynamic
Coordination with Individual Learning**,
with Amil
Dasgupta and
Jakub
Steiner,
*Games
and Economic Behavior*,
Vol. 74 (1), January 2012, 83–101

We
study coordination in dynamic global games with private learning.
Players choose whether and when to invest irreversibly in a project
whose success depends on its quality and the timing of investment.
Players gradually learn about project quality. We identify conditions
on temporal incentives under which, in sufficiently long games,
players coordinate on investing whenever doing so is not dominated.
Roughly speaking, this outcome occurs whenever players' payoffs are
sufficiently tolerant of non-simultaneous coordination. We also
identify conditions under which players coordinate on the
risk-dominant action. We provide foundations for these results in
terms of higher order beliefs.**Nonmanipulable**
**Bayesian
Testing**,
*Journal
of Economic Theory*,
Vol. 146 (5), September 2011, 2029–2041

This
paper considers the problem of testing an expert who makes
probabilistic forecasts about the outcomes of a stochastic process. I
show that, as long as uninformed experts do not learn the correct
forecasts too quickly, a likelihood test can distinguish informed
from uninformed experts with high prior probability. The test rejects
informed experts on some data-generating processes; however, the set
of such processes is topologically small. These results contrast
sharply with many negative results in the literature.**Communication,
Timing, and Common Learning**,
with Jakub
Steiner, *Journal
of Economic Theory*,
Vol. 146 (1), January 2011, 230–247

We
study the effects of stochastically delayed communication on common
knowledge acquisition (common learning). If messages do not report
dispatch times, communication prevents common learning under general
conditions even if common knowledge is acquired without
communication. If messages report dispatch times, communication can
destroy common learning under more restrictive conditions. The
failure of common learning in the two cases is based on different
infection arguments. Communication can destroy common learning even
if it ends in finite time, or if agents communicate all of their
information. We also identify conditions under which common learning
is preserved in the presence of communication. This paper
largely supercedes our earlier note, "Communication
Can Destroy Common Learning."**Contagion
through Learning**,
with Jakub
Steiner,
*Theoretical
Economics*,
Vol. 3 (4), December 2008, 431–458

Previously titled "Learning by Similarity in Coordination Problems."

We
study learning in a large class of complete information normal form
games. Players continually face new strategic situations and must
form beliefs by extrapolation from similar past situations. The use
of extrapolations in learning may generate contagion of actions
across games even if players learn only from games with payoffs very
close to the current ones. Contagion may lead to unique long-run
outcomes where multiplicity would occur if players learned through
repeatedly playing the same game. The process of contagion through
learning is formally related to contagion in global games, although
the outcomes generally differ. We characterize the long-run outcomes
of learning in terms of iterated dominance in a related incomplete
information game with subjective priors, which clarifies the
connection to global games.**Testing
Multiple Forecasters**,
with Yossi
Feinberg,
*Econometrica*,
Vol. 76 (3), May 2008, 561–582

We consider a
*cross-calibration* test of predictions by multiple potential
experts in a stochastic environment which tests whether each expert
is calibrated conditional on the predictions made by other experts.
We show that this test is good in the sense that a true expert –
one informed of the true distribution of the process – is
guaranteed to pass the test no matter what the other potential
experts do, and false experts will fail the test on all but a small
(category one) set of true distributions. Furthermore, even when
there is no true expert present, a test similar to cross-calibration
cannot be simultaneously manipulated by multiple false experts, but
at the cost of failing some true experts.

**Robust
Conventions and the Structure of Social Networks****Universal
Deformations, Rigidity, and Ihara's Cocycle**,
*Communications
in Algebra*,
Vol. 31(2), 2003, pp. 901–943.**Goldbach's
Conjecture for Z[x]**
(with Amarpreet
Rattan),

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