DIMENSIONALITY AND DISAGREEMENT: ASYMPTOTIC BELIEF DIVERGENCE IN RESPONSE TO COMMON INFORMATION

• Date: 01 November 2019
• Published date: 01 November 2019
• Author: Gregory Phelan,Isaac Loh
• DOI: http://doi.org/10.1111/iere.12406

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 4, November 2019 DOI: 10.1111/iere.12406

DIMENSIONALITY AND DISAGREEMENT: ASYMPTOTIC BELIEF DIVERGENCE

IN RESPONSE TO COMMON INFORMATION∗

BYISAAC LOH AND GREGORY PHELAN1

Northwestern University, U.S.A.;Williams College, U.S.A.

We provide a model of boundedly rational, multidimensional learning and characterize when beliefs will

converge to the truth. Agents maintain beliefs as marginal probabilities instead of joint probabilities, and

agents’ information is of lower dimension than the model. As a result, for some observations, agents may

face an identification problem affecting the role of data in inference. Beliefs converge to the truth when these

observations are rare, but beliefs diverge when observations presenting an identification problem are frequent.

Robustly, two agents with differing priors who observe identical, unambiguous information may disagree forever,

with stronger disagreement the more information received.

1. INTRODUCTION

People disagree and sometimes in big, persistent ways: People disagree about which poli-

cies best achieve outcomes, there is substantial disagreement among professional forecasters

and central bankers regarding outlooks for macroeconomic variables,2and many patterns in

financial markets strongly suggest investor disagreement (Hong and Stein, 2007). Furthermore,

disagreements sometimes grow as people see the same information and continue to disagree.

There is ample empirical evidence that people sometimes interpret evidence differently, whether

they are Bayesian updaters or otherwise.3And yet, in many cases disagreements do not grow,

and people appear to perceive evidence in similar ways (Gerber and Green, 1999). In this

article, we provide a model of multidimensional learning that can explain why disagreements

can persist or even grow as agents observe identical information. Specifically, an agent’s be-

liefs may differ from the true probabilities even after observing arbitrarily large amounts of

information, and what beliefs converge to may depend on an agent’s initial beliefs. As a result,

observing common information can lead to permanent divergence in beliefs given arbitrarily

small initial disagreements.

Our analysis requires two ingredients. First, agents sometimes face an identification problem

for doing inference. We consider a multidimensional learning problem in which the world

is of higher dimensionality than the signals (or information) people observe. In the sense of

Benoˆ

ıt and Dubra (2019), some signals are “equivocal,” meaning that the same signal can

∗Manuscript received December 2017; revised January 2019.

1We are grateful for feedback from Matthew Chao, Eddie Dekel, Yingni Guo, Asim Khwaja, Rachel Kranton, Bruce

Sacerdote, Larry Samuelson, Arunava Sen, Rakesh Vohra (the editor), and anonymous referees. Please address cor-

respondence to: Gregory Phelan, Department of Economics, Williams College, 24 Hopkins Hall Drive, Williamstown,

MA 01267. Phone: 413-597-6284. E-mail: gp4@williams.edu.

2Andrade et al. (2016) find that disagreements among FOMC members about projections for the Fed Funds Rate and

other variables are even greater than the disagreements among forecasters in the Survey of Primary Dealers. The point

is not that central bankers and professional forecasters disagree, but that there is disagreement even within category

(in which presumably there are similar information sets and objectives). Furthermore, economists rarely switch from

hawks to doves (Malmendier et al., 2017).

3For example, Hirshleifer and Teoh (2003) document how the presentation of accounting information affects its

interpretation, and Malmendier et al. (2017) show that personal experiences of inflation strongly influence the hawkish

or dovish leanings of central bankers, which is evidence that priors influence how FOMC members interpret the same

information. See Benoˆ

ıt and Dubra (2019) for a thorough discussion of the literature.

1861

C

(2019) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1862 LOH AND PHELAN

rationally be used to update beliefs in different ways depending on people’s priors. Second,

agents update beliefs about marginal probabilities, not the full joint distribution. Because of

the high dimensionality of the problem, boundedly rational agents reduce the dimensionality

of their learning problem by maintaining beliefs over marginal distributions and reconstructing

joint distributions from the marginals using a fixed correlation (namely, independence).

Our modification is driven by the observation that in multidimensional environments, main-

taining a full joint prior distribution may be cognitively taxing. Fully rational agents would use

Bayes’ rule to update the joint distribution over the full state space instead of just considering

the marginals and multiplying appropriately to form a joint distribution. Hence, beliefs about

states would rarely be independent even if the underlying states are. Indeed, developing and

maintaining a prior over the whole joint distribution of states of the world may be regarded

as extremely computationally expensive, as suggested by Kominers et al. (2018). Additionally,

Enke and Zimmermann (2017) provide strong experimental evidence that many people effec-

tively ignore the need to update correlation when updating beliefs, and this effect is driven by

complexity in the environment instead of the computational skills of the agents.

These two ingredients interact in important ways, and we discuss them in turn. First, equivocal

observations may lead to greater disagreement because agents use their current beliefs to update

marginal beliefs. How might this mechanism look in reality? Consider two examples:

(1) A liberal and a conservative are watching the news together under a conservative gov-

ernment. After several reports that the country is weakening, the liberal says, “The gov-

ernment is failing us.” The conservative, responds, “No, more evidence of the media’s

liberal bias.”

(2) Two economists, a Keynesian and a Neoclassical, are discussing the results of a recent

stimulus package. The new GDP results are sluggish. The Neoclassical says, “Goes to

show that stimulus doesn’t work.” The Keynesian replies, “Oh no, goes to show that the

economy is much worse than we thought” (perhaps later adding that the stimulus was

poorly designed).

In these cases, there are a number of underlying factors that contribute to the observed

signals, but the signals are of lower dimensionality than the world. For the liberal and the

conservative, the politics of news reflect the state of the country and how credibly that is

reported. For the economists, GDP is a function of the fundamental strength of the economy,

the effectiveness of stimulus, and potentially how well designed that stimulus was. But the

observations do not identify those underlying variables. In each case, the exact same signal is

interpreted in completely different ways, but the observers are not rejecting the information,

nor is the information ambiguous or unclear: They are simply using the evidence in different,

rational ways. Each observer uses the information to make inference about different underlying

variables. The Keynesian infers that weak GDP means something about the economy, whereas

the Neoclassical infers that weak GDP means the fiscal multiplier is low.

As we formalize below, the previous examples illustrate how agents reason when updating

marginal, not joint, probabilities. The news watchers are (together) correct that the news reports

suggest either that “country weak +media truthful”or“country strong +media biased.” But

agents updating marginal beliefs effectively consider changing only one “dimension” of their

beliefs at a time. So the liberal, believing “weak +truthful,” would compare that belief to “one-

dimensional perturbations”—namely, “strong +truthful” and “weak +biased.” The liberal

rightly concludes that these beliefs are bad explanations of the news reports and thus chooses

to stick with and even reinforce the initial belief. But the conservative, starting with different

priors, would reason in precisely the same way to reach the opposite conclusion! However, if

the two people instead considered the joint distribution, they would recognize that their beliefs

about the country and the media should be correlated: The reporting suggests “weak +true”or

“strong +biased.” If they recognized this correlation and focused on only these two states, then

they would have a chance of reaching agreement.