Persistent Disagreement and Polarization in a Bayesian Setting

For two ideally rational agents, does learning a finite amount of shared evidence necessitate agreement? No. But does it at least guard against belief polarization, the case in which their opinions get further apart? No. OK, but are rational agents guaranteed to avoid polarization if they have access to an infinite, increasing stream of shared evidence? No

In search of characterization of the preference for safety under the Choquet model

Victor prefers safety more than Ursula if whenever Ursula prefers some constant to some uncertain act, so does Victor. This paradigm, whose Expected Utility version takes the form of Arrow & Pratt's more risk averse concept, will be studied in the Choquet Uncertainty model, letting u and μ (v and ν) be Ursula's (Victor's) utility and capacity. A necessary and sufficient condition (A) on the pairs (u, μ) and (v, ν) will be presented for dichotomous weak increased uncertainty aversion, the preference by Victor of a constant over a dichotomous act whenever such is the preference of Ursula. This condition, pointwise inequality between a function defined in terms of v (u-1(⋅)) and another defined purely in terms of the capacities, preserves the flavor of the "more pessimism than greediness" characterization of monotone risk aversion by Chateauneuf, Cohen & Meilijson in the Rank-dependent Utility Model and its extension by Grant & Quiggin to the Choquet Utility Model. A sufficient condition (B) in terms of the capacities only, satisfied in particular if ν (⋅) = f (μ (⋅)) for some convex f, will be presented for more simplicity seeking, the preference by Victor over any act for some dichotomous act, that leaves Ursula indifferent. Condition A is thus a characterization of weak increased uncertainty aversion for convex f. An example will be exhibited disproving the more far reaching conjecture under which the dichotomous case implies the general case.Choquet Utility, greediness, pessimism, Rank-dependent Utility, Risk aversion, uncertainty.

Essays on Learning and Induction

What is the correct way to respond to newly acquired information? What methods for updating beliefs and other attitudes are rational? And what makes them rational? This dissertation is a collection of independent essays, each of which addresses these questions. Among other things, I investigate the extent to which Bayesian learning can be considered objective, the circumstances in which rational learning reduces uncertainty and produces consensus, whether rational learning is compatible with disagreement and polarization, and the relationship between long-run and short-run norms for learning

In search of characterization of the preference for safety under the Choquet model

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2011.31 - ISSN : 1955-611XVictor prefers safety more than Ursula if whenever Ursula prefers some constant to some uncertain act, so does Victor. This paradigm, whose Expected Utility version takes the form of Arrow & Pratt's more risk averse concept, will be studied in the Choquet Uncertainty model, letting u and μ (v and ν) be Ursula's (Victor's) utility and capacity. A necessary and sufficient condition (A) on the pairs (u, μ) and (v, ν) will be presented for dichotomous weak increased uncertainty aversion, the preference by Victor of a constant over a dichotomous act whenever such is the preference of Ursula. This condition, pointwise inequality between a function defined in terms of v (u-1(⋅)) and another defined purely in terms of the capacities, preserves the flavor of the "more pessimism than greediness" characterization of monotone risk aversion by Chateauneuf, Cohen & Meilijson in the Rank-dependent Utility Model and its extension by Grant & Quiggin to the Choquet Utility Model. A sufficient condition (B) in terms of the capacities only, satisfied in particular if ν (⋅) = f (μ (⋅)) for some convex f, will be presented for more simplicity seeking, the preference by Victor over any act for some dichotomous act, that leaves Ursula indifferent. Condition A is thus a characterization of weak increased uncertainty aversion for convex f. An example will be exhibited disproving the more far reaching conjecture under which the dichotomous case implies the general case.Victor aime plus la sécurité que Ursula si, dès que Ursula préfère une constante à un acte incertain, il en est de même pour Victor. Ce paradigme, qui, dans le modèle EU, n'est autre que le concept d'Arrow-Pratt : "plus d'aversion pour le risque que", sera étudié dans le modèle CEU, modèle de Choquet de décision dans l'incertain, où on appelle u et μ (v et ν) l'utilité et la capacité d'Ursula (de Victor). Nous présentons une condition nécessaire et suffisante (A) sur les paires (u, μ) et (v, ν) pour l'accroissement faible d'aversion pour l'incertain dichotomique, la préférence de Victor pour une constante à un acte dichotomique dès que Ursula a cette préférence. Cette condition, inégalité ponctuelle entre une fonction en termes de v (u-1(⋅)) et une autre uniquement en termes de capacités, garde la forme de la caractérisation de l'aversion pour le risque monotone de Chateauneuf, Cohen et Meilijson et de son extension à l'incertain monotone de Grant et Quiggin dans le modèle de Choquet. Nous présentons une condition suffisante (B) de "plus de goût pour la simplicité" (préférence de Victor pour un acte dichotomique sur tout autre acte, qui laisse Ursula indifférente), uniquement en termes de capacités, satisfaite en particulier si ν (⋅) = f (μ (⋅)) pour une f convexe. Nous exposons un contre-exemple à la conjecture suivant laquelle le cas dichotomique impliquerait le cas général

Learning and Disagreement in an Uncertain World

Most economic analyses presume that there are limited differences in the prior beliefs of individuals, as assumption most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a simple model of Bayesian learning. Two individuals with different priors observe the same infinite sequence of signals about some underlying parameter. Existing results in the literature establish that when individuals are certain about the interpretation of signals, under very mild conditions there will be asymptotic agreement---their assessments will eventually agree. In contrast, we look at an environment in which individuals are uncertain about the interpretation of signals, meaning that they have non-degenerate probability distributions over the conditional distribution of signals given the underlying parameter. When priors on the parameter and the conditional distribution of signals have full support, we prove the following results: (1) Individuals will never agree, even after observing the same infinite sequence of signals. (2) Before observing the signals, they believe with probability 1 that their posteriors about the underlying parameter will fail to converge. (3) Observing the same sequence of signals may lead to a divergence of opinion rather than the typically presumed convergence. We then characterize the conditions for asymptotic agreement under "approximate certainty"---i.e., as we look at the limit where uncertainty about the interpretation of the signals disappears. When the family of probability distributions of signals given the parameter has "rapidly-varying tails" (such as the normal or exponential distributions), approximate certainty restores asymptotic agreement. However, when the family of probability distributions has "regularly-varying tails" (such as the Pareto, the log-normal, and the t-distributions), asymptotic agreement does not obtain even in the limit as the amount of uncertainty disappears. Lack of common priors has important implications for economic behavior in a range of circumstances. We illustrate how the type of learning outlined in this paper interacts with economic behavior in various different situations, including games of common interest, coordination, asset trading and bargaining.