Logics for modelling collective attitudes

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained

Incompleteness of Representation Theory: Hidden symmetries and Quantum Non-Integrability

Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of these hidden symmetries depends upon the realization of the Hamiltonian.Comment: 4 pages, Revtex, Phys. Rev. Lett. , July 27 (1997), in pres

The aggregation of propositional attitudes: towards a general theory

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. Our first theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the specific kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation.mathematical economics;

Learning and Pooling, Pooling and Learning

We explore which types of probabilistic updating commute with convex IP pooling (Stewart and Ojea Quintana 2017). Positive results are stated for Bayesian conditionalization (and a mild generalization of it), imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of (precise) externally Bayesian pooling operators due to Wagner (Log J IGPL 18(2):336--345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile

A Homological Approach to Belief Propagation and Bethe Approximations

We introduce a differential complex of local observables given a decomposition of a global set of random variables into subsets. Its boundary operator allows us to define a transport equation equivalent to Belief Propagation. This definition reveals a set of conserved quantities under Belief Propagation and gives new insight on the relationship of its equilibria with the critical points of Bethe free energy.Comment: 14 pages, submitted for the 2019 Geometric Science of Information colloquiu

Probabilistic Opinion Pooling with Imprecise Probabilities

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution (Madansky 44; Lehrer and Wagner 34; McConway Journal of the American Statistical Association, 76(374), 410--414, 45; Bordley Management Science, 28(10), 1137--1148, 5; Genest et al. The Annals of Statistics, 487--501, 21; Genest and Zidek Statistical Science, 114--135, 23; Mongin Journal of Economic Theory, 66(2), 313--351, 46; Clemen and Winkler Risk Analysis, 19(2), 187--203, 7; Dietrich and List 14; Herzberg Theory and Decision, 1--19, 28). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities (IP) by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi (Synthese, 62(1), 3--11, 39), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus

The Discursive Dilemma and Probabilistic Judgement Aggregation

Let S be a set of logically related propositions, and suppose a jury must decide the truth/falsehood of each member of S. A `judgement aggregation rule' (JAR) is a rule for combining the truth valuations on S from each juror into a collective truth valuation on S. Recent work has shown that there is no reasonable JAR which always yields a logically consistent collective truth valuation; this is referred to as the `Doctrinal Paradox' or the `Discursive Dilemma'. In this paper we will consider JARs which aggregate the subjective probability estimates of the jurors (rather than Boolean truth valuations) to produce a collective probability estimate for each proposition in S. We find that to properly aggregate these probability estimates, the JAR must also utilize information about the private information from which each juror generates her own probability estimate.discursive dilemma; doctrinal paradox; judgement aggregation; statistical opinion pool; interactive epistemology; common knowledge; epistemic democracy; deliberative democracy

Opinions and Preferences as Socially Distributed Attitudes

The dissertation focuses on how to best represent the consensus and attitude dynamic of a group given the attitudes of its individuals. This is done in the Bayesian epistemology framework using pooling with imprecise probabilities, and in utility theory extending Harsanyi's aggregation theorem to characterize other directed attitudes like spite and altruism. The final part of the dissertation considers attitudes within social networks and provides explanations and simulation models for online segregation and tribalism as well as the spread of rumors through contagion. The dissertation hopes to contribute to foundational issues like that of epistemic consensus, but also to new emerging phenomena in social epistemology

Optimal Auction Design and Irrelevance of Private Information

We consider the problem of mechanism design by a principal who has private information. We point out a simple condition under which the privacy of the principal's information is irrelevant in the sense that the mechanism implemented by the principal coincides with the mechanism that would be optimal if the principal's information were publicly known. This condition is then used to show that the privacy of the principal's information is irrelevant in many environments with private values and quasi-linear preferences, including the Myerson's classical auction environments in which the seller is privately informed about her cost of selling. Our approach unifies results by Maskin and Tirole, Tan, Yilankaya, Skreta, and Balestrieri. We also provide an example of a classical principal-agent environment with private values and quasi-linear preferences where a privately informed principal can do better than when her information is public.independent private values, optimal auction, resale, inverse virtual valuation function