Frederik Herzberg

The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates ☆

It is well known that the literature on judgement aggregation inherits the impossibility results from the aggregation of preferences that it generalises. This is due to the fact that the typical judgement aggregation problem induces an ultrafilter on the set of individuals. We propose a model-theoretic framework for the analysis of judgement aggregation and show that the conditions typically imposed on aggregators induce an ultrafilter on the set of individuals, thus establishing a generalised version of the Kirman–Sondermann correspondence. In the finite case, dictatorship then immediately follows from the principality of an ultrafilter on a finite set. This is not the case for an infinite set of individuals, where there exist free ultrafilters, as Fishburn already stressed in 1970. Following Lauwers and Van Liedekerke’s (1995) seminal paper, we investigate another source of impossibility results for free ultrafilters: the domain of an ultraproduct over a free ultrafilter extends the individual factor domains, such that the preservation of the truth value of some sentences by the aggregate model–if this is as usual to be restricted to the original domain–may again require the exclusion of free ultrafilters, leading to dictatorship once again.

Stochastic Calculus with Infinitesimals

1 Infinitesimal calculus, consistently and accessibly.- 2 Radically elementary probability theory.- 3 Radically elementary stochastic integrals.- 4 The radically elementary Girsanov theorem and the diffusion invariance principle.- 5 Excursion to nancial economics: A radically elementary approach to the fundamental theorems of asset pricing.- 6 Excursion to financial engineering: Volatility invariance in the Black-Scholes model.- 7 A radically elementary theory of Ito diffusions and associated partial differential equations.- 8 Excursion to mathematical physics: A radically elementary definition of Feynman path integrals.- 9 A radically elementary theory of Levy processes.- 10 Final remarks.

A definable nonstandard enlargement

This article establishes the existence of a definable (over ZFC), countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers (i.e. bounded with respect to the superstructure hierarchy). The underlying ultrafilter is the one constructed by Kanovei and Shelah [10]

Universal algebra for general aggregation theory: Many-valued propositional-attitude aggregators as MV-homomorphisms

This paper continues Dietrich and List’s [2011] work on propositionalattitude aggregation theory, which is a generalised unification of the judgement-aggregation and probabilistic opinion-pooling literatures. We first propose an algebraic framework for an analysis of (many-valued) propositional-attitude aggregation problems. Then we shall show that systematic propositional-attitude aggregators can be viewed as homomorphisms — algebraically structure-preserving maps — in the category of C.C. Chang’s [1958] MV-algebras. (Proof idea: Systematic aggregators are induced by maps satisfying certain functional equations, which in turn can be verified to entail homomorphy identities.) Since the 2-element Boolean algebra as well as the real unit interval can be endowed with an MV-algebra structure, we obtain as natural corollaries two famous theorems: Arrow’s theorem for judgement aggregation as well as McConway’s [1981] characterisation of linear opinion pools. Conceptually, this characterisation of aggregators can be seen as justifying a certain structuralist interpretation of social choice. Technically and perhaps more importantly, it opens up a new methodology to social choice theorists: the analysis of general aggregation problems by means of universal algebra.

Aggregation of Monotonic Bernoullian Archimedean Preferences: Arrovian Impossibility Results

Cerreia-Vioglio, Ghirardato, Maccheroni, Marinacci and Siniscalchi (Economic Theory, 48:341–375, 2011) have recently proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean (MBA) preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences — and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.

Internal laws of probability, generalized likelihoods and lewis' infinitesimal chances- : A response to adam elga

The rejection of an infinitesimal solution to the zero-fit problem by A. Elga ([2004]) does not seem to appreciate the opportunities provided by the use of internal finitely-additive probability measures. Indeed, internal laws of probability can be used to find a satisfactory infinitesimal answer to many zero-fit problems, not only to the one suggested by Elga, but also to the Markov chain (that is, discrete and memory-less) models of reality. Moreover, the generalization of likelihoods that Elga has in mind is not as hopeless as it appears to be in his article. In fact, for many practically important examples, through the use of likelihoods one can succeed in circumventing the zero-fit problem. 1. The Zero-fit Problem on Infinite State Spaces2. Elgas Critique of the Infinitesimal Approach to the Zero-fit Problem3. Two Examples for Infinitesimal Solutions to the Zero-fit Problem4. Mathematical Modelling in Nonstandard Universes?5. Are Nonstandard Models Unnatural?6. Likelihoods and DensitiesA. Internal Probability Measures and the Loeb Measure ConstructionB. The (Countable) Coin Tossing Sequence RevisitedC. Solution to the Zero-fit Problem for a Finite-state Model without MemoryD. An Additional Note on ‘Integrating over Densities’E. Well-defined Continuous Versions of Density Functions The Zero-fit Problem on Infinite State Spaces Elgas Critique of the Infinitesimal Approach to the Zero-fit Problem Two Examples for Infinitesimal Solutions to the Zero-fit Problem Mathematical Modelling in Nonstandard Universes? Are Nonstandard Models Unnatural? Likelihoods and Densities Internal Probability Measures and the Loeb Measure Construction The (Countable) Coin Tossing Sequence Revisited Solution to the Zero-fit Problem for a Finite-state Model without Memory An Additional Note on ‘Integrating over Densities’ Well-defined Continuous Versions of Density Functions

Addendum to “A definable nonstandard enlargement”

Łośs theorem for bounded D -ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [4], is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A combinatorial infinitesimal representation of Lévy processes and an application to incomplete markets

Starting from an R-valued Lévy process with infinitesimal generator ‘, an infinitesimal mesh h and an internal hyperfinite discretisation (lattice) L of the state space R, the reduced lifting is constructed—this is the unique right lifting that can be written as the hyperfinite sum of a generalised Andersonian random walk for mesh h and a hyperfinite sum of independent jumps in L, each allowed to occur at any time in h·*N0. By assigning each of these components a natural economic interpretation, a suitable internal notion of risk-neutrality is introduced. A reduced lifting for a process with solely positive jumps— modelling the stock price process of a conservatively managed company—can then be shown to describe, with respect to this notion of risk-neutrality, the logarithm of a weakly complete market model (complete in the sense that there is an essentially unique risk-neutral measure which preserves the structure of the model, that is the Markov property as well as the independence of the jumps and the lifted diffusion part), thereby circumventing the incompleteness typically entailed by a continuous Lévy market model.

The dialectics of infinitism and coherentism: inferential justification versus holism and coherence

This paper formally explores the common ground between mild versions of epistemological coherentism and infinitism; it proposes—and argues for—a hybrid, coherentist–infinitist account of epistemic justification. First, the epistemological regress argument and its relation to the classical taxonomy regarding epistemic justification—of foundationalism, infinitism and coherentism—is reviewed. We then recall recent results proving that an influential argument against infinite regresses of justification, which alleges their incoherence on account of probabilistic inconsistency, cannot be maintained. Furthermore, we prove that the Principle of Inferential Justification has rather unwelcome consequences—formally resembling the Sorites paradox—as soon as it is iterated and combined with a natural Bayesian perspective on probabilistic inferences. We conclude that strong versions of foundationalism and infinitism should be abandoned. Positively, we provide a rough sketch for a graded formal coherence notion, according to which infinite regresses of epistemic justification will often have more than a minimal degree of coherence.

Elementary non-Archimedean utility theory

A non-Archimedean utility representation theorem for independent and transitive preference orderings that are partially continuous on some convex subset and satisfy an axiom of incommensurable preference for elements outside that subset is proven. For complete preference orderings, the theorem is deduced directly from the classical von Neumann-Morgenstern theorem; in the absence of completeness, Aumanns [Aumann, R.J., 1962. Utility theory without the completeness axiom. Econometrica 30 (3), 445-462] generalization is utilized.