Judgment Aggregation. Gabriella Pigozzi
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Название: Judgment Aggregation
Автор: Gabriella Pigozzi
Издательство: Ingram
Жанр: Компьютерное Железо
Серия: Synthesis Lectures on Artificial Intelligence and Machine Learning
isbn: 9781681731780
isbn:
3Manipulation will be the topic of Chapter 5.
4It is worth observing, in passing, that these two landmark results can be viewed as stemming from two different ways of conceiving democracy: the first one sees democracy as based on preferences, while the second sees it as based on knowledge (epistemic conception, as called in [Coh86, CF86]).
5See [Sup05] for a reconstruction of the intellectual path that led Arrow to introduce the axiomatic method in economics, and in particular Alfred Tarski’s influence, of whom Arrow attended a course in the calculus of relations as an undergraduate student.
6In Chapter 4 we will come back to the subtle relationships between impossibility and possibility theorems.
7This, as we shall see, is a more controversial requirement.
8It is impossible to underestimate the influence that Arrow’s theorem had in the development and foundation of social choice as a formal discipline. His result generated a vast literature, including many other impossibility results, like [Bla57, Sen69, Sen70, Pat71, Gib73, Sat75], to quote only few of them. Political scientists (most notably, William Riker [Rik82]) argued that Arrow’s findings posed serious threat to the theory of democracy.
9On the relations between judgment aggregation and preference aggregation, see Sections 1.2.2 and 3.4.
10The premise-based procedure has been reconsidered later as one of the possible escape routes from the many impossibility results that plague the discipline (see Section 4.3.1 later in the book).
11We will come back later in Chapter 2 to another (logically simpler) formalization of the Condorcet paradox as a set of judgments about preferences (Example 2.15).
12Different procedures for judgment aggregation have been assessed with respect to their truth-tracking capabilities, see [BR06, HPS10].
13We will discuss this result in Chapter 3 (Section 3.4.1).
14See also [RGMT06] for an introduction to behavioral social choice.
15We will touch upon this topic in Chapter 5 (Section 5.3.3).
16We will discuss this topic in detail in Chapter 4 (Section 4.3.3) and Chapter 6.
CHAPTER 2
Basic Concepts
This chapter is devoted to an introduction of the basic framework of judgment aggregation based on propositional logic. Our presentation is based on the framework first proposed in [LP02] and later developed by Dietrich and List in a long series of works (e.g., [DL07a, DL07c] to name just a few).
Chapter outline: We start in Section 2.1 by introducing the notions of agenda, judgment set, judgment profile, and aggregation function. In the same section we will also define a number of concrete aggregation functions. Section 2.2 proceeds by defining some properties of agendas, which have to do with how ‘tightly’ the formulae in the agenda are logically related to one another. We will see later that the more interconnected an agenda is, the more difficult the aggregation problem becomes. In Section 2.3 we look into a set of natural properties that one might wish to impose on the aggregation function to guarantee its ‘good’ behavior. In the concluding section we refer the reader to alternative formal frameworks—not necessarily based on logic—that have been developed in the literature to cast the theory of judgment aggregation.
2.1 PRELIMINARIES
2.1.1 AGENDAS IN PROPOSITIONAL LOGIC
In this book we will only be concerned with the aggregation of judgments that are expressed in propositional logic, which has been the framework of choice for most of the literature.1 So we start by briefly recapitulating—for the readers unfamiliar with propositional logic—some basic notions from its syntax and semantics. For a comprehensive exposition the reader is referred to [vD80, Ch. 1].
Propositional logic
The language of propositional logic, which we denote by L, consists of all the formulae that can be defined inductively from a countable set At = {p, q, …} of atomic propositions (also called atoms) using the logical connectives ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (equivalence). The inductive definition goes as follows: [Base] all elements of At are formulae in L; [Step] if φ and ψ belong to L, then also ¬φ (“not φ”), φ ∧ ψ (“φ and ψ”), φ ∨ ψ (“φ or ψ”), φ → ψ (“if φ then ψ”), and φ ↔ ψ (“φ if and only if ψ”) belong to L, and nothing else belongs to L.2 We say that a formula is positive if its outermost connective is not a negation (e.g., p → q, ¬p ∨ q).3
The meaning of a formula φ ∈ At is its truth value as specified by a valuation function V : L → {0, 1} where 0 stands for “false” and 1 for “true.” Each valuation V is an extension of some valuation V : At → {0, 1} of truth values to atoms, which obeys the following constraints:
We conclude with some auxiliary terminology concerning special classes of propositional formulae. A formula φ is a tautology if, for any valuation V, V ⊨ φ; it is a contradiction if, for any valuation V, V ⊭ φ; it is contingent if it is neither a tautology nor a contradiction. A set of formulae Φ is consistent if it has a model, that is, if there exists a valuation V, such that V ⊨ φ for each φ ∈ Φ; a formula φ is a logical consequence of a set of formulae Φ (in symbols, Φ ⊨ φ) if for every valuation V such that V ⊨ Φ, it is the case that V ⊨ φ
Agendas
With the machinery of propositional logic in place, we can frame the problem of the aggregation of judgments СКАЧАТЬ