Binary Aggregation with Integrity Constraints
Umberto Grandi
This dissertation presents the framework of binary aggregation with integrity constraints, positioning it with respect to existing frameworks for the study of collective decision making developed in Social Choice Theory and Artificial Intelligence, and exploring theoretical results which may pave the way for its implementation.
In our setting, a set of individuals need to make a choice over a set of binary issues. Each individual submits a yes/no choice for each of the issues, and these choices are then aggregated into a collective choice by means of an aggregation procedure.
Individual choices are bound by a rationality assumption, specifying the range of answers that is considered to be rational.
We represent rationality assumptions with formulas in a simple propositional language, calling them integrity constraints.
This framework can be employed to model a variety of situations of collective decision making, such as multiple referenda, committee elections, as well as the problem of aggregating individual preferences or judgments.
A concept that is central to the whole dissertation is that of collective rationality: assuming that each individual satisfies a given integrity constraint, we are interested in finding out whether the output of a given aggregation procedure still satisfies the same integrity constraint.
We call a situation in which collective rationality is not satisfied a paradox, and we show that most of the classical paradoxes studied in the literature on Social Choice Theory can be seen as instances of our general definition.
We focus our analysis on the Condorcet paradox, the discursive dilemma, the Ostrogorski paradox and the multiple election paradox, identifying a common syntactic property in the integrity constraints that define these paradoxical situations.
We classify integrity constraints into syntactically defined languages, and define CR[L] as the class of collectively rational procedures with respect to all integrity constraints in a given language L.
For instance, we indicate with CR[cubes] the class of aggregation procedures that are collectively rational with respect to integrity constraints that can be expressed as conjunctions of literals (i.e., cubes).
On the other hand, classes of aggregation procedures are usually defined in axiomatic terms, and we indicate with $F_L[AX]$ the class of procedures satisfying a list of axiomatic properties AX.
We investigate the relation between these two definitions for several natural fragments of the language of propositional logic, and for several axiomatic properties from the literature on Social Choice Theory.
As an example, we prove that the class of collectively rational procedures with respect to cubes coincides with the class of unanimous procedures, i.e., those aggregators that accept (reject) a given issue if all individuals agree on accepting (rejecting) the issue: $CR[cubes]=F_L[Unanimity].
The frameworks of preference aggregation and judgment aggregation are the main settings developed in the literature to study problems related to the aggregation of individual expressions.
We provide an embedding from each of the two frameworks into binary aggregation by devising a suitable integrity constraint, and we provide alternative proofs of some of the classical results in these settings by making use of our characterisation results of collective rationality.
In the framework of judgment aggregation we focus on the novel problem of safety of the agenda: given a set of propositional formulas which constitute the objects of judgment, i.e., the agenda, how can we guarantee that the collective judgment will be consistent when all individual judgments are.
For several classes of procedures defined in axiomatic terms, we provide necessary and sufficient conditions for an agenda to be safe, and we show that the problem of checking such conditions is a highly intractable problem (Pi_2^p-complete) for all classes under consideration.
There are several examples of aggregation procedures that are collectively rational for every possible integrity constraint, and we conclude the dissertation by studying three such procedures.
We investigate the computational complexity of the two classical problems of winner determination and strategic manipulation, i.e., we compare the complexity of computing the winner of an election with the problem of determining whether individuals have incentives to misrepresent their vote in order to favour their own position.
Overall, this dissertation constitutes a systematic study of the problem of collective rationality in binary aggregation, a problem that is central to the literature in Social Choice Theory and that proved useful to gain insight into a variety of applications in Artificial Intelligence.