Aggregation is the conjunction of information, aimed at its compact represen tation. Any time when the totality of data is described in terms of general ized indicators, conventional counts, typical representatives and characteristic dependences, one directly or indirectly deals with aggregation. It includes revealing the most significant characteristics and distinctive features, quanti tative and qualitative analysis. As a result, the information becomes adaptable for further processing and convenient for human perception. Aggregation is widely used in economics, statistics, management, planning, system analysis, and many other fields. That is why aggregation is so important in data pro cessing. Aggregation of preferences is a particular case of the general problem of ag gregation. It arises in multicriteria decision-making and collective choice, when a set of alternatives has to be ordered with respect to contradicting criteria, or various individual opinions. However, in spite of apparent similarity the problems of multicriteria decision-making and collective choice are somewhat different. Indeed, an improvement in some specifications at the cost of worsen ing others is not the same as the satisfaction of interests of some individuals to the prejudice of the rest. In the former case the reciprocal compensations are considered within a certain entirety; in the latter we infringe upon the rights of independent individuals. Moreover, in multicriteria decision-making one usu ally takes into account objective factors, whereas in collective choice one has to compare subjective opinions which cannot be measured properly.

In this book, first published in 1990, leading theorists and applied economists address themselves to the key questions of aggregation. The issues are covered both theoretically and in wide-ranging applications. Of particular intrest is the optimal aggregation of trade data, the need for micro-modelling when imoprtant non-linearities are present (for example, tax exhaustion in modelling company behaviour) and the use of a micro-model to stimulate labour supply behaviour in a macro-model of the Netherlands.

Our interest in problems of aggregation originates from about seven years ago when we became involved in research in the field of applied microeconomics. To our astonishment a vast majority of researchers in this area took it for granted that their, mostly thoroughly derived, micro models could meaningfully be confronted with per capita data. Nany of them did not even realize - at least they gave no utterance to it - that applying macro data in micro models raises considerable problems. Those who did mention the difficulty, almost always belittled its importance. Fortunately, there are noteworthy exceptions. Thinking about aggregation raises at least two questions: "Why or why not aggregate?" and "How to aggregate and, in particular, to what degree?" General answers to these questions can only be given in uninformative wording (as many assertions in economics): one aggregates for the sake of tractability, because of the lack of (individual) data, to avoid or to reduce multicollineartiy, to save degrees of freedom; one abstains from aggregation to avoid loss of information, to avoid aggregation biases and one aggregates such and to such degree as to bypass or reduce the drawbacks mentioned above.

Aggregation of individual opinions into a social decision is a problem widely observed in everyday life. For centuries people tried to invent the `best' aggregation rule. In 1951 young American scientist and future Nobel Prize winner Kenneth Arrow formulated the problem in an axiomatic way, i.e., he specified a set of axioms which every reasonable aggregation rule has to satisfy, and obtained that these axioms are inconsistent. This result, often called Arrow's Paradox or General Impossibility Theorem, had become a cornerstone of social choice theory. The main condition used by Arrow was his famous Independence of Irrelevant Alternatives. This very condition pre-defines the `local' treatment of the alternatives (or pairs of alternatives, or sets of alternatives, etc.) in aggregation procedures. Remaining within the framework of the axiomatic approach and based on the consideration of local rules, Arrovian Aggregation Models investigates three formulations of the aggregation problem according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision. In other words, we study three aggregation models. What is common between them is that in all models some analogue of the Independence of Irrelevant Alternatives condition is used, which is why we call these models Arrovian aggregation models. Chapter 1 presents a general description of the problem of axiomatic synthesis of local rules, and introduces problem formulations for various versions of formalization of individual opinions and collective decision. Chapter 2 formalizes precisely the notion of `rationality' of individual opinions and social decision. Chapter 3 deals with the aggregation model for the case of individual opinions and social decisions formalized as binary relations. Chapter 4 deals with Functional Aggregation Rules which transform into a social choice function individual opinions defined as choice functions. Chapter 5 considers another model – Social Choice Correspondences when the individual opinions are formalized as binary relations, and the collective decision is looked for as a choice function. Several new classes of rules are introduced and analyzed.

This article proves a very general version of the Kirman-Sondermann [Journal of Economic Theory, 5(2):267-277, 1972] correspondence by extending the methodology of Lauwers and Van Liedekerke [Journal of Mathematical Economics, 24(3):217-237, 1995]. The paper first proposes a unified framework for the analysis of the relation between various aggregation problems and the social structure they induce, based on first-order predicate logic and model theory. Thereafter, aggregators satisfying Arrow-type rationality axioms are shown to be restricted reduced product constructions with respect to the filter of decisive coalitions; an oligarchic impossibility result follows. Under stronger assumptions, aggregators are restricted ultraproduct constructions, whence a generalized Kirman-Sondermann correspondence as well as a dictatorial impossiblity result follow. -- Arrow-type preference aggregation ; Judgment aggregation ; Systematicity ; Model theory ; First-order predicate logic ; Filter ; Ultrafilter