Domain conditions in social choice theory (Q2757759)

Consider a society of \(n\) individuals which must rank a set of options. Each individual has a ranking of these options (given by a complete preorder). A social welfare function associates a social ranking (also a complete preorder) to each \(n\)-list of individual rankings. Arrow's impossibility theorem asserts that the only social welfare function satisfying independence (the social ranking of two options depends only on the individual rankings of these two options) and unanimity (if every individual prefers some option to another option, so does the society) is dictatorship (there is an individual whose (strict) preference is reflected by (or, more precisely, included in) the social (strict) preference), given that the set of permissible \(n\)-lists of individual rankings is sufficiently large (in fact, one often takes the set of all \(n\)-lists of complete preorders). This result is probably one of the most famous results in the social sciences of the last century (that is the 20th century), and the publication of this theorem at the end of the forties marks the birth of what is now called social choice theory. Also in the late forties, a British economist, D. Black, showed, in a kind of geometrical way, that if individual preferences over a line are single-peaked (each individual has a most-preferred point in his ranking and his preference level decreases monotonously according to the distance from this most-preferred point to its left and to its right, when possible), then majority rule \((a\) is socially preferred to \(b\) if the number of individuals who prefer \(a\) to \(b\) is greater than the number of people who prefer \(b\) to \(a\)), given that the number of individuals is odd. K. J. Arrow translated Black's condition into his set-theoretic/binary relations framework making clear that if the set of \(n\)-lists of individual preferences is suitably restricted, majority rule was a (non-dictatorial!) social welfare function satisfying (of course) independence and unanimity.NEWLINENEWLINENEWLINEThe book under review is the last word on all kinds of conditions of this type and other conditions on the domain of the aggregation procedures guaranteeing that the social ranking or the social outcomes satisfy some rationality (or other) properties. This includes properties such as quasi-transitivity of the social ranking, acyclicity of the strict part of this ranking for aggregation rules such as qualified majority rules, rules based on simple games, multi-stage rules, but also properties such as strategy-proofness of social choice functions (which select a single option given \(n\)-lists of individual preferences) or social choice correspondences (which select a subset of options).NEWLINENEWLINENEWLINEI will briefly describe the book contents. A chapter is devoted to majority decision and extensions. Individual preferences are first supposed to be complete preorders. Conditions guaranteeing the transitivity and quasi-transitivity of the social ranking are given. Thus individual preferences are supposed to be only quasi transitive and similar results are presented. Special majority, multi-stage majority and simple games are considered. A remarkable feature is that, among the conditions which are introduced, single-peakedness appears to be not only rather convincing from an intuitive point of view, but also extremely robust.NEWLINENEWLINENEWLINEOne can also consider social welfare functions satisfying independence and unanimity and try to characterize the domains guaranteeing the existence of such functions, or to give sufficient conditions for this existence in specific situations (for instance, when the set of options has some mathematical structure, e.g. a Cartesian product structure). This is the object of Chapter 4. In this same chapter, we find domain conditions for the strategy-proofness of social choice functions or correspondence.NEWLINENEWLINENEWLINESuppose now that more than half of the total number of individuals have the same strict preference. Then, of course, majority rule will give this preference as the social ranking. Chapter 5 deals with (a lot more sophisticated) restrictions of this type, called restrictions on the distribution of individual preferences. NEWLINENEWLINENEWLINEChapter 6 is more mathematically-demanding. It considers spaces of options made of Euclidean spaces (or parts of) or more general spaces endowed with suitable topologies. Individual preferences can then be endowed with topological properties and the social welfare function be continuous. An important result in this topic is that one can obtain a necessary and sufficient condition for the existence of a continuous, anonymons (a symmetry condition on the individuals in the \(n\)-lists) and unanimous social welfare function, viz. that the space of preferences (individual and social) be contractible. Contractibility is a well-known concept of algebraic topology and homotopy theory, but seldom mentioned even in the most mathematical economic theory. Of course, as the author remarks, the continuity property is highly debatable in some social choice contexts (such as voting).NEWLINENEWLINENEWLINEThis too short summary is far from giving a complete description of the book. There are 45 theorems which are all important results and the pages between theorems are interspersed with many comments which could have a `theorem status'. Furthermore, many of these comments clarify and explain difficult theoretical issues.NEWLINENEWLINENEWLINEThis major work is an essential contribution to social choice theory. It is accordingly compulsory reading for social choice theorists. But it will also reveal useful to researchers in microeconomics and game theory. Mathematicians, moreover, will find an example of how simple structures in the social sciences can give rise to rather complex issues.