Axiomatic Concensus Theory in Group Choice and Biomathematics

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In his PhD thesis, which contributed to his Nobel Prize in Economics, Arrow characterized all decision rules that possess some obviously desired properties. A convenient way to describe his result to mathematicians is with the structure of a domain, range, and a mapping. I will call them, respectively, the "inputs," "outputs," and "decision or voting rule. For inputs, let each person have a complete, transitive ranking of the candidates: there is no restriction on what ranking a voter selects. This condition makes sense; to have rational outcomes, then the inputs, the voters' preferences, should be rational i.

The output is the societal ranking of the candidates: we require it to be transitive. All that remains is to specify the mapping, or decision rule. But rather than being specific, Arrow considers a class of voting rules that satisfy the following two rather innocuous appearing properties. The first condition is called the Pareto condition: think of it as an unanimity condition. Namely, if everyone ranks some pair of candidates in the same way, then that ranking should be the pair's societal ranking.

To introduce the second condition, why should a pair's societal ranking be determined only in the very special setting of unanimity? Why not always rank each pair in terms of what the voters think of these two alternatives? To illustrate the kind of problems we want to avoid, imagine a department forced to select only one of the three candidates, Barbara, Connie, and Donna, for a tenure track position where the top two choices are Barbara and Connie.

But imagine the departmental outcry if the committee admits that the ranking of Barbara and Connie was so close that if a couple of committee members had a better opinion of Donna , then Connie would have been top-ranked and selected. What is going on? What does Donna have to do with the relative ranking of Barbara and Connie and the ultimate choice of Barbara?

To avoid these kinds of troublesome outcomes, Arrow imposed the condition of binary independence. This condition requires the societal ranking of each pair to be strictly determined by how the voters rank this particular pair. In other words, when determining the societal ranking for a particular pair, all other information is irrelevant. Arrow then characterized all possible decision rules for three or more candidates that satisfy these conditions: there is only one, a dictatorship!

Namely, there is precisely one voter who is endowed with the powerful role where the societal ranking must always agree with his personal ranking: this dictator need not be benevolent because the societal outcome is totally independent of what the other voters want. A dictator! Outside of the influence imposed by some departmental chairs, rarely in our society or daily life do we use, or even consider, dictatorial decision rules.

As an immediate illustration, the widely used plurality vote is not dictatorial. What we learn from Arrow's Theorem about the plurality vote or any other non-dictatorial rule, then, is that there must exist situations where this rule violates one of the posed conditions. This observation brings us to the presidential elections in Florida: Bush won the election, but if fewer voters liked Nader, Gore would have won. In other words, in a head-to-head election Gore would have been the strong winner, which means that our plurality vote violates binary independence.

For references and further reading about Arrow's Theorem, let me suggest Arrow's seminal book and another one, Saari, [Decisions] , that for obvious reasons I particularly like. In [ Decisions ], I show that Arrow's theorem admits a far more benign interpretation than given above and that we can avoid Arrow's "no method is fair" implication. While [ Decisions ] is written in a manner to make the new results available to an audience beyond the mathematics community, the math reader probably will detect from the central arguments that Arrow's result occurs because a Z 2 x Z 2 x Z 2 orbit differs from an S 3 orbit.

A bit later I will weave Arrow's Theorem back into the discussion about the book under review, but for now, let's turn to scotch. Not any scotch, but fine, smooth single-malt scotch! An argument that must rage among connoisseurs, whether professionals or amateurs, is to determine which is the better one: this deep thirst for the truth requires perpetual experimentation and careful tasting. To put a cork on all of this fun, maybe an answer can be found from mathematics and existing opinions.

Users' Guide 6. John D. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. Day, F. ISBN pbk. Axiomatic set theory. Consensus Social sciences 4. Social choice. McMorris, F. The term "bioconsensus" was invented by Fred S. Apart from that, some fifteen years ago I conjectured that it ought to be possible to construct a mathematical framework in which many of the known versions of Arrow's impossibility theorem could be established.

That turned out to be a difficult and subtle project on which several of us have labored through the years. Progress has been made; much remains to be done. Since consensus theory has its roots in the theory of elections, many contributions have been and are being made by political scientists, sociologists and economists. In the context of human behavior, consensus research is carried out by psychologists.

Faced with conflicting evidence on evolutionary history, systematic biologists appeal to concepts of consensus, and molecular biologists attempt to apply consensus theory in areas of DNA research. Market researchers find the discipline relevant since it can be viewed as a theory of how large organizations make decisions based on possibly conflicting lines of evidence.

Contemporary applications of consensus theory involve airplane and missile navigational systems, methods to combat bioterrorism, drug development based on DNA research, marketing and manufacturing decisions of large companies, and stock market predictions. There is a demonstrated need for research in consensus theory, and indeed it is an active and lively area of endeavor in several disciplines.

But the situation is unusual because we have an essentially mathematical framework that was developed by people who were not formally trained as mathematicians. The literary style in the social and behavioral sciences can be quite different from the formalism that some mathematicians have grown to expect.

Definitions are sometimes embedded in discussions and not formally delineated; symbolism is not often what one expects, and implied parentheses may be ambiguous. Nevertheless, when one reads papers in different disciplines, one is struck by the similarity of the arguments. In consensus theory there is a pressing need for mathematicians to develop appropriate models in which all of the relevant concepts can be embedded, compared, and analyzed. I liken the situation to that of measure theory before it was developed as a mathematical discipline. The rudiments were there in the classical theory of real variables, but the subject was difficult because people used subtle results from the reals when they did not really need them.

The Lebesgue integral became easier to understand in terms of a measure algebra than it ever was as a branch of classical real analysis. What was a second-year graduate course could suddenly be treated at the advanced undergraduate level. Galois theory as a tool for xiii xiv Foreword understanding solvability of polynomials is another example. Indeed, twentieth-century mathematics is distinguished by the axiomatic approach.

It is replete with examples where abstract axiom systems make it easier to understand what is happening. Sometimes one uses unneeded machinery when a subject is embedded in a concrete framework; certainly this has been true for consensus theory. The mathematics discussed by Day and McMorris is sound and has a unique flavor. Often positive results are attained when something that looks unpleasant happens; often one encounters a situation where if X and Y happen, then some apparently desirable thing Z cannot happen, or some undesirable thing W must necessarily occur.

A collection of sets that is closed under unions and intersections may be undesirable, while closure under intersection but not union may turn out to be what is needed. For those of us trained in algebra this gives the subject a distinctive flavor. In group theory we are largely interested in types of subgroups, rather than subsets that are not subgroups. In consensus theory we get mileage out of ternary relations having the property that abc, bca, and cab cannot simultaneously occur.

The subject is in its infancy. Almost everywhere one looks there are open questions. For example, consensus methods often operate on collections of subsets of a fixed set. It has been useful to represent these collections in terms of ternary relations, to form the consensus in terms of these relations, and then to recapture the consensus object. Yet there is no axiomatic theory of these relational representations, and until very recently no one seems to have worried about just when such representations were injective.

Consensus theory involves good mathematics that is exciting, worthwhile, and potentially applicable to other areas. Yet our current course structure does not expose our students to it, and since the literature is so multidisciplinary, it is not so easy to learn on one's own. And there is a certain amount of intellectual snobbery that makes some mathematicians uncomfortable with the classical consensus literature. All of this cries out for a modern axiomatic treatment that will make things easier and more transparent to the mathematical community.

This is precisely what Day and McMorris have undertaken to present.

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  • I know them well: they have a deep knowledge of the subject matter, are careful workers, and are good expositors. This book describes the frontier of a new and important branch of applied mathematics, and with its extensive bibliography it provides an indispensable guide to the consensus literature. I feel a sense of wonderment when I see the parallels between results in voting theory and corresponding results for unrooted and rooted trees. There is something fundamental driving all of this, and Day and McMorris's book helps to reveal these fundamental concepts.

    As such, their book will be an invaluable resource for researchers in bioinformatics, for evolutionary biologists, for ecologists, for students of discrete mathematics, for computer scientists, and for workers in fields related to social welfare economics. Pears [, p. Solving this problem requires that we evaluate consensus rules in terms of the basic properties or axioms that they satisfy. But for this problem one can sometimes obtain a fundamental impossibility result by first listing a set of seemingly reasonable axioms that any consensus rule on X should satisfy, then proving that no such rule can exist.

    When it occurs, this contradiction encourages researchers to explore the extent to which the axioms can be weakened, while still maintaining the contradiction, and to explore how to alter the axioms so as to eliminate the contradiction. Desirable consequences of the latter analysis are characterizations of consensus rules in terms of axioms with which users may assess the appropriateness of the rules for given applications.

    Thus, although the axiomatic approach may seem to be abstract and purely technical, its practical and concrete aspects enable us to distinguish between what is realizable and what is not. An archetypal problem of aggregation concerns group choice, i. In K. Arrow pioneered the use of axiomatic paradigms in group choice where, for X a set of preference rankings, he established that there cannot exist a rational efficient democratic rule to obtain a consensus of such rankings.

    His paradigms have since been refined, extended, and applied in areas far removed from their origins in group choice and welfare economics. We review this axiomatic research first in the area of group choice, then in areas of biomathematics where the objects of interest represent, e. We establish the axiomatic consensus theory's basic results using standard terminology and notation and with uniform attention to rigor and detail. We cite both traditional and current literature, and we pose open problems that remain to be solved. Through the years we have benefited immeasurably from consensus-related discussions and collaborations with J.

    Barthe'lemy, M. Janowitz, B. Leclerc, B. G Mirkin, B.

    Journal of Applied Mathematics

    Monjardet, H. Mulder, R. Powers, and F. We view this book both as a xv xvi Preface culmination of these many beneficial interactions and as a consequence of our tutorial [] on 2 October in a meeting on bioconsensus [] at the Center for Discrete Mathematics and Theoretical Computer Science DIMACS , Rutgers University. We are particularly indebted to M. Leclerc, and R.

    Powers for their valuable critiques of the book in a preliminary version. William H. Day F. McMorris Chapter 1 Achieving Consensus In all these fields the axiomatic approach—inspired by Arrow's work—is the same. One chooses axioms that the consensus function should satisfy and one determines the functions if any satisfying the axioms.

    Monjardet [, p. G Mirkin and F. Roberts [, p. We will review the history and development of this field, emphasizing how axiomatic consensus theory has flourished in, and contributes to our understanding of, group choice and bioconsensus. We set the stage with typical examples. Example 1. A benevolent society wishes to adopt a rule to achieve consensus so that, when choosing from among various alternatives, the society's officials can apply the rule to members' rankings of the alternatives and so obtain a social ranking that reveals the society's collective will.

    After discussion the members identify properties of collective rationality, independence, optimality, and democracy that the rule should satisfy. The consensus rule should return a transitive social ranking for all tuples of members' rankings of the alternatives. When voting is imminent, some alternatives may be no longer available. If each member has ranked the available alternatives the same in one situation as in another, then no matter what is true about the rankings of unavailable alternatives, the social ranking of the available alternatives should be the same in both situations.

    Achieving Consensus 3. If every member prefers alternative x to alternative y, then the social ranking should prefer x to y.

    Download Axiomatic Consensus Theory In Group Choice And Biomathematics

    The social ranking should not always be the ranking of a dictatorial member. The committee evaluating rules for possible adoption learns that no consensus rule can satisfy these seemingly reasonable properties. To fathom why they are contradictory, the committee considers a natural way to achieve consensus by means of the majority rule: for each pair x, y of alternatives, x should be socially preferred to y if a majority of members would choose x over y if only those alternatives were available. But the social ranking so derived can violate the transitivity required by property 1 on the preceding page, so it does not exhibit the collective rationality required of an adopted consensus rule.

    Although Example 1. Furthermore the problems exhibited by the example arise not just in group choice but in all fields where aggregation problems arise. Suppose that each individual is an expert systematist, each preference is a hierarchical classification, and the social choice is a consensus classification: such problems are relevant in fields of data analysis, classification, and systematics. Consensus problems occur as well in bioinformatics or computational biology []. Waterman et al. The sequences are approximately aligned and contain unknown patterns that occur perhaps imperfectly in each sequence.

    Waterman's algorithm moves a window of fixed width along the aligned sequences. At each window position it searches for short candidate consensus patterns that, within the window, minimize the total number of mismatches to substrings of each sequence. From among the candidate consensus patterns identified at all window positions, the algorithm selects those consensus patterns with the minimum total number of mismatches. Since the algorithm is parameterized by the number of symbols from which the sequences are formed, the number of sequences, the length of the aligned set of sequences, the window width, the consensus pattern length, and the number of mismatches permitted when comparing consensus pattern to sequence substring, it is hard to visualize how the algorithm compares with those solving other consensus problems.

    Mirkin and Roberts [] show that well-known consensus rules of the social sciences, based on the median and mean, are special cases of Waterman's method for appropriate choices of its parameter values and that Waterman's recommended parameter values cause his method to be based on the median consensus rule. It might be useful to axiomatize Waterman's method or the median rule in the context of his method [, p. Bains [25, 26] describes a heuristic algorithm to align sets of either DNA or protein sequences. The algorithm iterates between an alignment step, which generates a 1.

    Consensus Rules 3 new sequence alignment using the current consensus sequence, and a consensus step, which generates a new consensus sequence using the current sequence alignment. These steps are unapologetically heuristic, e. To understand better the behavior of such algorithms, one might investigate the abstract properties required by each step to reveal the algorithm's theoretical bases, to ensure its convergence, and to achieve for it an acceptable computational complexity.

    This book stresses formal aspects of the problem of achieving consensus. We introduce the basic concepts, survey the literature, and suggest topics for further study; but for consensus methodology we describe neither its many applications in the biological sciences nor its many associated advances in algorithm design and computational complexity.

    We assume that readers have a certain familiarity with mathematical reasoning, a certain capacity for abstract thought, and a certain interest in how mathematics is used to model aspects of biological problems. We state many theorems, cite authors and sources, and prove the main results. Theoretical investigations of problems in group choice are not new: still cited today are contributions of Borda [] in , Condorcet [] in , and Pareto [] in But the recent origins of our interest in consensus lie in K.

    Arrow's influential doctoral thesis [11], which appeared in and launched him on a path to share with J. Arrow's brilliant insight was to visualize the preference aggregation problem axiomatically so as to prove that there is no rational, efficient, democratic rule of preference aggregation. We will study such abstract formulations of consensus. In this chapter we describe formal models of consensus rules and introduce axiomatic paradigms with which to investigate them.

    In Chapter 2 we use the paradigms to investigate Arrow's impossibility theorem for weak orders, which generalize the rankings in Example 1. In Chapter 3 we apply Arrow's paradigms to obtain impossibility results for basic problems of biological consensus. In Chapter 4 we apply Arrow's paradigms to obtain possibility results, i. In Chapter 5 we present formalizations of Arrow's paradigms that reveal general conditions under which the paradigms can be applied.

    Such studies reveal a shift of perspective: where previously our investigation of an object's internal structure may have yielded insights into the relationships among those objects, now we use object interrelationships to gain insights into the basic properties of sets of those objects. In Chapter 6 we extend the consensus concept to problems where consensus objects may include subsets or supersets of the alternatives associated with the individual objects; such problems arise in systematic biology [9,,] and may benefit from additional axiomatic investigations of possibilities and impossibilities.

    Although we may mention variants, we will use a standard terminology and notation beginning with the basic concept of consensus. Achieving Consensus k-tuples of X. The tuple x 1 , Convention 1. In the early chapters we consider consensus rules which map a profile of k individual objects to a single consensus object. Later we study consensus rules that may accept profiles of varying lengths or may return more than one object as a consensus result.

    Beginning in section 4.

    We view the generic elements of X as atomic indecomposable objects that are distinguishable from each other but otherwise structureless and featureless. However, in specific cases the objects of interest may be quite complex; typically we specify a finite nonempty set S of alternatives outcomes, bases, evolutionary units, sites, species, vertices on which the objects are based. For example, objects may be types of n-ary relations Chapter 2, sections 3.

    We define such object types as the need arises, but since Chapters 2 and 3 employ n-ary relations, here we give their common setting. Also we may shorten any ordered pair a, b to ab. Definition 1. A binary relation on S is a subset R c S2 of ordered pairs. To show membership in R we write a,b e R, ab 6 R, or aRb. Paradigms Table 1. Logical Notation. P and Q are logical sentences; v is a variable.

    Meaning It is not the case that P. P and Q. P logically implies Q. P and Q are logically equivalent. For every v, P. For some v, P. Paired delimiters used to prevent ambiguity of meaning. To investigate binary relations we use elementary logical notation Table 1. Parentheses always delimit the formula to which quantifiers apply; thus in 3x x 1. One might formulate a consensus rule to exhibit some desirable features, then analyze that rule to identify other strengths or weaknesses; or one might formulate a set of axioms or properties that many researchers would accept as desirable, then determine the set of consensus rules satisfying those axioms.

    Arrow used the latter paradigm and for the most part so will we, but its application is sensitive to the questions being asked and to the relative strengths of the axioms involved. Formulating a viable set of axioms is something of an art. Alone each axiom should be compelling. If the set is too strong, no consensus rule can satisfy all the axioms; if the set is too weak, the set of consensus rules satisfying the axioms may be too large or too unorganized to be useful.

    In Arrow's analysis, axioms of independence and optimality are separately compelling but, taken together, are so strong that they yield an unsatisfactory set of dictatorial consensus rules. The result can be expressed as in the following templates. Chapter 1. Achieving Consensus 6 Table 1. Types of Binary Relations R on Sn. Also given is a symbol for the set of all relations on Sn of this type. For the types of relations defined in Table 1. Tree cond. Axioms 7 Template 1. Template 1. No consensus rule can have the desirable properties X, Y, and Z. The impossibility results we describe include Theorems 2.

    But how might desirable rules if they exist be characterized? We might sequentially introduce relatively weak axioms of symmetry, neutrality, monotonicity, or the like until they yield a meaningful set of consensus rules. The result can be expressed as in the following theorem. Characterizing a set of many rules may be valuable.

    For example, the majority and strict consensus rules are often used to take the consensus of profiles of hierarchies. There exist parameterized sets of consensus rules for hierarchies that include the majority and strict rules as extremes. Characterizing the parameterized set might yield useful characterizations of its extremes. The possibility results we describe include characterizations of unique rules Theorems 2.

    But in a twilight zone between impossibility and possibility are ambiguous results as in the following template. A consensus rule has the desirable properties X, Y, and Z if and only if it also has the undesirable properties V and W. Such results include Theorems 3. We use logical notation Table 1. Our approach is pragmatic and informal: we use logical symbols if the result is easier to understand than the longer or less clear formulation without symbols.

    William Henry Day (Author of Axiomatic Concensus Theory in Group Choice and Biomathematics)

    The axioms are defined in tables that concern particular settings Table 1. An index Table A. Regrettably there has been little effort to develop, and less success in achieving, a standard nomenclature for consensus axioms; when browsing the consensus literature, the reader must be prepared to find axiomatic concepts disguised by a variety of sometimes perplexing aliases. For example, our axioms of decisive neutrality, neutrality, and 5-neutrality are also called [] neutrality, profile stability, and permutation compatibility, respectively.

    Achieving Consensus 8 Table 1. Axioms: Settings Table 2. Within each chapter, one series of integers names that chapter's tables, e. The tables in Appendix A are named A. Table 1. When works by several lead authors are cited, their names usually are ordered chronologically by year of first contribution. The modern mathematical treatment of group choice, with its focus on formal evaluations of alternative consensus rules, began in the Enlightenment with contributions by Borda [], Condorcet [, , ,], and their contemporaries.

    McLean's [] survey of such work in describes both axiomatic and probabilistic approaches to the design and analysis of voting procedures. McLean and London [] identify aspects of Borda's and Condorcet's contributions that were anticipated in medieval works by Ramon Lull c and Nicolas Cusanus More generally, McLean and Urken [] find contemporary issues of group choice in writings of Pliny the Younger 62? Lewis Carroll , 9 1. Notes Table 1.

    Black's [81, 85] history of the mathematical theory of committees and elections describes contributions of Borda, Condorcet, Laplace , Galton , Dodgson, and Nanson. McLean [] analyzes Nanson's work in social choice and electoral reform. Suzumura [] introduces the major lines of research in social choice theory and welfare economics during the twentieth century, research in part deeply influenced by Duncan Black [81,85, ,,]and Kenneth Arrow[ll, 13,18,19,20,21].

    The contemporary era of group choice began in with Black's [73, 74, 75, 76, 77, 78, 79, 86] creation of a multidimensional spatial theory of voting and with Arrow's [10] formulation and analysis of the celebrated impossibility theorem. Black's monograph [81], which consolidated his early research and historical investigations, was reprinted in [85] along with later papers, e. Arrow's doctoral dissertation on the impossibility theorem appeared in as a monograph [11]; the second edition [13], now usually cited, appends a commentary entitled "Notes on the Theory of Social Choice, Accessible to nonspecialists are Barbut's [33] elementary introduction to social choice theory, Riker's [] essay on the momentous contributions in the s to social choice theory, Arrow's [14] views on formal theories of social choice, and Plott's [] leisurely survey of axiomatic social choice theory.

    Sen [], Pattanaik [], and Campbell and Kelly [] give more advanced reviews of social choice research in the Arrovian framework. Saari [,,, ], Tanguiane [], and Stensholt [] view Arrow's theorem and group choice from geometric perspectives. Moulin [] and Barbera [32] stress the strategic theory of social choice, which concerns the conditions under which a sincere ballot is a voter's best strategy.

    Arrow, Sen, and Suzumura [20,21] give a comprehensive introduction to social choice and welfare with, in particular, reviews on Arrovian impossibility theorems [5, 32, , ], voting procedures [], and the structure of social choice rules [, ]. In France the contemporary era of group choice began in with the publication of a paper by Guilbaud [], which Arrow [13, p. Achieving Consensus of the theory of collective choice and the general problem of aggregation" and which helped to resurrect Condorcet's essay [] "from the deep oblivion where it had fallen" [].

    Monjardet [,] appraises the influence of Guilbaud's ideas on research in social choice theory, particularly [] at Guilbaud's center in Paris, now called the Centred'Analyse etde Mathematique Sociale at the Ecole des Hautes Etudes en Sciences Sociales. Representative of this tradition until the early s and unavailable in English translation are papers by Barbut [35,36], Guilbaud andRosenstiehl [,], Feldman [], Monjardet [,], and Barth61emy [39,40,41].

    Elementary logic, relations, and graphs are reviewed in most standard texts on discrete mathematics, e. More advanced treatments are by Suppes [] for logic, Suppes [] and Kaplansky [] for set theory, Harary [] and Berge [65] for graph theory, and Birkhoff [72], Crawley and Dilworth [], and Gratzer [] for lattice theory. Topics in bioinformatics are treated thoroughly by Stephen [], Waterman [],andGusfield[]. The theory of preference that Arrow uses Its first fruits were a series of articles in the journals, some of them dealing with fundamental aspects of the theory of committees.

    By axiomatizing the theory Arrow's work had blown a sudden energy into the subject. Black [84, p.

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    Black [73, p. The true grounds for disagreement are the conditions which it is reasonable to impose on the aggregation procedure, and even here it is possible to show that the limits of disagreement are not as wide as might be supposed from some of the more intemperate statements made. Arrow [13, p. In section 2. Arrow's impossibility theorem for weak orders, a result of outstanding significance in the theory of group choice.

    In later chapters we will apply the underlying paradigms to biological and data analysis problems having little to do with the theory of group choice. We introduce weak orders using the notational conventions of Chapter 1. Axiomatics in Group Choice Table 2. Incidence Matrix of Rz in Example 2. Ordered partitions are equivalent to weak orders as follows.

    From Z we can derive a binary relation Rz on S by the following rule: for all x, y e 5, xy is in Rz if x and y are in the same class of Z or if they are in distinct classes with the class of x preferred to the class of y. RZ can be depicted by its incidence matrix Table 2. Rz is complete Table 1. Transitivity Table 1. Since the whole argument reverses, a one-to-one correspondence exists between the sets of ordered partitions of 5 and weak orders on S: a problem on ordered partitions can be treated equivalently as a problem on weak orders. Table 2.

    Every weak order can be decomposed into useful subsidiary relations. Definition 2. For each weak order R on S let P be its strict preference relation 2. Impossibilities 13 so P is irreflexive and transitive; and let I be its indifference relation so I is an equivalence relation whose classes are ordered by P. Pattanaik [, p. Let 5 be a set of n alternatives. C is a social welfare function by the following definition. SWFs, as well as consensus rules on equivalence relations or tree quasi-orders Table 1. Convention 2. Axioms: Rules on Weak Orders. For notation see Definitions 2.

    We will formulate some axioms of SWFs Table 2. To begin there is the dilemma that, depending on how a SWF C is defined, profiles O, where Is the value of Maj defined and single valued for every profile, i. An example with cyclic majorities shows the problem. Impossibilities 15 Example 2. Paradox of Voting. Since that relation is not transitive, it is not a weak order, so MajQ is undefined and Q is inadmissible for Maj.

    On finding that the arithmetic was correct and the intransitivity persisted, my stomach revolted in something akin to physical sickness. Not only was the problem to which I had addressed myself more complicated than I had supposed, it was of a different kind. The problem of inadmissible profiles can be addressed in various ways. Or one could exclude from consideration certain profiles of individual orders on a priori grounds; but the extent of such exclusions should be limited lest the problem become trivial.

    A way to ensure the robustness of a SWF involves restricting a relation on 5 to a subset of 5 as in the following definition. If our a priori knowledge of the individual orders is incomplete to the extent that, for each set X of three alternatives, the weak order on X of every individual is completely unknown in advance, then it would seem to be inappropriate for some particular profile of weak orders on X never to occur by restriction from an admissible profile.

    Such is the motivation for the tree-triples axiom FT in Table 2. Clearly CR implies FT. Pareto optimality PO in Table 2. The axiom is named for Vilfredo Pareto [], an Italian economist, mathematician, and sociologist whose work formed the foundation of modern welfare economics and whose ideas formed the basis of Italian fascism.

    But autonomy Atn in Table 2. Clearly PO implies Atn. Let the members of a benevolent society conduct an election. Each member ranks a set S of alternatives. From the members' ballots the overall consensus ranking of S is calculated. When determining the consensus ranking of any subset X of S, 1 16 Chapter 2. Axiomatics in Group Choice Specifically, let two profiles of individual orders on S be such that when restricted to X c S every individual's weak orders are identical.

    Independence of irrelevant alternatives Ind in Table 2. Three axioms of Table 2. Decisive neutrality DN requires that if sets xy and zw of alternatives are used in the same way in profiles Q and Q', then the sets must be used in the same way in the social orders R and R'. Positive responsiveness PR requires that if the social order does not strictly prefer y to x and if the individual preferences remain the same except that one individual changes in a way favorable to x, then the new social order should strictly prefer x to y.

    Symmetry Sym requires that a SWF ensure the anonymity or equality of individuals: the social order should be determined only by the individual orders and not by the way the individuals or subscripts are associated with those orders. A 1-constant Csti SWF is uninformative since for every profile the social order is indifferent between every two alternatives. With anti-Pareto optimality APO , if every individual prefers x to y, then the social order perversely prefers y to x.

    For a dictatorial Dei SWF, the social order is based on the preferences of one individual: if the dictator prefers x to y, then so must society. For an inverse dictatorship ID , if the dictator prefers x to y, then the social order perversely prefers y to x. These axioms are related in basic ways. Lemma 2. Then Proof. Id easily follow from the definitions. Concerning 2. If Indb holds, then independence holds for all two-element subsets of any given X c S whence, by the restriction of weak orders, independence holds for X.

    Arrow [19, p. I already knew that majority voting, a plausible way of aggregating preferences, was unsatisfactory; a little experimentation suggested that no other method would work in the sense of defining an ordering. The development of the theorems and their proofs then required only about three weeks, although writing them as a monograph Social Choice and Individual Values took many months. Arrow [15, p. Theorem 2. Our proof of Arrow's theorem uses sets of individuals who are decisive in the sense that by acting together they could influence an election's result.

    Uc is the set of all decisive sets. Vc is the set of all inversely decisive sets. Example 2. Decisive sets for weak orders exhibit invariance properties, which are based on a technical property of binary relations. Imagine D on a grid S2 of points in the plane. Axiomatics in Group Choice Lemma 2. This requirement, which Sen [] calls invariant decisiveness, prevents the use of any information regarding particular features of alternatives when discriminating among them.

    K, then Direct: Inverse: Proof. Assume U'ab for ab e S2. If x — b, UIax is true by hypothesis. By Lemma 2. The inverse result follows similarly. D Lemma 2. This requirement, which Sen [] calls equivalent subsets, prevents the use of any information regarding the presence or absence of individuals who themselves do not form a decisive subset.

    If any nonsingleton set J c A' is decisive, then Lemma 2. Impossibilities 19 Table 2. Sen's Strategy to Prove Impossibility [, pp. Let C be a consensus rule on X for which axioms of independence, Pareto optimality, and dictatorship are specified. For pairs a, B e Sm of m-tuples, establish a property of invariant decisiveness. For pairs I, J c. K, establish a property of equivalent subsets or equivalent. Use these properties to prove by recursive partitioning that independence and Pareto optimality imply dictatorship. Proof of 'Arrow's Theorem 2. Since K is decisive by PO and finite, recursive partitioning by equivalent subsets using Lemma 2.

    The theorems that we obtain here are, of course, somewhat weaker than Arrow's Theorem, but the fact remains that Arrow's other conditions suffice to exclude all of the democratic social choice processes of interest. Wilson [, p. Unless such a SWF is 1-constant, it has at least one inversely decisive set. If C is not 1 -constant, then xPy for some x, y e Sand 20 Chapter 2. Axiomatics in Group Choice With Lemma 2. Proof of Wilson's Theorem 2. If K e Uc, then since K is finite, recursive partitioning by equivalent subsets using Lemma 2.

    Similarly, ID holds if K e Vc. Arrow's theorem follows easily from Wilson's theorem. Proof of Arrow's Theorem 2. Add to basket. Multigrid Methods Stephen F. Achieving Consensus; 2. Axiomatics in Group Choice; 3. Impossibilities in Bioconsensus; 4. Possibilities in Bioconsensus; 5. General Models of Consensus; 6. Rating details. Book ratings by Goodreads. Goodreads is the world's largest site for readers with over 50 million reviews.

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    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics
    Axiomatic Concensus Theory in Group Choice and Biomathematics Axiomatic Concensus Theory in Group Choice and Biomathematics

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