Echelons in Incomplete Relations

An efficient method of value assessment of a set of exchangeable alternatives A = {a ₁,a ₂, ,a _n} is presented. It particularly applies to situations where certain preferences may be easily evaluated or are already known, while other binary comparisons may not at once be available. Further applications are to ranking partial tournaments and the emergence and the characterisation of organisational hierarchy. By sequentially performing transitively efficient assessments of uncompared pairs, an initial weakly acyclical preference structure in A is transformed into an ordering of A in echelons. We call these nicely surveyable preference structures echelon orders. Theoretical properties of echelon orders are investigated, including a characterisation and a numerical representation.     

Inferring a linear ordering over a power set

An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are `accompanied' by `noise' components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let be a linear ordering on 2^X. The following condition is imposed on . For every quadruple (A,B,C,D)Y, where Y is some domain in (2^X)⁴, if and , then . The implications and interpretation of this condition for various domains Y are discussed.     

Rawls versus Bentham: An axiomatic examination of the pure distribution problem

The decision rules yielded respectively by the Rawlsian maximin conception of justice and by classical utilitarianism are compared and contrasted. The discussion is based on the assumption of a pure distribution problem and sharp differences are brought out. An axiomatic analysis of the two conceptions is undertaken, the result of which is that Rawls and utilitarianism both omit essential aspects of distributional welfare judgments: Rawls leaves out questions of welfare differences, utilitarianism leaves out questions of welfare levels. It is possible to pay attention to the ranking of welfare levels without concentrating exclusively on the welfare levels of worst off persons only, thereby departing from both Bentham and Rawls.I have benefited from the comments of Partha Dasgupta.     

On having the opportunity

There are least two general notions of having the opportunity to perform an action, one that concerns what is permitted on the basis of rules, regulations, or customs and another that concerns what is possible according to the physical or mental circumstances. The notion of having the opportunity within a set of circumstances is investigated and analysis in terms of physical possibility is proposed. A distinction is then made between a strong, person-dependent sense of opportunity and a weak, person-independent sense. It is also observed that care must be taken to distinguish the proposed analyses from criteria for determining whether someone has an opportunity.     

Reasonable beliefs

The theory of inductive inference developed by I. Levi in Gambling with Truth has two important limitations. In the first place, it makes acceptability of hypotheses question-dependent: any statement of the form It is reasonable to believe a hypothesis h on evidence e has to be expanded so as to include a reference to some specific question which h is supposed to answer, or, alternatively, to a set of possible answers which is associated with that question. Secondly, Levi's account is insensitive to modal distinctions: no difference is made between what is merely reasonable to believe and what is unreasonable not to believe. The aim of this paper is to construct a theory which utilizes Levi's main proposals and, at the same time, avoids the above-mentioned limitations.     

An analysis of simple counting methods for ordering incomplete ordinal data

Measurement in the social sciences often involves an attempt to completely order a set of entities on the basis of an underlying attribute. However, limitations of the measurement process often prevent complete empirical determination of the desired ordering. Nevertheless, the ordinal data obtained from the measurement process can be used in attempting to recover or construct more of the underlying order than is provided by the data. Previous research (Fishburn and Gehrlein, 1974a) has shown that a simple one-stage construction method, referred to as the cardinal rule, is fairly effective in correctly identifying ordered pairs in the underlying linear order that are not identified by the measurement process. The present paper re-examines the cardinal rule from the perspective of construction methods based on simple counting measures derived from the data, and argues that it is the best one-stage method in this class when a natural monotonicity assumption holds for the measurement process. The paper then examines two-stage construction rules that are based on the cardinal rule and the simple counting measures. It is shown that one of the two-stage rules gives better overall results than does the cardinal rule by itself.     

Rules of behavior and expected utility theory. Compatibility versus dependence

Conclusions We have seen that many decision rules which are intuitively and/or empirically supported and compatible with MEU, are compatible with it but not dependent on it.There are of course rules of behavior which are implied in MEU and also depend on it like this:If the hope of winning any of the prizes in a lottery motivates you to buy a ticket, and if you win half the amount of the highest prize, you should play double or nothing with your prize.Suppose you would prefer a one in a million chance of winning $2 million to a two in a million chance of winning $1 million, but your first choice is not available so you buy a ticket for $1 million. If you win, you should play 50–50 double or nothing with your prize. Generalize: put x in the place of $1 million and p in the place of 1/1 000000, and test yourself against this principle (Pf = preferred to): (p, 2x)Pf(2p, x) (0.5, 2x)Pf(1, x).See Friedmann and Savage (1968).In the real world lotteries are multiprize, i.e., composite games of elements like these. The same applies: If you would not have preferred the highest prize exchanged for a higher probability of some lower prize, then winning a lower prize would put you in the market for some simple bet like above. If you stand up to this test, you are a unique person because, as we know, such bets are not made.While in the process of finishing the final draft, I got hold of (Samuelson, 1983). He expresses grave doubts as to what he calls the dogma of Expected Utility maximizing. In a somewhat apologetic way, he preserves some formulations deriving behavior from EUM because, as he states in a general way, they do not depend on that particular dogma. More specifically: many models incompatible with EUM imply risk aversion, which would result also from maximizing the expectation of a concave utility function,In view of the authoritarian disposition of some of the strongest defenders of EUM and of Samuelson's (well deserved) authority and his leading role in the school of EUM theory, his open expression of doubt may well mark the beginning of the last chapter in the history of the rise and fall of the most powerful school that has so far been active in 20th century decision theory.     

Comparative accuracy of value solutions in non-sidepayment games with empty core

This article reports a test of the predictive accuracy of solution concepts in cooperative non-sidepayment n-person games with empty core. Six solutions were tested. Three of these were value solutions (i.e., -transfer value, -transfer nucleolus, and -transfer disruption value) and three were equilibrium solutions (deterrence set, stable set, and imputation set). The test was based on a laboratory experiment utilizing 5-person, 2-choice normal form games with empty core; other related data sets were also analyzed. Goodness-of-fit results based on discrepancy scores show that the three value solutions are about equally accurate in predicting outcomes, and that all three are substantially more accurate than the other solutions tested.     

Prisoner's dilemma from a moral point of view

In a recent issue of this journal, C. L. Sheng claims to havesolved andexplained the Prisoner's Dilemma (PD) by studying it from a moral point of view - i.e., by assuming that each player feels sympathy for the other. Sheng does not fully clarify this claim, but there is textual evidence that his point is this: PD's arise only for agents who feel little or no sympathy for each other; they cannot arise in the presence of a high degree of reciprocal sympathy. A high degree of such sympathysolves the PD in that it prevents PD's from arising, and a low degree of itexplains the PD in that it provides an essential condition for the occurrence of that game. This thesis is false, as some examples show. These examples are important; they prevent us from underestimating the problem posed by the PD.     

Inductive-nomological explanations and psychological laws

This paper explores some major conceptions of non-deductive explanations, with special reference to the ability of these conceptions to account for some important types of psychological explanations. The central focus is on a kind of tendency generalization and the nature of the explanations that are made possible by generalizations of this sort. It is argued that although these generalizations cannot plausibly be cast in the Hempelian form P(G, F) = r, they are lawlike, and the explanations they ground can have a degree of explanatory power commensurate with exhibiting the phenomenon to be explained as nomically expectable.     

Majority voting on orders

We characterize two lexicographic-type preference extension rules from a set X to the set of all orders on this set. Elements of X are interpreted as basic economic policy decisions, whereas elements of are conceived as political programs among which a collectivity has to choose through majority voting. The main axiom is called tournament-consistency, and states that whenever majority pairwise comparisons based on initial preferences on X define an order on X, then this order is also chosen by a majority among all other orders in . Tournament-consistency thus allows to predict the outcome of majority voting upon orders from the knowledge of majority preferences on their components.     

Preference extension rules for ranking sets of alternatives with a fixed cardinality

This paper analyzes the problem of deriving a ranking of fixed-cardinality subsets of a universal set from a given ranking of the elements of this universal set. Only subsets with a given number of elements are being ranked, which is where the approach in this paper differs from the literature on extension rules that establish preference relations on the power set of the universal set. Common examples for areas where such preferences on subsets with a fixed cardinality are needed are elections of committees of a given size, many-to-one matchings, and decision problems under ignorance. The main result of the paper is a characterization of a class oflexicographic rank-ordered rules by means of two axioms, namely, aresponsiveness condition used in the matching literature and a well-knownneutrality requirement which ensures that the names of the alternatives are irrelevant for the ranking of the sets.     

The general relevance of the impossibility teorem in smooth social choice

A complete classification theorem for voting processes on a smooth choice spaceW of dimensionw is presented. Any voting process is classified by two integersv ^* () andw(), in terms of the existence or otherwise of the optima set, IO(), and the cycle set IC().In dimension belowv ^* () the cycle set is always empty, and in dimension abovew() the optima set is nearly always empty while the cycle set is open dense and path con nected. In the latter case agenda manipulation results in any outcome.For admissible (compact, convex) choice spaces, the two sets are related by the general equilibrium result that IO() union IC() is non-empty. This in turn implies existence of optima in low dimensions. The equilibrium theorem is used to examine voting games with an infinite electorate, and the nature ofstructure induced equilibria, induced by jurisdictional restrictions.This material is based on work supported by a Nuffield Foundation grant.     

The Optimality of the Expert and Majority Rules Under Exponentially Distributed Competence

We study the uncertain dichotomous choice model. In this model a set of decision makers is required to select one of two alternatives, say support or reject a certain proposal. Applications of this model are relevant to many areas, such as political science, economics, business and management. The purpose of this paper is to estimate and compare the probabilities that different decision rules may be optimal. We consider the expert rule, the majority rule and a few in-between rules. The information on the decisional skills is incomplete, and these skills arise from an exponential distribution. It turns out that the probability that the expert rule is optimal far exceeds the probability that the majority rule is optimal, especially as the number of the decision makers becomes large.     

On inference from inconsistent premisses

The main object of this paper is to provide the logical machinery needed for a viable basis for talking of the consequences, the content, or of equivalences between inconsistent sets of premisses.With reference to its maximal consistent subsets (m.c.s.), two kinds of consequences of a propositional set S are defined. A proposition P is a weak consequence (W-consequence) of S if it is a logical consequence of at least one m.c.s. of S, and P is an inevitable consequence (I-consequence) of S if it is a logical consequence of all the m.c.s. of S. The set of W-consequences of a set S it determines (up to logical equivalence) its m.c.s. (This enables us to define a normal form for every set such that any two sets having the same W-consequences have the same normal form.) The W-consequences and I-consequences will not do to define the content of a set S. The first is too broad, may include propositions mutually inconsistent, the second is too narrow. A via media between these concepts is accordingly defined: P is a P-consequence of S, where P is some preference criterion yielding some of the m.c.s. of S as preferred to others, and P is a consequence of all of the P-preferred m.c.s. of S. The bulk of the paper is devoted to discussion of various preference criteria, and also surveys the application of this machinery in diverse contexts - for example, in connection with the processing of mutually inconsistent reports.     

Sequential voting by veto: Making the Mueller-Moulin algorithm more versatile

This paper shows that a relatively easy algorithm for computing the (unique) outcome of a sophisticated voting procedure called sequential voting by veto (SVV) applies to a more general situation than considered hitherto. According to this procedure a sequence of n voters must select s out of m + s options (s > 0, m 3 n 3 2). The ith voter, when his turn comes, vetoes k _i options (k _i 1, k _i = m). The s remaining non-vetoed options are selected. Every voter is assumed to be fully informed of all other voters total (linear) preference orderings among the competing options, as well as of the order in which the veto votes are cast. This algorithm was proposed by Mueller (1978) for the special case where s and the k _i are all equal to 1, and extended by Moulin (1983) to the somewhat more general case where the k _i are arbitrary but s is still 1. Some theoretical and practical issues of voting by veto are discussed.     

Complementary properties of binary relations

After defining the complementary relation R of a binary relation R on a set X, this paper constructs the binary relation C (is a complementary property of) on the set P of nine well known elementary properties that R might possess. It deduces some theorems about C; especially that symmetry is the only one of these possible properties of R on X that is possessed by C on P. The set P may be enlarged to contain other elementary properties of R on X without changing the truth of these theorems, when the symbols of sets are properly modified. Finally, the paper discusses the desirability of a general theory of elementary properties of binary relations for the further development of statistical decision theory.     

Utilities for distributive justice

This paper falls within the field of Distributive Justice and (as the title indicates) addresses itself specifically to the meshing problem. Briefly stated, the meshing problem is the difficulty encountered when one tries to aggregate the two parameters of beneficence and equity in a way that results in determining which of two or more alternative utility distributions is most just. A solution to this problem, in the form of a formal welfare measure, is presented in the paper. This formula incorporates the notions of equity and beneficence (which are defined earlier by the author) and weighs them against each other to compute a numerical value which represents the degree of justice a given distribution possesses. This value can in turn be used comparatively to select which utility scheme, of those being considered, is best.Three fundamental adequacy requirements, which any acceptable welfare measuring method must satisfy, are presented and subsequently demonstrated to be formally deducible as theorems of the author's system. A practical application of the method is then considered as well as a comparison of it with Nicholas Rescher's method (found in his book, Distributive Justice). The conclusion reached is that Rescher's system is unacceptable, since it computes counter-intuitive results. Objections to the author's welfare measure are considered and answered. Finally, a suggestion for expanding the system to cover cases it was not originally designed to handle (i.e. situations where two alternative utility distributions vary with regard to the number of individuals they contain) is made. The conclusion reached at the close of the paper is that an acceptable solution to the meshing problem has been established.I would like to gratefully acknowledge the assistance of Michael Tooley whose positive suggestions and critical comments were invaluable in the writting of this paper.     

A foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a probability distribution.     

A systems concept of society: Beyond individualism and holism

Three rival views of the nature of society are sketched: individualism, holism, and systemism. The ontological and methodological components of these doctrines are formulated and analyzed. Individualism is found wanting for making no room for social relations or emergent properties; holism, for refusing to analyze both of them and for losing sight of the individual.A systems view is then sketched, and it is essentially this: A society is a system of interrelated individuals sharing an environment. This commonsensical idea is formalized as follows: A society is representable as an ordered triple Composition of , Environment of , Structure of , where the structure of is the collection of relations (in particular connections) among components of . Included in the structure of any are the relations of work and of managing which are regarded as typical of human society in contrast to animal societies.Other concepts formalized in the paper are those of subsystem (in particular social subsystem), resultant property, and emergent or gestalt property. The notion of subsystem is used to build the notion of an F-sector of a society, defined as the set of all social subsystems performing a certain function F (e.g. the set of all schools). In turn, an F-institution is defined as the family of all F-sectors. Being abstractions, institutions should not be attributed a life and a mind of their own. But, since an institution is analyzable in terms of concrete totalities (namely social subsystems), it does not comply with the individualist requirement either.It is also shown that the systems view is inherent in any mathematical model in social science, since any such schema is essentially a set of individuals endowed with a certain structure. And it is stressed that the systems view combines the desirable features of both individualism and holism.