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Observations of the structure of closed groups are presented which give rise to four observable tendencies within either a sociogram or the catij matrix as developed by Bernard and Killworth (1973) . These observations are used to form theories of social structure of varying degrees of complexity; all these theories, however, are somewhat unusual in that intransitivity of interpersonal sentiment, far from being an artifact, is proposed as an essential part of the structure of human groups, whereas recent papers have stressed the occurrence of transitivity within a group. A statistical theory including the earlier theories shows that at least three subgroups must exist within a group, even if only two are forced by external organization. One of these subgroups is formed of mutually unknown individuals who serve as connecting links between other subgroups formed by strong interpersonal sentiment.

The conditions are found under which the removal of an intransitivity in one triad of a sociometric or cognitive structure may be accomplished without in the process creating new intransitivities elsewhere in the structure. The results are applied to modify and elaborate the theories holding intransitivity to be unstable so that sociometric or cognitive structures become more nearly transitive over time, and also to shed some light on the question of the extent to which observed intransitivities may properly be attributed to measurement error. Properties of tetrads play a central role in this theory, which thus challenges the widespread notion due to Simmel and others, that once group size exceeds three there are no significant new properties to consider. Empirical data document the relevance of this theory: “correction” of “errors” in the data or change in truly intransitive triads turns out not to significantly reduce the overall amount of observed intransitivity.

The present study investigated the extent to which intransitivity, as measured by the total circular triad (TCT) score in the method of paired comparisons, varies across different domains of content and across different groups of subjects. Three paired comparisons instruments were administered to 276 high school students and to 358 college students. Results indicated statistically significant ( p ≤ .001) correlations among the three TCT indices for both groups of subjects.

MR. ARMSTRONG'S extremely able 'Comment'2 has done much to clarify in my own mind the essence of his original article and the issues on which my position differed, or should have differed, from his own. I am grateful for the clarity of his presentation, and thank him for providing me with the opportunity to reformulate my position in what I hope will be a more revealing way. My 'Rejoinder' will parallel Armstrong's 'Comment' and deal in the same order with the three subjects he discusses there: (1) a criticism of one of his demonstrations; (2) the question of whether the basic conclusions of his article follow from his axioms; and (3) the significance for his thesis of my psychological interdictions against the concept of marginal preference. I Armstrong defends his demonstration purporting to show that the assumption of intransitivity of indifference itself (i.e. without the concept of marginal preference) suffices for a more powerful analysis of groupwelfare scales than under the traditional assumption of transitivity. It might be worth while to reproduce the issue. Consider the two man group consisting of individuals ot and A. The problem is to rank social states A and B on a group-welfare scale. Individual ox prefers A to B, while individual P prefers B to A. By traditional analysis, no ranking is defined for such a situation. Armstrong demonstrates that incorporation of the assumption of intransitivity of indifference into the analysis succeeds in achieving a ranking. He introduces 'situations' C and D. Individual a prefers A to C and C to B; P is indifferent between A and D, and between D and B (while, consistently, preferring B to A). He then speaks about 'situation C for a combined with situation D for A'. oz prefers A to such a 'combination' and prefers the combination to B; P is indifferent between the 'combination' and A, and also between the 'combination' and B. A is therefore group-preferred to the 'combination', and the 'combination' is group-preferred to B. So, since preferenceeven if not indifference-is transitive, A is group-preferred to B.

In a note critical of Robert Paul Wolff's conclusions concerning the reasonableness of majority rule in the light of the voter's paradox,' David Sobers2 incorrectly argues that the famed paradox can only be derived from an inconsistent set of premises. By displaying the putative inconsistency, Sobers believes that he has dissolved the puzzle and thereby disarmed Wolff's criticism. In this counter note I aim to show that the voter's paradox is genuine when correctly understood and that Wolff's criticism of majority rule is a proper one. The voting paradox, apparently first discovered by Condorcet,3 is simply stated: given a society of voters each of whom individually has a consistent set of preferences concerning some set of at least three alternative choices open to society, if the method of pairwise simple majority decision is used to generate a preference ranking of the alternatives for society, the resulting ranking may be intransitive. That is, majority rule cannot guarantee consistent social choices. For example, to take the Wolff case (p. 61), if we have three members of a society, I, II, and III, faced with a choice from among three alternatives, A, B, and C, such that the individual preferences are as shown in table 1, then simple majority rule will yield an intransitivity. In the issue of A against B

INVARIANT SUBSPACES OF ABSTRACT MULTIPLICATION OPERATORS by Hermann Flaschka We describe a class of operators on a Banach space ft whose members behave, in a sense, like multiplication operators, and consequently leave invariant a proper closed subspace of IB. One of the sufficient conditions for an operator to be such an "abstract multiplication" bears a striking resemblence to an assumption made by j. Wermer, who approached the invariantsubspace problem from a very different point of view. INVARIANT SUBSPACES OF ABSTRACT MULTIPLICATION OPERATORS by Hermann Flaschka §1. We want to describe a class of operators on a Banach space IB whose members behave, in a sense, like multiplication operators, and consequently leave invariant a proper closed subspace of ft -that is, they are intransitive. One of the sufficient conditions for an operator to be such an "abstract multiplication" bears a striking resemblance to an assumption made by J. Wermer [1], who approached the invariant-subspace problem from a very different (and rather more sophisticated) point of view. Our comments are presented in the hope that this connection, as well as the more general pattern which appears to emerge, may be more than superficial. Wermer considered operators whose deviation from being an isometry is limited. More precisely, he assumed that T and T~ are both bounded, and that either (A) ||T | | = 0 ( e ( n ) , n = 0 , + l , 0 < a < 1, and the spectrum of T contains at least two points, or (B) ||T|| = 0(|n| ), for some fixed k < oo . Such a. T jus intransitive. If a = 0 in (A) , or k = 0 in (B) , IIT || < K for all n; the space may then be renormed to make T an isometry, and intransitivity follows from a theorem of Godement [2] . The crucial estimate is that in (A); it is a reformulation of the requirement that

In this paper it is shown that neither the Scitovsky nor the double index number criterion for an increase in welfare is transitive. The consistent use of either could therefore lead to inconsistent judgements. The relationship of these results to Arrow's general theorem concerning the impossibility of constructing a social welfare function satisfying certain restrictions is discussed. The author feels sure that many of these results must be familiar to individual economists, but they do not appear to be generally known.

Hempel's qualitative criteria of converse consequence and special consequence for confirmation are examined, and the resulting paradoxes traced to the general intransitivity of confirmation. Adopting a probabilistic measure of confirmation, a limiting form of transitivity of confirmation from evidence to predictions is derived, and it is shown to what extent its application depends on prior probability judgments. In arguments involving this kind of transitivity therefore there is no necessary “convergence of opinion” in the sense claimed by some personalists. The conditions of application of the limiting transitivity theorem are most perspicuously described in terms of De Finetti's notion of exchangeability, which leads to a suggested revaluation of the function of theories in relation to confirmation and explanation.

In a transitive group of order g and of degree n the subgroup G, composed of all the substitutions of G which omit a given letter is of order gin. The properties of G, frequently throw light on the possible properties of G. In particular, when G1 is transitive and of degree n-1, G must be multiply transitive, but when G, is of a lower degree than n-1, there must be at least one substitution besides identity which is commutative with every substitution of G. If the order of such a substitution is less thani n its systems of intransitivity are systems of imprimitivity of G. Hence it results that, when G is primitive and G, is not of degree n-1, G must be the regular group of prime order p. While a number of such properties relating to G, have been known for a long time little has been done towards determining the possible substitution groups which involve a particular G,. For instance, when G, is the symmetric group of degree mn, G must be the symmetric group of degree m 4+1, except in the special case when m 2. In this case it may also be the octic group of degree 4, as is well known, but it cannot be any other group. This theorem results directly from the facts that there is no substitution on the letters of the symmetric group of degree mn> 2 which is commutative with every substitution of this symmetric group and that a regular group of degree m can always be used as the G, for at least one transitive group of degree 2in and of order 2 mn. In a similar manner it results that every alternating group, except the alternating group of degree 3, can appear as the G, of only the alternating group of the next larger degree, while the alternating group of degree 3 is also the G, of a group of order 18 and of degree 6. As a special case of the theorems just stated there results the well known theorem, first proved by Ruffini, that there is no four-valued rational function on five variables. In fact, if such a function were possible there would be a tranisitive substitution group of order 30 and of degree 5. The G, of this group would have to be the symmetric group of degree 3 since there is no other group of order 6 on less than 5 letters. The fact

A RECENT discussion of welfare economics 'summarizes the question of transitivity as follows. There is not now conclusive reason to believe that the assumption of transitivity of preference in individual choice is deeply suspect. There are, however, grounds for being willing to make deep empirical investigations into the validity of the assumption'.' At another point the same author says: 'My conclusions indicate that we currently possess no strong evidence to warrant dropping (the assumption of transitivity). '2 Given the role that transitivity plays in much of the modern literature, and in Rothenberg's book, his attitude towards it seems very defensive. We do not have strong evidence against it; there is no conclusive reason to suspect deeply the assumption. In sum, the evidence against transitivity is not sufficient to condemn it, but Rothenberg is clearly doubtful. In this he is more or less in accord with the bulk of current opinion on the matter. It is the purpose of this article to present a contrary view. It will be argued that the assumption of transitivity is not particularly doubtful or dubious. In view of the apparent suspicion of the assumption by the bulk of modern welfare economists this might seem to be a revolutionary position, but the practice of the same economists has indicated relatively little real doubt of the transitive nature of preference orderings. Article after article has appeared in which this assumption is made, something which would seem to indicate that the writers have at least some confidence in its reliability. Further, the chains of reasoning based upon this assumption do not appear to have led to demonstrably false conclusions about the real world. This fact, in itself, is powerful although not conclusive evidence for the truth of the transitivity assumption. The debate about the assumption of transitivity turns upon the interpretation of certain real world experiences. If an individual is asked to take his choice among a large number of pairs of alternatives, with each alternative appearing as one member of quite a number of the pairs,3 or a large number of people are asked to judge between three alternatives

Following Smedslund’s (1960) findings concerning the intransitivity of preschoolers’ inferences about the preferences of others, three age levels of public school children were tested on an inference-preference task, using colors as stimuli. The median ages of groups tested were 6;3, 9;2, 11;2. Appreciable transitivity of response appeared only in the eldest group. Verbal explanations for transitive responses were required for an additional group with a median age of 7;0. The data suggest that the nontransitive hypotheses described by Smedslund (1963) do contribute to the inferences of children, but indicate that egocentrism is not, as Smedslund (1960) contended, an adequate explanation for the intransitivity of children’s inferences about preferences.

When society uses the process of majority division to choose between three or more possibilities it has long been noted that an intransitive ordering of preferences may result. For this reason majority rule has been considered a theoretically (as opposed to practically) unacceptable method of making society's choices. When a collective choice mechanism is required to bring forth a transitive ordering of alternatives, there can be little doubt that an important and serious restriction has been imposed. Yet, the justification for this obviously crucial condition is often found couched in vague references to the irrationality of intransitive preference orderings. Because such an ordering is intuitively irrational there has been little, if any, investigation into the rational basis for intransitivity. Consequently, on the logical rather than intuitive level, the question of the rationality of intransitive preference orderings is far from straightforward, much less resolved. By contrasting cardinal analysis with majority division I hope to determine the cause of majoritarian intransitivity and lay the foundation for the acceptance of such intransitivity as logical and rational. I shall argue that

Three methodological issues related to the measurement of intransitivity in social networks are analyzed. Indices of measurement are defined and criteria established for constructing a sociometric questionnaire to be used for obtaining data to test hypotheses about transitivity of sentiment. Empirical analysis is presented to support the proposed methodology, which is then used to examine a sample of 51 sociograms obtained from junior high school classes. The results reveal a very low incidence of intransitivity in friendship choices in every group. The intransitive triads which do exist are examined at different choice levels to determine if the intensity of the relationships affect the structure of the triad. The findings indicate that sentiment choices are less likely to be intransitive when the relationships are intense than when the bonds are weak. Patterns are detected in the kinds of intransitive triads which appear most frequently and variation is observed in the number of intransitive triads in which individual group members are involved.

Individuals are often asked to make choices from all possible pairs of a set of stimuli. For each pair they may have to select the alternative which is heavier, more preferable, more favorable to Blacks, etc. Intransitive or circular choice behavior occurs when an individual selects stimulus A over B, B over C, and then C over A. Such a pattern of choices is referred to as a circular triad. Since most theories of choice assume transitivity as a basic principle of human choice behavior, explanation of the occurrence of intransitivities is a crucial problem in the behavioral sciences. Three types of explanation have been posited for the existence of circular choices. One suggests that the subject is not taking the necessary effort to make careful choices; that is, intransitivities occur due to the individual's sloppiness or carelessness. Since subjects typically seem to make choices in a conscientious manner, this outlook is not very plausible. Another explanation is that intransitivities arise from the close proximity of stimuli on an underlying linear continuum. Thus, even though the subject is conscientious, the stimuli may in some cases be too similar to distinguish. This explanation implies that there is really no difference between intransitivity and inconsistency - the reversal of responses by a subject over two administrations of the same pairs of choices. The third type of explanation concerns the development of a choice model which might account for intransitive behavior. Some attempts at such a model focus on the multidimensional nature of the stimuli, circularities being caused by the way in which a subject combines ratings on the separate dimensions to yield an overall choice between the pair of alternatives. Specifically, the additive difference model developed by Tversky (1969), drawing on earlier work by May (1954) and Morrison (1962) suggests that stable circular triads (i.e., intransitivities occurring in exactly the same way) would be expected from two administrations of the same choice pairs. Accordingly, five questionnaires were administered to a group of subjects on two separate occasions. In each case the subjects for the second administration were the same individuals who had completed the questionnaire the first time, so that data could be obtained on the responses of the same subject to the same pairs of choices on two distinct administrations. The principal concerns were the following: To find evidence for the existence of stable (i.e., over testings) circular triads. To investigate the appropriateness of the additive difference model, if such stable circularities are found. If such evidence is not found, to investigate the intrasitivity-as-inconsistency explanation by exploring the relationship between circular triads and changes in response to individual items over the two administrations. With the exception of one subject on one questionnaire, no such evidence for stable intransitivities was found. Consequently, the inconsistency explanation was investigated by a factor analysis of nine variables derived from the subjects' choices. These variables were measures of consistency, transitivity and discrimination among stimuli. Results of this analysis suggest that the assumption of circular choices as synonymous with inconsistency is upheld for three of the five questionnaires.

I. The state of affairs, 507. — II. The deficiency of the Complete Ordering Axiom, 508. — III. Redefinition of preference and the choice mechanism, 510. — IV. Intransitivity, 518. — V. Expected behavior under price and income changes, 525. — VI. Conclusion, 536.

as an attempt to maximize something.1 The first rationality assumption will be treated as a truism, and this study will examine the second assumption: "Transitivity of Preference Orderings." Since this psychological assumption, which is basic to economic theory, is rejected by most psychologists, there is reason for the economist to show concern. Since most empirical tests of choice behavior appear to reflect intransitive preference patterns,2 there is reason for the economist to investigate the nature of rationality. Transitivity might be considered synonymous with internal consistency. If a set of preferences is transitive, the alternatives are ordered in a progression, from most preferred to least preferred. For purposes of exposition herein, strong algebraic transitivity will be discussed before weak. A strongly ordered group is a group in which there is no ambiguity of position, each element of the group having a unique position in the ordering. Preference will be

This article is a technical note on a problem frequently encountered in assessing consumer preferences. Intransitivity is seen in situations where Product a is preferred to Product b, b to c, but c to a. Inadequate preference transitivity casts suspicion on rank orders of products derived from series of paired comparisons. This problem can be given graph theoretic treatment which, although it offers no absolute solution, does make the problem clearer and emphasizes the area of the analyst's judgment.