Testing transitivity in graphs

The situation is considered whether a graph can be assumed to have been generated by a random model capturing more transitivity than a simple uniform model. Three different test quantities based on induced triad counts and local densities are used. A simulation study is made in order to estimate critical values of the tests for different significance levels. The powers of the tests are estimated against the Bernoulli triangle model, a simple random graph model in which the clustering and transitivity is higher than in the uniform model. The test based on the proportion of transitive triads has the highest power in most cases, but the test based on density difference (the difference between mean local density and overall graph density) is more powerful against models with high transitivity. The tests are applied to a large set of school class sociograms. In this situation, uniform randomness is rejected in favor of transitivity most frequently when the test based on the proportion of transitive triads out of the non-vacuously transitive triads is used. It is concluded that this test, which also performed reasonably well when applied to random data, is the best at detecting transitivity. Although the Bernoulli triangle model fits to the empirical data set better than the uniform model, there are fewer truly intransitive triads in the data than could be expected under either of the models.     

A latent space model for cognitive social structures data

This paper introduces a novel approach for modeling a set of directed, binary networks in the context of cognitive social structures (CSSs) data. We adopt a relativist approach in which no assumption is made about the existence of an underlying true network. More specifically, we rely on a generalized linear model that incorporates a bilinear structure to model transitivity effects within networks, and a hierarchical specification on the bilinear effects to borrow information across networks. This is a spatial model, in which the perception of each individual about the strength of the relationships can be explained by the perceived position of the actors (themselves and others) on a latent social space. A key goal of the model is to provide a mechanism to formally assess the agreement between each actors’ perception of their own social roles with that of the rest of the group. Our experiments with both real and simulated data show that the capabilities of our model are comparable with or, even superior to, other models for CSS data reported in the literature.     

Acyclic choice and group veto

The Arrow Impossibility Theorem assumes transitivity. Acyclicity is a more appropriate rationality condition for collective decision procedures. Many impossibility theorems for acyclic choice need to assume that there are more alternatives than individuals. In this paper we show that by considering circumstances under which groups (or coalitions) have veto it is possible to prove impossiblity theorems without making such an assumption. Our results extend recent theorems by Blau and Deb, and Blair and Pollak.I would like to thank my research supervisor Amartya Sen, the referee and the editor of this journal for their comments on earlier drafts of this paper.     

Arrow��s theorem and max-star transitivity

In the literature on social choice with fuzzy preferences, a central question is how to represent the transitivity of a fuzzy binary relation. Arguably the most general way of doing this is to assume a form of transitivity called max-star transitivity. The star operator in this formulation is commonly taken to be a triangular norm. The familiar max- min transitivity condition is a member of this family, but there are infinitely many others. Restricting attention to fuzzy aggregation rules that satisfy counterparts of unanimity and independence of irrelevant alternatives, we characterise the set of triangular norms that permit preference aggregation to be non-dictatorial. This set contains all and only those norms that contain a zero divisor.     

An agent-based random-utility-maximization model to generate social networks with transitivity in geographic space

Stochastic actor-based approaches receive increasing interest in the generation of social networks for simulation in time and space. Existing models however cannot be readily integrated in agent-based models that assume random-utility-maximizing behavior of agents. We propose an agent-based model to generate social networks explicitly in geographic space which is formulated in the random-utility-maximizing (RUM) framework. The proposed model consists of a friendship formation mechanism and a component to simulate social encounters in a population. We show how transitivity can be incorporated in both components and how the model can be estimated based on data of personal networks using likelihood estimation. In an application to the Swiss context, we demonstrate the estimation and ability of the model to reproduce relevant characteristics of networks, such as geographic proximity, attribute similarity (homophily), size of personal networks (degree distribution) and clustering (transitivity). We conclude that the proposed social-network model fits seamlessly in existing large-scale micro-simulation systems which assume RUM behavior of agents.     

ASYMMETRIC VALUATION OF GAINS AND LOSSES AND PREFERENCE ORDER ASSUMPTIONS

Recent findings of large differences between people's valuations of gains and losses imply that the common preference order axioms of completeness, transitivity, and dominance, on which economic analyses and predictions of consumer behavior are largely based, may not be consistent with actual choices. This paper reports the results of an experimental test of the preference order assumptions using real exchanges of two goods and money. The findings show consistent differences in the valuation placed on goods and money depending on whether the entitlements are being acquired or given up, and provide direct evidence of preference assumption violations.     

The structure of fuzzy preferences: Social choice implications

It has been shown that, with an alternative factorization of fuzzy weak preferences into symmetric and antisymmetric components, one can prove a fuzzy analogue of Arrow's Impossibility Theorem even when the transitivity requirements on individual and social preferences are very weak. It is demonstrated here that the use of this specification of strict preference, however, requires preferences to also be strongly connected. In the absence of strong connectedness, another factorization of fuzzy weak preferences is indicated, for which nondictatorial fuzzy aggregation rules satisfying the weak transitivity requirement can still be found. On the other hand, if strong connectedness is assumed, the fuzzy version of Arrow's Theorem still holds for a variety of weak preference factorizations, even if the transitivity condition is weakened to its absolute minimum. Since Arrow's Impossibility Theorem appeared nearly half a century ago, researchers have been attempting to avoid Arrow's negative result by relaxing various of his original assumptions. One approach has been to allow preferences – those of individuals and society or just those of society alone – to be “fuzzy.” In particular, Dutta [4] has shown that, to a limited extent, one can avoid the impossibility result (or, more precisely, the dictatorship result) by using fuzzy preferences, employing a particularly weak version of transitivity among the many plausible (but still distinct) definitions of transitivity that are available for fuzzy preferences. Another aspect of exact preferences for which the extension to the more general realm of fuzzy preferences is ambiguous is the factorization of a weak preference relation into a symmetric component (indifference) and an antisymmetric component (strict preference). There are several ways to do this for fuzzy weak preferences, all of them equivalent to the traditional factorization in the special case when preferences are exact, but quite different from each other when preferences are fuzzy (see, for example, [3]). A recent paper in this journal [1], by A. Banerjee, argues that the choice of definitions for indifference and strict preference, given a fuzzy weak preference, can also have “Arrovian” implications. In particular, [1] claims that Dutta's version of strict preference presents certain intuitive difficulties and recommends a different version, with its own axiomatic derivation, for which the dictatorship results reappear even with Dutta's weak version of transitivity. However, the conditions used to derive [1]'s version of strict preference imply a restriction on how fuzzy the original weak preference can be, namely, that the fuzzy weak preference relation must be strongly connected. Without this restriction, I will show that the rest of [1]'s conditions imply yet a third version of strict preference, for which Dutta's possibility result under weak transitivity still holds. On the other hand, if one accepts the strong connectedness required in order for it to be valid, I show that [1]'s dictatorship theorem can in fact be strengthened to cover any version of transitivity for fuzzy preferences, no matter how weak, and further, that this dictatorship result holds for any “regular” formulation of strict preference, including the one originally used by Dutta. Received: 13 May 1996 / Accepted: 13 January 1997     

Connecting and resolving Sen's and Arrow's theorems

It is shown that the source of Sen's and Arrow's impossibility theorems is that Sen's Liberal condition and Arrow's IIA counter the critical assumption that voters have transitive preferences. But if the procedures are not permitted to treat the transitivity of individual preferences as a valued input, then we cannot expect rational outputs. Once this common cause for these perplexing conclusions is understood, these classical conclusions end up admitting quite benign interpretations where it becomes possible to propose several resolutions. Received: 2 April 1996 / Accepted: 15 October 1996     

EMPLOYEE EVALUATION OF PENSION CLAIMS AND THE IMPACT OF INDEXING INITIATIVES

There is discussion in both Canada and the United States of proposals which require private pension plans to provide indexed pension benefits. This paper employs both an auction and an implicit contract model to identify the compensating wage differentials required of possible policy initiatives. The contract model, motivated by the prevalence (especially in Canada) of ad hoc cost-of-living adjustments to pensions in pay, presumes that workers have a call option on the investment earnings in excess of the interest rate assumption used to value the plan. The analysis indicates that wage offsets are potentially quite large, may be difficult to predict, and may vary substantially across firms. In addition, the case for policy action would appear to rest largely on the implicit assumption that workers misperceive the value of their pension benefits.     

If you do not know who knows what: Advice seeking under changing conditions of uncertainty after an acquisition

In this study we develop a model to explain the dynamics of advice seeking after an acquisition. We build on a theory of advice seeking that draws from prospect theory and expectancy theory. We theorize that immediately after an acquisition there is uncertainty about who knows what, but over time individuals become more aware of the expertise within the organization and they change their advice networks based upon this increased awareness. Our model examines four micro-processes of advice seeking: reciprocity, preferential attachment, transitivity, and legacy-firm tie preferences. To test our hypotheses we use post-acquisition data over four time periods in a recruitment consulting firm. Our longitudinal analysis uses a stochastic actor-orientated model and our results indicate that immediately after the acquisition individuals have a tendency to seek advice based upon reciprocity and preferential attachment. However, over time these tendencies diminish. Surprisingly, transitivity does not play a significant role, which suggests that other micro-processes such as reciprocity are dominant. In addition, individuals in the acquired firm have a tendency to make more ties and there is a preference for same firm ties in both legacy firms, with the tendency being higher in the acquired firm. Our findings add to theories on the process of advice seeking under conditions of uncertainty, on knowledge transfer processes in mergers and acquisitions, and the knowledge based view of the firm.     

A systematic approach to the construction of non-empty choice sets

Suppose a strict preference relation fails to possess maximal elements, so that a choice is not clearly defined. I propose to delete particular instances of strict preferences until the resulting relation satisfies one of a number of known regularity properties (transitivity, acyclicity, or negative transitivity), and to unify the choices generated by different orders of deletion. Removal of strict preferences until the subrelation is transitive yields a new solution with close connections to the “uncovered set” from the political science literature and the literature on tournaments. Weakening transitivity to acyclicity yields a new solution nested between the strong and weak top cycle sets. When the original preference relation admits no indifferences, this solution coincides with the familiar top cycle set. The set of alternatives generated by the restriction of negative transitivity is equivalent to the weak top cycle set.     

Reciprocity,transitivity, and the mysterious three-cycle

Reciprocity and transitivity are the two most important structural mechanisms underlying friendship network evolution. While on their own they are understood in great detail, the relation between them is rarely studied systematically. Are friendships outside of social groups more or less likely to be reciprocated than friendships embedded in a group? Using a theoretical framework that focusses on the situations in which friends interact and the social structures that stabilise one-sided friendships, I propose that the tendency towards reciprocation of friendships within transitive groups is usually lower than outside of transitive groups. In a meta-analysis of two datasets including 29 friendship networks using stochastic actor-oriented models (SAOMs), the interaction between reciprocity and transitivity is analysed. Supporting the theoretical reasoning, the interaction is consistently negative. Second, the tendency against forming three-cycles in friendship networks, which was consistently found in previous studies, is shown to be spurious and a result of neglecting to control for the tendency against reciprocation in transitive groups. The tendency against three-cycles is commonly seen as an indicator that unreciprocated friendships indicate local hierarchy differences between individuals; this proposition has to be re-evaluated in light of the findings of this study. Future studies that analyse the evolution of friendship networks should consider modelling reciprocation in transitive triplets and potentially omit modelling three-cycles.     

Avoidance,antipathy, and aggression: A three-wave longitudinal network study on negative networks,status, and heteromisos

Our aim is to explain negative networks in Dutch high schools, using three-wave stochastic actor oriented models (SAOMs). We differentiate between avoidance, antipathy, and aggression based on how costly and visible these behaviours are. Our results show that pupils’ ethnicity does not explain negative ties. Moreover, we do not find that negative ties form archetypical social hierarchies, formed by networks that are asymmetrical and transitive. Instead, we find positive effects of reciprocity on avoidance, antipathy, and aggression, and we find no effects of transitivity. Rather than allowing themselves to be dominated by their classmates, pupils fight back and reciprocate negative behaviour. We further show that some pupils behave negatively to a lot of their classmates, and that some pupils are treated negatively by many classmates. These results require us to reconsider what status hierarchies look like. Finally, we explore the extent to which the avoidance, antipathy, and aggression networks co-evolve.     

Quasi-transitive and Suzumura consistent relations

We examine properties of binary relations that complement quasi-transitivity and Suzumura consistency in the sense that they, in conjunction with the original axiom(s), are equivalent to transitivity. In general, the conjunction of quasi-transitivity and Suzumura consistency is weaker than transitivity but in the case of collective choice rules that satisfy further properties, this conjunction implies transitivity of the social relation. We prove this observation by characterizing the Pareto rule as the only collective choice rule such that collective preference relations are quasi-transitive and Suzumura consistent, and standard social choice axioms are satisfied.     

Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models

Social network data often involve transitivity, homophily on observed attributes, community structure, and heterogeneity of actor degrees. We propose a latent cluster random effects model to represent all of these features, and we develop Bayesian inference for it. The model is applicable to both binary and non-binary network data. We illustrate the model using two real datasets: liking between monks and coreaderships between Slovenian publications. We also apply it to two simulated network datasets with very different network structure but the same highly skewed degree sequence generated from a preferential attachment process. One has transitivity and community structure while the other does not. Models based solely on degree distributions, such as scale-free, preferential attachment and power-law models, cannot distinguish between these very different situations, but the latent cluster random effects model does.     

WHY CANNOT POOR COUNTRIES UTILIZE EXISTING KNOWLEDGE? EXPANSION OF FIRMS AND HUMAN CAPITAL ACCUMULATION BY TRAINING

Current growth literature does not explain the inability of poor countries to utilize existing knowledge. This study presents a stylized development model based on firms' expansion through on‐the‐job training. Since trained workers can supervise and train more workers, a development process in which output and productivity increase up to a full utilization of existing technology is generated. Unlike schooling‐based models, which require an assumption regarding a smaller local stock of knowledge, this study relies on the size of the industrial sector and the limited supply of training opportunities, therefore highlights barriers to development, even when technology and capital are available. (JEL O1, O4, J2)     

Fuzzy preferences and Arrow-type problems in social choice

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.For comments on an earlier version of the paper I am indebted to an anonymous referee, an anonymous member of the Board of Editors and to participants in the 1991 Annual Conference of the Indian Econometric Society at North Bengal University, India. However, I retain sole responsibility for any error(s) that the paper may contain.     

A characterization of utilitarianism without the transitivity axiom

We show that d'Aspremont and Gevers' (1977) characterization of utilitarianism can be strengthened by dropping the condition of transitivity of the social preference relation while slightly strengthening the Anonymity condition. The Anonymity condition we use is weaker, however, than the combination of transitivity and d'Aspremont and Gevers' Anonymity condition. Received: 24 January 1996/Accepted: 1 December 1997     

Multiple Agoras: Local and Regional Environmental Policies between Globalization and European Pathways of Transformation

Bell's well-known assumption that the nation-state is becoming too small for the big problems of life and too big for the small problems of life, has become commonplace in the contemporary debate on the nation-state in a global age. Against the background of the upcoming EU East-enlargement, the article investigates some of the environmental policy implications behind this catchphrase. Drawing on Beck's thesis of an 'unbinding of politics' it is argued that a globalizing civil society continues to be embedded across various cultural and regulatory spaces. Attempts to (re)vitalize bottom-up policy input therefore require special attention to the dynamics of regional culture and local society, where the political and the sub-political actually intersect. This, however, is not an argument for a crude bottom-up model of governance, but rather a refined understanding of multi-level governance. The argument is supported by case study material form the German-Polish border region.     

Rationalizability of choice functions on general domains without full transitivity

The rationalizability of a choice function by means of a transitive relation has been analyzed thoroughly in the literature. However, not much seems to be known when transitivity is weakened to quasi-transitivity or acyclicity. Such weakenings are particularly relevant in the context of social choice. We describe the logical relationships between the different notions of rationalizability involving, for example, the transitivity, quasi-transitivity, or acyclicity of the rationalizing relation. Furthermore, we discuss sufficient conditions and necessary conditions for rational choice on arbitrary domains. Transitive, quasi-transitive, and acyclical rationalizability are fully characterized for domains that contain all singletons and all two-element subsets of the universal set.