On the extension of additive utilities to infinite sets

Independence condition C is known as necessary and sufficient for the existence of an additive utility on a finite subset X of a Cartesian product. A stronger necessary condition, H, interpreted as both an independence and Archimedean condition, is derived. It is shown to be sufficient when X is countable by constructing an additive utility as the limit of a sequence of additive utilities on finite subsets of X. When X is not countable, but is a Cartesian product, another necessary condition, the existence of A, a countable perfectly (order-) dense subset of X, is added to H; an additive utility is constructed by extension to X of an additive utility on a countable set linked to A. An application to a no-solvability case is given.     

Interdependent preferences on finite sets

The degree of interdependence of a preference relation ? on a finite subset X of a product set X₁ × X₂ × … × X_n is defined in terms of the highest order of preference interaction among the X_i that must be taken into account in a real-valued, interdependent additive representation for ?. The degree is zero when indifference holds throughout X, and zero or one in the additive conjoint measurement case. A degree of n signifies complete preference interdependence among the X_i.     

Trees with structure

We investigate M-trees, that is, trees with structure possible at each node or level. M is a mathematical structure such as a set or a Cartesian product. An extension of Pólya's theorem is proved which allows the number of M-trees for a given number of nodes to be counted. The special case of componential trees is investigated. Here M is a Cartesian product of 0's and 1's. A componential analysis is a componential tree of depth 1, that is, there is no hierarchy. We prove that for any componential tree there exists a componential analysis which makes the same predictions on a triad test of judged similarity. A brief empirical example is given, in which a componential tree is applied as a model of a sernantic domain.     

Additive Utilities on Densely Ordered Sets

This paper shows sufficient conditions for the existence of additive utilities without a restricted solvability axiom. Our conditions require that each essential component of the underlying Cartesian product be densely ordered.     

Geometric analogue of holographic reduced representation

Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1, one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on ‘projected products’, a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.     

What is the Question to Which Husserl’s Fifth Cartesian Meditation is the Answer?

Interpreters generally agree that the Fifth Cartesian Meditation fails to achieve its task, but they do not agree on what that task is. In my essay, I attempt to formulate the question to which the Fifth Cartesian Meditation gives the answer. While it is usually assumed that the text poses a rather ambitious question, I suggest that the text asks, “How is the Other given to me on the most basic level?” The answer would be that the Other is given as accessible in the mode of inaccessibility. Husserl’s failure to convey this question clearly seems rooted in ambiguities concerning the concepts of solipsism and the sphere of ownness.     

Technical note on the joint receipt of quantities of a single good

The joint receipt of x and y is the fact of receiving them both. If x and y are objects that are valued, their joint receipt is valued as well. Assumming joint receipt is a binary operation that satisfies the conditions of extensive measurement, there is a numerical representation that is additive over joint receipt. We consider the case where x and y are quantities of the same infinitely divisible good. Different sets of assumptions are explored. Invariance with respect to multiplication proves to be interesting. Invariance with respect to addition yields a linear form. A relaxation of the latter yields an approximately linear form. Finally, we consider a non-commutative but bisymmetric joint-receipt operation with a representation arising from preferences over gambles.     

Cardinal coordinate independence for expected utility

A representation theorem for binary relations on $\text{R}$ⁿ is derived. It is interpreted in the context of decision making under uncertainty. There we consider the existence of a subjective expected utility model to represent a preference relation of a person on the set of bets for money on a finite state space. The theorem shows that, for this model to exist, it is not only necessary (as has often been observed), but it also is sufficient, that the appreciation for money of the person has a cardinal character, independent of the state of nature. This condition of cardinal appreciation is simple and thus easily testable in experiments. Also it may be of help in relating the neo-classical economic interpretation of cardinal utility to the von Neumann-Morgenstern interpretation.     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

Multilinear representations for ordinal utility functions

An ordinal utility function u over two attributes X₁, X₂ is additive if there exists a strictly monotonic function ϕ such that ϕ(u) = v₁(x₂) + v₂(x₂) for some functions v₁, v₂. Here we consider the class of ordinal utility functions over n attributes for which each pair of attributes is additive, but not necessarily separable, for any fixed levels of the remaining attributes. We show that while this class is more general than those that are ordinally additive, the assessment task is of the same order of difficulty, and involves a hierarchy of multilinear rather than additive decompositions.     

An interaction model for odor quality and intensity

The perceived intensity and quality of binary odor mixtures are studied in relation to bow their components are perceived when presented separately. Subjects judged the perceived intensity and quality of 6 concentrations of pyridine, 6 concentrations ofn-butanol, and their 36 possible combinations. The results show that the perceived intensity of the mixture can be predicted from the perceived intensity of its components presented separately (R _A andR _B) by the Euclidian arithmetic model. The maximum probability of identifying a mixture as a “mixture” is reached whenR _A andR _B are equally strong. An interaction model for mixtures that relates perceived intensity and quality is presented and tested. The test reveals that both the perceived intensity of a mixture and the probability that it will be identified as the one or the other component can, by a simple interaction model, be successfully predicted fromR _A andR _B.     

Social motives among Anglo American and Mexican American children: Experimental and projective measures

The measurement approaches to social motives, cooperation-competition games and projective tests, have revealed apparently similar findings: Anglo American children are more competitive and higher in n Achievement compared to Mexican American children, who are more cooperative and higher in n Affiliation. The present experiment tested the extent to which (1) competitiveness is related to n Achievement, (2) cooperativeness is related to n Affiliation, and (3) the cultural differences in cooperation-competition are related to those in n Affiliation and n Achievement. Results indicate that social motives inferred from a cooperation-competition game have a meaningful relation to those inferred from a projective test, but that the cultural differences in cooperation-competition are larger and are not explained by cultural differences in n Affiliation and n Achievement. Social motives inferred from each measurement methodology, while somewhat related, are distinct.     

Utility-uncertainty trade-off structures

A qualitative axiomatization of a generalization of expected utility theory is given in which the expected value of simple gambles is not necessarily the product of subjective probability and utility. Representation and uniqueness theorems for these generalized structures are derived for both Archimedean and nonarchimedean cases. It is also shown that a simple condition called distributivity is necessary and sufficient in the case of simple gambles for one of these generalized expected utility structures to have simultaneously an additive subjective probability function and a multiplicative combining rule for expected values.     

Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

Coordination in Market Entry Games with Symmetric Players

We report the results of two experiments designed to study tacit coordination in a class of market entry games with linear payoff functions, binary decisions, and zero entry costs, in which each of n = 20 players must decide on each trial whether or not to enter a market whose capacity is public knowledge. The results show that although the subjects differ considerably from one another in their decision policies, tacit coordination emerges quickly on the aggregate level and is accounted for most successfully by the Nash equilibrium solution for noncooperative n-person games.     

Nearest neighbor analysis of point processes: Simulations and evaluations

Tversky, Rinott, and Newman (Journal of Mathematical Psychology, 1983, 27, 000) examine the asymptotic behavior of a measure of the centrality of the nearest neighbor relation. The applicability of their conclusions when the number of dimensions (d) and the number of points (n) take on the small-to-moderate values commonly encountered in the analysis of proximity data is investigated. The results suggest that convergence is fast when n is large relative to d and slow when d is large relative to n.     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.     

Matching Topological and Frame Products of Modal Logics

The simplest combination of unimodal logics \({\mathrm{L}_1 \rm and \mathrm{L}_2}\) into a bimodal logic is their fusion, \({\mathrm{L}_1 \otimes \mathrm{L}_2}\), axiomatized by the theorems of \({\mathrm{L}_1 \rm for \square_1 \rm and of \mathrm{L}_2 \rm for \square_{2}}\). Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product\({\mathrm{L}_1 \times \mathrm{L}_2 \rm of \mathrm{L}_1 \rm and \mathrm{L}_2}\). Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product\({\mathrm{L}_1 \times_{t}\mathrm{L}_2}\), using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \({\mathrm{S}4: \mathrm{L}_1 \times_t \mathrm{L}_2 = \mathrm{L}_1 \times \mathrm{L}_2 \rm iff \mathrm{L}_1 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_2 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_1, \mathrm{L}_2 = \mathrm{S}5}\).     

On finding the bidimension of a relation

A method is presented for evaluating the bidimension of a finite binary relation, i.e., the number of biorders (Guttman relations) needed to yield the relation as their intersection. In case the relation is induced by a binary data matrix, the bidimension equals the minimal number of dimensions needed for a representation of the data matrix according to the conjunctive model of C. H. Coombs and R. C. Kao (Nonmetric factor analysis, Engineering Research Bulletin No. 38, Univ. of Michigan Press, Ann Arbor, 1955). Central to the evaluation of the bidimension is its characterization, provided by J.-P. Doignon, A. Ducamp, and J.-C. Falmagne (Journal of Mathematical Psychology, 28, 73–109, 1984), as the chromatic number of some associated hypergraph. A procedure is described to reduce hypergraphs of this kind to subhypergraphs with the same chromatic number. This reduction can be used throughout in applying a recurrence relation that expresses the chromatic number of a hypergraph in terms of the chromatic numbers of some of its subhypergraphs.     

On the construction of fuzzy preference structures

In this paper, we analyse the generalization of the classical method of the construction of a preference structure from a reflexive binary relation to the case of fuzzy binary relations. According to our approach, there are two interesting fuzzy preference structures we can construct from a given reflexive fuzzy binary relation. These two fuzzy preference structures correspond to the two extreme solutions of the system of functional equations in the method of Fodor and Roubens. We also prove that only one of two fuzzy preference structures allows its fuzzy relation of strict preference to be transitive with respect to the φ-transform of the Lukasiewicz t-norm when the reflexive fuzzy relation it is constructed from is also transitive with respect to the same t-norm. © 1998 John Wiley & Sons, Ltd.