Symmetry Axiom Research Articles

An efficient solidarity value for interval cooperative games

This paper proposes an efficient interval solidarity value that operates well for interval cooperative games. In addition to the axioms of symmetry, efficiency, and additivity, this value also satisfies two new axioms, namely, interval-egalitarian A-null player and interval differential marginality. The interval-egalitarian A-null player axiom equally divides the result of the difference between the grand coalition value and the sum of the solidarity value of players in the degenerate interval game among A-null players. The interval differential marginality axiom is an interval version of the Casajus differential marginality axiom. This property states that the difference in the interval solidarity value of two players is determined by the difference between their average marginal contributions in the degenerate interval game. Eventually, the efficiency results and applicability of the proposed approach are compared with those of the other methods.

METHODS FOR DETERMINING SIMILARITY OF CATEGORICAL ORDERED DATA

Context. The development of effective distance metrics and similarity measures for categorical features is an important task in data analysis, machine learning, and decision theory since a significant portion of object properties is described by non-numerical values. Typically, the dependence between categorical features may be more complex than simply comparing them for equality or inequality. Such attributes can be relatively similar, and to construct an effective model, it is necessary to consider this similarity when calculating distance or similarity measures.
 Objective. The aim of the study is to improve the efficiency of solving practical data analysis problems by developing mathematical tools for determining the similarity of objects based on categorical ordered features.
 Method. A distance based on weighted Manhattan distance and a similarity measure for determining the similarity of objects based on categorical ordinal features (i.e. a linear order with scales of preference considering the problem domain can be specified on the attribute value set) are proposed. It is proven that the distance formula satisfies the axioms of non-negativity, symmetry, triangle inequality, and upper bound, and therefore is a distance metric in the space of ranked categorical features. It is also proven that the similarity measure presented in the study satisfies the axioms of boundedness, symmetry, maximum and minimum similarity, and is described by a decreasing function.
 Results. The developed approach has been implemented in an applied problem of determining the degree of similarity between objects described by ordered categorical features.
 Conclusions. In this study, mathematical tools were developed to determine similarity between structured data described by categorical attributes that can be ordered based on a specific priority in the form of a ranking system with preferences. Their properties were analyzed. Experimental studies have shown the convenience and “intuitive understanding” of the logic of data processing in solving practical problems. The proposed approach can provide the opportunity to conduct new meaningful research in data analysis. Prospects for further research lie in the experimental use of the proposed tools in practical tasks and in studying their effectiveness.

A non-lexicographic rule in the preference-approval setting

We present the first non-lexicographic social ranking rule in the preference-approval setting. Our rule is obtained using three well-known axioms and a novel symmetry axiom, which balances preferences and approval information, thereby avoiding the extreme outcomes seen in existing lexicographic methods. Other non-lexicographic alternatives are discussed.

Extension of semi-Lipschitz maps on non-subadditive quasi-metric spaces: new tools for Artificial Intelligence

The axioms of symmetry and indiscernibility of a metric function can be eliminated and still define a topological space with good enough properties. However, among all the axioms that constitute the notion of a metric, the subadditivity or triangular inequality is the one that has been more difficult to relax. Even so, and due to the potential applications in other fields such as Artificial Intelligence, several efforts have been made in this direction. The notion of quasi-Banach space or the notion of b-metric space, are examples of structures where the subadditivity has been replaced by a weaker inequality. Following this line, in this paper we consider the non-symmetric version of the so called strong b-metric spaces. We focus our attention on the possibility of extending semi-Lipschitz maps, and still preserve (a uniform multiple of) the Lipschitz constant. As our main result, we prove that the possibility of obtaining such extensions characterizes the spaces satisfying the proposed weak versions of the triangular inequality. We also show how our results can be applied in Machine Learning to solve some technical issues, in particular, some problems related to the extension of Lipschitz functions with respect to the cosine distance.

∀ or ∃?

This paper shows that in some axioms regarding the mixture of random variables, the requirement that the conclusions hold for all values of the mixture parameter can be weakened by requiring the existence of only one nontrivial value of the parameter, which need not be fixed. This is the case for the independence, betweenness, and mixture symmetry axioms. Unlike the standard axioms, these weaker versions cannot be refuted by experimental methods.

Quaternion model of Pythagorean fuzzy sets and its distance measure

Pythagorean fuzzy set (PFS), one of several non-standard fuzzy sets, introduces a number pair a,b that satisfies the condition a2+b2≤1 to express membership grade. The above expression suggests that PFS provides a more extensive description space for describing fuzzy information, thereby attracting much attention in scientific research and engineering practice. Many studies have attempted to further exploit the potential of PFS in a reasonable manner. Consequently, this paper proposes a quaternion model of Pythagorean fuzzy set (QPFS). In QPFS, membership, non-membership, and hesitation function are expressed in the form of quaternion. A major advantage of QPFS over PFS is that its description space of fuzzy information stretches from the real plane to the hypercomplex plane. This is useful for capturing the composite features of fuzzy information as well as describing multi-dimensional fuzzy information. We then define the basic logical operations of QPFS, including complement, union, and intersection, and also derive the properties of these operations. As an additional contribution, this paper presents a distance measure (QPFSD) in the quaternion model of Pythagorean fuzzy sets (QPFSs). QPFSD is a strict distance measure that satisfies the axioms of distance measure, i.e., nonnegativity, symmetry, and triangle inequality. When QPFSs collapse into Pythagorean fuzzy sets (PFSs), QPFSD becomes the distance measure of real space. It has been demonstrated through some numerical examples that QPFSD is an effective method for measuring difference between QPFSs. Furthermore, we apply QPFSD to domains such as expert evaluation and data-driven environments in the QPFS framework. Experimental results related to expert evaluation suggest that QPFSD and the models involving it can provide an intuitive decision-making. Experimental results based on Iris data set demonstrate that with the increase of the number of training sets, the recognition rate of methods associated with QPFSD also increases for unknown targets. With the help of iris data, based on QPFSD, we investigate the performance of QPFS, CPFS, PFS and FS, respectively. The average recognition rate associated with QPFS is the highest among these. Through the use of iris dataset experiments and numerical example, this paper compares QPFSD and other distance measures in the CPFS and PFS frameworks, respectively. As a result of the dataset experiment, the recognition of methods pertaining to QPFSD is highest within CPFS and PFS frameworks. The numerical experiments indicate that QPFSD is highly sensitive to the differences between fuzzy information compared with other distance measures.

Approach to modeling in the metric space of the energy exchange of two media

The work is devoted to the development and analysis of a topological model of the interaction of two energy media in a metricspace. The Hausdorff metric space is chosen as the initial set,which obeys the axioms of identity, symmetry and triangle. The real elementof the system in the space of energy measures,designed to transfer energy from one medium to another,is represented in correspondence with its cellular image,defined as a virtual Grassmannian. When energy is transferred from a medium with a higher energy to a medium with a lower energy, energy measures determine the organization of processes in the designed heat exchangeequipment. Informative components of the Grassmannian are also its area and perimeter. An analysis of the structural model, using the modified Heron formula and the Cayley-Merger determinant, showed that,assuming equilibrium at the Grassmannian nodes, its area in the space of energy measures should be equal to zero. At equilibrium, the semi perimeterin terms of energy measures is the energy potential applied to the element under conditions of its minimum. Relationships between the area of the Grassmannian and the potential applied to the element, the temperature efficiency, and the mixing efficiency of two flows are obtained. The study of this relationship shows that the Grassmannian perimeter has an extremum at an area equal to zero, at which the perimeter is equal to theapplied potential. From a design point of view, this means that when specifying flows and inlet temperatures,the requirements for the apparatus are formulated in terms of energy or final temperatures. This essentially determines the required potential and thecorresponding energy exchange efficiency.In this case, the potential takes the minimum required value corresponding to the requirement of the project, and the amount of transferred energy corresponds to the required one at fixed values of flows and energy exchange efficiency.

The Unique Characterization of the Shapley Value for Bi-cooperative Games

The aim of the present paper is to study the unique characterization for bicooperative games. Shapley value is the expected marginal contribution of the alliance. We will introduce some properties for bicooperative games. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced excluded null axiom. In our second axiomatization we use structural axiom and a zero excluded axiom instead of effective axiom in classical cooperative games. Finally, We provide here linearity, anonymity, dummy and efficiency and structural axioms to study a one-point solution concept for Bi-cooperative games.

A new axiomatic approach to the impartial nomination problem

In this paper we introduce a new set of axioms that characterize uniform random dictatorship (URD) as a randomized impartial nomination rule. Unlike earlier work we use a variable population model which allows us to employ axioms that reflect consistency and proportionality–axioms that ensure that the rule behaves well with respect to the combinatorial structure of the nomination profile. Earlier work characterizing URD employed strong symmetry axioms and so it is surprising that our axioms characterize the same rule.

Axiomatizations of the proportional division value

We present axiomatic characterizations of the proportional division value for TU-games, which distributes the worth of the grand coalition in proportion to the stand-alone worths of the players. First, a new proportionality principle, called proportional-balanced treatment, is introduced by strengthening Shapley’s symmetry axiom, which states that if two players make the same contribution to any nonempty coalition, then they receive the amounts in proportion to their stand-alone worths. We characterize the family of values satisfying efficiency, weak linearity, and proportional-balanced treatment. We also show that this family is incompatible with the dummy player property. However, we show that the proportional division value is the unique value in this family that satisfies the dummifying player property. Second, we propose appropriate monotonicity axioms, and obtain axiomatizations of the proportional division value without both weak linearity and the dummifying player property. Third, from the perspective of a variable player set, we show that the proportional division value is the only one that satisfies proportional standardness and projection consistency. Finally, we provide a characterization of proportional standardness.

Axiomatizations of the $$\beta $$ and the score measures in networks

Monotonicity is an appealing principle of relational power measures in social networks. It states that as the influence of an individual changes, the relational power measure should change in the same direction. We axiomatically characterize the $$\beta $$ - and score-measures on directed networks using three axioms: two different monotonicity axioms, the same symmetry axiom and two different efficiency axioms, respectively. For every individual, the $$\beta $$ -measure refers to a weighted sum over all dominated individuals by itself, in which for each dominated individual the weight is given by the reciprocal of the number of its dominators, while the score-measure refers simply to the number of dominated individuals by itself. We also extend the result for the score-measure to undirected networks.

Probabilities of first-order sentences on sparse random relational structures: An application to definability on random CNF formulas

Abstract We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma $ and a first-order theory $T$ for $\sigma $ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite $\sigma $-structures that satisfy $T$ and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in $\sigma $ is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation $R$ in the vocabulary $\sigma $. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random $k$-CNF formula on $n$ variables, where each possible clause occurs with probability $\sim c/n^{k-1}$, independently any first-order property of $k$-CNF formulas that implies unsatisfiability does almost surely not hold as $n$ tends to infinity.

Quantifying Asymmetry in Gait: The Weighted Universal Symmetry Index to Evaluate 3D Ground Reaction Forces.

Though gait asymmetry is used as a metric of functional recovery in clinical rehabilitation, there is no consensus on an ideal method for its evaluation. Various methods have been proposed to analyze single bilateral signals but are limited in scope, as they can often use only positive signals or discrete values extracted from time-scale data as input. By defining five symmetry axioms, a framework for benchmarking existing methods was established and a new method was described here for the first time: the weighted universal symmetry index (wUSI), which overcomes limitations of other methods. Both existing methods and the wUSI were mathematically compared to each other and in respect to their ability to fulfill the proposed symmetry axioms. Eligible methods that fulfilled these axioms were then applied using both discrete and continuous approaches to ground reaction force (GRF) data collected from healthy gait, both with and without artificially induced asymmetry using a single instrumented elbow crutch. The wUSI with a continuous approach was the only symmetry method capable of identifying GRF asymmetry differences in different walking conditions in all three planes of motion. When used with a continuous approach, the wUSI method was able to detect asymmetries while avoiding artificial inflation, a common problem reported in other methods. In conclusion, the wUSI is proposed as a universal method to quantify three-dimensional GRF asymmetries, which may also be expanded to other biomechanical signals.

Probabilistic characterisation of models of first-order theories

We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms T and a structure M which we only know is a model of T, we are interested in the probability that M would satisfy a sentence ψ. Answering this question for all sentences in the language will give a probability distribution over the set of sentences which can be regarded as the probabilistic characterisation of the model M. We investigate defining these probabilistic characterisations as the limit of probability functions imposed on the set of finite models of T. We show how a symmetry axiom can uniquely specify the probability function over finite models and will study the existence of the limit in terms of the quantifier complexity of T.

Duality of gauges and symplectic forms in vector spaces

A gauge $$\gamma$$ in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces $$(X, \gamma )$$ extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then the fixation of a symplectic form yields an identification between X and its dual space $$X^*$$ . The image of the polar body $$K^{\circ }\subseteq X^*$$ under this identification yields a (skew-)dual gauge on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.

A Perspective-Invariant Approach to Nash Bargaining

The Nash axioms lead to different results depending on whether the negotiation is framed in terms of gains relative to no agreement or in terms of sacrifices relative to an ideal. We look for a solution that leads to the same result from both perspectives. To do so, we restrict the application of Nash’s IIA axiom to bargaining sets where all options are individually rational and none exceed either party’s ideal point. If we normalize the bargaining set so that the disagreement point is (0, 0) and maximal gains are (1, 1), then any perspective-invariant bargaining solution must lie between the Utilitarian solution and the maximal equal-gain (minimal equal-sacrifice) solution. We show that a modified version of Nash’s symmetry axiom leads to the Utilitarian solution and that a reciprocity axiom leads to the equal-gain (equal-sacrifice) solution, both of which are perspective invariant. This paper was accepted by Joshua Gans, business strategy.

Differential geometry of spatial curves for gauges

We derive Frenet-type results and invariants of spatial curves immersed in 3-dimensional generalized Minkowski spaces, i.e., in linear spaces which satisfy all axioms of finite dimensional real Banach spaces except for the symmetry axiom. Further on, we characterize cylindrical helices and rectifying curves in such spaces, and the computation of invariants is discussed, too. Finally, we study how translations of unit spheres influence invariants of spatial curves.

Axiomatic characterizations of the egalitarian solidarity values

We define and characterize the class of all egalitarian solidarity values, which are convex combinations of the solidarity value and the equal division solution. First, assuming that the parameter is given exogenously, we provide two alternative characterizations of the corresponding value: Either with the classical axioms of efficiency, additivity and symmetry,and a new axiom called α-egalitarian A-null player axiom; or replacing the additivity and theα-egalitarian A-null playeraxioms by a new axiom called α-average marginality. Finally, we give two characterizations of the whole family of egalitarian solidarity values in such a way that the parameter α is obtained endogenously by the set of axioms.

Curvature types of planar curves for gauges

In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this, also the notions of evolutes and involutes to gauge planes.

Axiomatizations of the Proportional Division Value

We present axiomatic characterizations of the proportional division value for TU-games, a value that distributes the worth of the grand coalition in proportion to the stand-alone worths of its members. First, a new proportionality principle, called balanced treatment, is introduced by strengthening Shapley's symmetry axiom, which states that if two players make the same contribution to any nonempty coalition, then they receive the amounts in proportion to their stand-alone worths.We characterize the family of values satisfying efficiency, weak linearty, and balanced treatment. We also show that this family is incompatible with the dummy player property. However, we show that the proportional division value is the unique value in this family that satisfies the dummifying player property. Second, we propose three appropriate monotonicity axioms by considering two games in which the stand-alone worths of all players are equal or in the same proportion to each other, and obtain three axiomatizations of the proportional division value without both weak linearity and the dummifying player property. Third, from the perspective of a variable player set, we show that the proportional division value is the only one that satisfies proportional standardness and projection consistency. Finally, we provide characterizations of proportional standardness.