A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package

The application of multi-attribute utility theory whose aggregation process is based on the Choquet integral requires the prior identification of a capacity. The main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed. All the reviewed methods have been implemented within the Kappalab R package. Their application is illustrated on a detailed example.     

Minimum variance capacity identification

In the framework of multi-criteria decision making whose aggregation process is based on the Choquet integral, we present a maximum entropy like method enabling to determine, if it exists, the “least specific” capacity compatible with the initial preferences of the decision maker. The proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity. The application of the proposed approach is illustrated on two examples.     

Indecomposable totally positive numbers in real quadratic orders

In a totally real number field, every totally positive integral number is a finite sum of (additively) indecomposable totally positive integral numbers, and up to multiplication by totally positive units, there exist only finitely many indecomposables. In the paper it is shown that in quadratic fields all these numbers can be listed in a very efficient way by using the so-called intermediate convergents of a certain quadratic irrationality. The method can be viewed as a simple extension of the standard method of calculating the fundamental unit by using continued fractions. As an application it is shown that for instance in Z|√d| a number is decomposable if its norm is >d. It is remarkable that this bound does not depend on the size of the fundamental unit.     

Continuity properties of Hilbert space valued martingales

Necessary and sufficient conditions for Hölder continuity of Hilbert space valued martingales are given in terms of the associated quadratic variation. As an application one obtains a sufficient condition for a mild solution of a stochastic evolution equation to have a continuous version if the semigroup governing this equation is analytic. Further we derive Levy's modulus of continuity for the Hilbert space valued stochastic integral with the Wiener process as integrator and obtain a generalization of the loglog law for that integral.     

Constructing a monotonic quadratic objective function in n variables from a few two-dimensional indifferences

We develop a model for constructing quadratic objective functions in n target variables. At the input, a decision maker is asked a few simple questions about his ordinal preferences (comparing two-dimensional alternatives in terms `better', `worse', `indifferent'). At the output, the model mathematically derives a quadratic objective function used to evaluate n-dimensional alternatives.Thus the model deals with some imaginary decisions (criteria aggregates) at the input, and disaggregates the decision maker's preference into partial criteria and their cross-correlations (=a quadratic objective function). Therefore, the model provides an approximation step which is next to the disaggregation of a preference into additively separable linear criteria with weight coefficients.The model is based on least squares fitting a quadratic indifference hypersurface (if n=2, indifference curve) to several alternatives which are supposed to be equivalent in preference. The resulting ordinal preference is independent of the cardinal utility scale used in intermediate computations which implies that the model is ordinal. The monotonicity of the quadratic objective function is implemented by means of a finite number of linear constraints, so that the computational model is reduced to restricted least squares.In illustration, we construct a quadratic objective function of German economic policy in four target variables: inflation, unemployment, GNP growth, and increase in public debt. This objective function is used to evaluate the German economic development in 1980–1994.In another application, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations to the South of Stuttgart.In Appendix A we provide an original fast algorithm for restricted least squares and quadratic programming used in the main model.     

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.     

Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

Positive definite <Emphasis Type="Italic">n</Emphasis>-regular quadratic forms

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.     

Quadratic integrals and the reducibility of the equations of motion of a complex mechanical system in a central field

A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4].     

Integral points for groups of multiplicative type

We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As an application, the strong approximation theorem for linear algebraic groups of multiplicative type is established. Moreover, the sum of two integral squares over some quadratic fields is discussed.     

Mean–variance approximations to expected utility

It is often asserted that the application of mean–variance analysis assumes normal (Gaussian) return distributions or quadratic utility functions. This common mistake confuses sufficient versus necessary conditions for the applicability of modern portfolio theory. If one believes (as does the author) that choice should be guided by the expected utility maxim, then the necessary and sufficient condition for the practical use of mean–variance analysis is that a careful choice from a mean–variance efficient frontier will approximately maximize expected utility for a wide variety of concave (risk-averse) utility functions. This paper reviews a half-century of research on mean–variance approximations to expected utility. The many studies in this field have been generally supportive of mean–variance analysis, subject to certain (initially unanticipated) caveats.     

Monotonicity properties of the superposition operator and their applications

We establish some properties of the superposition operator which are associated with monotonicity. Those properties are expressed in terms of the notion of degree of decrease or degree of increase. An application of the obtained results to the study of solvability of a quadratic Volterra integral equation is also derived.     

Existence of positive continuous solutions of perturbed hammerstein equations

We improve previous existence results for a class of perturbed Hammerstein integral equations, where the relevant Hammerstein operator is decreasing on positive functions. We strongly weaken the assumptions on the nonlinearities involved, and obtain existence of positive continuous solutions, even on noncompact domains, applying the Schauder-Tychonoff fixed-point theorem, via the generalized Ascoli theorem and the regularizing effect of the integral operator. An application to perturbed quadratic integral equations of interest in transport theory is also given     

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

We consider the Euler equations on the Lie algebra so(4, ℂ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.     

Quadratic integral games and causal synthesis

The game problem for an input-output system governed by a Volterra integral equation with respect to a quadratic performance functional is an untouched open problem. In this paper, it is studied by a new approach called projection causality. The main result is the causal synthesis which provides a causal feedback implementation of the optimal strategies in the saddle point sense. The linear feedback operator is determined by the solution of a Fredholm integral operator equation, which is independent of data functions and control functions. Two application examples are included. The first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables. The second is the standard linear-quadratic differential games, for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential Riccati operator equation.

     

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on preferences (name hereafter preference for interior points) that characterizes preferences favouring compromise solutions, a natural attitude in various contexts such as multicriteria optimisation, robust optimisation and optimisation with multiple agents. Within Choquet expected utility theory, this condition amounts to using a submodular capacity and a convex utility function. Under these assumptions, we focus on the fast determination of Choquet-optimal paths and spanning trees. After investigating the complexity of these problems, we introduce a lower bound for the Choquet integral, computable in polynomial time. Then, we propose different algorithms using this bound, either based on a controlled enumeration of solutions (ranking approach) or an implicit enumeration scheme (branch and bound). Finally, we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes.     

Structure condition under initial enlargement of filtration

We address the question of how the structure condition is affected when one possesses some additional information at the very beginning of the investment period.The structure condition represents essentially an alternative to non-arbitrage conditions for the Markowitz’s portfolio optimization framework,and is crucial for the existence of the optimal portfolio in quadratic utility settings.Herein,we provide practical assumption on the initial market model and the additional information to preserve the structure condition.The stochastic tools that drive this result are a generalization of the Lazaro-Yor representation by Lazaro and Yor(1978)and optional stochastic integral.     

Constructing a quasi-concave quadratic objective function from interviewing a decision maker

A model for constructing quadratic objective functions (=utility functions) from interviewing a decision maker is considered. The interview is designed to guarantee a unique non-trivial output of the model and to enable estimating both cardinal and ordinal utility, depending on interview scenarios. The model is provided with operational restrictions for the monotonicity of the objective function (=either only growth, or only decrease in every variable) and its quasi-concavity (=convexity of the associated preference). Thereby constructing a monotonic quasi-concave quadratic objective function is reduced to a problem of non-linear programming. To support interactive editing of a quadratic objective function, the stability of the model (the continuous dependence of the output ordinal preference on the input data) is proved. In illustration, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations in the south of Stuttgart.     

Representation of preference orderings on Lp -spaces by integral functionals: discounting,continuity and TAS utility

This paper presents an axiomatic approach in a continuous time framework for representing preference orderings on L^p -spaces in terms of integral functionals. We show that if preference orderings on L^p -spaces satisfy continuity, separability, sensitivity, substitutability, additivity and lower boundedness, then there exists a utility function for the preference orderings such that the utility function is an integral functional with an upper semicontinuous integrand satisfying the growth condition. Moreover, if the preference orderings satisfy the continuity with respect to the weak topology of L^p -spaces, then the integrand is a concave integrand. As a result, time additive separable (TAS) utility functions with constant discount rates are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Represented value sets for integral binary quadratic forms and lattices

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.