Decision making under uncertainty comprising complete ignorance and probability

This paper investigates a model of decision making under uncertainty comprising opposite epistemic states of complete ignorance and probability. In the first part, a new utility theory under complete ignorance is developed that combines Hurwicz–Arrow's theory of decision under ignorance with Anscombe–Aumann's idea of reversibility and monotonicity used to characterize subjective probability. The main result is a representation theorem for preference under ignorance by a particular one-parameter function – the τ-anchor utility function. In the second part, we study decision making under uncertainty comprising an ignorant variable and a probabilistic variable. We show that even if the variables are independent, they are not reversible in Anscombe–Aumann's sense. This insight leads to the development of a new proposal for decision under uncertainty represented by a preference relation that satisfies the weak order and monotonicity assumptions but rejects the reversibility assumption. A distinctive feature of the new proposal is that the certainty equivalent of a mapping from the state space of uncertain variables to the prize space depends on the order in which the variables are revealed. Explicit modeling of the order of variables explains some of the puzzles in multiple-prior model and the models for decision making with Dempster–Shafer belief function.     

On minimax and Pareto optimal security payoffs in multicriteria games

In this paper, we characterize minimax and Pareto-optimal security payoff vectors for general multicriteria zero-sum matrix games, using properties similar to the ones that have been used in the single criterion case. Our results show that these two solution concepts are rather similar, since they can be characterized with nearly the same sets of properties. Their main difference is the form of consistency that each solution concept satisfies. We also prove that both solution concepts can transform into each other, in their corresponding domains.     

On fuzzy goals and maximizing decisions in stochastic optimal control

Fuzzy set theory has developed significantly in a mathematical direction during the past several years but few applications have emerged. This paper investigates the role of fuzzy set theory in certain optimal control formulations. In particular, it is shown that the well-known quadratic performance criterion in deterministic optimal control is equivalent to the exponential membership function of a certain fuzzy decision (set). In a stochastic setting, similar equivalences establish new definitions for “confluence of goals” and “maximizing decision” in fuzzy set theory. These and other definitions could lead to the development of a more applicable theory of fuzzy sets.     

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

We consider fair allocation of indivisible items under additive utilities. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. The result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Approximation of expected returns and optimal decisions under uncertainty using mean and mean absolute deviation

We consider economic decision problems under uncertainty consisting of choosing an optimal decisionX, so as to maximize to expected value of an objective function depending on a stochastic parameterp. The paper establishes an optimal policy intervalX _A X ¹ X _B, where the boundsX _A,X _B are given in terms of simple parameters of the distribution ofp, in particular the mean, and the mean absolute deviationd=E ¦p– ¦. The convexity assumptions needed to establish such bounds are shown to hold naturally in some classical problems of production under uncertainty.

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Zusammenfassung Wir betrachten wirtschaftliche Entscheidungsprobleme mit Unsicherheit, in denen eine optimale EntscheidungX so getroffen werden soll, daß der Erwartungswert einer Zielfunktion, abhängig von einem stochastischen Parameterp, maximiert werden soll. In dieser Arbeit wird ein IntervallX _A X ¹ X _B für die optimale Politik angegeben, wobei die SchrankenX _A,X _B durch einfache Größen der Verteilung vonp ausgedrückt werden, im besonderen durch den Mittelwert und die mittlere absolute Abweichungd=E ¦p– ¦. Ferner wird gezeigt, daß die für die Herleitung der Schranken benötigten Konvexitätsannahmen in natürlicher Weise für einige klassische Produktionsprobleme mit Unsicherheit gelten.

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Supported by BARD Project No. I-10-79 and by Technion VPR Fund-Lawrence Deutsch Research Fund.     

Decision with Dempster-Shafer belief functions: Decision under ignorance and sequential consistency

This paper examines proposals for decision making with Dempster-Shafer belief functions from the perspectives of requirements for rational decision under ignorance and sequential consistency. The focus is on the proposals by Jaffray & Wakker and Giang & Shenoy applied for partially consonant belief functions. We formalize the concept of sequential consistency of an evaluation model and prove results about sequential consistency of Jaffray-Wakker’s model and Giang-Shenoy’s model under various conditions. We demonstrate that the often neglected assumption about two-stage resolution of uncertainty used in Jaffray-Wakker’s model actually disambiguates the foci of a belief function, and therefore, makes it a partially consonant on the extended state space.     

On the new distribution generalizing the Gibbs, Bose-Einstein, and Pareto distributions

A distribution corresponding to a parastatistic and generalizing the Gibbs, Bose-Einstein, and Pareto distributions as well as rank distributions is given.     

A necessary and sufficient condition to a pareto optimal without optimizing the weighting criterion

The easiest thecnique to reduce the classical multiple criteria decision problem into a reasonable single criterion decision problem is the weighting method. Po-Lung Yu (1985) gives a well known necessary condition fory ⁰ to be a Pareto optimal, namelyy ⁰ maximizes λ^ty overY, for some λ ∈ℝ ^p, such that λ_i≥0 for alli and some λ_j>0. In this brief note we generalize the necessary condition of Po-Lung Yu.     

A new risk criterion in fuzzy environment and its application

Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue–Stieltjes (L–S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method.     

A new stability criterion for discrete-time neural networks: Nonlinear spectral radius

In this paper, the exponential stability of nonlinear discrete-time systems is studied. A novel notion of nonlinear spectral radius is defined. Under the assumption of Lipschitz continuity for the activation function, the developed approach is applied to stability analysis of discrete-time neural networks. A series of sufficient conditions for global exponential stability of the neural networks are established and an estimate of the exponential decay rate is also derived for each case.     

A novel technique of object ranking and classification under ignorance: An application to the corporate failure risk problem

This paper exposits a novel technique for the ranking and classification of objects to a particular state. Each object is described by measurements from a number of variables which may offer different levels of support for the individual objects to be associated with the two states, a given hypothesis and not the hypothesis. The Dempster–Shafer theory of evidence is a central component of this technique. This allows for a measure of concomitant ignorance, which may encompass the precision of the individual measurements as well as the possible ambiguity of their influence in the subsequent classification of objects. The level of ignorance present influences the utilisation of the technique as a tool for the ranking or classification of objects. A simplex plot method of representing data allows a clear visual representation (interpretation) to the degree of interaction of the support from the variables to the ranking or classification of the objects. To illustrate the proposed technique, the application considered here is the elucidation of the risk of corporate failure of a number of companies. Subsequently, each variable (financial and non-financial) may offer support for the ranking and classification of companies to between the extreme states of being a failed or non-failed company. A comparison on the ranking and classification of companies is made with a traditional multivariate discriminant analysis.     

A new criterion for the logarithmic Sobolev inequality and two applications

We give a criterion for the logarithmic Sobolev inequality (LSI) on the product space X₁×?×XN. We have in mind an N-site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H. It is optimal for Gaussians with positive covariance matrix. To illustrate, we give two applications: one with weak interactions and one with strong interactions and a decay of correlations condition.     

The Banzhaf index in complete and incomplete shareholding structures: A new algorithm

In this global world, many firms present a complex shareholding structure with indirect participation, such that it may become difficult to assess a firm’s controllers. Furthermore, if there are numerous dominant shareholders, the control can be shared between them. Determining who has the most influence is often a difficult task. To measure this influence, game theory allows the modeling of voting games and the computing of the Banzhaf index. This paper firstly offers a new algorithm to compute this index in all structures and then suggests some modelisations of the floating shareholder. Then, our model is applied to a real case study: the French group Lafarge. This exemplary case demonstrates how the float’s structure and hidden coalition can impact the power relationship between dominant shareholders.     

On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion

Optimal schedules in the job shop problem with preemption and with the objective of minimizing an arbitrary regular function of operation completion times are studied. It is shown that for any instance of the problem there always exists an optimal schedule that meets several remarkable properties. Firstly, each changeover date coincides with the completion time of some operation, and so, the number of changeover dates is not greater than the total number of operations, while the total number of interruptions of the operations is no more than the number of operations minus the number of jobs. Secondly, every changeover date is “super-integral”, which means that it is equal to the total processing time of some subset of operations. And thirdly, the optimal schedule with these properties can be found by a simple greedy algorithm under properly defined priorities of operations on machines. It is also shown that for any instance of the job shop problem with preemption allowed there exists a finite set of its feasible schedules which contains at least one optimal schedule for any regular objective function (from the continuum set of regular functions).     

A new pinching theorem for complete self-shrinkers and its generalization

In this paper,we firstly verify that if M~n is an n-dimensional complete self-shrinker with polynomial volume growth in R~(n+1),and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in R~(n+1) with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-β_λ≤1/18,then S≡β_λ,λ 0 and M is a cylinder.Here β_λ=1/2(2+λ~2+|λ|(λ~2+4)~(1/2)).