Higle, J. L. and Sen, S. (eds.), Stochastic Decomposition. A Statistical Method for Large Scale Stochastic Linear Programming. Kluwer Academic Publ., Dordrecht. Boston. London 1996, 248 pp., $112.00, ISBN 0-7923-3840-5. Horst, R., Pardalos, P. M. and Thoai, N. V., Introduction to Global Optimization. Nonconvex Optimization and its Applications 3. Kluwer Academic Publ., Dordrecht. Boston. London 1995, 318 pp., & HB 109.00, ISBN 0-7923-3556-2; f PB 46.00, ISBN 0-7923-3557-0 Ledyard, J. 0 . (ed.), The Economics of Informational Decentralization: Complexity, Efficiency and Stability. Essays in Honor of Stanley Reiter. Kluwer Academic Publ., Dordrecht. Boston. London 1995, 456 pp., & 129.25, ISBN 0-7923-9502-6. Mizrahi, A., Sullivan, M., Finite Mathematics. An Applied Approach. Seventh Edition. J. Wiley & Sons New York, Chichester, Brisbane, Toronto, Singapore 1996, 599 pp. and 53 pp. Appendix, £ 23, 50, ISBN 0-47 1 - 10700-X.
In this paper we introduce a new approach to representing both TU-games and NTU-games as special economic structures.Instead of representing a game as a market -an exchange economy with concave utility functionsas in the extant literature, we represent an arbitrary game as a coalition production economy with a public good.The economy provides an indirect description of the game.Our model uses the idea of a social planner or arbitrager who, through arbitrage on productive activities, seeks to minimize the net cost of providing the public good subject to the constraint that the society satises its reservation welfare level.Note that in constrast to the prior literature on representing games as economic structures, we require neither that the game be balanced nor that there are large numbers of players.Our approach to representing a game as an economic structure uses techniques from consumer demand theory and, in particular, the notion of a compensated demand correspondence.The main results of this paper exhibit a relationship between cooperative game theory and consumer demand theory.
Given the Bregman distance induced by a differentiable strictly convex function and a set of points, called sites, the farthest Bregman Voronoi cell of a particular site is the set of points for which the farthest site with respect to the considered Bregman distance is the given site. We improve some results of an earlier paper and obtain new expressions for farthest Bregman Voronoi cells, as well as conditions for their nonemptiness. We also give conditions for a closed convex set to be a cell. Moreover, using a minimax theorem due to B. Ricceri, we extend his result on the existence of two different intersecting cells, using essentially the same approach.
Abstract We review the application of some fundamental concepts of discrete convex analysis to cooperative game theory, as well as several representations of n -person cooperative games by convex functions of n variables.
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This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems.
This note determines the precise connection between an agent`s attitude towards income risks and his attitude over risks in the underlying consumption space. Our results follow a general mathematical theory connecting the curvature properties of an objective function with the ray-curvature properties of its dual.
In this note, we prove that the set of maximal monotone operators between a normed linear space X and its continuous dual X ∗ can be identified as some subset of the set Γ(X × X∗) of all lower semicontinuous convex proper functions on X ×X∗. Throughout this note, X will stand for a Banach space, X for its continuous dual and 〈·, ·〉 for the duality pairing between X and X and defined by 〈x, x〉 = x(x). By the symbol T : X −−→ −→ X ∗ we denote a set-valued mapping T that associates with each x ∈ X a (possibly empty) set T (x) in X. Recall that T is declared monotone if x1 ∈ T (x1), x ∗ 2 ∈ T (x2) =⇒ 〈x1 − x2, x ∗ 1 − x ∗ 2〉 ≥ 0. A monotone mapping T which cannot be properly extended to another monotone mapping is called maximal monotone. Such mappings are very important in variational analysis and optimization. Equivalently, denoting by gph T := {(x, x) ∈ X ×X | x ∈ T (x)} the graph of the set-valued mapping T , maximal monotone operators are those monotone operators satisfying the property : (1) If 〈x− y, x − y〉 ≥ 0 for all (x, x) ∈ gph T, then (y, y) ∈ gph T. A typical example of maximal monotone operator is the subdifferential of a lower semicontinuous convex function; see Rockafellar [1] and the monograph by Simons [3]. Following the standard notations in Convex Analysis (for instance [2]), Γ(X×X) is the set of lower semicontinuous convex functions φ : X×X → R = [−∞,+∞], which are proper, i.e., φ is not identically equal to +∞. Given φ ∈ Γ(X ×X), define b(φ) = {(x, x) ∈ X ×X| φ(x, x) ≤ 〈x, x〉} 1991 Mathematics Subject Classification. 47H05, 49J53 .
