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 functions – as 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 satis…es 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
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.
A weighted average worth per capita formula is presented for any semivalue of a TU game. Further, this formula is used to derive a characterisation of the class of games with the property that a given semivalue belongs to the power core of the game, by means of a linear system of inequalities. It is shown that for the Shapley value, the only efficient semivalue, this system reduces to the system already obtained by Inarra and Usategui. The potential approach is also used even for the more general case of values possessing a potential. A direct proof shows that for a value possessing a potential, the value of a game is in the power core relative to this value, if and only if the potential game is weak average convex. From this result, it follows that for a game and each of its subgames the value possessing a potential is in the corresponding power cores, if and only if the potential game relative to the value is average convex. This is an extension of the result obtained by Marin–Solano and Rafels for the Shapley value, proved by using the dividend form of the game.
Abstract Given a set $$T\subseteq {\mathbb {R}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> and a nonnegative function r defined on T , we consider the power of $$x\in {\mathbb {R}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> with respect to the sphere with center $$t\in T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> and radius $$r\left( t\right) ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> that is, $$ {p_r\left( x,t\right) }:=\left\| x-t\right\| ^{2}-r^{2}\left( t\right) ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mfenced> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mfenced> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mfenced> <mml:mi>t</mml:mi> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> with $$\left\| \cdot \right\| $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mo>·</mml:mo> </mml:mfenced> </mml:math> denoting the Euclidean distance. The corresponding power cell of $$s\in T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> is the set $$\begin{aligned} C_{T}^{r}(s):=\{x\in {\mathbb {R}}^{n}:{ p_r}(x,s)\le {p_r}(x,t),\ \text{ for } \text{ all }\ t\in T\}. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mspace/> <mml:mtext>for</mml:mtext> <mml:mspace/> <mml:mspace/> <mml:mtext>all</mml:mtext> <mml:mspace/> <mml:mspace/> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>T</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> We study the structure of such cells and investigate the assumptions on r that allow for generalizing known results on classical Voronoi cells.
Janssen, R. and van Herwijnen, M., Definite: A System to Support Decisions on a Finite Set of Alternatives. Vol. 1: Software package, Vol. 2: Multiobjective Decision Support for Environmental Management, Vol. 3: Definite User Manual. Kluwer Academic Publishers, Dordrecht.Boston.London 1994, Vol. 1: 4 disks, Vol. 2: 232 pp., Vol. 3: 219 pp. $ 1750.00, ISBN 0-7923-2696-2. Florenzano, M., Guddat, J., Jimenez, M., Jongen, H.Th., Lopez Lagomasino, G. and Marcellan, F. (eds.), Approximation and Optimization in the Caribbean 11. Proc. of the Second Intern. Conf. on Approximation and Optimization in the Caribbean, Havana, Cuba, September 26-October 1, 1993. Approximation and Optimization 8. Peter Lang GmbH, Frankfurt/M. Berlin Bern New York Paris Wien 1995, 682 pp., DM 103.00, ISBN 3-631-49071-2.
In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron $P$ attains its infimum on $P$. In this work we search for larger classes of sets $F$ with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymptotic properties, and investigate stability of the Frank-and-Wolfe class under various operations.
Minimax fractional programming problems are analyzed from the view- point of lower subdifferentiability, obtaining Kuhn-Tucker type optimality conditions. Multiobjective optimization problems with fractional objectives are also studied.
