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Historia problemu Webera

100%
Młodak A.
Mathematica Applicanda
|
2009
|
tom 37
|
nr 51/10
PL
W pracy zajmujemy się zagadnieniem, które znane jest powszechnie jako konstrukcja mediany Webera. Chodzi mianowicie o znalezienie takiego punktu w przestrzeni Ên , że suma jego odległości od m danych punktów w tejże przestrzeni jest najmniejsza. Prezentujemy historię badań w tej dziedzinie począwszy od najprostszej formytego problemu, tj. minimalizacji sumy odległości od wierzchołków trójkąta, którym zajmowano się w XVII i XVIII wieku aż po współczesne wyniki w tym zakresie i jego dalsze uogólnienia. Wskazujemy także na możliwości jego zastosowań w statystyce i ekonometrii.
2
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New axiomatizations of values of TU-games using reduction properties

100%
Młodak A.
Applicationes Mathematicae
|
2013
|
tom 40
|
nr 4
483-502
EN
We propose new axiomatizations of values of cooperative games where traditional properties connected with special players (dummy, null or zero) are replaced with weaker properties relating to such participants of the game. We assume that the change of payoff of a player when combining the game with another game where this player is special is constant. Using such axioms with an additional assumption that a value is odd and-if necessary-the fairness axioms holds, one can obtain axiomatizations without additivity where not only classical dummy, null or zero players axioms but even equal treatment can be redundant. These properties are used to construct new axiomatizations of the Shapley, Banzhaf and Deegan-Packel values. Some of them contain a new mirror game axiom.
3
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Some values for constant-sum and bilateral cooperative games

100%
Młodak A.
Applicationes Mathematicae
|
2007
|
tom 34
|
nr 3
359-371
EN
We prove new axiomatizations of the Shapley value and the Banzhaf value, defined on the class of nonnegative constant-sum games with nonzero worth of the grand coalition as well as on nonnegative bilateral games with nonzero worth of the grand coalition. A characteristic feature of the latter class of cooperative games is that for such a game any coalition and its complement in the set of all players have the same worth. The axiomatizations are then generalized to the entire class of constant-sum or bilateral games, respectively. Moreover, a new axiomatization of the Deegan-Packel value on the set of all cooperative games is presented and possibilities of creation of its version in those special cases are discussed.
4
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Axiomatization of values of cooperative games using a fairness property

100%
Młodak A.
Applicationes Mathematicae
|
2005
|
tom 32
|
nr 1
69-86
EN
We propose new systems of axioms which characterize four types of values of cooperative games: the Banzhaf value, the Deegan-Packel value, the least square prenucleolus and the least square nucleolus. The common element used in these axiomatizations is a fairness property. It requires that if to a cooperative game we add another game in which two given players are symmetric, then their payoffs change by the same amount. In our analysis we will use an idea applied by R. van den Brink (2001) to obtain an axiomatic characterization of the Shapley value.
5
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Three additive solutions of cooperative games with a priori unions

100%
Młodak A.
Applicationes Mathematicae
|
2003
|
tom 30
|
nr 1
69-87
EN
We analyze axiomatic properties of three types of additive solutions of cooperative games with a priori unions structure. One of these is the Banzhaf value with a priori unions introduced by G. Owen (1981), which has not been axiomatically characterized as yet. Generalizing Owen's approach and the constructions discussed by J. Deegan and E. W. Packel (1979) and L. M. Ruiz, F. Valenciano and J. M. Zarzuelo (1996) we define and study two other solutions. These are the Deegan-Packel value with a priori unions and the least square prenucleolus with a priori unions.
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