TY - CHAP

T1 - Generalized proportional solutions

AU - Naumova, Natalia I.

PY - 2014/4/1

Y1 - 2014/4/1

N2 - The paper is a survey of results on goal programming problems, which were obtained in Leningrad-St.Petersburg state university. Goal programming problem is determined by a feasible set of vectors and a goal point ; its aim is to select a feasible point the most close to the goal point in certain sense. A particular case of the goal programming problem is the claim problem, when a given amount of resource must be distributed among agents with unequal claims on the resource. The paper describes axioms on a selection rule that guarantee the solution of the claim problem under the selection rule of the goal programming problem to be equal sacrifice with respect to a certain family of utility functions of agents. For problems with convex relatively closed feasible sets, the general selection rule minimizes the proximity measure to the goal point given by the sum of anti-derivatives of differences between utility functions. The least square solution and maximal weighted entropy solution are obtained under additional axioms. For generalized claim problem, each member of a fixed collection of coalitions of agents A has its claim. Several generalizations of the proportional method for claim problems are examined. Conditions on A that provide existence, coincidence, and inclusion results of generalizations of the proportional solution are obtained. Cooperative games are treated as generalized claim problems, where A consists of all nonempty coalitions of agents. This permits to apply results on goal programming problem to justify special solutions of cooperative games: the weighted entropy solution, the least square solution, the Shapley value.

AB - The paper is a survey of results on goal programming problems, which were obtained in Leningrad-St.Petersburg state university. Goal programming problem is determined by a feasible set of vectors and a goal point ; its aim is to select a feasible point the most close to the goal point in certain sense. A particular case of the goal programming problem is the claim problem, when a given amount of resource must be distributed among agents with unequal claims on the resource. The paper describes axioms on a selection rule that guarantee the solution of the claim problem under the selection rule of the goal programming problem to be equal sacrifice with respect to a certain family of utility functions of agents. For problems with convex relatively closed feasible sets, the general selection rule minimizes the proximity measure to the goal point given by the sum of anti-derivatives of differences between utility functions. The least square solution and maximal weighted entropy solution are obtained under additional axioms. For generalized claim problem, each member of a fixed collection of coalitions of agents A has its claim. Several generalizations of the proportional method for claim problems are examined. Conditions on A that provide existence, coincidence, and inclusion results of generalizations of the proportional solution are obtained. Cooperative games are treated as generalized claim problems, where A consists of all nonempty coalitions of agents. This permits to apply results on goal programming problem to justify special solutions of cooperative games: the weighted entropy solution, the least square solution, the Shapley value.

KW - Bargaining problem

KW - Claim problem

KW - Cooperative games

KW - Goal programming

KW - Proportional solution

KW - Weighted entropy

UR - http://www.scopus.com/inward/record.url?scp=84956779677&partnerID=8YFLogxK

M3 - Chapter

AN - SCOPUS:84956779677

SN - 9781631170737

SP - 123

EP - 143

BT - Advances in Economics and Optimization

PB - Nova Science Publishers, Inc.

ER -