On the Difficulty of Budget Allocation in Claims Problems with Indivisible Items and Prices

First of all, the authors thank the Associate Editor and three anonymous Reviewers for their time to review our paper and also for their for their incisive comments and suggestions which have been very helpful to improve the contents of the paper. The authors gratefully acknowledge financial support from the Ministerio de Ciencia, Innovacion y Universidades (MCIU), the Agencia Estatal de Investigacion (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-097965-B-I00 and the Spanish Ministry of Science under Project ECO2017-86245-P, as well as Junta de Andalucia under Projects Grupos PAIDI SEJ426 and project P18-FR-2933.In this paper we study the class of claims problems where the amount to be divided is perfectly divisible and claims are made on indivisible units of several items. Each item has a price, and the available amount falls short to be able to cover all the claims at the given prices. We propose several properties that may be of interest in this particular framework. These properties represent the common principles of fairness, efficiency, and non-manipulability by merging or splitting. Efficiency is our focal principle, which is formalized by means of two axioms: non-wastefulness and Pareto efficiency. We show that some combinations of the properties we consider are compatible, others are not.Ministerio de Ciencia, Innovacion y Universidades (MCIU)Agencia Estatal de Investigacion (AEI)European Commission PGC2018-097965-B-I00Spanish Government ECO2017-86245-PJunta de Andalucia PAIDI SEJ426 P18-FR-293

On horizontal cooperation in linear production processes with a supplier that controls a limited resource

In this paper we consider a two-echelon supply chain with one supplier that controls a limited resource and a finite set of manufacturers who need to purchase this resource. We analyze the effect of the limited resource on the horizontal cooperation of manufacturers. To this end, we use cooperative game theory and the existence of stable distributions of the total profit among the manufacturers as a measure of the possibilities of cooperation. The game theoretical model that describes the horizontal cooperation involves externalities, which arise because of the possible scarcity of the limited resource and the possible coalition structures that can be formed. Furthermore, manufacturers do not know how the supplier will allocate the limited resource, therefore, how much of this resource they will obtain is uncertain for all concerned. Nevertheless, when the limited resource is not scarce for the grand coalition, the existence of stable distributions of the total profit is guaranteed and consequently the collaboration among the manufacturers is profitable for them all. In the event that the limited resource is insufficient for the grand coalition, we introduce a new cooperative game that assesses the expectations of each coalition of manufacturers regarding the amount of the limited resource they can obtain. We analyze two extreme expectations: the optimistic and the pessimistic. In the optimistic case, we cannot reach a conclusion regarding the full cooperation of the manufacturers. In the pessimistic case, with one reasonable assumption, the existence of stable distributions of the total profit is guaranteed and as a result the collaboration among manufacturers is a win–win deal.Ministerio de Economía y Competitividad | Ref. MTM2014-54199-PMinisterio de Economía y Competitividad | Ref. MTM2014-53395-C3-3-PFundación Séneca | Ref. 19320/PI/1

Allocating costs in set covering problems

This paper deals with the problem of allocating costs in set covering situations. In particular, we focus on set covering situations where the optimal covering is given in advance. Thus, we take into account only the facilities that have to be opened and look for rules distributing their cost. We define a cooperative game and study the core and the nucleolus. We also introduce two new rules: the equal split rule on facilities and the serial rule. We axiomatically characterize the core, the nucleolus, and the two rules. Finally, we study several monotonicity properties of the rules

Linear (semi-)infinite programs and cooperative games

In 1975 Stef Tijs defended his Ph.D. thesis entitled “Semi-infinite and infinite matrix games and bimatrix games”. Following this, his paper “Semi-infinite linear programs and semi-infinite matrix games” was pub- lished in 1979. Both these works deal with programs and noncoopera- tive games in a (semi-)infinite setting. Several decades later these works and Stef Tijs himself inspired some researchers from Italy, Spain and The Netherlands to study cooperative games arising from linear (semi) infinite programs. These studies were performed under the inspiring supervision of Stef Tijs

Games Arising from Infinite Production Situations

OWEN (1975) introduced linear production (LP) situations and TIMMER, BORM and SUIJS (1998) introduced more general situations involving the linear transformation of products (LTP). They showed that the corresponding LTP games are totally balanced. In this paper we look at LTP situations with an infinite number of transformation techniques. The linear program that calculates the maximal profit, is a semi-infinite program. We show that an optimal solution of the dual program exists and that it is a core-element of the corresponding LTP game. Journal of Economic Literature Classification Number: C71, C61. 1991 Mathematics Subject Classification Number: 90D12, 90C05. Keywords: linear semi-infinite programs, linear transformation, cooperative games, balancedness. 1 Introduction OWEN (1975) introduced linear production (LP) situations. These are production situations where there is a finite set of producers, each of them owns a bundle of resources and all producers can use the same finite s..

Semi-Infinite Assignment Problems and Related Games

In 1972 Shapley and Shubik introduced assignment games associated to finite assignment problems in which two types of agents were involved and they proved that these games have a non-empty core. In this paper we look at the situation where the set of one type is infinite and investigate when the core of the associated game is non-empty. Two infinite programming problems arise here, which we tackle with the aid of finite approximations. We prove that there is no duality gap and we show that the core of the corresponding game is non-empty. Finally, the existence of optimal assignments is discussed. Keywords: Infinite programs, assignment, cooperative games, balancedness. 1 Introduction Nowadays many markets and transactions are bilateral, so 'two-sided' market models have become widely used in economic theory. Since 1972, when Shapley and Shubik ([9]) introduced finite assignment games, much work related to these games has been developed. We point out the book of Rothand Sotomayor ([7])..

Semi-infintite assignment problems and related games

In this paper we look at semi-infinite assignment problems. These are situations where a finite set of agents of one type has to be assigned to an infinite set of agents of another type. This has to be done in such a way that the total profit arising from these assignments is as large as possible. An infinite programming problem and its dual arise here, which we tackle with the aid of finite approximations. We prove that there is no duality gap and we show that the core of the corresponding game is nonempty. Finally, the existence of optimal assignments is discussed