Pure bargaining problems with a coalition structure

The final publication is available at Springer via http://dx.doi.org/10.1007/s41412-016-0007-2We consider here pure bargaining problems endowed with a coalition structure such that each union is given its own utility. In this context we use the Shapley rule in order to assess the main options available to the agents: individual behavior, cooperative behavior, isolated unions behavior, and bargaining unions behavior. The latter two respectively recall the treatment given by Aumann–Drèze and Owen to cooperative games with a coalition structure. A numerical example illustrates the procedure. We provide criteria to compare any pair of behaviors for each agent, introduce and axiomatically characterize a modified Shapley rule, and determine its natural domain, that is, the set of problems where the bargaining unions behavior is the best option for all agents.Peer ReviewedPostprint (author's final draft

A note about rappels and uniform discounts

Consideramos los descuentos comerciales por compras basados en rappels. Si se plantea un descuento de tipo uniforme aparece una paradoja no deseada: el “truco del comprador”. Para eludirla se proponen dos variantes que respetan el principio básico de “a mayor pedido, mayor descuento”. La primera es la de los descuentos continuos. La segunda, la de los descuentos graduales.We consider commercial discounts for buying based on rappels. If a uniform type discount is taken, then an unwanted paradox appears: the “buyer trick”. To avoid this paradox, two proposals are presented. Both of them observe the basic principle “to a greater order, a greater discount”. The first one is that of continuous discounts. The second consists of following a table of graded discounts.Peer Reviewe

Coalition formation and stability

This paper aims to develop, for any cooperative game, a solution notion that enjoys stability and consists of a coalition structure and an associated payoff vector derived from the Shapley value. To this end, two concepts are combined: those of strong Nash equilibrium and Aumann--Dr\`{e}ze coalitional value. In particular, we are interested in conditions ensuring that the grand coalition is the best preference for all players. Monotonicity, convexity, cohesiveness and other conditions are used to provide several theoretical results that we apply to numerical examples including real--world economic situations.Peer ReviewedPostprint (author's final draft

Semivalues: power, potential and multilinear extensions

The notions of power and potential, both defined for any semivalue, give rise to two endomorphisms of the vector space of all cooperative games on a given player set. Several properties of these linear mappings are stated and their action on unanimity games is emphasized. We also relate in both cases the multilinear extension of the image game to the multilinear extension of the original game

A note on multinomial probabilistic values

This is a post-peer-review, pre-copyedit version of an article published in "TOP". The final authenticated version is available online at: https://doi.org/10.1007/s11750-017-0464-1Multinomial values were previously introduced by one of the authors in reliability and extended later to all cooperative games. Here, we present for this subfamily of probabilistic values three new results, previously stated only for binomial semivalues in the literature. They concern the dimension of the subspace spanned by the multinomial values and two characterizations: one, individual, for each multinomial value; another, collective, for the whole subfamily they form. Finally, an application to simple games is providedPeer ReviewedPostprint (author's final draft

On the axiomatic characterization of the coalitional multinomial probabilistic values

The coalitional multinomial probabilistic values extend the notion of multinomial probabilistic value to games with a coalition structure, in such a way that they generalize the symmetric coalitional binomial semivalues and link and combine the Shapley value and the multinomial probabilistic values. By considering the property of balanced contributions within unions, a new axiomatic characterization is stated for each one of these coalitional values, provided that it is defined by a positive tendency profile, by means of a set of logically independent properties that univocally determine the value. Two applications are also shown: (a) to the Madrid Assembly in Legislature 2015–2019 and (b) to the Parliament of Andalucía in Legislature 2018–2022.This research project was partially supported by funds from the Spanish Ministry of Science and Innovation under grant PID2019-104987GB-I00.Peer ReviewedPostprint (published version

Dimension, egalitarianism and decisiveness of European voting systems

An analysis of three major aspects has been carried out that may apply to any of the successive voting systems used for the European Union Council of Ministers, from the first one established in the Treaty of Rome in 1958 to the current one established in Lisbon. We mainly consider the voting systems designed for the enlarged European Union adopted in the Athens summit, held in April 2003 but this analysis can be applied to any other system. First, it is shown that the dimension of these voting systems does not, in general, reduce. Next, the egalitarian effects of superposing two or three weighted majority games (often by introducing additional consensus) are considered. Finally, the decisiveness of these voting systems is evaluated and compared.Peer ReviewedPostprint (author's final draft

Decisiveness of decisive symmetric games

Reseach supported by Grant SGR 2009-01029 of the Catalonia Government (Generalitat de Catalunya) and Frants MTM 2006-06064 and MTM 2009-08037 of the Science and Innovation Spanish Ministry and the European Regional Development Fund.Binary voting systems, usually represented by simple games, constitute a main DSS topic. A crucial feature of such a system is the easiness with which a proposal can be collectively accepted, which is measured by the "decisiveness index" of the corresponding game. We study here several functions related to the decisiveness of any simple game. The analysis, including the asymptotic behavior as the number n of players increases, is restricited to decisive symmetric gammes and their compositions, and it is assumed that all players have a common probability p to vote for the proposal. We show that, for n large enough, a small variation, either positive or negaive, in p when p=1/2 takes the decisiveness to quickly approach , respectively, 1 or 0. Moreover, we analyze the speed of the decisiveness convergence.Preprin

The proportional partitional Shapley value

A new coalitional value is proposed under the hypothesis of isolated unions. The main difference between this value and the Aumann–Drèze value is that the allocations within each union are not given by the Shapley value of the restricted game but proportionally to the Shapley value of the original game. Axiomatic characterizations of the new value, examples illustrating its application and a comparative discussion are provided.Peer ReviewedPostprint (author’s final draft

Multinomial probabilistic values

Multinomial probabilistic values were introduced by one of us in reliability. Here we define them for all cooperative games and illustrate their behavior in practice by means of an application to the analysis of a political problem.Peer ReviewedPostprint (author’s final draft