The Owen and the Owen-Banzhaf values applied to the study of the Madrid Assembly and the Andalusian Parliament in legislature 2015-2019

This work focuses on the Owen value and the Owen-Banzhaf value, two classical concepts of solution defined on games with structure of coalition blocks. We provide a computation procedure for these solutions based on a method of double-level work obtained from the multilinear extension of the original game. Moreover, two applications to several possible political situations in the Madrid Assembly and the Andalusian Parliament (legislatures 2015-2019) are also given.Peer ReviewedPostprint (published version

Two classes of weighted values for coalition structures with extensions to level structures

In this paper we introduce two new classes of weighted values for coalition structures with related extensions to level structures. The values of both classes coincide on given player sets with Harsanyi payoffs and match therefore adapted standard axioms for TU-values which are satisfied by these values. Characterizing elements of the values from the new classes are a new weighted proportionality within components property and a null player out property, but on different reduced games for each class. The values from the first class, we call them weighted Shapley alliance coalition structure values (weighted Shapley alliance levels values), satisfy the null player out property on usual reduced games. By contrast, the values from the second class, named as weighted Shapley collaboration coalition structure values (weighted Shapley collaboration levels values) have this property on new reduced games where a component decomposes in components of lower levels (these are singletons in a coalition structure) if one player of this component is removed from the game. The first class contains the Owen value (Shapley levels value) and the second class includes a new extension of the Shapley value to coalition structures (level structures) as a special case

Two classes of weighted values for coalition structures with extensions to level structures

In this paper we introduce two new classes of weighted values for coalition structures with related extensions to level structures. The values of both classes coincide on given player sets with Harsanyi payoffs and match therefore adapted standard axioms for TU-values which are satisfied by these values. Characterizing elements of the values from the new classes are a new weighted proportionality within components property and a null player out property, but on different reduced games for each class. The values from the first class, we call them weighted Shapley alliance coalition structure values (weighted Shapley alliance levels values), satisfy the null player out property on usual reduced games. By contrast, the values from the second class, named as weighted Shapley collaboration coalition structure values (weighted Shapley collaboration levels values) have this property on new reduced games where a component decomposes in the components of the next lower level if one player of this component is removed from the game. The first class contains as a special case the Owen value (Shapley levels value) and the second class includes a new extension of the Shapley value to coalition structures (level structures) as a special case

Two classes of weighted values for coalition structures with extensions to level structures

In this paper we introduce two new classes of weighted values for coalition structures with related extensions to level structures. The values of both classes coincide on given player sets with Harsanyi payoffs and match therefore adapted standard axioms for TU-values which are satisfied by these values. Characterizing elements of the values from the new classes are a new weighted proportionality within components property and a null player out property, but on different reduced games for each class. The values from the first class, we call them weighted Shapley alliance coalition structure values (weighted Shapley alliance levels values), satisfy the null player out property on usual reduced games. By contrast, the values from the second class, named as weighted Shapley collaboration coalition structure values (weighted Shapley collaboration levels values) have this property on new reduced games where a component decomposes in the components of the next lower level if one player of this component is removed from the game. The first class contains as a special case the Owen value (Shapley levels value) and the second class includes a new extension of the Shapley value to coalition structures (level structures) as a special case

Bisemivalues for bicooperative games

We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues in the context of cooperative games. Moreover, the notion of induced bisemivalues on lower cardinalities also makes sense and an adaptation of Dragan’s recurrence formula is obtained. For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.Peer ReviewedPostprint (author's final draft

Owen coalitional value without additivity axiom

We show that the Owen value for TU games with coalition structure can be characterized without additivity axiom similarly as it was done by Young for the Shapley value for general TU games. Our axiomatization via four axioms of efficiency, marginality, symmetry across coalitions, and symmetry within coalitions is obtained from the original Owen's one by replacement of the additivity and null-player axioms via marginality. We show that the alike axiomatization for the generalization of the Owen value suggested by Winter for games with level structure is valid as well

Supply chain collaboration

In the past, research in operations management focused on single-firm analysis. Its goal was to provide managers in practice with suitable tools to improve the performance of their firm by calculating optimal inventory quantities, among others. Nowadays, business decisions are dominated by the globalization of markets and increased competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in operations management has shifted its focus from single-firm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms’ decisions ask for alignment and coordination of actions. Therefore, game theory, the study of situations of cooperation or conflict among heterogenous actors, is very well suited to deal with these interactions. This has been recognized by researchers in the field, since there are an ever increasing number of papers that applies tools, methods and models from game theory to supply chain problems

Coalitional power indices applied to voting systems

We describe voting mechanisms to study voting systems. The classical power indices applied to simple games just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions among parties, players or voters. We measure the power of each party, player or voter when there are coalitions among them. In particular, we study real situations of voting systems using extended Shapley–Shubik and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to compute them.Peer ReviewedPostprint (published version

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.Cooperative game; Semivalue; Power; Potential; Multilinear extension

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes