The core of games on distributive lattices : how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, it is not obvious to define a suitable notion of core, reflecting the team structure, and previous attempts are not satisfactory in this respect. We propose a new notion of core, which imposes efficiency of the allocation at each level of the hierarchy, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.Cooperative game, feasible coalition, core, hierarchy.

The core of games on ordered structures and graphs

In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

ED EPSInternational audienceFinding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness

Games on concept lattices: Shapley value and core

We introduce cooperative TU-games on concept lattices, where a concept is a pair (S, S′ ) with S being a subset of players or objects, and S ′ a subset of attributes. Any such game induces a game on the set of players/objects, which appears to be a TU-game whose collection of feasible coalitions is a lattice closed under intersection, and a game on the set of attributes. We propose a Shapley value for each type of game, axiomatize it, and investigate the geometrical properties of the core (nonemptiness, boundedness, pointedness, extremal rays).Agence Nationale de la Recherche ANR-13-BSHS1-001

A new approach to the core and Weber set of multichoice games

Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that the set of imputations may be unbounded, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex compact set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their coincidence in the convex case remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.multichoice game ; lattice ; core

A new approach to the core and Weber set of multichoice games

ED EPSInternational audienceMultichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that the set of imputations may be unbounded, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex compact set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their coincidence in the convex case remain valid. A last section makes a comparison with the core defined by van den Nouweland et al

Lattices and discrete methods in cooperative games and decisions

Questa tesi si pone l'obiettivo di presentare la teoria dei giochi, in particolare di quelli cooperativi, insieme alla teoria delle decisioni, inquadrandole formalmente in termini di matematica discreta. Si tratta di due campi dove l'indagine si origina idealmente da questioni applicative, e dove tuttavia sono sorti e sorgono problemi più tipicamente teorici che hanno interessato e interessano gli ambienti matematico e informatico. Anche se i contributi iniziali sono stati spesso formulati in ambito continuo e utilizzando strumenti tipici di teoria della misura, tuttavia oggi la scelta di modelli e metodi discreti appare la più idonea. L'idea generale è quindi quella di guardare fin da subito al complesso dei modelli e dei risultati che si intendono presentare attraverso la lente della teoria dei reticoli. Ciò consente di avere una visione globale più nitida e di riuscire agilmente ad intrecciare il discorso considerando congiuntamente la teoria dei giochi e quella delle decisioni. Quindi, dopo avere introdotto gli strumenti necessari, si considerano modelli e problemi con il fine preciso di analizzare dapprima risultati storici e solidi, proseguendo poi verso situazioni più recenti, più complesse e nelle quali i risultati raggiunti possono suscitare perplessità. Da ultimo, vengono presentate alcune questioni aperte ed associati spunti per la ricerca

Subcalculus for set functions and cores of TU games.

This paper introduces a subcalculus for general set functions and uses this framework to study the core of TU games. After stating a linearity theorem, we establish several theorems that characterize mea- sure games having finite-dimensional cores. This is a very tractable class of games relevant in many economic applications. Finally, we show that exact games with Þnite dimensional cores are generalized linear production games.TU games; non-additive set functions; subcalculus; cores