A comparison of the average prekernel and the prekernel

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral setting

A COMPARISON OF THE AVERAGE PREKERNEL AND THE PREKERNEL

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral settings

Transfer Rate Rules and Core Selections in NTU Games.

Dans cette note, on propose quelques applications directes d’un résultat d’existence de Bonnisseau et Iehlé (2003). Ces auteurs ont montré l’existence d’allocations du cœur dans les jeux NTU qui satisfont un équilibre de taux de transfert sous une condition de balancement dépendant. Il s’avère que la notion de balancement dépendant procure en fait un outil manipulable pour sélectionner le cœur. Pour illustrer ce fait, nous montrons que cette notion permet d’obtenir des résultats d’existence dans des modèles de cœur avec partenariat, cœur socialement stable, prekernel moyen intersecté avec le cœur et de cœur interne faible.Different kinds of asymmetries between players can occur in core allocations, in that case the stability of the concept is questioned. One remedy consists in selecting robust core allocations. We review, in this note, results that all select core allocations in NTU games with different concepts of robustness. Within a unified approach, we deduce the existence of allocations in: the partnered core, the social stable core, the core intersected with average prekernel, the weak inner core. We use a recent contribution of Bonnisseau and Iehle (2003) that states the existence of core allocations with a transfer rate rule equilibrium under a dependent balancedness assumption. It shall turn out to be manipulable tools for selecting the core.Sélections du coeur dans les jeux NTU; Balancement dépendant; Jeux coopératifs; Core Selections in NTU Games; Dependent balancedness; Cooperative game;

Transfer Rate Rules and Core Selections in NTU Games

Different kinds of asymmetries between players can occur in core allocations, in that case the stability of the concept is questioned. One remedy consists in selecting robust core allocations. We review, in this note, results that all select core allocations in NTU games with different concepts of robustness. Within a unified approach, we deduce the existence of allocations in: the partnered core, the social stable core, the core intersected with average prekernel, the weak inner core. We use a recent contribution of Bonnisseau and Iehlé (2003) that states the existence of core allocations with a transfer rate rule equilibrium under a dependent balancedness assumption. It shall turn out to be manipulable tools for selecting the core.Cooperative games, dependent balancedness, core selections in NTU games.

A comparison of the average prekernel and the prekernel.

We propose positive and normative foundations for the average prekernel of NTU games, and compare them with the existing ones for the prekernel. In our non-cooperative analysis, the average prekernel is approximated by the set of equilibrium payoffs of a game where each player faces the possibility of bargaining at random against any other player. In the cooperative analysis, we characterize the average prekernel as the unique solution that satisfies a set of Nash-like axioms for two-person games, and versions of average consistency and its converse for multilateral settings

The Value for Actions-Set Games

Action-Set games are transferable utility games where the set of players is finite, every player has a finite set of actions, and the worth of the game is a function of the actions taken by the players. In this setting a rule has to determine individual payoffs at each combinations of actions. Following an axiomatic approach, we define the set of Consistent Bargaining Equilibria.Action-set games, Shapley value, Prekernel, Consistent Bargaining Equilibria

An axiomatization of the prekernel of nontransferable utility games

We characterize the prekernel of NTU games by means of consistency, converse consistency, and five axioms of the Nash type on bilateral problems. The intersection of the prekernel and the core is also characterized with the same axioms over the class of games where the core is nonempty.Prekernel, NTU games, consistency, converse consistency

Payoffs-dependent Balancedness and Cores

We provide a result for non-emptiness of the core in NTU games. We use a payoffs-dependent balancedness condition, based on transfer rate mappings. Going beyond the non-emptiness of standard core, existence of some refined solution is proved, including specific core allocations and equilibrium-core allocations in parameterized collection of cooperative games. The proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non convexities. Applications to various extant results taken from game theory and economic theory are given.Cooperative games, Core solutions, Non-emptiness

The Nucleolus, the Kernel, and the Bargaining Set: An Update

One of David Schmeidler’s many important contributions in his distinguished career was the introduction of the nucleolus, one of the central single-valued solution concepts in cooperative game theory. This paper is an updated survey on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. As a first approach to these concepts, we refer the reader to the great survey by Maschler (1992); see also the relevant chapters in Peleg and Sudholter (2003). Building on the notes of four lectures on the nucleolus and the kernel delivered by one of the authors at the Hebrew University of Jerusalem in 1999, we have updated Maschler’s survey by adding more recent contributions to the literature. Following a similar structure, we have also added a new section that covers the bargaining set. The nucleolus has a number of desirable properties, including nonemptiness, uniqueness, core selection, and consistency. The first way to understand it is based on an egalitarian principle among coalitions. However, by going over the axioms that characterize it, what comes across as important is its connection with coalitional stability, as formalized in the notion of the core. Indeed, if one likes a single-valued version of core stability that always yields a prediction, one should consider the nucleolus as a recommendation. The kernel, which contains the nucleolus, is based on the idea of “bilateral equilibrium” for every pair of players. And the bargaining set, which contains the kernel, checks for the credibility of objections coming from coalitions. In this paper, section 2 presents preliminaries, section 3 is devoted to the nucleolus, section 4 to the kernel, and section 5 to the bargaining set.Iñarra acknowledges research support from the Spanish Government grant ECO2015-67519-P, and Shimomura from Grant-in-Aid for Scientific Research (A)18H03641 and (C)19K01558

Payoff-dependent balancedness and cores (revised version)

We prove the non-emptiness of the core of an NTU game satisfying a condition of payoff-dependent balancedness, based on transfer rate mappings. We also define a new equilibrium condition on transfer rates and we prove the existence of core payoff vectors satisfying this condition. The additional requirement of transfer rate equilibrium refines the core concept and allows the selection of specific core payoff vectors. Lastly, the class of parametrized cooperative games is introduced. This new setting and its associated equilibrium-core solution extend the usual cooperative game framework and core solution to situations depending on an exogenous environment. A non-emptiness result for the equilibrium-core is also provided in the context of a parametrized cooperative game. Our proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non convexities. Applications to extant results taken from game theory and economic theory are given.balancedness, cooperative game, core, parametrized game