On the reduced game property for and the axiomatization of theT-value of TU-games

The positive prekernel, a solution of cooperative transferable utility games, is introduced. We show that this solution inherits many properties of the prekernel and of the core, which are both sub-solutions. It coincides with its individually rational variant, the positive kernel, when applied to any zero-monotonic game. The positive (pre)kernel is a sub-solution of the reactive (pre)bargaining set. We prove that the positive prekernel on the set of games with players belonging to a universe of at least three possible members can be axiomatized by non-emptiness, anonymity, reasonableness, the weak reduced game property, the converse reduced game property, and a weak version of unanimity for two-person games.

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Reduced Game Property of Linear Values with Equal Treatment Property

Operations Research/Management Science at Work - International Series in Operations Research & Management Science ◽

10.1007/978-1-4615-0819-9_20 ◽

2002 ◽

pp. 317-332

Author(s):

Tsuneyuki Namekata ◽

Theo S.H. Driessen

Keyword(s):

Equal Treatment ◽

Reduced Game ◽

Game Property

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NONEMPTY CORE-TYPE SOLUTIONS OVER BALANCED COALITIONS IN TU-GAMES

International Game Theory Review ◽

10.1142/s0219198912500181 ◽

2012 ◽

Vol 14 (03) ◽

pp. 1250018 ◽

Cited By ~ 4

Author(s):

JUAN C. CESCO

Keyword(s):

Equal Opportunity ◽

Axiomatic Characterization ◽

Individual Rationality ◽

Transferable Utility ◽

Tu Game ◽

Tu Games ◽

Additional Axiom ◽

Game Property ◽

Minimality Property ◽

New Framework

In this paper we introduce two related core-type solutions for games with transferable utility (TU-games) the [Formula: see text]-core and the [Formula: see text]-core. The elements of the solutions are pairs [Formula: see text] where x, as usual, is a vector representing a distribution of utility and [Formula: see text] is a balanced family of coalitions, in the case of the [Formula: see text]-core, and a minimal balanced one, in the case of the [Formula: see text]-core, describing a plausible organization of the players to achieve the vector x. Both solutions extend the notion of classical core but, unlike it, they are always nonempty for any TU-game. For the [Formula: see text]-core, which also exhibits a certain kind of "minimality" property, we provide a nice axiomatic characterization in terms of the four axioms nonemptiness (NE), individual rationality (IR), superadditivity (SUPA) and a weak reduced game property (WRGP) (with appropriate modifications to adapt them to the new framework) used to characterize the classical core. However, an additional axiom, the axiom of equal opportunity is required. It roughly states that if [Formula: see text] belongs to the [Formula: see text]-core then, any other admissible element of the form [Formula: see text] should belong to the solution too.

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A Reduced Game Property for the Proportional Solution for Claims Problems

American Journal of Mathematical and Management Sciences ◽

10.1080/01966324.2000.10737522 ◽

2000 ◽

Vol 20 (3-4) ◽

pp. 403-413

Author(s):

Somdeb Lahiri

Keyword(s):

Claims Problems ◽

Reduced Game ◽

Proportional Solution ◽

Game Property

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CONSISTENCY FOR PROPORTIONAL SOLUTIONS

International Game Theory Review ◽

10.1142/s0219198902000744 ◽

2002 ◽

Vol 04 (03) ◽

pp. 343-356 ◽

Cited By ~ 1

Author(s):

ELENA YANOVSKAYA

Keyword(s):

Cooperative Game ◽

Solution Set ◽

Original Game ◽

Payoff Vector ◽

Tu Games ◽

Reduced Game ◽

Reduced Games ◽

Consistency Property ◽

Finite Set

One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual Davis–Maschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.

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