Converse Reduced Game Property Research Articles

EGALITARIAN SOLUTION FOR GAMES WITH DISCRETE SIDE PAYMENT

In this paper, we study the egalitarian solution for games with discrete side payment, where the characteristic function is integer-valued and payoffs of players are integral vectors. The egalitarian solution, introduced by Dutta and Ray in 1989, is a solution concept for transferable utility cooperative games in characteristic form, which combines commitment for egalitarianism and promotion of indivisual interests in a consistent manner. We first point out that the nice properties of the egalitarian solution (in the continuous case) do not extend to games with discrete side payment. Then we show that the Lorenz stable set, which may be regarded a variant of the egalitarian solution, has nice properties such as the Davis and Maschler reduced game property and the converse reduced game property. For the proofs we utilize recent results in discrete convex analysis on decreasing minimization on an M-convex set investigated by Frank and Murota.

Characterizations of solutions for games with precedence constraints

We generalize the characterizations of the positive core and the positive prekernel to TU games with precedence constraints and show that the positive core is characterized by non-emptiness (NE), boundedness (BOUND), covariance under strategic equivalence, closedness (CLOS), the reduced game property (RGP), the reconfirmation property (RCP) for suitably generalized Davis–Maschler reduced games, and the possibility of nondiscrimination. The bounded positive core, i.e., the union of all bounded faces of the positive core, is characterized similarly. Just RCP has to be replaced by a suitable weaker axiom, a weak version of CRGP (the converse RGP) has to be added, and CLOS can be deleted. For classical games the prenucleolus is the unique further solution that satisfies the axioms, but for games with precedence constraints it violates NE as well as the prekernel. The positive prekernel, however, is axiomatized by NE, anonymity, reasonableness, the weak RGP, CRGP, and weak unanimity for two-person games (WUTPG), and the bounded positive prekernel is axiomatized similarly by requiring WUTPG only for classical two-person games and adding BOUND.

The Positive Core for Games with Precedence Constraints

We generalize the characterizations of the positive core and the positive prekernel to TU games with precedence constraints and show that the positive core is characterized by non-emptiness (NE), boundedness (BOUND), covariance under strategic equivalence, closedness (CLOS), the reduced game property (RGP), the reconfirmation property (RCP) for suitably generalized Davis-Maschler reduced games, and the possibility of nondiscrimination. The bounded positive core, i.e., the union of all bounded faces of the positive core, is characterized similarly. Just RCP has to be replaced by a suitable weaker axiom, a weak version of CRGP (the converse RGP) has to be added, and CLOS can be deleted. For classical games the prenucleolus is the unique further solution that satisfies the axioms, but for games with precedence constraints it violates NE as well as the prekernel. The positive prekernel, however, is axiomatized by NE, anonymity, reasonableness, the weak RGP, CRGP, and weak unanimity for two-person games (WUTPG), and the bounded positive prekernel is axiomatized similarly by requiring WUTPG only for classical two-person games and adding BOUND.

The bounded core for games with precedence constraints

An element of the possibly unbounded core of a cooperative game with precedence constraints belongs to its bounded core if any transfer to a player from any of her subordinates results in payoffs outside the core. The bounded core is the union of all bounded faces of the core, it is nonempty if the core is nonempty, and it is a continuous correspondence on games with coinciding precedence constraints. If the precedence constraints generate a connected hierarchy, then the core is always nonempty. It is shown that the bounded core is axiomatized similarly to the core for classical cooperative games, namely by boundedness (BOUND), nonemptiness for zero-inessential two-person games (ZIG), anonymity, covariance under strategic equivalence (COV), and certain variants of the reduced game property (RGP), the converse reduced game property (CRGP), and the reconfirmation property. The core is the maximum solution that satisfies a suitably weakened version of BOUND together with the remaining axioms. For games with connected hierarchies, the bounded core is axiomatized by BOUND, ZIG, COV, and some variants of RGP and CRGP, whereas the core is the maximum solution that satisfies the weakened version of BOUND, COV, and the variants of RGP and CRGP.

Flow sharing and bankruptcy games

We discuss the relationship between bankruptcy problems and flow sharing problems, respectively, and show that the latter type of problem can be interpreted as a generalization of the former. The corresponding flow sharing game is convex, hence we can use the converse reduced game property to characterize the nucleolus and the constrained egalitarian solution. Our main contribution is thus to introduce the nucleolus as an alternative solution concept for flow sharing problems, and to offer a game-theoretic interpretation of the traditional egalitarian solution.

A Note on an Axiomatization of the Core of Market Games

As shown by Peleg (1993), the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TU game, it is shown that weak symmetry is redundant in this result. Hence, the core on market games is axiomatized by the remaining five properties, if the universe of players contains at least four members. (This abstract was borrowed from another version of this item.)

A Note on an Axiomatization of the Core of Market Games

As shown by Peleg (1993), the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TU game, it is shown that weak symmetry is redundant in this result. Hence, the core on market games is axiomatized by the remaining five properties, if the universe of players contains at least four members.

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set.

Axiomatizations of the core on the universal domain and other natural domains

We prove that the core on the set of all transferable utility games with players contained in a universe of at least five members can be axiomatized by the zero inessential game property, covariance under strategic equivalence, anonymity, boundedness, the weak reduced game property, the converse reduced game property, and the reconfirmation property. These properties also characterize the core on certain subsets of games, e.g., on the set of totally balanced games, on the set of balanced games, and on the set of superadditive games. Suitable extensions of these properties yield an axiomatization of the core on sets of nontransferable utility games.

THE POSITIVE PREKERNEL OF A COOPERATIVE GAME

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.

An axiomatization of the core for finite and continuum games

We provide a new axiomatization of the core of games in characteristic form. The games may have either finite sets of players or continuum sets of players and finite coalitions. Our research is based on Peleg's axiomatization for finite games and on the notions of measurement-consistent partitions and the f-core introduced by Kaneko and Wooders. Since coalitions are finite in both finite games and in continuum games, we can use the reduced game property and the converse reduced game property for our axiomatization. Both properties are particularly appealing in large economies.

An Axiomatization of the Core of Market Games

We provide an axiomatization of the core of market games (totally balanced coalitional games). We prove that the core is fully characterized by the following five independent properties: (i) nonemptiness; (ii) individual rationality; (iii) superadditivity; (iv) the weak reduced game property; and (v) the converse reduced game property. Also, we consider a possible alternative characterization.

An axiomatization of the core of cooperative games without side payments

Let M be a set of m players, m≧3, and let Γ be the set of all (finite) games (without side payments) that have a non-empty core. When M is finite, the following four (independent) axioms fully characterize the core on Γ: (i) non-emptiness, (ii) individual rationality, (iii) the reduced game property, and (iv) the converse reduced game property. If M is infinite, then the converse reduced game property is redundant.