All assignment games with the same core have the same nucleolus

Universitat de Barcelona. Espai de Recerca en Economia, Working Papers in Economics 01/2004;
Source: RePEc

ABSTRACT There exist coalitional games with transferable utility which have the same core but different nucleoli. We show that this cannot happen in the case of assignment games. Whenever two assignment games have the same core, their nucleoli also coincide. To show this, we prove that the nucleolus of an assignment game coincides with that of its buyerseller exact representative.

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    ABSTRACT: Assignment games with side payments are models of certain two-sided markets. It is known that prices which competitively balance supply and demand correspond to elements in the core. The nucleolus, lying in the lexicographic center of the nonempty core, has the additional property that it satisfies each coalition as much as possible. The corresponding prices favor neither the sellers nor the buyers, hence provide some stability for the market.An algorithm is presented that determines the nucleolus of an assignment game. It generates a finite number of payoff vectors, monotone increasing on one side, and decreasing on the other. The decomposition of the payoff space and the lattice-type structure of the feasible set are utilized in associating a directed graph. Finding the next payoff is translated into determining the lengths of longest paths to the nodes, if the graph is acyclic, or otherwise, detecting the cycle(s). In an (m,n)-person assignment game withm = min(m,n) the nucleolus is found in at most 1/2m(m + 3) steps, each one requiring at mostO(mn) elementary operations.
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    ABSTRACT: In the assignment game framework, we try to identify those assignment matrices in which no entry can be increased without changing the core of the game. These games will be called buyer-seller exact games and satisfy the condition that each mixed-pair coalition attains the corresponding matrix entry in the core of the game. For a given assignment game, a unique buyer-seller exact assignment game with the same core is proved to exist. In order to identify this matrix and to provide a characterization of those assignment games which are buyer-seller exact in terms of the assignment matrix, attainable upper and lower core bounds for the mixed-pair coalitions are found. As a consequence, an open question posed in Quint (1991) regarding a canonical representation of a “45o-lattice” by means of the core of an assignment game can now be answered.
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