Resumen de Buyer-seller exactness in the assignment game

Marina Núñez , Carles Rafels Pallarola

• Assignment games associated to different assignment matrices may have the same core (no examples will be found in 2 × 2 matrices). Nevertheless, among all matrices leading to the same core, we are interested in those which have maximal entries. We say an assignment game is buyer-seller exact if each mixed-pair coalition attains the corresponding matrix entry in the core of the game. When an assignment game is buyer-seller exact, no matrix entry can be increased without modifying the core.

  A representation result is obtained: for a given assignment game (N,w) defined by matrix A, a unique buyer-seller exact assignment game Ar with the same core is proved to exist. The proof makes use of the characterization of the extreme core allocations of the assignment game (Núñez and Rafels, 2001).

  Once existence is proved, we would like to characterize those assignment games which are buyer-seller exact in terms of the matrix entries, and, at the same time, provide a method to compute Ar for a given matrix A. In Solymosi and Raghavan (2001) exact assignment games are characterized in terms of matrix properties: an assignment game is exact if its matrix is dominant diagonal and doubly dominant diagonal. But for a given assignment game, the exact game with the same core which does exist, might not be an assignment game, this is why buyer-seller exactness seems more suitable for assignment games.

  In order to identify the matrix of the buyer-seller exact game related to a given assignment game, expressions for attainable upper and lower core bounds for mixed pair coalitions are found: the maximum joint core payoff of buyer i and seller j is w(N) - w(N{i, j}).

  A new property called strongly dominant diagonal is introduced. All strongly dominant diagonal matrices are doubly dominant diagonal. Finally, an assignment game is proved to be buyer-seller exact if and only if its matrix is strongly dominant diagonal.

  Some consequences will be deduced. First an open question posed by Quint (1991) searching for a canonical representation of a 45 degrees lattice by means of the core of an assignment game can now be answered. The second consequence is a characterization of those assignment matrices such that there is no other one leading to the same core. Finally we analyze when a buyer-seller exact game is exact: this happens when the correspondent matrix, in addition to being strongly dominant diagonal, is dominant diagonal.