On properties of the set of awards vectors for a claims problem

• Miguel Ángel Mirás Calvo ^[1] ; Iago Núñez Lugilde ^[1] ; Carmen Quinteiro Sandoming ^[1] ; Estela Sánchez Rodríguez ^[1]
  1. [1] Universidade de Vigo

     Universidade de Vigo

     Vigo, España

     
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 32, Nº. 1, 2024, págs. 137-167
• Idioma: inglés
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• Resumen

  • We study the geometric structure of a particular type of nonempty convex polytopes that are the intersection of an n-rectangle with a hyperplane x1 + ⋯ + xn = E, E > 0. This type of polytopes arise naturally when studying, for instance, the set of awards vectors for a claims problem, the core of the game associated with a bankruptcy problem, the core-cover set of a game, or the class of two-bound core games. We explore in detail the geometry of such a polytope and provide explicit expressions to compute its volume and its centroid. In particular, we describe a procedure to compute the average-of-awards rule for a claims problem directly from the parameters of the problem. We show that computing the average-of-awards rule is # P-complete.