On some balanced, totally balanced and submodular delivery games

Abstract

On some balanced, totally balanced and submodular delivery games Granot, D.; Hamers, Herbert; Tijs, S.H.

On games arising from multi-depot Chinese postman problems

“…Granot et al (1999) show that all weakly cyclic graphs are globally 1-CP submodular, but for the case of k > 1, a weakly cyclic graph is not necessarily globally k-CP submodular, as the following example shows. This example also illustrates how different choices of Q with the same cardinality may lead to different games.…”

“…Follows by the definition of locally and globally k-CP balanced graphs. (iii) → (i): Granot et al (1999) show that any locally 1-CP balanced graph is weakly Eulerian. We therefore consider |Q| > 1.…”

“…We find that the class of k-CP balanced graphs coincide with the class of CP-balanced graphs characterized in Granot et al (1999). Define an allocation x ∈ R N as:…”

“…Then since c {v} (N ) = c Q (N ) and c {v} (S) ≥ c Q (S), we may infer that Core(c Q ) ⊆ Core(c {v} ). From Granot et al (1999), there exists a t such that Core(c {v} ) = ∅, and therefore Core(c Q ) = ∅. Thus, G is not locally k-CP balanced.…”

“…Hamers et al (1999) introduced CP games and showed that weakly Eulerian graphs are CP-balanced, while Hamers (1997) showed that weakly cyclic graphs are CP-submodular. Full characterizations of the classes of CP-balanced and CP-submodular graphs were given in Granot et al (1999). In Granot et al (2004), the distinction between global and local requirements was made, and the authors characterized the classes of locally CP-submodular graphs and digraphs.…”