Characterizing convexity of games using marginal vectors

Abstract

In this paper we study the relation between convexity of TU games and marginal vectors. We show that if specific marginal vectors are core elements, then the game is convex. We characterize sets of marginal vectors satisfying this property, and we derive the formula for the minimum number of marginal vectors in such sets.

Core-based criterion for extreme supermodular functions

“…for every π ∈ Υ and every i ∈ N. For any π ∈ Υ, the row-vector x m (π, * ) ∈ R N is nothing but what is named in game-theoretical literature the marginal vector of m with respect to π, despite different notation; compare (7) with the definition from [37]. Thus, the entry x m (π, i) can be interpreted as the payoff to the player i ∈ N provided that the distribution of the overall worth m(N) is based on the ordering of players given by the enumeration π ∈ Υ.…”

The cone of supermodular games on finite distributive lattices

“…In particular, the implication from 2. to 1. was proved by Ichiishi in [11]. The necessary and sufficient conditions involving specific marginal vectors can be found in [21].…”

Cooperative Lot Sizing Games in Supply Chains

“…The Bondareva-Shapley Theorem signifies that the core of the profit game v is nonempty if and only if for every minimal balanced collection fS 1 ; : : : ; S k g with balancing weights 1 ; : : : ; k the inequality P k j D1 j v.S j / Ä v.N / holds (see Bondareva 1963 andShapley 1967 for the proof). The studies about marginal vectors and convexity of profit games are advanced by van Velzen et al (2004). Apart from the non-emptiness of the core, another part of literature deals with proving concavity or convexity of specific games, respectively, and other core properties.…”