arXiv subjects

Alexandre Skoda

Publications and source records attributed to Alexandre Skoda.

4 recordsLinked to original sources

arXiv2022

Weighted Average-convexity and Cooperative Games

We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted average-convexity and cooperative games. First, we prove that if a game is weighted average-convex, then the corresponding weighted Shapley value is in the core. Second, we exhibit necessary conditions for a communication TU-game to preserve the weighted average-convexity. Finally, we provide a complete characterization when the underlying graph is a priority decreasing tree.

cs.GT↗

arXiv2020

Inheritance of Convexity for the $\tilde{\mathcal{P}}_{\min}$-Restricted Game

We consider a restricted game on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components obtained with a specific partition $\tilde{\mathcal{P}}_{\min}$. This partition relies on the same principle as the partition $\mathcal{P}_{\min}$ introduced by Grabisch and Skoda (2012) but restricted to connected coalitions. More precisely, this new partition $\tilde{\mathcal{P}}_{\min}$ is induced by the deletion of the minimum weight edges in each connected component associated with a coalition. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with $\tilde{\mathcal{P}}_{\min}$.

cs.DM↗

arXiv2019

Inheritance of Convexity for the $\mathcal{P}_{\min}$-Restricted Game

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the subcomponents corresponding to a minimum partition. This minimum partition $\mathcal{P}_{\min}$ is induced by the deletion of the minimum weight edges. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with Pmin. Moreover, we prove that these graphs can be recognized in polynomial time.

cs.DM↗

arXiv2016

Complexity of inheritance of $\mathcal{F}$-convexity for restricted games induced by minimum partitions

Let $G = (N,E,w)$ be a weighted communication graph (with weight function $w$ on $E$). For every subset $A \subseteq N$, we delete in the subset $E(A)$ of edges with ends in $A$, all edges of minimum weight in $E(A)$. Then the connected components of the corresponding induced subgraph constitute a partition of $A$ that we call $P_{\min}(A)$. For every game $(N, v)$, we define the $P_{\min}$-restricted game $(N, \bar{v})$ by $\bar{v}(A) = \sum_{F \in P_{\min}(A)} v(F)$ for all $A \subseteq N$. We prove that we can decide in polynomial time if there is inheritance of $\mathcal{F}$-convexity from $(N, v)$ to the $P_{\min}$-restricted game $(N, \bar{v})$ where $\mathcal{F}$-convexity is obtained by restricting convexity to connected subsets.

cs.DM↗