Pathwise smooth splittable congestion games and inefficiency

Very recently the inefficiency of Nash equilibria has been analyzed in the context of SplittableCongestionGames. These games are like the congestion games but allow the players to use convex combinations of subsets of resources. A new notion has been introduced in order to give bounds on the inefficiency or Price of Anarchy; such a notion has been termed the local smoothness. We present a unified framework where local smoothness and smoothness, a previously introduced notion, are presented as particular cases of a more general approach which we term pathwise smoothness. Such an approach is based partially on the Hadamard's Lemma, which shows that it is possible to present any function, linear or not, by means of families of linear functions.