The indirect function of an NTU game (Q1417349)

A game with transferable utility (TU game) is a pair \((N;v)\) where \(N\) is a finite set of players \(v: 2^N \rightarrow \mathbb R\) with \(v(\emptyset)=0\) is a given function on the set of coalitions. A nontransferable utility game (NTU game) is a pair \((N;V)\) where \(N\) is a finite set of players and \(V\) is a mapping that assigns to a coalition \(S\subseteq N,\) a nonempty closed convex set \(V(S)\) in \(\mathbb R^S\) with the property that the set \(\{x\mid x\in V(S)\) and \(x_i\geq y_i\) \(\forall i\in S\), \(\forall y_i\in V(\{i\})\}\) is bounded. The notion of an indirect function for TU games has earlier been introduced in: [\textit{J. E. Martínez-Legaz}, `Dual representation of cooperative games based on Fenchel-Moreau conjugation', Optimization 36, 291--319 (1996; Zbl 0854.90148)]. The present paper generalizes this notion to a class of NTU games.