The Shapley value and average convex games
Several reasonable solutions have been proposed for cooperative games (Shapley value, NM solution, core, etc.). These solutions are not always consistent with each other. Hence, it is interesting to know when they are consistent. In particular, Shapely value does not always belong to the core. Shapley himself has found a condition (he called it convexity) when it does [\textit{L. S. Shapley}, Internat. J. Game Theory 1, 11-26 (1971; Zbl 0222.90054)]. Convexity stems naturally from the so-called superadditivity, the following natural condition on a game: when two disjoint coalitions decide to act together, they can guarantee themselves at least the same amount as when they acted separately. In mathematical terms, this means that \(v(A\cup B)\geq v(a)+v(B)\). In other words, that the gain \(g(A,B)=v(A\cup B)-v(A)-v(B)\) is non-negative. Convexity means that this gain increases when \(B\) increases, i.e., that \(g(A,B)\geq g(A,B-\{i\})\) for all \(i\in B\). Convex games are rare. The authors find a more general class of games, for which Shapley value is still in the core: games for which the gain increases ``in the average'', i.e., for which \(g(A,B)\geq | B|^{-1}\sum g(A,B-\{i\})\). They call such games average convex. As an example, they take games in which \(v(S)\) is described by a Cobb- Douglas law \(v(S)=A\lambda(| S|)^ \alpha \eta(| S|)^ \beta\), where \(\lambda(s)=\lambda s\) is the total amount of capital owned by coalition \(S\), \(\eta(S)=\sum_{i\in S}\eta_ i\) is the total amount of labor owned by \(S\), \(0<\alpha\leq 1\leq\beta\). A similar result is proved for an even more general class of games.
scientific article; zbMATH DE number 399312