Minimum norm solutions for cooperative games

Abstract

We show that to each linear solution that has the inessential game property, there is an inner product on the space of games such that the solution to each game is the best additive approximation of the game (w.r.t. the norm derived from this inner product). If the space of games has an inner product, then the function that to each game assigns the best additive approximation of this game (w.r.t. to the norm derived from this inner product) is a linear solution that has the inessential game property. Both claims remain valid also if solutions are required to be efficient. JEL Classification: C78

Hodge decomposition and the Shapley value of a cooperative game

“…We have also shown that this game decomposition may be understood in terms of the least-squares solution to a linear problem, where the solution is exact if and only if the game is inessential. In this sense, our decomposition may be considered as an edge-based (rather than vertex-based) variant of the least-squares and minimum-norm solution concepts of Ruiz et al [23] and Kultti and Salonen [21]. The normal equations for this linear problem yield another, equivalent characterization of the game decomposition in terms of the well-studied graph Laplacian.…”

“…This characterization of the game components and the Shapley value also implies two equivalent characterizations: one in terms of the least-squares solution to a linear problem, whose solution is exact if and only if the game is inessential; the other in terms of the graph Laplacian. The first of these two characterizations is related to the least-square and minimum-norm solution concepts of Ruiz et al [23] and Kultti and Salonen [21]. Furthermore, since the combinatorial Hodge decomposition holds for arbitrary weighted graphs, this decomposition of cooperative games also generalizes to cases where edges of the hypercube graph are weighted or removed altogether.…”

“…This characterization of the game components and the Shapley value also implies two equivalent characterizations: one in terms of the least-squares solution to a linear problem, whose solution is exact if and only if the game is inessential; the other in terms of the graph Laplacian. The first of these two characterizations is related to the least-square and minimum-norm solution concepts of Ruiz et al [23] and Kultti and Salonen [21].…”

A solution concept for network games: The role of multilateral interactions

“…This idea is similar to most of the semi-values for cooperative games (e.g., the minimum norm solutions,Kultti & Salonen, 2007; least square values,Ruiz, Valenciano, & Zarzuelo, 1996, 1998 Banzhaf value, Banzhaf, 1965 for simple games and their extensions to general cooperative games i.e., TU games byOwen, 1975;Dragan, 1996; Marichal & Mathonet, 2011 etc.) and may be thought of as describing the power of players.…”

A Hodge theoretic extension of Shapley axioms

“…Extensive research on the cooperative and noncooperative games have been inspired by and evolved from the pioneering study by Shapley [16][17][18][19], and various concepts of solutions have been proposed, e.g., Kalai and Samet [8], Ruiz et al [15] and Kultti and Salonen [12]. In particular, the combinatorial Hodge decomposition has recently been applied to game theory in various contexts, including noncooperative games (Candogan et al [1]), cooperative games (Stern and Tettenhorst [20]), and also other interesting problems in economics, e.g., ranking of social preferences (Jiang et al [7]).…”