Abstract
Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.
Highlights
Cooperative games are among the most studied classes of problems in game theory, with plenty of applications in economics, mathematics, and computer science
It is not surprising that some researchers have investigated whether it is possible to give an efficient characterization of submodular instances, for prominent cooperative games that are in general non-convex
This paper focuses on one of the most fundamental cooperative games, namely the spanning tree game
Summary
Cooperative games are among the most studied classes of problems in game theory, with plenty of applications in economics, mathematics, and computer science. We refer to [11,13] for other interesting properties of submodular games involving other crucial solution concepts Given these observations, it is not surprising that some researchers have investigated whether it is possible to give an efficient characterization of submodular instances, for prominent cooperative games that are in general non-convex. An important step forward was made by Kobayashi and Okamoto [9], who gave a characterization of submodularity for instances of the spanning tree game where the edge weights are restricted to take only two values. We can efficiently test the submodularity of our instance by checking whether the inequality (∗) is satisfied on this family of subsets of vertices Combining these two ingredients yields a polynomial-time characterization of submodularity for spanning tree games, as described in Sect.
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