Tensor approximation of cooperative games and their semivalues

Abstract

We propose an algorithm to approximate the semivalues of general transferable-utility cooperative games that involve a large set of players 1,…,|N| and possibly depend on uncertain parameters. We first encode the game's utility function using a low-rank tensor decomposition, namely the tensor train (TT) model, which requires a limited number of function evaluations. The TT format casts the utility as a compressed tensor of shape 2|N| and makes it possible to efficiently work with the exponentially-sized set of all possible coalitions of players. Given a game compressed in this manner, the proposed algorithm obtains arbitrary semivalues without incurring additional error, in particular the Shapley values and Banzhaf-Coleman indices, which are two of the most important allocation rules in cooperative game theory. Our algorithm takes O(|N|R2) operations per semivalue, where R is the game's TT rank. We show experimentally that many classical games can be compressed at low error with a moderate TT rank, making our algorithm more sample-efficient than Monte Carlo-based estimation. We also give a theoretical bound for the error of the semivalues obtained through our algorithm. Last, when the game depends on randomly distributed parameters, we are able to compute the expected semivalues efficiently.