Least Square Approximations and Linear Values of Cooperative Games

TL;DR

This paper establishes a general optimization framework showing that linear values in cooperative games can be derived from least square approximation problems, unifying and extending previous results in the field.

Contribution

It demonstrates that every linear value corresponds to a least square approximation problem and vice versa, providing a unified theoretical foundation for these concepts.

Findings

Every linear value arises from a least square approximation problem.

Every least square approximation with linear constraints yields a linear value.

Explicit formulas for solutions follow from the general framework.

Abstract

Many important values for cooperative games are known to arise from least square optimization problems. The present investigation develops an optimization framework to explain and clarify this phenomenon in a general setting. The main result shows that every linear value results from some least square approximation problem and that, conversely, every least square approximation problem with linear constraints yields a linear value. This approach includes and extends previous results on so-called least square values and semivalues in the literature. In particular, is it demonstrated how known explicit formulas for solutions under additional assumptions easily follow from the general results presented here.