The Shapley Value in the Knaster Gain Game

TL;DR

This paper analyzes the Shapley value in a cooperative game modeling collusion among risk-averse agents, revealing how it can measure collusion attitudes and influence coalition formation.

Contribution

It introduces the use of the Shapley value as a measure of collusion attitude and characterizes coalitions that maximize collusion gains.

Findings

Shapley value indicates players' collusion attitudes.

Sign of Shapley value relates to player rankings.

Certain players consistently deter collusion.

Abstract

In Briata, Dall'Aglio and Fragnelli (2012), the authors introduce a cooperative game with transferable utility for allocating the gain of a collusion among completely risk-averse agents involved in the fair division procedure introduced by Knaster (1946). In this paper we analyze the Shapley value (Shapley, 1953) of the game and propose its use as a measure of the players' attitude towards collusion. Furthermore, we relate the sign of the Shapley value with the ranking order of the players' evaluation, and show that some players in a given ranking will always deter collusion. Finally, we characterize the coalitions that maximize the gain from collusion, and suggest an ad-hoc coalition formation mechanism.