Simultaneous penalization and subsidization for stabilizing grand cooperation

Abstract

In this paper we propose a new instrument, a simultaneous penalization and subsidization, for stabilizing the grand coalition and enabling cooperation among all players of an unbalanced cooperative game. The basic idea is to charge a penalty z from players who leave the grand coalition, and at the same time provide a subsidy ω to players who stay in the grand coalition. To formalize this idea, we establish a penalty-subsidy function ω(z) based on a linear programming model, which allows a decision maker to quantify the trade-off between the levels of penalty and subsidy. By studying function ω(z), we identify certain properties of the trade-off. To implement the new instrument, we design two algorithms to construct function ω(z) and its approximation. Both algorithms rely on solving the value of ω(z) for any given z, for which we propose two effective solution approaches. We apply the new instrument to a class of machine scheduling games, showing its wide applicability.