Stabilizing Grand Cooperation via Cost Adjustment: An Inverse Optimization Approach

For an unbalanced cooperative game, its grand coalition can be stabilized by some instruments, such as subsidization and penalization, that impose new cost terms to certain coalitions. In this paper, we study an alternative instrument, referred to as cost adjustment, that does not need to impose any new coalition-specific cost terms. Specifically, our approach is to adjust existing cost coefficients of the game under which (i) the game becomes balanced so that the grand coalition becomes stable, (ii) a desired way of cooperation is optimal for the grand coalition to adopt, and (iii) the total cost to be shared by the grand coalition is within a prescribed range. Focusing on a broad class of cooperative games, known as integer minimization games, we formulate the problem on how to optimize the cost adjustment as a constrained inverse optimization problem. We prove N P-hardness and derive easy-to-check feasibility conditions for the problem. Based on two linear programming reformulations, we develop two solution algorithms. One is a cutting-plane algorithm, which runs in polynomial time when the corresponding separation problem is polynomial time solvable. The other needs to explicitly derive all the inequalities of a linear program, which runs in polynomial time when the linear program contains only a polynomial number of inequalities. We apply our models and solution algorithms to two typical unbalanced games, including a weighted matching game and an uncapacitated facility location game, showing that their optimal cost adjustments can be obtained in polynomial time.