Lagrangian heuristic for simultaneous subsidization and penalization: implementations on rooted travelling salesman games

This work examines the problem of stabilizing the grand coalition of an unbalanced cooperative game under the concept of simultaneous subsidization and penalization (S&P). We design a generic framework for developing heuristic algorithms to evaluate the trade-off between subsidy and penalty in the S&P instrument. By incorporating some Lagrangian relaxation techniques, we develop an approach for computing feasible subsidy–penalty pairs under which the grand coalition is stabilized in unbalanced cooperative games. This approach is particularly applicable when the characteristic functions of a cooperative game involve intractable integer programmes. To illustrate the performance of the Lagrangian relaxation based approach, we investigate the rooted travelling salesman game, and the computational results obtained show that our new approach is both efficient and effective.