Atsushi Hori

Incorporating an assurance region (AR) into the slacks-based measure (SBM) improves practicality; however, its efficiency measure may not have desirable properties, such as monotonicity. We incorporate a closer target setting approach into the SBM with AR and a variant of the SBM with AR. We demonstrate that the efficiency measure with the hybrid approach has the same desirable properties as that without AR, and we also show that the efficiency scores can be computed by solving linear programming problems. Our proposed approach can handle zeros in the observed input-output data without any data transformation or additional model modification.

The multi-leader--multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader--multi-follower game (SLMFG). Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers' strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers' noncooperative game that is a function with leaders' strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers' response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments and confirm its validity.