Games of ordered preference (GOOPs) model multi-player equilibrium problems in which each player maintains a distinct hierarchy of strictly prioritized objectives. Existing approaches solve GOOPs by deriving and enforcing the necessary optimality conditions that characterize lexicographically constrained Nash equilibria through a single-level reformulation. However, the number of primal and dual variables in the resulting KKT system grows exponentially with the number of preference levels, leading to severe scalability challenges. We derive a compact reformulation of these necessary conditions that preserves the essential primal stationarity structure across hierarchy levels, yielding a "reduced" KKT system whose size grows polynomially with both the number of players and the number of preference levels. The reduced system constitutes a relaxation of the complete KKT system, yet it remains a valid necessary condition for local GOOP equilibria. For GOOPs with quadratic objectives and linear constraints, we prove that the primal solution sets of the reduced and complete KKT systems coincide. More generally, for GOOPs with arbitrary (but smooth) nonlinear objectives and constraints, the reduced KKT conditions recover all local GOOP equilibria but may admit spurious non-equilibrium solutions. We introduce a second-order sufficient condition to certify when a candidate point corresponds to a local GOOP equilibrium. We also develop a primal-dual interior-point method for computing a local GOOP equilibrium with local quadratic convergence. The resulting framework enables scalable and efficient computation of GOOP equilibria beyond the tractable range of existing exponentially complex formulations.

翻译：偏好有序博弈（GOOP）刻画了多参与者均衡问题，其中每个参与者维护着严格排序的优先级目标层级结构。现有方法通过推导并实施描述字典序约束纳什均衡的最优性条件，将其转化为单层重构形式来求解GOOP。然而，由此生成的KKT系统中原始变量与对偶变量的数量随偏好层级数呈指数增长，导致严重的可扩展性挑战。我们提出了一种紧凑的必要条件重构形式，在保持层级间原始平稳性核心结构的前提下，构建了规模随参与者数量与偏好层级数呈多项式增长的"精简"KKT系统。该精简系统虽构成完整KKT系统的松弛形式，但仍是局部GOOP均衡的有效必要条件。对于具有二次目标函数与线性约束的GOOP，我们证明了精简与完整KKT系统的原始解集完全一致。更一般地，对于具有任意（光滑）非线性目标函数与约束的GOOP，精简KKT条件可恢复全部局部GOOP均衡，但可能引入虚假的非均衡解。我们引入二阶充分条件以判定候选点是否为局部GOOP均衡，并开发了具有局部二次收敛性的原始-对偶内点法用于计算局部GOOP均衡。该框架使得在现有指数复杂度模型的可解范围之外，仍能实现GOOP均衡的可扩展高效计算。