一句话：证明定量可达性博弈中 SPE 约束存在问题是 PSPACE 完全的，并给出约束不动点算法。
本文研究有限有向图上的多人定量可达性博弈，每名玩家的目标是以最快速度到达各自的目标顶点集。作者不采用 Nash 均衡，而关注其精化——子博弈完美均衡（SPE）。已知此类博弈总存在 SPE，且约束存在问题可判定；本文确定其精确复杂度为 PSPACE-complete。为证明该结果，作者提出一种新算法，迭代地构造刻画 SPE 结局集合的约束集：通过反复应用强化约束的算子直至不动点，并用至多指数大小的有限图表示 SPE 结局。对该计算过程进行细致分析后可得到 PSPACE 成员性，从而完成复杂度刻画。
We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership.
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@article{BrihayeBGRB20,
title = {The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games},
author = {Thomas Brihaye and Véronique Bruyère and Aline Goeminne and Jean-François Raskin and Marie van den Bogaard},
journal = {Logical Methods in Computer Science},
volume = {Volume 16, Issue 4},
year = {2020},
doi = {10.23638/lmcs-16(4:8)2020}
}