AI 中文总结
本文提出支持收缩过程，给出计算双矩阵博弈1/2-WSNE的确定性多项式时间算法及相关两方协议，改进了收益查询算法的复杂度依赖。
AI 中文摘要
我们提出了支持收缩，这是一种用于计算双矩阵博弈中良好支持纳什均衡（WSNE）的结构化过程。在公共行动矩形上，该过程交替保留行收益矩阵的行极大极小策略的支持，以及列收益矩阵的列极大极小策略的支持。每次限制都会保留所选矩阵博弈的值，且仅能增大另一博弈的值。在稳定矩形处，两种最大化策略的全支持与互补松弛性使得每个幸存动作都是紧的。随后，交叉两种极小极大策略可得到保留子博弈的精确纳什均衡，而两个零和值则限定了对已删除动作的偏差。支持收缩给出了一种确定性多项式时间算法，可计算所有收益在[0,1]内的有理双矩阵博弈的1/2-WSNE，且无附加松弛；还给出了一种确定性O(ε⁻²log²n)比特的两方协议，用于计算(1/2+ε)-WSNE。相同的支持收缩证书，在之前加入随机单侧定位后，可得到一种O(ε⁻²n log n)收益查询算法，用于实现相同的保证，将原有的ε⁻⁴依赖改进为ε⁻²。
英文摘要
We introduce the support contraction, a structural procedure for computing well-supported Nash equilibria (WSNE) in bimatrix games. On a common action rectangle, the procedure alternately retains the support of a row maximin strategy for the row-payoff matrix and the support of a column maximin strategy for the column-payoff matrix. Each restriction preserves the value of the matrix game that selected it and can only increase the other value. At a stable rectangle, full support of the two maximizing strategies and complementary slackness make every surviving action tight. Crossing the two minimax strategies then gives an exact Nash equilibrium of the retained subgame, while the two zero-sum values bound deviations to deleted actions. Support contraction gives a deterministic polynomial-time algorithm that computes a $1/2$-WSNE of every rational bimatrix game with payoffs in $[0,1]$, with no additive slack, and a deterministic $O(ε^{-2}\log^2 n)$-bit two-party protocol for a $(1/2+ε)$-WSNE. The same support-contraction certificate, preceded by randomized one-sided localization, gives an $O(ε^{-2}n\log n)$ payoff-query algorithm for the same guarantee, improving the $ε^{-4}$ dependence to $ε^{-2}$.
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