PPAD Lemke验证线性互补问题的双矩阵对策化
A direct reduction of PPAD Lemke-verified linear complementarity problems to bimatrix games
摘要：
线性互补问题LCP（q，M）定义如下。对于给定的M，q找到z，使得q+Mz>=0，z>=0，z（q + Mz）=0，或者证明不存在这样的z。双矩阵博弈（2-NASH）的纳什均衡求解问题可以归结为线性互补问题（LCP）。此外，2-NASH在复杂度类PPAD（Polynomial-time Parity Argument Directed）中是完备的。然而，巧妙构造的约简（为任何PPAD问题设计的）非常复杂，因此虽然具有重要的理论意义，但实际上通过2-NASH解决LCP并不实用，并且它可能无法提供从研究从LCP问题（例如市场均衡）中获得的游戏中获得的潜在洞察力。本文的主要目标是构造一个简单的显式约简的任何LCP（q，M），可以验证为属于PPAD通过图诱导的通用Lemke算法与一些积极的覆盖向量d，对称2-NASH。特别地，该图的任何端点（除了算法的初始点之外）对应于解或所谓的次级射线。因此，如果任何次级射线可以用于在多项式时间内构造不存在问题的解决方案的证书，则LCP问题被验证为属于PPAD。我们实现了我们的目标表明，对于任何M，q和一个积极的d满足一定的非退化假设的M，我们可以简单，直接地构造一个对称的2-NASH的纳什均衡对应一对一的端点图诱导LCP（q，M）和Lemke算法与覆盖向量d。我们注意到，对于给定的M，约简对所有正d都有效，除了测度为0的子集。
英文摘要：
The linear complementarity problem, LCP(q,M), is defined as follows. For given M,q find z such that q+Mz>=0, z>=0, z(q + M z)=0,or certify that there is no such z. It is well known that the problem of finding a Nash equilibrium for a bimatrix game (2-NASH) can be formulated as a linear complementarity problem (LCP). In addition, 2-NASH is known to be complete in the complexity class PPAD (Polynomial-time Parity Argument Directed). However, the ingeniously constructed reduction (which is designed for any PPAD problem) is very complicated, so while of great theoretical significance, it is not practical for actually solving an LCP via 2-NASH, and it may not provide the potential insight that can be gained from studying the game obtained from a problem formulated as an LCP (e.g. market equilibrium). The main goal of this paper is the construction of a simple explicit reduction of any LCP(q,M) that can be verified as belonging to PPAD via the graph induced by the generic Lemke algorithm with some positive covering vector d, to a symmetric 2-NASH. In particular, any endpoint of this graph (with the exception of the initial point of the algorithm) corresponds to either a solution or to a so-called secondary ray. Thus, an LCP problem is verified as belonging to PPAD if any secondary ray can be used to construct, in polynomial time, a certificate that there is no solution to the problem. We achieve our goal by showing that for any M,q and a positive d satisfying a certain nondegeneracy assumption with respect to M, we can simply and directly construct a symmetric 2-NASH whose Nash equilibria correspond one-to-one to the end points of the graph induced by LCP(q,M) and the Lemke algorithm with a covering vector d. We note that for a given M the reduction works for all positive d with the exception of a subset of measure 0.
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