Schultz (J Optim Theory Appl 73(1):89–99, 1992) formulated 2-person, zero-sum, discounted switching control stochastic games as a linear complementarity problem (LCP) and discussed computational results. It remained open to prove or disprove Lemke-processibility of this LCP. We settle this question by providing a counter example to show that Lemke’s algorithm does not always successfully process this LCP.We propose a new LCP formulation with the aim of making the underlying matrix belong to the classes R $$_{0}$$ and E $$_{0}$$ , which would imply Lemke processibility. While the underlying matrix in the new formulation is not $$E_0$$ , we show that it is an R $$_{0}$$ -matrix. Successful processing of Lemke’s algorithm depends on the choice of the initial vector d. Because of the special structure of the LCP in our context, we may, in fact, be able to find a suitable d such that our LCPs are processible by Lemke’s algorithm. We leave this open.

中文翻译：

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用于解决折扣切换控制随机博弈的 LCP 公式的 Lemke 可加工性

Schultz (J Optim Theory Appl 73(1):89–99, 1992) 将 2 人、零和、折扣切换控制随机博弈表述为线性互补问题 (LCP) 并讨论了计算结果。证明或反驳此 LCP 的 Lemke 可加工性仍然是开放的。我们通过提供一个反例来解决这个问题，以表明 Lemke 的算法并不总是成功地处理这个 LCP。我们提出了一个新的 LCP 公式，目的是使基础矩阵属于类 R $$_{0}$$ 和E $$_{0}$$ ，这意味着 Lemke 可加工性。虽然新公式中的基础矩阵不是 $$E_0$$ ，但我们证明它是 R $$_{0}$$ -matrix。Lemke 算法的成功处理取决于初始向量 d 的选择。由于 LCP 在我们上下文中的特殊结构，我们实际上可以 能够找到合适的 d 使得我们的 LCP 可以被 Lemke 算法处理。我们保持开放。