论文标题
关于有限的两人紧密矢量游戏表格的纳什可分解性
On Nash-Solvability of Finite Two-Person Tight Vector Game Forms
论文作者
论文摘要
我们考虑有限的两人正常表格游戏。其游戏形式的以下四个属性等效：（i）nash可辨式性，（ii）零和可溶解度，（iii）Win-lose-solvicalsion和（iv）紧密度。对于（ii，iii，iv），这是由Edmonds和Fulkerson在1970年表现出来的。然后，在1975年，（i）添加到此列表中，还显示这些结果不能以$ n $ n> 2 $的$ n $ person案例进行推广。在1990年，紧密度扩展到矢量游戏表格（$ v $ - forms），结果表明，这种$ v $ -Tighness和零和零可溶性仍然等效，但并不意味着NASH可溶性。这些结果适用于具有完美信息的几类随机游戏。在这里，我们建议另外一个紧密的扩展，引入$ v^+$ - 紧密的矢量游戏表单（$ v^+$ - 表单）。我们表明，在弱矩形游戏形式和正面成本功能的情况下，这种$ V^+$ - 紧密度和NASH可溶性是等效的。这个结果使我们能够将所谓的双路径猜想降低到$ v^+$ - $ v^+$ - 表格的紧密度。但是，这两个（同等）陈述保持开放。
We consider finite two-person normal form games. The following four properties of their game forms are equivalent: (i) Nash-solvability, (ii) zero-sum-solvability, (iii) win-lose-solvability, and (iv) tightness. For (ii, iii, iv) this was shown by Edmonds and Fulkerson in 1970. Then, in 1975, (i) was added to this list and it was also shown that these results cannot be generalized for $n$-person case with $n > 2$. In 1990, tightness was extended to vector game forms ($v$-forms) and it was shown that such $v$-tightness and zero-sum-solvability are still equivalent, yet, do not imply Nash-solvability. These results are applicable to several classes of stochastic games with perfect information. Here we suggest one more extension of tightness introducing $v^+$-tight vector game forms ($v^+$-forms). We show that such $v^+$-tightness and Nash-solvability are equivalent in case of weakly rectangular game forms and positive cost functions. This result allows us to reduce the so-called bi-shortest path conjecture to $v^+$-tightness of $v^+$-forms. However, both (equivalent) statements remain open.
