We know that the Nash equilibria of a game cannot be computed efficiently unless $P = PPAD$. But can they be learned? Are there dynamics that (1) can be computed efficiently by the players at each strategy profile and (2) are guaranteed to converge to the Nash equilibria? This is a question as ancient as the Nash equilibrium itself, and antedates by many decades current complexity considerations about it. It was recently proved in MPPS23 that no such dynamics can exist in general; however, the game used in that proof is degenerate, and a strong assumption of uniform convergence to a continuum of Nash equilibria is employed. We point out that both assumptions are necessary for that proof, because Nash-convergent dynamics do exist, which converge to all Nash equilibria in non-degenerate games; in fact we describe two very different families of such dynamics. However, we show that both of these families are intractable to compute locally, unless complexity classes collapse. We formulate a complexity theoretic Impossibility Conjecture: if a locally tractable Nash-convergent dynamic exists then $P=PPAD$. This is a novel kind of conjecture, combining topology and complexity, because the dynamic is only required to be tractable locally -- it could take exponentially many steps to converge, so long as the work done at each step is polynomial. Next, we show that any locally tractable dynamic which possesses a locally tractable Lyapunov function cannot exist unless $PPAD = CLS$. Finally, for the most general impossibility conjecture, we provide a complexity-theoretic explanation why it may be difficult to settle: we introduce a Proving Game to demonstrate that black-box reductions cannot distinguish between convergent and non-convergent dynamics in polynomial time.

翻译：我们知道，除非 $P = PPAD$，否则无法高效计算博弈的纳什均衡。但它们能否被学习？是否存在这样的动态过程：（1）每个策略组合下玩家都能高效计算，并且（2）保证收敛到纳什均衡？这个问题与纳什均衡本身一样古老，并且比当前关于其复杂性的考虑早了几十年。最近在 MPPS23 中证明，一般而言不存在这样的动态过程；然而，该证明中使用的博弈是退化的，并且采用了收敛到纳什均衡连续统的强一致收敛假设。我们指出，这两个假设对该证明都是必要的，因为确实存在非退化博弈中收敛到所有纳什均衡的纳什收敛动态过程；事实上，我们描述了两种截然不同的此类动态过程族。然而，我们表明，除非复杂性类发生坍塌，否则这两个动态过程族在局部计算上都是难解的。我们提出了一个复杂性理论上的不可能性猜想：如果存在一个局部易解的纳什收敛动态过程，那么 $P = PPAD$。这是一个新型猜想，结合了拓扑与复杂性，因为该动态过程仅需在局部易解——它可能需指数步数才能收敛，只要每一步的工作量是多项式的。接下来，我们证明，任何拥有局部易解李雅普诺夫函数的局部易解动态过程都不可能存在，除非 $PPAD = CLS$。最后，针对最一般的不可能性猜想，我们提供了一个复杂性理论解释，说明为何可能难以解决：我们引入了一个证明博弈，以证明黑盒归约无法在多项式时间内区分收敛与非收敛动态过程。