Does the failure of learning dynamics to converge globally to Nash equilibria stem from the geometry of the game or the complexity of computation? Previous impossibility results relied on game degeneracy, leaving open the case for generic, nondegenerate games. We resolve this by proving that while Nash-convergent dynamics theoretically exist for all nondegenerate games, computing them is likely intractable. We formulate the Impossibility Conjecture: if a locally efficient Nash-convergent dynamic exists for nondegenerate games, then $P=PPAD$. We validate this for three specific families of dynamics, showing their tractability would imply collapses such as $NP=RP$ or $CLS=PPAD$. En route, we settle the complexity of finding Nash equilibria of a given game that lie on a given affine subspace. Finally, we explain why the general conjecture remains open: we introduce a Proving Game to demonstrate that black-box reductions cannot distinguish between convergent and non-convergent dynamics in polynomial time. Our results suggest the barrier to Nash learning is not the non-existence of a vector field, but the intractability of computing it.

翻译：学习动力学无法全局收敛至纳什均衡，这一现象究竟源于博弈的几何结构，还是计算复杂性？先前的不可能性结果依赖于博弈的退化性，对于一般的非退化博弈情形则悬而未决。我们通过证明以下结论解决了这一问题：尽管从理论上存在对所有非退化博弈均能收敛至纳什均衡的动力学，但计算这类动力学很可能是难解的。我们提出“不可能性猜想”：若存在对非退化博弈局部高效的纳什收敛动力学，则 $P=PPAD$。我们针对三类具体的动力学族验证了这一猜想，证明其可计算性将导致诸如 $NP=RP$ 或 $CLS=PPAD$ 的复杂性类塌缩。在此过程中，我们确定了在给定仿射子空间中寻找博弈纳什均衡的复杂度。最后，我们解释了为何一般性猜想仍保持开放：通过引入一个“证明博弈”，我们论证了黑盒归约无法在多项式时间内区分收敛与非收敛动力学。我们的结果表明，纳什学习的主要障碍并非向量场的不存在性，而是其计算的难解性。