Nash-convergence of Game Dynamics and Complexity

TL;DR

This work investigates whether globally Nash-convergent learning dynamics exist for nondegenerate games and whether such dynamics can be computed efficiently. It introduces four families of Nash-convergent dynamics (Types 1–4), showing that for Types 1–3, local tractability would imply major complexity-class collapses ($P=PPAD$, $NP=RP$, and $CLS=PPAD$ respectively), while Type 4 remains open and tied to an Impossibility Conjecture $P=PPAD$. The paper also proves a nuanced affine-subspace Nash-equilibrium result: finding equilibria on an affine subspace is polynomial-time solvable for small dimension $d=O(rac{\,\log|G|}{\log\log|G|})$ but NP-hard for larger $d$, and it introduces the Proving Game to argue that black-box proofs of the conjecture are unlikely. Overall, the results suggest the barrier to Nash learning stems from computational intractability of encoding and computing convergent dynamics rather than the nonexistence of such dynamics, pointing to the need for new complexity-topology approaches. The discussion highlights implications for adopting alternative solution concepts and motivates future work on replicate dynamics and attractor computation under complexity constraints.

Abstract

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.

Figures (1)

• Figure 1: An illustration of the Type 2 dynamic. The space is divided into an odd number of polytopes $P_i$, each containing a Nash equilibrium $z_i$. The odd-numbered polytopes have dimension $d$, while the even-numbered polytopes have dimension $d-1$.

Theorems & Definitions (18)

• Proposition 3.1
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof
• Conjecture 4.5: The Impossibility Conjecture