On the Geometry of Games and their Solvers

2026-05-28

AI Summary

Existing equilibrium-finding methods for heterogeneous games lack a unified framework capable of adapting to diverse game structures. This work proposes a structure-aware solver composition framework that maps games into a low-dimensional, continuous representation space aligned with solver behavior, enabling dynamic assembly of base solver primitives tailored to local game regions. By introducing a geometric perspective on game solvability, the approach transcends the limitations of traditional discrete categorization and integrates a structure identification network, a strategy mapping mechanism, and bounded residual correction. Experiments reveal that fixed primitives suffer from systematic regional mismatches, whereas learned representations construct a structured game spectrum that effectively uncovers the continuous boundaries of algorithmic performance validity.

Abstract

A central challenge in game theory and learning systems such as GANs is understanding which algorithms can efficiently compute equilibria across the heterogeneous landscape of games. Equilibrium computation is typically studied solver by solver and game class by game class, yielding strong local guarantees but a fragmented view of solver behaviour. Existing discrete taxonomies often provide an incomplete account of where algorithms succeed. We study this problem through a solver-game map linking games to effective solver dynamics. Classical theory identifies isolated regions of this map but provides limited insight into intermediate or overlapping regimes, suggesting that solvability is governed by latent structural properties defining a continuous solver-aligned geometry of games. We formalise this perspective through structure-aware solver synthesis. A learned structure recogniser maps each game to a low-dimensional solver-aligned representation, and a policy maps this representation to effective primitive mechanisms, adapting solver behaviour across regimes. This reveals regions where particular solver dynamics are effective and where mixtures of primitives are required rather than a single dominant solver. A bounded residual acts as a local corrector and diagnostic signal for incomplete solver bases or representations. The framework yields both an adaptive solver and an analytical lens: games with similar optimisation dynamics cluster together, revealing continuous regions of algorithmic validity and overlapping solver behaviour. Empirically, we show that fixed primitives exhibit systematic regime mismatch, while the learned representation organises game space into a structured cartography aligned with solver behaviour. These results suggest viewing equilibrium computation as the joint problem of learning solver mechanisms and mapping the geometry of solvability.

Problem

• equilibrium computation
• solver dynamics
• game geometry
• algorithmic validity
• structural properties

Innovation

• solver-game geometry
• structure-aware synthesis
• adaptive solver dynamics
• equilibrium computation
• learned game representation