On the Geometry of Games and their Solvers

Understanding where learning algorithms converge, cycle, or otherwise change behaviour is fundamental to solving games in economics and multi-agent systems. Existing game taxonomies are largely derived from analytical structure, such as potential, zero-sum, or monotone regimes, and provide powerful guarantees in restricted settings, but their boundaries need not coincide with realised solver dynamics or predict finite-time behaviour. We introduce behavioural cartography, a low-dimensional representation of games organised directly by the learning dynamics they induce. Rather than assigning only a structural class, it identifies which games elicit similar solver behaviour, where algorithmic regimes change, and how solver competence varies across game space. We construct the atlas from mechanistically distinct primitives spanning gradient, optimistic/extragradient, and fictitious-play families, evaluated across solver settings and initialisations. Games with similar induced dynamics are placed nearby in an oracle atlas z ⋆ (G), while a payoff-based model learns to locate unseen games in the same space, respecting strategic invariances including action relabelling and player exchange. Crucially, the atlas generalises beyond its construction: it predicts withheld properties such as recurrence and reversal and transfers to solver primitives excluded entirely from atlas construction. Across heterogeneous solvers, the learned atlas strongly preserves which games produce similar dynamics (pairwise-distance Spearman 0.917); when a solver is withheld, its behaviour can still be inferred from limited observations, reducing error by 30% relative to a constant reference at 205 labels. The resulting cartography reveals coherent solver-specific convergence and sensitivity fields, continuous behavioural variation along controlled structural paths, and distinctions not fully captured by conventional game classifications. Together, these results provide a predictive empirical atlas of game solving that complements analytical taxonomy by showing how algorithms behave across game space, how their capabilities relate, and how newly introduced solvers can be characterised.