The Symmetry Trap: Parametric Equilibria and the Welfare Cost of Architectural Monoculture

When multiple AI agents share the same parameterization---as occurs when deploying several instances of the same large language model or reinforcement learning policy---gradient-based learning exhibits a structural symmetry-preservation property that forces convergence to a specific, often Pareto-inferior equilibrium. We formalize this phenomenon as the \textit{Monoculture Equilibrium} and provide a rigorous theoretical characterization of its existence, stability, and welfare costs. Concretely, we prove that in symmetric anti-coordination games, identically initialized agents are confined to a symmetric invariant manifold from which they converge to the unique symmetric mixed Nash equilibrium---a saddle point of the full gradient dynamics that is globally unstable to perturbations in the heterogeneous direction. We quantify the resulting welfare loss via the Price of Monoculture (PoM), derive a symmetry-breaking threshold $\delta^*$ below which architectural diversity fails to escape the trap, and extend these results to $N$-player and continuous-action settings. Experiments with parametric policy gradient agents, large language models in repeated Hawk-Dove play, and $N$-firm Cournot competition corroborate the theory, revealing sharp phase transitions at predicted thresholds and significant welfare gaps under monoculture conditions ($p < 10^{-8}$ across all comparisons).