Parametric Open Source Games

Open-source game theory studies strategic settings in which agents may condition their behavior on the internal decision procedures of others, but existing formulations are typically discrete or symbolic and therefore do not align well with modern parametric learning systems. We introduce parametric open-source games, a continuous framework in which each player selects a parameter vector, and a semantics map transforms the full parameter profile into a mixed action in an underlying finite game, allowing behavior to depend explicitly on opponents' parameters. We first formalize the induced parametric game and establish basic equilibrium existence results, and then study a canonical open-source sigmoid semantics model in symmetric $2 \times 2$ games, deriving an exact threshold governing when selfish gradient dynamics shift from defection toward cooperation. We further present a boundary characterization for verifying parametric program Nash equilibria through one-dimensional best-response conditions, and extend the framework to a neural semantics class that preserves the same first-order characterization through the ratio of cross-player to self-player sensitivity. Across canonical games and social dilemmas, the resulting framework shows how access to internal parameterizations can qualitatively reshape learning dynamics and equilibrium structure, and when appropriately initialized, can steer selfish optimization toward cooperative outcomes.