# Parametric Open Source Games

arXiv: https://arxiv.org/abs/2606.27068 · HTML: https://mlsanity.com/paper/2606.27068/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 17 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [On Gradient-Based Learning in Continuous Games](https://mlsanity.com/paper/1804.05464/) (2018)
- [Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games](https://mlsanity.com/paper/2002.09806/) (2020)
- [Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic](https://mlsanity.com/paper/1401.5577/) (2014)
- [Cooperative and uncooperative institution designs: Surprises and problems in open-source game theory](https://mlsanity.com/paper/2208.07006/) (2022)

## Abstract (arXiv)

Open-source game theory studies agents whose behavior may depend on one another's decision procedures, but most existing models use discrete or symbolic programs. We introduce parametric open-source games, a continuous analogue of program equilibria in which players choose parameter vectors and semantics maps convert the full parameter profile into mixed actions in an underlying finite game. We establish equilibrium existence results, derive an exact coupling threshold at which selfish gradient ascent in symmetric 2×2 games switches from defection toward cooperation, and give a one-dimensional boundary test for parametric program Nash equilibria. We further extend the framework to a neural semantics class whose first-order cooperation condition is governed by the ratio of cross-player to self-player sensitivity. Across canonical games, the framework shows how access to internal parameterizations can qualitatively reshape learning dynamics and equilibrium structure, and how sufficiently strong open-source coupling can steer selfish optimization toward cooperative outcomes.