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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.17607 |
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| _version_ | 1866914575783297024 |
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| author | Bichler, Martin Hoehener, Jan-Sebastian |
| author_facet | Bichler, Martin Hoehener, Jan-Sebastian |
| contents | Autonomous pricing agents are widely deployed in online marketplaces, making algorithmic pricing a prominent application of multi-agent learning. Experimental studies often report collusive outcomes, but these findings typically rely on Q-learning in complete-information environments and lack rigorous convergence guarantees. In this paper, we study the stochastic learning dynamics of Regularized Robbins-Monro (RRM) algorithms in a Bayesian Bertrand competition with private costs. We show that this setting violates standard stability conditions, including monotonicity and the Minty variational inequality, rendering classical convergence results for gradient-based learning inapplicable. Despite this, we prove that Euclidean RRM algorithms converge almost surely to the unique, efficient Bayes-Nash equilibrium within a finite-dimensional approximation of the strategy space. By analyzing symmetric piecewise-linear pricing strategies in a duopoly, we explicitly construct a global Lyapunov function for the projected primal dynamics and establish global asymptotic stability of the equilibrium. Our analysis yields rigorous convergence guarantees for stochastic first-order learning algorithms in Bayesian Bertrand competition and provides a principled counterpoint to widespread claims of algorithmic collusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17607 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of Stochastic First-Order Algorithms in Bertrand Competition Under Incomplete Information Bichler, Martin Hoehener, Jan-Sebastian Computer Science and Game Theory Autonomous pricing agents are widely deployed in online marketplaces, making algorithmic pricing a prominent application of multi-agent learning. Experimental studies often report collusive outcomes, but these findings typically rely on Q-learning in complete-information environments and lack rigorous convergence guarantees. In this paper, we study the stochastic learning dynamics of Regularized Robbins-Monro (RRM) algorithms in a Bayesian Bertrand competition with private costs. We show that this setting violates standard stability conditions, including monotonicity and the Minty variational inequality, rendering classical convergence results for gradient-based learning inapplicable. Despite this, we prove that Euclidean RRM algorithms converge almost surely to the unique, efficient Bayes-Nash equilibrium within a finite-dimensional approximation of the strategy space. By analyzing symmetric piecewise-linear pricing strategies in a duopoly, we explicitly construct a global Lyapunov function for the projected primal dynamics and establish global asymptotic stability of the equilibrium. Our analysis yields rigorous convergence guarantees for stochastic first-order learning algorithms in Bayesian Bertrand competition and provides a principled counterpoint to widespread claims of algorithmic collusion. |
| title | Convergence of Stochastic First-Order Algorithms in Bertrand Competition Under Incomplete Information |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2605.17607 |