{"paper":{"title":"Convergence of Stochastic First-Order Algorithms in Bertrand Competition Under Incomplete Information","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Euclidean RRM algorithms converge almost surely to the unique efficient Bayes-Nash equilibrium in finite-dimensional approximations of Bayesian Bertrand competition.","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Jan-Sebastian Hoehener, Martin Bichler","submitted_at":"2026-05-17T19:08:32Z","abstract_excerpt":"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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The strategy space admits a finite-dimensional approximation by symmetric piecewise-linear pricing functions for which a global Lyapunov function can be explicitly constructed (abstract, paragraph on duopoly analysis).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Euclidean RRM algorithms converge almost surely to the unique efficient Bayes-Nash equilibrium in a finite-dimensional approximation of Bayesian Bertrand competition with private costs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Euclidean RRM algorithms converge almost surely to the unique efficient Bayes-Nash equilibrium in finite-dimensional approximations of Bayesian Bertrand competition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"61097d1bae2af27059781e48b21e083cb7b368c92d424ec0ae235e63ddd2793a"},"source":{"id":"2605.17607","kind":"arxiv","version":1},"verdict":{"id":"9913a176-bf21-4766-9c97-0fc128345d85","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:14:16.199865Z","strongest_claim":"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.","one_line_summary":"Euclidean RRM algorithms converge almost surely to the unique efficient Bayes-Nash equilibrium in a finite-dimensional approximation of Bayesian Bertrand competition with private costs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The strategy space admits a finite-dimensional approximation by symmetric piecewise-linear pricing functions for which a global Lyapunov function can be explicitly constructed (abstract, paragraph on duopoly analysis).","pith_extraction_headline":"Euclidean RRM algorithms converge almost surely to the unique efficient Bayes-Nash equilibrium in finite-dimensional approximations of Bayesian Bertrand competition."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.17607/integrity.json","findings":[{"note":"Identifier '10.1016/0893-9659(90)90018-h' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. 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