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A practical consequence of these dynamics is the\n                    <jats:italic>performatively stable equilibrium<\/jats:italic>\n                    , where each player\u2019s strategy is a best response under the induced distribution. Prior work relies on\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\beta $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -smoothness, assuming Lipschitz continuity of loss function gradients with respect to data distributions, which is impractical as the data distribution maps, i.e., the relationship between joint decision and the resulting distribution shifts, are typically unknown, rendering\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\beta $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    unobtainable. To overcome this limitation, we propose a gradient-based\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\hat{\\varepsilon }_i$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -sensitivity measure. It directly quantifies the impact of decision-induced distribution shifts on decision-making and is calculable for arbitrary data distribution maps. Leveraging this measure, we derive convergence guarantees for performatively stable equilibria under a practically feasible assumption of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -strong monotonicity. Notably, we establish a linear convergence rate in finite sample scenarios when\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha &gt; 2 \\sqrt{\\sum _{i = 1}^n \\hat{\\varepsilon }_i^2}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , with a probability depending on sample complexity. Accordingly, we develop a sensitivity-informed repeated retraining algorithm that adjusts players\u2019 loss functions based on the sensitivity measure to achieve the strong monotonicity. This approach ensures the game satisfies the derived convergence condition and thus guarantees convergence to performatively stable equilibria for arbitrary data distribution maps. Experiments with various data distribution maps on prediction error minimization game, Cournot competition, and revenue maximization game show that our approach outperforms state-of-the-art baselines, achieving lower losses and faster convergence, validating the theoretical convergence conditions and confirming the effectiveness of the proposed algorithm.\n                  <\/jats:p>","DOI":"10.1007\/s10994-026-07115-w","type":"journal-article","created":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T11:15:07Z","timestamp":1786014907000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Towards Performatively Stable Equilibria in Decision-Dependent Games for Arbitrary Data Distribution Maps"],"prefix":"10.1007","volume":"115","author":[{"given":"Guangzheng","family":"Zhong","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiming","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,6]]},"reference":[{"issue":"8","key":"7115_CR1","doi-asserted-by":"publisher","first-page":"2267","DOI":"10.1007\/s10994-021-06037-z","volume":"110","author":"R Akrour","year":"2021","unstructured":"Akrour, R., Atamna, A., & Peters, J. 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