Learning Against Nature: Minimax Regret and the Price of Robustness
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917277765468160 |
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| author | Che, Yeon-Koo Li, Longjian Luo, Tianling |
| author_facet | Che, Yeon-Koo Li, Longjian Luo, Tianling |
| contents | We study how a decision-maker (DM) learns from data of unknown quality to form robust, ''general-purpose'' posterior beliefs. We develop a framework for robust learning and belief formation under a minimax-regret criterion, cast as a zero-sum game: the DM chooses posterior beliefs to minimize ex-ante regret, while an adversarial Nature selects the data-generating process (DGP). We show that, in large samples of $n$ signal draws, Nature optimally induces ambiguity by choosing a process whose precision converges to the uninformative signals at the rate $1/\sqrt{n}$. As a result, learning against the adversarial DGP is nontrivial as well as incomplete: the DM's ex-ante regret remains strictly positive even with an infinite amount of data. However, when the true DGP is fixed and informative (even if only slightly), our DM with a robust updating rule eventually learns the state with enough data. Still, learning occurs at a sub-exponential rate -- quantifying the asymptotic price of robustness -- and it exhibits ''under-inference'' bias. Our framework provides a decision-theoretic dual to the local alternatives method in asymptotic statistics, deriving the characteristic $1/\sqrt{n}$-scaling endogenously from the signal ambiguity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15246 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning Against Nature: Minimax Regret and the Price of Robustness Che, Yeon-Koo Li, Longjian Luo, Tianling Theoretical Economics We study how a decision-maker (DM) learns from data of unknown quality to form robust, ''general-purpose'' posterior beliefs. We develop a framework for robust learning and belief formation under a minimax-regret criterion, cast as a zero-sum game: the DM chooses posterior beliefs to minimize ex-ante regret, while an adversarial Nature selects the data-generating process (DGP). We show that, in large samples of $n$ signal draws, Nature optimally induces ambiguity by choosing a process whose precision converges to the uninformative signals at the rate $1/\sqrt{n}$. As a result, learning against the adversarial DGP is nontrivial as well as incomplete: the DM's ex-ante regret remains strictly positive even with an infinite amount of data. However, when the true DGP is fixed and informative (even if only slightly), our DM with a robust updating rule eventually learns the state with enough data. Still, learning occurs at a sub-exponential rate -- quantifying the asymptotic price of robustness -- and it exhibits ''under-inference'' bias. Our framework provides a decision-theoretic dual to the local alternatives method in asymptotic statistics, deriving the characteristic $1/\sqrt{n}$-scaling endogenously from the signal ambiguity. |
| title | Learning Against Nature: Minimax Regret and the Price of Robustness |
| topic | Theoretical Economics |
| url | https://arxiv.org/abs/2602.15246 |