Bayesian dithering for learning: Asymptotically optimal policies in dynamic pricing
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Bibliographic record
Abstract
We consider a dynamic pricing and learning problem where a seller prices multiple products and learns from sales data about unknown demand. We study the parametric demand model in a Bayesian setting. To avoid the classical problem of incomplete learning, we propose dithering policies under which prices are probabilistically selected in a neighborhood surrounding the myopic optimal price. By analyzing the effect of dithering in facilitating learning, we establish regret upper bounds for three typical settings of demand model. We show that the dithering policy achieves an upper bound of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mrow> <mml:mi>log</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="">$\log T$</mml:annotation> </mml:semantics> </mml:math> when the parameter set is finite. It can be modified to achieve a constant regret bound under an additional assumption. We also prove an upper bound of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:msqrt> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>log</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:msqrt> <mml:annotation encoding="">$\sqrt {T\log T}$</mml:annotation> </mml:semantics> </mml:math> when the parameter set is compact and convex. Each bound matches (up to a logarithmic factor) the existing lower bound of any pricing policy. In this way, we show that dithering policies achieve asymptotically optimal performance in three different parameter settings, which demonstrates dithering as a unified approach to strike the balance between exploration and exploitation.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it