Optimal Shopping When the Sales Are on—a Markovian Full-Information Best-Choice Problem
Why this work is in the frame
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Bibliographic record
Abstract
We study a full-information best-choice problem viewed in a shopping context. A certain commodity can be found at certain random times with stochastically fluctuating prices. While the prices may have a tendency to decrease, the instants at which items are offered become less frequent and it is possible that the item currently found will be the last one. The prospective customer's objective is to buy at the right time so as to minimize the expected price of the acquired item. We propose a two-dimensional Markov chain model with a rather general continuous-time point process structure and dependence of the random prices on the availability times of the items. The value function v of the associated optimal stopping problem is characterized as the smallest solution of a two-dimensional integral equation; this allows us to find the optimal policy under certain conditions. In particular, we consider a nonhomogeneous Poisson model for which more specific results can be obtained. We derive a differential equation of which v is the uniformly smallest nonnegative solution. This way v is determined up to a boundary condition at infinity. We provide criteria for identifying a solution as the value function and also for the natural stopping rule to be optimal. Several examples are given.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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