{
  "abstract": "We study prophet inequalities for $n$ i.i.d. base rewards subject to multiplicative discounts across $S$ phases. For equal-length phases and a common discount factor $\\widetilde\\gamma$, we characterize the optimal competitive ratio of single-quantile policies as $n\\to\\infty$ with $S$ and $\\widetilde\\gamma$ fixed. The resulting sharp guarantee describes the dependence on the phase parameters: it recovers $1-1/e$ without discounting and approaches $1/2$ in the worst-case regime. We also establish a distribution-free comparison that bounds the discounted prophet by a fractional i.i.d. maximum with effective horizon $B$, the sum of the discount weights across all observations. The comparison yields acceptance probability $1/B$, determined only by the phase profile. This calibration attains the canonical optimum even among single-quantile policies that may choose their acceptance probability using the known base distribution. The comparison also gives guarantees for heterogeneous discounts and unequal phase lengths. A complementary optimal-stopping upper bound establishes the same limiting optimum of $1/2$ for arbitrary stopping rules in the canonical model. Finally, the same effective-horizon calibration sets acceptance probability $1-\\widetilde\\gamma$ under infinite-horizon geometric discounting. This explicit quantile rule attains Boshuizen's (1994) known optimal worst-case ratio $1/(1+\\widetilde\\gamma)$ without solving a Bellman equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.30118",
  "author_total": 2,
  "authors": [
    {
      "name": "Jung-Hun Kim"
    },
    {
      "name": "Vianney Perchet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkc1wwavwtypw.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "dkc1wwavwtypw",
  "issns": [],
  "landing_url": "https://hal.science/hal-05855084v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ],
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      "hal",
      "1.info.info-dm"
    ],
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      "hal",
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  "openalex": null,
  "pdf_url": "https://hal.science/hal-05855084/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkc1wwavwtypw.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T12:20:25Z",
      "native_id": "hal-05855084",
      "source": "hal",
      "url": "https://hal.science/hal-05855084v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-oc",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Effective Horizons for Prophet Inequalities with Discounted I.I.D. Rewards",
  "title_text": "Effective Horizons for Prophet Inequalities with Discounted I.I.D. Rewards",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:25Z",
  "url": "https://research.draconitic.com/paper/dkc1wwavwtypw",
  "venue": "HAL",
  "versions": []
}