The Gittins Index for a Transparent One-Armed Bandit with a Continuous Payoff Spectrum in Equilibrium States
Abstract
We present an analogue of the classical Gittins index for a one-armed decision problem with a continuous spectrum of payoffs. The model assumes that the decision-maker observes independent realizations of a random variable and, at each step, decides whether to accept the current opportunity or continue observing. We show that the optimal strategy takes the form of a threshold rule, while the corresponding reservation index is determined by a one-dimensional fixed-point equation with a direct decision-theoretic interpretation. The model is illustrated with an example of bookmaker betting related to horse racing and the Kelly criterion. This perspective allows the proposed index to be viewed as a threshold of informational advantage. This, in turn, points to potential applications in optimal stopping problems and decision-making under uncertainty.
Versions
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BibTeX
@misc{makowski2026gittins,
title = {{The Gittins Index for a Transparent One-Armed Bandit with a Continuous Payoff Spectrum in Equilibrium States}},
author = {Marcin Makowski and Edward W. Piotrowski and Jan L. Cieśliński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30150},
eprint = {2610.30150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30150},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Gittins Index for a Transparent One-Armed Bandit with a Continuous Payoff Spectrum in Equilibrium States
AU - Marcin Makowski
AU - Edward W. Piotrowski
AU - Jan L. Cieśliński
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.30150
UR - https://arxiv.org/abs/2610.30150
AB - We present an analogue of the classical Gittins index for a one-armed decision problem with a continuous spectrum of payoffs. The model assumes that the decision-maker observes independent realizations of a random variable and, at each step, decides whether to accept the current opportunity or continue observing. We show that the optimal strategy takes the form of a threshold rule, while the corresponding reservation index is determined by a one-dimensional fixed-point equation with a direct decision-theoretic interpretation. The model is illustrated with an example of bookmaker betting related to horse racing and the Kelly criterion. This perspective allows the proposed index to be viewed as a threshold of informational advantage. This, in turn, points to potential applications in optimal stopping problems and decision-making under uncertainty.
ER -