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arxiv: 2605.16601 · v1 · pith:R3VRS2G4new · submitted 2026-05-15 · 💻 cs.GT

Online Contract Selection for Continual Coverage

Qinge Chi , Sebastian Perez-Salazar This is my paper

Pith reviewed 2026-05-19 21:06 UTC · model grok-4.3

classification 💻 cs.GT
keywords online algorithmscompetitive analysiscontract selectioncontinual coveragei.i.d. pricesquantile thresholdsdeferred modelconcurrent model
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The pith

For i.i.d. prices, quantile-threshold algorithms achieve an exact asymptotic competitive ratio of 2.472 in the deferred contract model and improve the concurrent-model bounds to 4.179.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how to keep a system covered at every time step by buying contracts of arbitrary length online, when each period's price is drawn from a known distribution. In the deferred model contracts queue back-to-back without overlap; in the concurrent model they can start immediately and overlap. When prices are i.i.d., the authors give quantile-based threshold rules whose worst-case ratio against the offline optimum is exactly characterized for the deferred case and asymptotically equals roughly 2.472. The same rules yield a matching lower bound and an asymptotic upper bound of 4.179 for the concurrent case, tightening earlier numbers. When prices are merely independent but not identically distributed, no finite ratio is possible at all.

Core claim

For i.i.d. prices the deferred model admits an exact characterization of the optimal competitive ratio, which approaches ζ* ≈ 2.472 as the horizon grows; the concurrent model has a lower bound of ζ* and an asymptotic upper bound of 4.179, both obtained from quantile-based threshold algorithms whose analysis uses linear-programming duality in quantile space. The same setting yields no finite competitive ratio when prices are independent but not identically distributed.

What carries the argument

Quantile-based threshold algorithms that set contract lengths according to the price quantile and are analyzed by LP duality in quantile space.

If this is right

  • The deferred model has a tight asymptotic competitive ratio of ζ* ≈ 2.472.
  • The concurrent model has an asymptotic competitive ratio between ζ* and 4.179.
  • The algorithms translate directly into practical threshold rules that work for any known distribution.
  • No finite competitive ratio exists when prices are independent but not identically distributed.
  • The bounds improve the previous lower bound of 2.148 and upper bound of 6.052.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Procurement systems that know the price distribution can replace ad-hoc rules with simple quantile thresholds that guarantee a constant factor of the hindsight optimum.
  • The sharp separation between i.i.d. and merely independent prices may appear in other continual online resource problems such as repeated energy procurement or cloud-instance leasing.
  • Learning the distribution on the fly would be a natural next question, since the current results collapse without distributional knowledge.

Load-bearing premise

The price distribution is known in advance and successive prices are i.i.d.

What would settle it

For a concrete i.i.d. distribution such as uniform on [0,1], exhibit an online policy whose ratio against the offline optimum stays below 2.472 for arbitrarily long horizons, or show that some non-i.i.d. independent price sequence admits a finite competitive ratio.

Figures

Figures reproduced from arXiv: 2605.16601 by Qinge Chi, Sebastian Perez-Salazar.

Figure 2
Figure 2. Figure 2: LP objective for j = 55, . . . , 65 6 Competitive Ratio Upper Bound for OSCC We now extend the analysis to any CDF F. For this, we consider ALG with the same family of inputs described at the beginning of Section 5.1. We require that a and q to satisfy a/q ≥ 2 so that t0 ≤  a · q 0  , which simplifies the expression of t0 and thus part of the analysis (see details in Lemma 6.6). We then assume n → ∞ and … view at source ↗
read the original abstract

Motivated by applications where a system must remain operational via continual procurement of contracts, we study two online contract selection problems under uncertain prices. At each time step, a price drawn from a known distribution is revealed online, and the decision-maker may initiate a contract of arbitrary duration, incurring a cost equal to the product of the price and the contract length; moreover, every time period must be covered by at least one active contract. We consider two models depending on how contracts cover time: a \emph{deferred model}, in which contracts are queued back-to-back, and a \emph{concurrent model}, in which contracts become active immediately and may overlap. In both settings, we seek online algorithms that minimize their competitive ratio, i.e., the ratio between the expected cost incurred by the online algorithm and the expected offline optimal cost when all prices are known in advance. We first focus on the case where prices are independent and identically distributed (i.i.d.). For the deferred model, we characterize exactly the worst-case optimal competitive ratio, which is asymptotically $\zeta^* \approx 2.472$ as the time horizon grows. For the concurrent model, we prove a lower bound of $\zeta^*$ on the optimal competitive ratio and an asymptotic competitive ratio of at most $4.179$. These bounds improve upon the current lower bound of $2.148$ and upper bound of $6.052$ on the optimal competitive ratio. For both models, our algorithms are quantile-based that can be easily translated into practical threshold-based algorithms for any distribution. Our proofs follow from linear programs and duality arguments in quantile spaces. Lastly, we show that, in both models, no finite competitive ratio exists when the prices are still independent but not necessarily identically distributed, proving a striking division in the two price settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies two online contract selection problems for continual coverage under prices drawn from known distributions: a deferred model where contracts queue back-to-back and a concurrent model allowing overlaps. For i.i.d. prices, it exactly characterizes the optimal competitive ratio for the deferred model (asymptotically ζ* ≈ 2.472) and gives a lower bound of ζ* together with an asymptotic upper bound of 4.179 for the concurrent model, improving prior bounds of 2.148 and 6.052. Both results rely on quantile-based threshold algorithms obtained from LP formulations and duality arguments in quantile space. The paper also proves that no finite competitive ratio exists when prices are independent but not identically distributed.

Significance. If the LP-duality derivations hold, the work supplies the first exact asymptotic characterization for the deferred model and materially tighter bounds for the concurrent model, together with a clean impossibility result that separates the i.i.d. and non-i.i.d. regimes. The quantile-space LP technique is a methodological strength that yields practical threshold algorithms for arbitrary distributions and avoids parameter fitting. These contributions advance competitive analysis for coverage-constrained procurement problems.

major comments (2)
  1. [§4] §4 (deferred-model LP): the exact asymptotic value ζ* ≈ 2.472 is obtained by solving an infinite-horizon LP in quantile space; the manuscript should supply either a closed-form expression for the limit or a rigorous convergence argument showing that the finite-horizon optima approach this value, because the central claim of an 'exact characterization' rests on this limit.
  2. [§5] §5 (concurrent-model upper bound): the algorithm achieving the asymptotic ratio 4.179 is constructed from the dual of a quantile LP; it is unclear whether the same LP can be used to prove a matching upper bound or whether the gap between ζ* and 4.179 is an artifact of the particular dual relaxation chosen.
minor comments (2)
  1. [Model section] The definition of the competitive ratio (expected online cost over expected offline optimum) is stated clearly in the abstract but should be repeated verbatim in the model section to avoid any ambiguity when the horizon is finite versus asymptotic.
  2. [Throughout] Table summarizing the four competitive-ratio results (deferred i.i.d., concurrent i.i.d. lower/upper, non-i.i.d. impossibility) would improve readability; currently the bounds are scattered across the abstract and main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive feedback on our manuscript. We address each of the major comments in detail below. The suggested clarifications can be incorporated with minor revisions.

read point-by-point responses

  1. Referee: [§4] §4 (deferred-model LP): the exact asymptotic value ζ* ≈ 2.472 is obtained by solving an infinite-horizon LP in quantile space; the manuscript should supply either a closed-form expression for the limit or a rigorous convergence argument showing that the finite-horizon optima approach this value, because the central claim of an 'exact characterization' rests on this limit.

    Authors: We agree that a rigorous justification for the asymptotic limit is necessary to support the claim of an exact characterization. In the revised version, we will add a convergence theorem proving that the optimal objective values of the finite-horizon quantile LPs converge to that of the infinite-horizon LP as the horizon length tends to infinity. This is established by showing that the sequence of finite optima is monotone and bounded, hence convergent, and that any limit point satisfies the infinite LP. While we do not have a closed-form expression for ζ*, the convergence argument rigorously justifies the asymptotic value. We will also include numerical evidence of rapid convergence for large horizons. revision: yes

  2. Referee: [§5] §5 (concurrent-model upper bound): the algorithm achieving the asymptotic ratio 4.179 is constructed from the dual of a quantile LP; it is unclear whether the same LP can be used to prove a matching upper bound or whether the gap between ζ* and 4.179 is an artifact of the particular dual relaxation chosen.

    Authors: The 4.179 upper bound is obtained by solving the dual of the quantile LP to derive threshold parameters for an online algorithm, then proving that this algorithm achieves a competitive ratio of at most 4.179 against the offline optimum. The LP relaxation is used to find a good dual solution that yields a feasible algorithm, but it does not directly provide a matching lower bound for the concurrent model. The lower bound of ζ* is shown separately by reducing the concurrent problem to the deferred model. We acknowledge that the gap may be due to the relaxation and do not claim optimality of 4.179; however, it represents a significant improvement over the previous upper bound of 6.052. In the revision, we will explicitly state that the bound is not necessarily tight and discuss the role of the dual relaxation. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central results follow from explicit LP formulations and duality arguments defined over quantile space for the i.i.d. case, together with a direct adversarial construction for the non-i.i.d. impossibility result. These steps are self-contained, use the stated modeling assumptions (known distribution, coverage requirement) as inputs without redefining the target competitive ratio inside the same equations, and do not rely on self-citations or fitted parameters that are later renamed as predictions. The quantile-based threshold algorithms are derived from the dual solutions rather than presupposed by them.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard online-algorithm assumptions rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Prices are drawn independently from a known distribution
    Enables quantile-based threshold design and LP formulation in quantile space.
  • domain assumption Every time period must be covered by at least one active contract
    Defines the feasibility constraint for both deferred and concurrent models.

pith-pipeline@v0.9.0 · 5857 in / 1529 out tokens · 58153 ms · 2026-05-19T21:06:01.862528+00:00 · methodology

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