Sequential Pricing with Deadlines and Correlated Buyers

Abstract: We study sequential posted pricing for selling a single item under an exogenous deadline: a seller makes take-it-or-leave-it price offers to buyers one at a time, stopping when some buyer accepts or the selling opportunity expires. We consider both deterministic deadlines and random deadlines drawn from a known distribution, and allow buyer valuations to be either independent or arbitrarily correlated. Despite its practical relevance, this collection of deadline-constrained sequential pricing models has received limited algorithmic study beyond special cases. We develop the first approximation algorithms for revenue maximization across these settings.

• The paper introduces a structured approach where non-adaptive pricing policies are optimal by leveraging order-induced submodularity.
• It develops tailored approximation strategies, achieving (1-1/e) guarantees for deterministic deadlines and MILP-based methods for stochastic horizons.
• The work rigorously proves NP-hardness limits and delineates computational bounds across varying correlation and deadline structures.

Sequential Pricing with Deadlines and Correlated Buyers: A Technical Essay

Problem Setting and Motivation

This paper studies the computational and algorithmic aspects of sequential posted-pricing for selling a single item under deadline constraints, in environments where buyers’ valuations may be arbitrarily correlated. The model is highly relevant for practical scenarios including negotiated sales (e.g., off-market home sales), sequential ad mediation, and other contexts where a seller presents take-it-or-leave-it offers, ceasing the process upon acceptance or when an exogenous deadline (which may be deterministic or stochastic) is reached.

Distinctively, unlike classic mechanism design settings, this work allows (1) both deterministic and random deadlines (horizons), and (2) buyer valuations that need not be independent, but can be drawn from an arbitrary joint distribution. Adaptivity is shown to be irrelevant: the optimal policy is non-adaptive, i.e., a fixed plan suffices, due to the only information revealed along the execution being whether previous offers have been accepted or not.

It is already established that the problem is NP-hard even with independent values and a deterministic horizon [xiao2020complexity]. Therefore, approximation algorithms are the only tractable route.

Main Algorithmic Contributions and Approximation Guarantees

The work systematically investigates several regimes defined by the nature of the horizon and correlation structure of buyer valuations, developing tailored polynomial-time approximation algorithms with matching hardness results in the strongest settings.

Known (Deterministic) Deadline

In the setting where the total number of offers, $n$, is fixed and buyer valuations are arbitrarily correlated, the authors establish that the expected revenue maximization objective can be cast as the maximization of a monotone, submodular function subject to a matroid constraint. Submodularity emerges only after imposing a canonical execution order: for any set of buyer–price pairs, it is optimal to offer them in decreasing order of price. This structure enables the application of continuous greedy and pipage rounding [calinescu2011maximizing], yielding an optimal $(1-1/e)$-approximation.

Hardness is tight: it is shown to be NP-hard to achieve a factor better than $(1-1/e)$, even with independent values, by reduction from weighted Max-$k$-Cover [feige1998threshold].

Unknown (Stochastic) Horizon, Independent Valuations

For a random number of sequential opportunities, the canonical order approach is not generally extendable, as the optimal order of offers may depend non-globally on the subset of offers under consideration.

The contributions are as follows:

• Deterministic $1/4$-Approximation: By extending the time-indexed ground set and defining an appropriate pruning function $G(\cdot)$ on offer sets, the authors show that while the objective is not submodular, it possesses a submodular order in reverse execution time. This property enables the use of local-search techniques for a $1/4$-approximation [pmlr-v202-udwani23a].
• Randomized $(1/2 - \varepsilon)$-Approximation: Introducing a mixed-integer linear program (MILP) relaxation that decouples horizon and valuation uncertainty, the authors derive a threshold structure for optimal solutions. By solving this relaxation via an EPTAS and designing a randomized, OCRS-style attenuation scheme inspired by contention resolution [alaei2014bayesian], they construct an implementable policy achieving $(1/2-\varepsilon)$-approximation. The analysis is shown to be tight due to a provable factor-2 gap between the MILP relaxation and the true optimum in stochastic settings.
• Memoryless (Geometric) Horizons: When the deadline is geometric (i.e., the probability of surviving to the next offer is constant), a global score order exists [brubach2023onlinematchingframeworksstochastic], and the revenue function is monotone and submodular. This yields $(1-1/e)$-approximation as in the deterministic case.

Unknown Horizon, Correlated Valuations

Much of the algorithmic structure facilitating approximation in the independent-value case collapses under arbitrary correlation. In this more general and difficult regime:

• If the horizon distribution has Increasing Failure Rate (IFR) structure, a median-based reduction yields a $(1-1/e)$0-approximation: optimizing the deterministic-horizon algorithm at the median horizon length captures, up to a constant, the stochastic-horizon optimal revenue. The analysis leverages a scaling property for deterministic optima and a geometric decay in the survival function of the IFR random variable.
• For general horizon distributions (no structure), the best algorithm is to try deterministic-horizon greedy for all possible horizon values and choose the best plan. Due to the potential for optimal revenue being spread across multiple time scales, this method incurs a logarithmic approximation loss: the guarantee is $(1-1/e)$1, where $(1-1/e)$2 is the number of buyers. This is proven tight via an explicit construction.

The comprehensive results are summarized below:

Horizon	Valuations	Approximation Factor
Known	Independent	$(1-1/e)$3 [PTAS]
Known	Correlated	$(1-1/e)$4 (tight)
Unknown	Independent	$(1-1/e)$5 (det.), $(1-1/e)$6 (rand.)
Unknown	Correlated	$(1-1/e)$7
Geometric	Independent	$(1-1/e)$8
IFR	Correlated	$(1-1/e)$9

Technical Insights

The paper’s algorithmic innovations fundamentally rely on uncovering latent structural principles:

• Order-Induced Submodularity: The key to achieving strong approximation for correlated buyers in the deterministic horizon case is that, when buyer–price pairs are executed in decreasing price order, the expected maximum-of-acceptances objective becomes monotone submodular, reducing to a matroid-constrained instance.
• Submodular Order & Pruning: For stochastic horizons, conventional submodularity fails. However, by imposing a reverse-time order and defining the pruning operator (which discards dominated offers within time steps), the authors show diminishing returns along this order are sufficient for directed local search to afford tractable guarantees.
• MILP Relaxation and Attenuation: For powerful relax-and-round guarantees, the separation of horizon and valuation uncertainty allows for a MILP upper bound whose solution has a threshold (knapsack) structure. The contention-resolution attenuation step bridges the relaxation and implementable policies.

Critically, all attempts to extend the known submodularity-based or relaxation-based algorithms to the fully general (correlated, arbitrary horizon) setting fail due to constructed counterexamples and proven gaps. Thus, the best known algorithm in that regime is logarithmic.

Numerical and Complexity Results

• Tight Approximation for Known Horizon, Correlated: $(1-1/e)$0, NP-hard to do better.
• Randomized Attenuation Algorithm: For independent values, the $(1-1/e)$1-approximation cannot be exceeded due to an inapproximability gap between the MILP and realizable sequential policies.
• Negative Results: For correlated buyers and arbitrary random horizons, both the submodular order and relaxation approaches exhibit arbitrarily large gaps, demonstrating that constant-factor approximation is unachievable unless further structure is imposed.

Implications and Future Directions

Practically, the results delineate the boundaries of tractable sequential pricing under deadlines with and without independence. For markets where buyer values are driven by few common shocks (marketwide covariates), it is crucial to recognize that algorithmic approximation guarantees deteriorate unless the correlation can be structured or otherwise bounded.

Theoretically, the paper leaves open significant problems:

1. Constant-Factor Approximation for Correlated Values & Arbitrary Horizon: No such guarantee is known or ruled out, but existing techniques do not suffice.
2. Correlation Structure: Understanding what types of realistic correlation (for instance, correlations induced by multiple offers to the same buyer, or Markovian buyers) admit efficient approximation remains a major direction.
3. Beyond Revenue Maximization: Extensions to welfare or profit functions, as well as integration with online learning or adaptive exploration in incomplete information settings, offer broad ground for further research.

Conclusion

This work provides a thorough, technically deep treatment of the algorithmic aspects of deadline-constrained, sequential posted pricing under both independent and correlated buyer models. By rigorously characterizing the nature of approximation guarantees, elucidating the role of value correlation and horizon randomness, and developing reduction- and relaxation-based frameworks for maximizing expected revenue, it both advances the tractable frontiers of this class of dynamic mechanism design problems and clearly maps the challenges that remain for algorithmic pricing in complex, deadline-driven environments.