📈 economics

Optimal selling time with evolving private information

This paper analyzes the optimal selling time for an indivisible object when the buyer's value follows a privately observed Markov process, characterizing dynamic incentive compatibility and demonstrating that under specific conditions, incomplete information leads to a threshold-based selling rule that weakly delays the sale compared to the complete information case.

Original authors: Kiho Yoon

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kiho Yoon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a seller holding a single, unique item—perhaps a rare piece of land, a specialized machine, or a license for a new technology. This item cannot be split or shared; it is all or nothing. The seller's goal is simple: sell it at the right moment to get the most money possible. The complication is that the item's value to the buyer is not fixed. Instead, the buyer's willingness to pay changes over time, shifting up and down like a tide. In a perfect world where the seller could see exactly what the buyer is thinking at every moment, the decision would be a straightforward calculation of whether to sell now or wait for a potentially better offer later. But in reality, the buyer's changing value is a secret. The buyer knows their own current desire for the item, but the seller does not. The buyer must report their value, and the seller must decide whether to trust that report and sell, or to wait and see if the value changes. This creates a delicate game of trust and timing, where the seller must design a system that encourages the buyer to be honest while still maximizing the seller's profit.

This is the core puzzle tackled by Kiho Yoon in a study of how to sell a single object when the buyer's value evolves secretly over time. The research moves beyond simple pricing to a dynamic strategy where the seller commits to a set of rules for when to sell and how much to charge, knowing that the buyer will try to manipulate the timing to their own advantage. The central discovery is that when the seller cannot see the buyer's true value, the optimal strategy changes in a specific and predictable way: the seller should wait longer before selling than they would if they had full knowledge. This delay is not a mistake or a failure of the market; it is a necessary adjustment to the lack of information. The study proves that the seller must set a higher bar for the buyer's reported value before agreeing to a sale. This higher bar acts as a filter, ensuring that the buyer is not just claiming a high value to get a deal, but actually has a strong, genuine interest that justifies the immediate transfer of the object.

The researchers found that this problem can be understood by looking at two competing forces. On one hand, selling immediately brings in cash right now. On the other hand, waiting offers the chance for the value to rise, which could lead to a bigger payout later. When the seller knows everything, they simply compare the current value against the potential future value. When information is hidden, a new factor enters the equation: the "information rent." This is the extra value the buyer holds simply because they know more than the seller. To get the buyer to reveal their true value, the seller must effectively pay a price in the form of a lower sale price or a delayed sale. The study shows that this hidden cost of information makes the seller more cautious. The seller raises the threshold for selling, meaning they require a higher reported value to trigger a sale. This higher threshold naturally leads to a delay in the transaction. The research demonstrates that this delay happens in every possible scenario, not just on average. Even if the buyer's value is rising rapidly, the seller still waits longer under incomplete information than they would if they could see the value directly.

To solve this complex problem, the author developed a mathematical framework that tracks how the buyer's past reports influence the seller's future decisions. They showed that the optimal strategy takes the form of a simple rule: if the buyer's reported value is above a certain number, sell; if it is below, wait. This number, or threshold, is not fixed; it changes based on how much "information rent" the buyer has accumulated from previous interactions. The more the buyer has been able to hide their true value in the past, the higher the threshold becomes in the present. The study also identified specific conditions under which this strategy works perfectly. For instance, if the way the buyer's value changes over time follows certain predictable patterns, the seller can rely on a straightforward rule that looks only at the current period's value and the immediate future, without needing to calculate complex long-term scenarios.

The paper also explored what happens when the buyer's value changes in different ways. In some cases, the effect of the buyer's initial private information fades away quickly, and the seller's strategy reverts to the standard "wait or sell" calculation after just one period. In other cases, the influence of that initial secret lingers, constantly shifting the threshold for sale. The researchers provided examples to show that while a simple rule often works, it is not always guaranteed to be the best solution. They found that in certain complex situations, the standard rules for ensuring honesty might fail, requiring the seller to check more carefully whether the buyer is truly being honest. However, in most realistic scenarios, the simple threshold rule holds up. The study concludes that while the seller cannot eliminate the delay caused by hidden information, they can manage it effectively by setting a clear, rising bar for when a sale is acceptable. This ensures that the seller does not sell too early to a buyer who is merely pretending to be eager, nor do they wait so long that they miss a good opportunity. The result is a balanced approach that protects the seller's revenue while respecting the buyer's private knowledge.

Technical Summary: Optimal Selling Time with Evolving Private Information

Problem Formulation
This paper addresses the optimal stopping problem faced by a monopolistic seller selling a single indivisible object to a buyer. The buyer's value for the object, denoted $\theta_t$ , evolves over time according to a privately observed Markov process. Unlike standard optimal stopping problems where the state is public, the seller faces incomplete information and must elicit the buyer's value in every period while ensuring dynamic incentive compatibility (truthful reporting) and participation. The seller commits to a dynamic mechanism to maximize expected discounted revenue. The core constraint is that the object can be allocated at most once; a sale in period $t$ terminates all future allocation opportunities.

Methodology and Framework
The analysis proceeds by first establishing a complete-information benchmark (Section 2) and then formulating the incomplete-information problem as a dynamic mechanism design problem (Section 3).

Dynamic Incentive Compatibility: Building on Pavan et al. (2014), the paper characterizes incentive compatibility using an envelope formula and an integral monotonicity constraint. The buyer's continuation utility is expressed as the integral of the "marginal type" effect, which includes both the direct allocation effect and the indirect effect of current reports on future types via impulse responses.
Relaxed Problem and Virtual Valuations: Following standard mechanism design practice, the author first solves a "relaxed" problem that ignores the integral monotonicity constraint. By applying the envelope formula, the seller's revenue is expressed in terms of dynamic virtual valuations:
$\psi_t = \theta_t - r_t$
Here, $\theta_t$ is the current value, and $r_t$ is the inherited information rent. The rent $r_t$ is defined recursively as the initial information rent term ( $\lambda(\theta_1)$ ) propagated through the realized impulse responses of the type process.
Augmented State Space: The problem is transformed into an optimal stopping problem over an augmented state space $S_t = (\theta_t, r_t)$ . The stopping payoff is $x - r$ , where $x$ is the current value and $r$ is the current inherited rent.
Implementation Conditions: After solving the relaxed problem, the paper derives conditions under which the resulting threshold rule satisfies the previously ignored integral monotonicity constraint. This involves checking "on-path" single crossing and the behavior of the marginal effect of reports on future allocations.

Key Results

Threshold Structure of the Relaxed Problem: Under Dynamic Single Crossing (the gain from selling now versus later is increasing in the augmented state) and Stochastic Monotonicity (a higher current state leads to a stochastically higher future augmented state), the optimal rule for the relaxed problem is a threshold rule. The seller sells if and only if the current value $\theta_t$ exceeds a threshold $c_t(r_t)$ that is increasing in the inherited information rent $r_t$ .
Comparison with Complete Information: The complete-information benchmark corresponds to the case where $r_t = 0$ . The paper proves that incomplete information weakly delays the sale relative to the complete-information stopping time along every realization of the value process. This delay occurs because the inherited information rent lowers the immediate revenue from selling, an effect that dominates the reduction in the continuation value (the value of waiting).
Dynamic Implementability: The paper provides sufficient conditions for the threshold rule of the relaxed problem to be dynamically implementable:
- On-path single crossing: The stopping set must be an upper set in the current report.
- Discounted impulse bound: The discounted one-period impulse response must be bounded by 1.
- Monotone information rent propagation: Higher earlier reports must not increase later stopping thresholds.
  The paper also notes that these are sufficient but not necessary conditions; an endogenous test based on the specific structure of the impulse responses can verify implementability even when these primitive conditions fail.
One-Step Look-Ahead (1-SLA) Rule: A condition is provided under which a simple one-step look-ahead rule (comparing current payoff to expected next-period payoff) is optimal for the relaxed problem. This occurs when the stopping regions are closed under continuation.

Examples and Applications
Section 4 illustrates the theory through several cases:

Independent Values: The inherited information rent vanishes after the first period, reducing the problem to a complete-information stopping problem thereafter.
Autoregressive Process: The inherited rent decays geometrically. The stopping threshold shifts up by exactly the amount of the current rent ( $c_t(r) = c^0_t + r$ ), and the virtual valuation follows the same transition law as the buyer's value.
Scaling Process: Demonstrates that the theory applies to state-dependent impulse responses that telescope.
Two-Period Counterexamples: Illustrate that future allocations need not be increasing in past reports (a standard requirement in screening) because a sale truncates the future. They also show that the sufficient conditions for implementability are not necessary.

Significance and Contribution
The paper claims to be the first to study the revenue-maximizing sale period for an indivisible object where the buyer's value follows a general privately observed Markov process. Its primary contributions are:

Characterization of the Relaxed Problem: It identifies the optimal stopping structure (threshold rule) and explicitly quantifies how incomplete information (via inherited information rent) distorts the stopping threshold relative to the efficient complete-information benchmark.
Dynamic Implementation: It addresses the specific feasibility issue of stopping problems, where a sale eliminates future allocations, rendering standard monotonicity conditions on future allocations inappropriate. It provides both primitive sufficient conditions and an endogenous test for implementability.
Synthesis: It bridges dynamic mechanism design (specifically the work of Pavan et al., 2014) with optimal stopping theory, showing how the "option value" of waiting interacts with information rents in a dynamic setting.

The paper concludes by noting that while the sufficient conditions for implementation are useful, a full characterization of transition kernels where the endogenous test holds automatically remains an open question, as does the analysis of specific seller instruments (e.g., posted prices vs. menus).

Technical Summary: Optimal Selling Time with Evolving Private Information

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