Bayesian Sequential Search with Censored Observations

Abstract: This paper studies how information censoring enables a myopic cutoff rule in Bayesian sequential search. Under full information, Bayesian learning generally destroys the monotonicity of continuation values, preventing simple cutoff rules. We show that one-sided censoring restores monotonicity by limiting posterior fluctuations, thereby making a myopic cutoff rule optimal. By decomposing the intertemporal change in the marginal value of search into a fallback-value effect and a learning effect, we derive necessary and sufficient conditions for monotonicity under lower censoring and characterize the optimal cutoff rule. In contrast, under full revelation, monotonicity requires highly restrictive conditions. We further show that expected monotonicity (i.e., the supermartingale property) is characterized by the same conditions under both lower censoring and full revelation, owing to Bayes plausibility and the affine structure of the problem. Thus, censoring restores monotonicity not by altering expected learning, but by reducing posterior volatility. Finally, we apply our framework to job search, consumer price search, and product experimentation.

• The paper characterizes when lower censoring makes the one-step marginal search value decrease along every feasible path, restoring the optimality of myopic reservation-cutoff stopping rules.
• The analysis separates fallback-value and learning effects, showing that censoring suppresses posterior volatility while expected learning remains identical to full revelation, and provides necessary and sufficient monotonicity conditions.
• The results apply to recruitment screening and consumer price search, demonstrate that full revelation can destroy cutoff tractability, and establish finite stopping under mild conditions through likelihood-ratio dynamics.

Lehrer and Li study Bayesian sequential search with free recall when the payoff distribution is unknown and observations are endogenously censored by the decision maker's (DM) current fallback value. The paper's central claim is that one-sided censoring restores the monotonicity of the one-step marginal search value, and with it the optimality of myopic reservation-cutoff stopping rules, whereas full revelation generically destroys monotonicity. The analysis proceeds by decomposing the intertemporal change in the marginal search value into a fallback-value effect and a learning effect, deriving sharp necessary and sufficient conditions under censoring, and contrasting them with the highly restrictive conditions required under full revelation. A notable secondary finding is that expected monotonicity is characterized by identical conditions under censoring and full revelation, so censoring stabilizes search by suppressing posterior volatility rather than by altering the average drift of beliefs.

The model and the role of monotonicity

A DM searches through ex ante identical boxes with free recall, paying cost $c$ per inspection. The reward distribution $L$ is unknown but belongs to a finite set $\{L_1,\ldots,L_\ell\}$; the state is the pair $(Y_t,\boldsymbol\alpha_t)$ of the fallback value (the best payoff observed so far) and the posterior belief. The paper's baseline is the two-distribution case with finite support, and its focal information structure is lower censoring: an outcome $X_{t+1}\le Y_t$ is pooled into the coarse signal $\{X\le Y_t\}$, while an outcome $X_{t+1}>Y_t$ is fully revealed. This structure captures settings such as recruitment screening, product experimentation against a benchmark, and (in its dual form) consumer price search.

The key object is the one-step marginal search value $U(y,\boldsymbol\alpha)=\mathbb E_\alpha[(X-y)_+]$, which is affine in the posterior. A search problem is monotone if $U$ weakly decreases after every feasible state transition. Under pathwise monotonicity, the paper proves that the myopic rule—stop iff $U(Y_t,\boldsymbol\alpha_t)\le c$—is optimal, because once the marginal value falls below cost it can never recover along any continuation path. This is the sense in which monotonicity delivers tractability: without it, a DM may rationally accept a negative one-step gain as an investment in future belief states, and cutoff rules fail.

Monotonicity under lower censoring

The decomposition writes the realized change in the marginal search value as a learning effect $L$0 plus a fallback-value effect $L$1, where $L$2. Under the single-crossing order $L$3 (with $L$4 inferior), the fallback-value effect is always non-positive, and the learning effect is positive only after a fully revealed outcome $L$5, where $L$6 is the crossing point. The main characterization (Theorem 1) states that the problem is monotone if and only if, for every state and every $L$7, the exact fallback-value effect $L$8 dominates the exact learning effect $L$9. The paper also provides a state-free sufficient condition depending only on the primitive distributions, bounding the learning effect by maximizing the posterior shift over $\{L_1,\ldots,L_\ell\}$0 (which occurs at $\{L_1,\ldots,L_\ell\}$1 with $\{L_1,\ldots,L_\ell\}$2). The extension to continuous distributions parallels the discrete result.

The mechanism behind the stabilizing role of censoring is that the censored posterior is a martingale projection of the full-information posterior: pooling the lower tail compresses posterior dispersion, and by Jensen's inequality bounds favorable belief revisions. The paper is explicit that the single-crossing assumption cannot be weakened to first-order stochastic dominance: it constructs an FOSD-ordered pair for which a mid-support revelation induces an optimistic shift that dominates the fallback effect and breaks monotonicity.

Two structural results refine the characterization. First, the stopping cutoff $\{L_1,\ldots,L_\ell\}$3 lies between the known-distribution benchmarks $\{L_1,\ldots,L_\ell\}$4 and $\{L_1,\ldots,L_\ell\}$5, is weakly decreasing in $\{L_1,\ldots,L_\ell\}$6 and $\{L_1,\ldots,L_\ell\}$7, and admits an equivalent belief-cutoff representation $\{L_1,\ldots,L_\ell\}$8 with explicit form $\{L_1,\ldots,L_\ell\}$9 when interior. Second, under mild genericity assumptions, the optimal policy stops after a uniformly bounded number of searches on every feasible history, with a recursive pathwise bound expressed through likelihood-ratio dynamics: consecutive censored events multiply the odds ratio by $(Y_t,\boldsymbol\alpha_t)$0, driving the DM to stop after finitely many rejections.

Applications

The job-search application maps recruitment screening onto lower censoring with an endogenous threshold: firms screen applicants against the worker's current fallback requirement, revealing exact match values only when the prospective offer strictly exceeds it. A worked example with $(Y_t,\boldsymbol\alpha_t)$1, $(Y_t,\boldsymbol\alpha_t)$2, $(Y_t,\boldsymbol\alpha_t)$3, and $(Y_t,\boldsymbol\alpha_t)$4 illustrates the mechanics: two consecutive rejections at fallback value $(Y_t,\boldsymbol\alpha_t)$5 multiply the odds ratio by $(Y_t,\boldsymbol\alpha_t)$6 each, moving it from $(Y_t,\boldsymbol\alpha_t)$7 past the stopping cutoff $(Y_t,\boldsymbol\alpha_t)$8 and inducing exit. The consumer price-search section develops the upper-censoring dual, where quotes above the current best price are censored, and derives the analogous necessary and sufficient condition plus a state-free sufficient condition for monotonicity, with the stopping cutoff $(Y_t,\boldsymbol\alpha_t)$9 weakly decreasing in optimism.

Full revelation: a negative benchmark

Under full revelation, the paper proves that monotonicity holds if and only if the single-crossing point satisfies $X_{t+1}\le Y_t$0: every non-maximal observation must shift the posterior toward the inferior distribution, leaving only the maximal realization as a potential source of optimism. This is a starkly restrictive requirement—for the running example with $X_{t+1}\le Y_t$1, the problem is monotone under both lower- and upper-censoring but not under full revelation. Monotonicity under full revelation moreover implies monotonicity under any garbling, a consequence of the affinity of $X_{t+1}\le Y_t$2 in $X_{t+1}\le Y_t$3 and the law of total expectation. The paper draws the structural conclusion that informational coarsening is not merely a modeling convenience but a necessary ingredient for preserving cutoff tractability, even though full revelation weakly raises the DM's search payoff.

Expected monotonicity and the volatility channel

Expected (supermartingale) monotonicity requires only that $X_{t+1}\le Y_t$4 decrease on average. The paper shows the condition is equivalent across lower censoring and full revelation: both regimes share the same fallback transition, and by Bayes plausibility plus the affinity of $X_{t+1}\le Y_t$5 in $X_{t+1}\le Y_t$6, the expected learning effect is identical. Censoring therefore restores pathwise monotonicity purely by reducing realized posterior volatility, not by changing expected learning. The paper is careful about the decision-theoretic status of expected monotonicity: it is strictly weaker than pathwise monotonicity (an explicit example demonstrates a problem that is expected monotone but not pathwise monotone) and does not guarantee myopic cutoff optimality, since favorable realizations can push $X_{t+1}\le Y_t$7 back above $X_{t+1}\le Y_t$8. Its operational value lies in delegation: a non-attending principal, such as an automated recruitment algorithm, can commit ex ante to a fixed stopping time that is optimal within the class of predetermined stopping times.

Dynamic programming foundation

Because the search problem is undiscounted, the Bellman operator is not a contraction and the Bellman equation admits multiple fixed points (the constant function equal to $X_{t+1}\le Y_t$9 is a spurious fixed point). In the online appendix, the paper provides a self-contained proof, adapted from Bertsekas's abstract dynamic programming, that the optimal value function is the pointwise limit of finite-horizon value functions and hence the smallest fixed point of the Bellman operator, with the induced stopping time optimal for the infinite-horizon problem. The construction extends to a variant with an exogenous outside option.

Limitations and open questions

The sharp monotonicity characterizations are confined to the two-distribution framework; in general multi-distribution settings the dynamic programming machinery applies, but monotonicity need not hold. The single-crossing order is a scalar condition, and the authors note that extending to continuous or multidimensional state spaces will likely require shape restrictions such as log-concavity or MLRP on posterior predictive distributions. The censoring rule is taken as exogenous; how a principal would optimally design the censoring threshold in a mechanism design or persuasion framework is left unaddressed. The finite pathwise bound relies on a genericity assumption ruling out the knife-edge case $\{X\le Y_t\}$0 and on finite support. Finally, the paper's separation between pathwise and expected monotonicity generates testable implications for reservation-cutoff adjustment after coarse failures, but no empirical analysis is conducted.

Conclusion

The paper establishes that endogenous one-sided censoring disciplines posterior dynamics sufficiently to restore the classical reservation-cutoff structure of Bayesian sequential search, characterizes exactly when this restoration occurs, and shows that the tractability hierarchy across information structures is driven by posterior volatility rather than expected learning. The resulting framework connects search with learning, censored-data Bayesian updating, and bandit-style experimentation, while leaving the multi-distribution, endogenous-censoring, and empirical extensions open.