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A Ranking Representation of Optimal Sequential Search
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In markets with a proliferation of product variety, consumers often lack sufficient information to determine which alternative best aligns with their preferences. To reduce the impact of this uncertainty, they can engage in search to acquire product information and more accurately assess the utility of searched products, enabling better-informed purchases. Consumers’ search process, hence, reflects trade-offs between the expected gains from extra product information and the associated search costs. Structurally analyzing the sequence of actions recording the search process can reveal consumer preferences, characterize market demand, and also help assess how market strategies, such as platform design chen2017sequential,gu2022consumer,donnelly2024welfare, advertising chan2015consumer, product ranking ursu2018power, compiani2024online, and recommendation systems kim2011mapping, wan2024product, interact with the search process to influence firm profitability and consumer welfare.
Sequential search models are among the most commonly applied frameworks in empirical studies. The setup of these models is that, when facing product-level uncertainty, consumers acquire product information step by step through a series of costly actions that are subject to feasibility constraints;\footnote{For example, entering a product page reveals product details and enables checkout, while entering the checkout page reveals shipping costs and enables purchase.} at each step, they select the next action based on all information obtained from previous actions, until selecting a purchase to a product whose information is fully revealed.\footnote{Two well-known models that stand in contrast to the sequential search models are the simultaneous search model stigler1961economics and the learning model erdem1996decision. The former assumes that consumers determine the products to be searched a priori and purchase one only after search is completed. The latter assumes that consumers repeatedly learn about products and update their beliefs, and may make a purchase at any time.} In a milestone paper, weitzman1979optimal studies sequential search in a process involving two action types, inspections (obtaining full product information) and purchases, and proves that its optimal solution is characterized by a set of policies, called optimal rules. Accordingly, conducting a sequential search process optimally is equivalent to following these rules in making all search and purchase decisions throughout the process. This representation has been widely adopted in subsequent research to describe the optimality of other sequential search models.
However, the policy-based representation poses substantial difficulties for empirical applications, primarily because it introduces a sequential dependence issue: under optimal rules, consumers’ decisions at each step depend on information from all previous search actions, often including private evaluations that researchers cannot observe or identify.\footnote{The original paper considers search only over identifiable product attributes, such as price. While analytically convenient, such a setting struggles to explain divergence in subsequent decisions arising from identical prior search processes. Empirical work, therefore, often allows for unobserved idiosyncratic post-inspection evaluations.} Thus, without further assumptions, later search and purchase decisions cannot be analyzed unless all previous search outcomes are simulated. This imposes a large dimensionality requirement on simulation and a substantial implementation burden, and makes the model for the process (the Weitzman model hereafter) dependent on a restricted specification and complete information about the search process, thereby limiting its practical simplicity and applicability.
To address these empirical challenges, we propose a novel representation of optimal sequential search that departs from the optimal rules. Our main conclusion is that, under two foundational assumptions that are already made in the classic Weitzman model, Independence and Invariance, an optimal sequential search process is observationally equivalent to a partial ranking of all actions feasible throughout the search process. That is, for an action sequence that can be interpreted as the sequence of selected actions in an optimal sequential search process, we can equally interpret it as a partial ranking of all actions the consumer can select during that process. The two interpretations have identical probabilities and are empirically indistinguishable. Hence, an optimal sequential search process admits a ranking representation.
The intuition for this equivalence is that, although the private information underlying each search decision is not directly observed, subsequent decisions in the search process can be used to infer which information is relevant at earlier stages. This generates a sequence of pairwise comparisons that, when combined, induces a ranking of feasible actions. Consider a Weitzman-style sequential search process in which a consumer inspects $A$, $B$, and $C$, and then purchases $A$. The decision to inspect $C$ depends on whether its expected gain exceeds the highest utility among $A$ and $B$. Since utilities are unobserved, an analyst must consider both cases in which $A$ is preferred to $B$ and vice versa. However, the final purchase of $A$ reveals that $A$’s utility exceeds that of $B$, so the decision to inspect $C$ should depend only on $A$. This yields a local ranking in which inspecting $C$ is preferred to purchasing $A$, which in turn is preferred to purchasing $B$.
The ranking representation alleviates the complexities induced by sequential dependence and offers two main advantages for empirical application. First, it leads to a simple empirical strategy: implementing a sequential search model via its ranking representation. Compared with the policy-based, the ranking representation provides an equally complete but more parsimonious set of inequality conditions to capture optimal sequential search. Moreover, as it shares the probability structure of ranking models, researchers can leverage the latter’s established econometric tools to streamline the application of sequential search models. Using the Weitzman model as a benchmark, we show that its likelihood function can be expressed in a value-difference form. Building on this expression, we provide direct identification arguments and develop a GHK-style simulator, termed the rank-based GHK simulator, for simulated maximum likelihood estimation.\footnote{The GHK-style simulator geweke1989bayesian, hajivassiliou1998method, keane1994computationally was originally proposed as an importance sampling method for simulating choice probabilities in multivariate probit models, and it subsequently became a primary tool for estimating ranking models.} Compared with the existing policy-based simulator, the proposed approach allows the direct computation of the conditional distribution of search outcomes for products not purchased, thereby reducing the required simulation dimension without loss of efficiency. Monte Carlo experiments show that our method preserves the accuracy advantages of policy-based GHK simulators while substantially lowering their high implementation complexity.
Second, the ranking representation broadens the empirical applicability of sequential search models. It is not limited to the classic Weitzman-style process, but applies to any sequential search environment that satisfies Independence and Invariance and in which selection-revealed pairwise comparisons among feasible actions form a unique directed acyclic graph. This generality ensures that the representation remains widely applicable and can accommodate more realistic search processes in modern digital markets. We consider two illustrative extensions. The first concerns partially observed search data, where the inspection order, the set of searched products, or the final purchase may not be fully recorded. The second concerns search processes with multi-stage information acquisition, such as sequential search with discovery greminger2022optimal. In these settings, conventional approaches often rely on ad hoc treatments, such as imputing missing data, discarding partially observed observations, or re-deriving optimal rules. By contrast, our empirical strategy based on the ranking representation avoids these issues, allowing researchers to exploit the data in a complete, parsimonious, and unified way. Monte Carlo experiments further show that, using modified rank-based GHK methods, both extensions can be implemented efficiently without the practical difficulties associated with traditional approaches.
Our findings hold importance for both theory and empirical practice. On the theoretical side, although sequential search has been widely considered closely related to discrete choice problems, a formal linkage has been limited to its purchase outcome.\footnote{Prior studies choi2018consumer show that the final purchase outcome in an optimal sequential search process can be viewed as the result of a simple static discrete choice problem. We demonstrate that such results follow directly from the equivalence between sequential search and ranking.} We extend the connection by establishing a complete equivalence between sequential search and ranking models that applies to a broad class of model settings. This link provides a rigorous theoretical foundation for treating consumers’ search processes as directly informative choice data in demand analysis. On the empirical side, the ranking representation breaks the long-standing constraints on empirical strategies in this literature, allowing researchers to fully exploit the rich information contained in search processes in a simple and transparent way. Although with the policy-based representation, the baseline Weitzman model remains solvable with numerical methods, it often proves difficult in extended settings that reflect complex market environments and diverse data structures. The ranking representation provides a robust, flexible, and operational foundation. Drawing on the extensive literature on the tractability of discrete-choice and ranking models, we can apply sequential search models straightforwardly to the analysis of complex consumer search.
The remainder of the paper is organized as follows. Sections (ref) through (ref) establish the ranking representation within the baseline Weitzman-style sequential search process. Specifically, Section (ref) formalizes the process and illustrates the empirical bottleneck inherent in the current policy-based representations. Section (ref) introduces the ranking representation. Section (ref) details the rank-based empirical strategy, covering the probability function, the identification arguments, and the rank-based GHK simulator. Section (ref) generalizes the ranking representation to a broader class of model variations, including those with partial observability or with multi-stage information acquisitions, and provides formal conditions for applicability. Section (ref) discusses the case when the Independence or Invariance assumption fails. Section (ref) concludes.
This paper contributes to three strands of literature. The first concerns the interpretation of consumers' decision-making under the sequential search setting. A large body of work follows weitzman1979optimal, treating observed actions as outcomes of an optimal sequential search process governed by optimal rules. Some recent studies interpret the sequence as the outcome of an optimal multi-stage discrete-choice problem, as proposed by keller2003branching. In addition to these two policy-based and choice-based representations, this paper proposes a rank-based interpretation in which the observed sequence of actions reveals the consumer’s ranking of feasible actions, offering a new theoretical perspective on consumer search.
Second, this paper develops a simple empirical strategy for sequential search models that fully exploits search data while simplifying identification and estimation. It extends the identification arguments in morozov2021estimation and ursu2024sequential from subsets of search decisions to the full set of decisions, with results following directly from standard discrete choice models. In estimation, It provides a novel and simple construction for the likelihood function, which ensures that high-dimensional integrals can be naturally decomposed into iteratively computable low-dimensional integrals, thereby reducing the simulation dimensionality and simplify the computation issue faced by all simulation-based maximum likelihood estimation methods, including the Crude Frequency Simulator chen2017sequential, ghose2019modeling, the Kernel-smoothed Frequency Simulator honka2017simultaneous, ursu2018power, ursu2020search, elberg2019dynamic, yavorsky2021consumer, ursu2023search, zhang2024product, ursu2025online, and the GHK-style simulator jiang2021consumer, chung2025simulated.\footnote{Other estimation approaches have been developed for specific contexts. For example, morozov2021estimation applies importance sampling to handle high-dimensional preference heterogeneity; morozov2023measuring and onzo2025bayesian employ Bayesian MCMC methods to capture unobserved post-search uncertainty. } Drawing on econometric tools for ranking models, we employ a GHK-style simulator to compute the newly constructed likelihood function and obtain a substantial performance improvement. The empirical approach most closely related to ours is the “double logit” method used in compiani2024online. Compared with theirs, our method advances by not relying on logit assumptions, not requiring fully observed sequence of actions, and having broader applicability in extended settings.
Lastly, this paper develops a unified empirical framework applicable to a broad class of sequential search models, including the classic weitzman1979optimal model and its various extensions. These extensions include settings with partial search data, such as when only the final purchase is used moraga2023consumer or when only the set of searched products is observed jolivet2019consumer\footnote{moraga2023consumer exploits individual-level search information as moment conditions to disentangle the utility component from the search cost related disutility effects.}, as well as settings with richer action types and multiple stages gibbard2023search, greminger2024heterogeneous, zhang2024product. We show that in each case, the underlying search process admits a ranking representation. This unifying property allows a common class of empirical strategies to be applied naturally across all these models.\footnote{Appendix (ref) provides a detailed comparison across models in recent empirical studies in terms of assumptions, data observability, and estimation methods.}
We begin with the sequential search process studied in weitzman1979optimal, and the model developed to analyze it, which already entails the empirical challenges addressed in this paper and is shared by a broad class of related settings. We first introduce the process and show how it delivers an observed sequence of consumer actions. Then, we summarize the optimal rules that characterize its optimal solution and discuss the difficulties in empirical applications.\footnote{For a comprehensive empirical review of the Weitzman model, see ursu2024sequential.}
Consider a representative consumer $i$ who plans to buy at most one from a set of available products in the market denoted by $\mathcal{M}_i$. While the consumer is aware of the existence of all items $\mathcal{M}_i$, she has only partial information on each product, which makes her evaluations on products uncertain and prevents her from determining the exact utility of any product.
To fully resolve uncertainty about product information, a consumer can inspect a product to disclose its details and determine its utility. A product can only be purchased after its utility has been fully revealed through inspection. After each step, the consumer can decide to stop searching. The set of all inspected products at the time of stopping is denoted by $\mathcal{S}_i$, and $J_i$ denotes its size. Upon stopping, the consumer then selects one from $\mathcal{S}_i$ for purchase. The set of uninspected products is denoted by $\bar{\mathcal{S}}_i = \mathcal{M}_i \backslash \mathcal{S}_i$.
Inspections are costly. Consumers must weigh the cost of inspections against their potential benefits to make optimal decisions, including which products to inspect or purchase and when to stop searching. As shown in Figure (ref), a Weitzman-style sequential search process is captured by a multi-stage decision-making process, with each decision (green boxes) made based on all information obtained at that moment. The candidate actions (white boxes) capture, at each stage, the set of feasible alternatives and the consumer's sequential selections.
The order observed in consumer $i$'s search process is denoted by $\mathcal{R}_i$. Following this order, the products in $\mathcal{S}_i$ are indexed sequentially, with the $j$-th inspected product labeled as Product $j$. Let $J_i$ denote the index of the last inspected product, so that products with $j \leq J_i$ are exactly those inspected and thus belong to $\mathcal{S}_i$. Accordingly, the inspected set can be written as $\{1,2,\dots,J_i\}_{\mathcal{R}_i}$. Let $\mathcal{H}_i$ denote the singleton containing the purchased product, and let $h_i$ denote its index in $\mathcal{R}_i$. Since only inspected products can be purchased, $\mathcal{H}_i \subset \mathcal{S}_i$ and $h_i \leq J_i$ always hold.
The tuple $\{\mathcal{H}, \mathcal{S}, \mathcal{R}, \mathcal{M}\}_i$ summarizes the complete set of actions taken by consumer $i$ in a Weitzman-style search process and constitutes an action sequence. For simplicity, we omit $\mathcal{R}_i$ when products are indexed numerically, and suppress the subscript $i$ when referring to sets or indices in isolation.
We now formalize how weitzman1979optimal maps observed action sequences into quantifiable relationships in the setting above. Let $u_{ij}$ denote the purchase value, defined as the utility that consumer $i$ obtains from purchasing product $j$. These values are initially unknown to the consumer, who instead holds knowledge about their distributions. By incurring a search cost $c_{ij} > 0$, the consumer can inspect product $j$ and learn $u_{ij}$. After inspecting the $(j-1)$th product, given that stopping search yields a utility of $\bar{u}$, the consumer solves the Bellman equation below: {
} Here, $\bar{S}_{ij}$ is the set of uninspected products at the beginning of step $j$. $F^u_{ij}(\cdot)$ and $f^u_{ij}(\cdot)$ are the cumulative distribution and probability density functions of $u_{ij}$. At every step $j$, the consumer chooses between stopping search and accepting $\bar{u}$, or inspecting the uninspected product with the highest expected inspection gain.
To make the problem tractable, weitzman1979optimal imposes two key assumptions:
Under these assumptions, weitzman1979optimal shows that the optimal solution of Equation (ref) reduces to a static decision problem: given a fallback value $\bar{u}$ available for final selection, should the consumer inspect an additional product? The decision depends on the trade-off between the expected incremental benefit of inspection and the associated search cost. For any product $j$, consumer $i$ is indifferent to inspection when $\bar{u}$ satisfies:
Equation (ref) defines $\bar{u}$ implicitly. Since the right-hand side is monotonically decreasing in $\bar{u}$, ranging from infinity to zero, a unique solution exists, denoted by $z_{ij}$ and defined as the reservation value of product $j$ for consumer $i$. This reservation value depends only on the search cost $c_{ij}$ and the pre-inspection distribution of $u_{ij}$.
With purchase and reservation values defined and Assumptions 1 and 2 made, weitzman1979optimal proposes a set of stepwise policies, known as “Pandora’s Rules,” to describe the optimal solution to Equation (ref). We restate them below using the terminology of this paper:
SELECTION RULE: If a product is to be inspected, it should be the uninspected product with the highest reservation value.
STOPPING RULE: Terminate search whenever the maximum purchase value of inspected products exceeds the reservation value of every uninspected product.
Existing studies often reformulate “Pandora’s Rules” into four optimal rules, each expressed as a set of inequalities involving reservation and purchase values. Together, these rules capture how decisions are made sequentially in an optimal Weitzman-style sequential search process.\footnote{For simplicity and consistency, we assume that there are no ties among actions or values throughout the paper.}
In Figure (ref), we illustrate how the optimal rules operate in a Weitzman-style sequential search process using an action sequence example: $\{\mathcal{H}_i=\{3\}, \mathcal{S}_i = \{1, 2, 3, 4\}, \mathcal{R}_i = \{1 \succ 2 \succ 3 \succ 4\}, \mathcal{M}_i = \{1,2,3,4,5\} \}$. In Step 1, consumer $i$ inspects product 1, which, under Optimal Ranking, implies that $z_{i1}$ exceeds the reservation values of all other products (red line). In Step 2, she inspects product 2, which requires $z_{i2}$ to exceed the reservation values of the remaining uninspected products by Optimal Ranking (red line) and, by Optimal Continuing, to exceed the highest inspected purchase value (blue line), which is $u_{i1}$ in this step. Both rules also apply in Steps 3 and 4, except that the maximum purchase values among inspected products are not observable in these steps. In Step 5, a purchase occurs, indicating that $u_{i3}$ exceeds both the purchase values of all previously inspected products, as required by Optimal Purchasing, and the highest reservation value among uninspected products, $z_{i5}$, as required by Optimal Stopping.
If a Weitzman-style sequential search process is optimal, then all decisions throughout the search process must satisfy Equations (ref) to (ref). Therefore, the probability that a search process described by an action sequence is optimal is the joint probability that all of these inequalities hold throughout the process. For the example shown in Figure (ref), the probability function takes the following form:
Here, $\mathbb{I}(\cdot)$ denotes the indicator function, $\mathcal{F}_{j}^c(u_{i,j-1}, z_{ij})$ denotes the joint cumulative distribution function of $u_{i,j-1}$ and $z_{ij}$ conditional on all previous steps, and $\mathcal{F}_{1}(z_{i1})$ denotes the cumulative distribution function of $z_{i1}$. The joint integrals of $u_{i,j-1}$ and $z_{ij}$ provide a general representation: when $z_{ij}$ is deterministic under a given specification, the associated measure with respect to $z_{ij}$ degenerates to a Dirac measure.
Equation (ref) involves a high-dimensional integral over the reservation values of all products and the purchase values of all inspected products. Equation (ref) presents a stepwise decomposition that highlights the main empirical challenge under optimal rules. At any step $j$ and thereafter, the consumer’s optimal decision depends on $\max_{\ell = 1}^{j-1}{u_{i\ell}}$, the highest purchase value among previously inspected products. Because this threshold is often unobserved (e.g., at Step 3 in Figure (ref)), whether the Optimal Continuing condition holds depends on the entire history of prior search outcomes, which must be treated as stochastic. We refer to this structural link between past unobserved outcomes and future search decisions as the model's sequential dependence inherent in the optimal rules.
Sequential dependence introduces cross-stage correlation to Equation (ref). As a result, computing the full likelihood requires computing the distributions of $\max_{\ell = 1}^{j-1}\{u_{i\ell}\}$ at all steps $j$, or at least up to the inspection of the purchased product. This leads to two main difficulties.
First, computation is demanding. The stepwise decomposition provided in Equation (ref) does not effectively reduce the computational burden. In the absence of a closed-form solution, the purchase values of all $J$ inspected products must be simulated. If reservation values are also stochastic, their realizations for these $J$ products must also be simulated. Moreover, because the likelihood contribution of a given action sequence can be very small, achieving adequate simulation accuracy typically requires a large number of draws in a high-dimensional space.
Second, empirical applicability is fragile. The probability function is highly sensitive to both the Weitzman specification and the completeness of the data. Missingness to the search process or the introduction of additional actions alters the distribution of $\max_{\ell = 1}^{j-1}\{u_{i\ell}\}$ in subsequent steps. Even minor specification changes, such as adding an outside option, can change these distributions throughout the process. As a result, empirical extensions often require different ways to construct the likelihood function under new specifications and, in some cases, modifying the optimal rules, which can be difficult or impossible.\footnote{Outside consumer search, some homogeneous sequential search models have been studied in contexts such as financial product search and labor search mccall1970economics. Subjects in these models evaluate only the currently sampled option, assuming that previously rejected options cannot be recalled and therefore do not involve sequential dependence. Therefore, they fall outside the scope of our discussion.}
To overcome the empirical challenges posed by sequential dependence, this paper proposes a novel representation of the optimal solution to sequential search, which leads to a new empirical strategy for a broad class of sequential search models.
Consider the example in Figure (ref). Taking the process as optimal, then in Step 5, Optimal Purchasing implies that $u_{i3} > \max \{ u_{i1}, u_{i2}, u_{i4} \}$, and Optimal Stopping implies that $u_{i3} > z_{i5}$.
Next, consider Step 4. Optimal Ranking implies that $z_{i4} > z_{i5}$, and Optimal Continuing implies that $z_{i4} > \max \{ u_{i1}, u_{i2}, u_{i3} \}$. Given that Optimal Purchasing and Optimal Stopping are satisfied in Step 5, we already know that $u_{i1}, u_{i2}, z_{i5}$ are all no greater than $u_{i3}$. Therefore, among all conditions in Step 4, it suffices that $z_{i4} > u_{i3}$, as the remaining conditions then follow trivially.
Now suppose the optimal rules in Steps 4 and 5 hold. Consider Step 3. The optimal rules imply that $z_{i3} > \max \{ z_{i4}, z_{i5} \}$ and $z_{i3} > \max \{ u_{i1}, u_{i2} \}$. Since Steps 4 and 5 establish that $z_{i4}$ is the largest among $z_{i4}, z_{i5}, u_{i1}, u_{i2}$, the only additional condition required at Step 3 is $z_{i3} > z_{i4}$.
Proceeding backward to Step 1 yields the relation shown in Figure (ref):
This defines a partial ranking over all feasible actions across the sequence, capturing the full empirical content implied by the search process in Figure (ref). Formally,
We verify Equation (ref) using a Monte Carlo simulation. In a 5-product market as depicted in Figure (ref), we assume that the reservation and purchase values of each product follow normal distributions with means $\mu = [5, 4, 3, 2, 1]^\top$ and a standard deviation of $\sigma=2$. We simulate 50 million consumers search following the optimal rules, compute the brute-force frequencies of all 1240 possible action sequences, and compare the top 400 sequences, which account for 99.52% of all observations, with the theoretical probabilities of their corresponding rankings\footnote{The probabilities are computed with the GHK-style simulator introduced in Section (ref).}. The results show close agreement between brute-force frequencies and computed probabilities, with a cosine similarity of 0.999999 (see Appendix (ref) for more comparison details). For the process in Figure (ref), the brute-force frequency is $1.2148 \times 10^{-4}$, and the computed probability of the ranking in Figure (ref) is $1.1726 \times 10^{-4}$, with a marginal relative error of 3.48%.
Compared to the optimal rules, we do not induce any changes in the data but simply exploit ex post information that is already contained in the data. The optimal rules are established from the consumer’s perspective, in which each decision depends only on information revealed in previous steps. However, in an action sequence, researchers observe subsequent decisions in the same search process, which provide additional information on the ranking of actions. As long as these ranking relations hold in the stages preceding the subsequent decisions, researchers can use ex post ranking information to identify constraints in earlier decisions that are already implied by other conditions and exclude them when constructing the probability function.
A direct echo of this intuition is that, among the optimal rules, Optimal Continuing before step $J$ is effectively redundant. Given that a Weitzman-style sequential search process is known to continue until step $J$ and that Optimal Stopping and Optimal Ranking hold, for any step $j < J$, the inequality $\max_{s=1}^{j}\{u_{is}\} < \max_{s=1}^{J-1}\{u_{is}\} < z_{iJ} < z_{ij}$ is satisfied by construction.
We next formalize the theory illustrated by the example and state the conditions under which it holds. Moving away from the optimal rules, we view each step of a Weitzman-style sequential search process as a discrete choice problem among current feasible actions. At each step, the consumer chooses either to inspect an uninspected product $k$ ($I_k$) or to purchase an inspected product $j$ ($P_j$). Each observed action can be represented as a discrete choice outcome $(a^*, \mathcal{A})$, where $a^*$ denotes the selected action and $\mathcal{A}$ the current feasible action set.
We restate Assumptions 1 and 2 in terms of actions, consistent with the rest of the paper.
With the two assumptions held, we present the following theorem:
We refer to Theorem (ref) as the ranking representation of optimal Weitzman-style sequential search. It implies that, under the Independence and Invariance assumptions, for any observed action sequence, the two events below are informationally equivalent and occur with the same probability. First, the consumer conducts an optimal Weitzman-style sequential search process, and her sequence of actions coincides with the observed sequence. Second, the consumer ranks all feasible actions according to their values, with the top $J+1$ positions corresponding to the observed sequence in the same order, except for the last-selected action if it is the purchase of the last inspected product.
As an illustration, consider the process shown in Figure (ref), in which the feasible actions include inspecting products 1 through 5 ($I_1$ to $I_5$), as well as actions that become feasible in subsequent steps, namely purchasing products 1 through 4 ($P_1$ to $P_4$). We assume that the nine actions constitute a unified set and the consumer ranks them according to their associated values. Then the ranking in Figure (ref) holds if and only if the process in Figure (ref) is optimal.
To prove Theorem (ref), having characterized each step of the Weitzman-style sequential search process as an action-level discrete choice problem, we further represent the entire process as a multi-stage discrete choice process $\{(a^*_j, \mathcal{A}_j)\}_{j=1}^J$, where $J$ denotes the length of the action sequence. Formally, this process is referred to as a branching project keller2003branching. In a branching project, the decision-maker chooses one action from a set of feasible actions at each stage and obtains an immediate payoff. The payoffs of feasible actions at the initial stage are fully known to the consumer, whereas those of infeasible actions are only partially revealed. Once an action is selected, its payoff is realized immediately, and the action is removed from the feasible set. The selected action may render some previously infeasible actions feasible in subsequent stages and fully reveal their payoffs, thereby inducing parent-child relationships among actions. At the same time, it may partially reveal the payoff information of actions that remain infeasible. Under the Independence assumption, selecting an action only changes the information on payoff of actions that can become feasible only if that action is selected. Under the Invariance assumption, true payoffs are determined ex-ante, and the feasibility of actions is not affected by any exogenous factors other than the decision-maker’s own selections.
Figure (ref) shows a branching project representation corresponding to the search process as in Figure (ref), which can be taken as a depth-two branching project satisfying both the Independence and Invariance assumptions, as all inspection actions are initially feasible with known and given search costs, while each purchase action becomes feasible and its underlying payoff is revealed only after the corresponding inspection being conducted.
Next, building on Luce’s Choice Axioms and Ranking Postulates luce1959individual, we present the following equivalence lemma.\footnote{Note that the Axioms and Ranking Postulates are applied algebraically at the level of individual utility realizations, where rankings are observed but not probabilistic. Hence, the lemma remains distribution-agnostic.}
We illustrate how the lemma applies in a Weitzman-style sequential search process with a toy example. Consider a consumer searching between two products, 1 and 2. In the first step, her choice set $\mathcal{A}_1$ consists of two alternative actions, “inspect product 1” ($I_1$) and “inspect product 2” ($I_2$). Suppose she selects $I_1$, that is, $a_1$ is $I_1$. In the second step, the choice set $\mathcal{A}_2 \setminus \{a_1\}$ contains $I_2$ and “purchase product 1” ($P_1$), so that $\mathcal{A}_2 = \{I_1, I_2, P_1 \}$.
Now consider the only two possible cases in the second step. If $\rho_1 = \{I_2 \succ P_1\}$, since $I_2$ already belongs to $\mathcal{A}_1$, adding $P_1$ to $\mathcal{A}_1$ does not change the probability that $I_1$ is selected in the first stage. The reason is that $P_1$ is strictly dominated by a feasible alternative, so including it in the choice set neither leads to its selection nor affects the selection probabilities of other alternatives.\footnote{Led by the Independence assumption, this irrelevance is consistent with Choice Axiom 1(ii), luce1959individual. } Therefore, this two-step process is probabilistically equivalent to the consumer selecting $I_1$ and then $I_2$ from $\mathcal{A}_2$, which corresponds to a ranking of $\{I_1 \succ I_2 \succ P_1\}$. The probability of this ranking is given by the first term on the right-hand side of Equation (ref), while the second term does not apply. In the other case, if $\rho_1 = \{P_1 \succ I_2\}$, the consumer selects $P_1$ in the second step, which is not in $\mathcal{A}_1$. This yields a partial ranking over $\mathcal{A}_2$ where $I_1$ and $P_1$ both rank above $I_2$, but their relative order is unspecified. Adding $P_1$ to $\mathcal{A}_1$ thus corresponds to two possible rankings, namely $\{I_1 \succ P_1 \succ I_2\}$ and $\{P_1 \succ I_1 \succ I_2\}$, which correspond to the two terms in the right-hand side of Equation (ref).
Hence, Lemma (ref) proves that the joint probability of two adjacent stages in a branching project is equal to the probability of a ranking over a unified choice set. By repeatedly applying Lemma (ref), this equivalence extends backward from the final two stages, the last inspection and final purchase, to the entire branching project corresponding to a sequential search process. This establishes Theorem (ref). Other proof details are provided in Appendix (ref).
We emphasize the central role of the Independence and Invariance assumptions in establishing Theorem (ref). Independence requires that a consumer’s information about the value of any action remain unaffected by prior selections, except in cases where the action would be infeasible without such a selection. Invariance requires that neither the structure of the branching project nor the value of any action vary with external conditions. Together, these assumptions imply that the binary dominance relations revealed at any local step hold globally across the entire search process. For example, reconsider the two-product example where $I_1$ and $I_2$ are sequentially selected. Under both assumptions, the ranking $\{I_{1} \succ I_{2} \succ P_{1}\}$ holds across stages. If either assumption fails, only stage-specific relations ${I_{1} \succ I_{2}^{pre}}$ and ${I_{2}^{post} \succ P_{1}}$ can be inferred, where $I_{2}^{pre}$ and $I_{2}^{post}$ denote inspecting product 2 before and after inspecting product 1. Whether these two actions are equivalent, and thus whether a consistent ranking exists, may depend on the selection of $I_1$ or on external conditions.
Finally, we establish value measures for actions for quantitative applications. Under the same Independence and Invariance assumptions, keller2003branching prove that with bounded payoffs, the optimality of a branching project is fully characterized by a Gittins index policy. In the Weitzman model, this index corresponds to the purchase value for purchase actions and to the reservation value for inspection actions. Therefore, an optimal branching project is equivalent to selecting actions sequentially based on these values. Since a branching project process is equivalent to a corresponding ranking of feasible actions, the resulting action ranking is thus equivalent to the ranking of action values under optimality. Hence, similar to Pandora’s Rule, Theorem (ref) can also be expressed as a system of inequalities over the values of all feasible actions, which is equivalent to the optimal rules summarized in the following proposition.
Proposition (ref) is no longer constrained by sequential dependence, as all conditions are conditioned on the lowest selected action value, $u_{ih}$ or $z_{iJ}$. Given this value, the three restrictions excluding the Distribution Condition correspond respectively to the reservation values of inspected products, the purchase values of inspected products, and the reservation values of uninspected products. These components are conditionally independent, which allows the probability function to be factorized and evaluated separately.
The ranking representation improves the tractability of the Weitzman model. Theorem (ref), together with its subsequent extensions, shows that in empirical applications, researchers can treat the ranking representation of an optimal sequential search process as an alternative empirical foundation. Econometrically, ranking models and standard discrete choice models exhibit a close structural correspondence: their probability functions can both be expressed in terms of utility differences across alternatives, implying that identification arguments and estimation methods largely carry over. Through this connection, these empirical strategies extend naturally to the Weitzman model. By leveraging tools from the discrete choice literature, the implementation complexity of a Weitzman model can be substantially reduced without sacrificing informational richness provided by the observed search process.\footnote{Figure (ref) illustrates that a sequential search process provides a richer set of inequality conditions than standard discrete choice data. While a purchase observation identifies that the selected alternative has higher utility than the remaining alternatives, a search sequence can be translated to a partial ranking among selected actions and relative to unselected actions. This increased number of inequality conditions per observation allows for sharper identification of model primitives, such as the covariance structures, without relying on functional form assumptions.}
Meanwhile, note that the ranking representation is established by Lemma (ref), which rests solely on the basic assumptions of Independence and Invariance, and not on any distributional assumptions or functional forms. This feature implies that the ranking representation is not tied to the baseline Weitzman model and can be applied in more general settings.
This section develops a novel, simple empirical strategy for the sequential search model by implementing it via its ranking representation, using the Weitzman setting as an example. In this section, we confine our discussion to an additive specification of purchase values, which combines the expected value determined by observed attributes with other components that may be stochastic and driven by unobservables.
Different from Equation (ref), the probability function of the Weitzman model based on Proposition (ref) is given as follows:
where $y_i \equiv \min\{u_{ih}, z_{iJ}\}$.
We consider a general baseline specification as follows:
Here, $\delta_{i}^u(X_{ij}^u)$ and $\delta_{i}^z(X_{ij}^z)$ denote the data-specified components of the purchase value and the reservation value, respectively. $X_{ij}^u$ and $X_{ij}^z$ represent the factors affecting consumer $i$’s evaluation of purchasing and inspecting product $j$. The two vectors may be identical; however, if certain factors influence the reservation value but not the purchase value, such as advertising exposure, they may enter only $X_{ij}^z$, allowing the two vectors to differ.\footnote{A variable that enters $X_{ij}^u$ but not $X_{ij}^z$ is generally inconsistent with the basic assumption of a rational consumer.}
We introduce three components that can be stochastic to account for unobserved factors in the model. The first is a pre-inspection taste shock, denoted by $\xi^u_{ij}$, which affects both inspection and purchase decisions for product $j$. It captures the part of the purchase value known to the consumer before inspection but unobserved by the researcher. The second is the post-inspection taste shock, denoted by $\varepsilon_{ij}$, which represents the portion of the purchase value revealed only upon inspection. We model $\varepsilon_{ij}$ as a one-dimensional stochasticity added to the deterministic component. In alternative formulations, it may also include product attributes that become known only through search honka2017simultaneous, kaye2024personalization, compiani2024online, greminger2024heterogeneous. Without loss of generality, we assume that $\varepsilon_{ij}$ is independently and identically distributed across consumers and products, with mean zero.
The third component, $\xi^z_{ij}$, is referred to as the inspection propensity. It enters only the reservation value and influences the consumer's inspection decisions.\footnote{We use the term “propensity” to describe the difference between the reservation value and the conditional expectation of the purchase value, following morozov2023measuring and onzo2025bayesian.} Search cost $c_{ij}$ is typically considered the primary determinant of inspection propensity. When $c_{ij}$ is independent and known to consumer, its impact to the inspection propensity depends solely on the distribution of $\varepsilon_{ij}$ and the magnitude of $c_{ij}$, expressed as $m_\varepsilon(c_{ij})$, a strictly decreasing function derived from Equation (ref).\footnote{The additivity of inspection propensity and the monotonicity of $m_\varepsilon(\cdot)$ are proved in Appendix (ref).} Beyond search costs, $\xi^z_{ij}$ may also capture other sources of stochasticity that are not directly tied to cost but influence consumers' inclination to inspect.\footnote{Existing empirical studies often do not distinguish between these two components, particularly for observable factors that influence search actions, such as list-page positions ursu2018power, store distances yavorsky2021consumer, and search refinement tools chen2017sequential. Some motives, however, are less appropriately attributed to search costs, such as searches undertaken for entertainment or leisure moe2003buying.}
We make the following assumptions, including Independence and Invariance:
To express the probability function, we divide the observed sequences into two cases. We first consider the case where the purchased product $h$ is not the last inspected product $J$. Let $\vec{\bm{z}}_i^k$ denote the reservation values of inspected products, $\vec{\bm{z}}_i^n$ the reservation values of uninspected products, and $\vec{\bm{u}}_i^{k^\prime}$ the purchase values of inspected but unpurchased products, ordered as follows; each vector is then decomposed into component vectors, with variables sharing the same superscript ($k, n, k^\prime$) arranged in the same order.
Following Equation (ref), the probability of the sequence is:
The differencing matrix $\hat{D}$ consists of four blocks: {
}
Hence, $\hat{D}$ has rank $J + |\mathcal{M}| - 1$, with its structure determined by the observed sequence.
For the case where the purchased product $h$ is the last inspected, we follow the vectorized form in the previous case and establish the probability function as below:
The differencing matrix $\tilde{D}$ is also of rank $J + |\mathcal{M}| - 1$, which consists of six parts: {
}
In the remainder of this paper, we denote the differencing matrix consistently as $D$, an element in $\{\hat{D}, \tilde{D}\}$, based on $\mathcal{R}_i$. Accordingly, the model's probability is expressed as follows:
Equation (ref) explicitly expresses the probability in a value-differencing form. While the formulation follows the general structure of a discrete choice probability train2009discrete, it employs a data-specified full-rank differencing matrix to account for the nature of sequential search. Equation (ref) therefore bridges the entire sequential search process and the discrete choice model family. To our knowledge, this connection has not been formally established in previous studies. Compared to Equation (ref), Equation (ref) separates the variation explained by the data (the right-hand side) from the variation implied by the stochastic structure (the left-hand side), allowing the high-dimensional integral in the probability function to be decomposed in a tractable manner. As a result, researchers can estimate sequential search models using the empirical strategy of discrete choice models with flexibility and computational efficiency, without sacrificing their rich informational content.
While most empirical studies discuss identification of a Weitzman model heuristically by linking the data variations to model parameters, a formal discussion remains necessary for broader applicability. Contributions in this direction are provided by morozov2021estimation and ursu2024sequential.\footnote{Other work explores identification under partial specifications. For instance, abaluck2025method studies the identification of preference parameters when it is uncertain which attributes are known to consumers before inspections, and onzo2025bayesian investigates nonparametric identification of parameter distributions.} However, these arguments rely on conditional probabilities of a subset of decisions rather than the full decision set. This leaves ambiguity because variation in a single decision may be driven by multiple parameters, while a single parameter may affect multiple decisions. For example, the stopping decision is jointly determined by preferences, search costs, and the scale of uncertainty, while preferences may also be inferred from the inspection order.
Once the probability function is rewritten as in Equation (ref), many identification arguments under linear specifications become straightforward. Since the differencing matrix $D$ is always of full rank, preference parameters can be identified separately from search costs, following arguments similar to those in standard discrete choice models (e.g., berry2014identification). However, because the stochastic components in reservation values and purchase values are dependent, it is essential to reconsider whether specific distributional assumptions are required for normalization. While the necessity of such assumptions depends on the specification, our discussion focuses on two fundamental principles: “only differences in utility matter,” and “the scale of utility is arbitrary.” These principles determine location and scale normalizations in discrete choice models, and remain critical for identification in sequential search models.
Let us examine the first principle. In the linear specification, the absolute levels of reservation and purchase values are irrelevant. As long as the probability function can be reformulated as Equation (ref), any constant added to all values cancels out with the differencing matrix.
We point out that the value-differencing form holds only if both reservation and purchase values include a conditionally independent stochastic component, which ensures the decomposition in the second equality of Equation (ref). Such randomness in purchase values is guaranteed by $\varepsilon_{ij}$, whereas reservation values require additional stochasticity from $\xi^u_{ij}$ or $\xi^z_{ij}$. In the full absence of these terms, the search process is determined entirely by observed heterogeneity, and consumers with identical preferences and search costs always follow the same process. This leads to degenerate likelihood functions and a perfect separation issue, resulting in estimation failure for likelihood-based methods.\footnote{An alternative is to increase the dimensionality of preference heterogeneity, but this typically introduces high-dimensional heteroskedasticity and additional identification difficulties.} Hence, in empirical applications, incorporating a conditionally independent stochastic component in action values is therefore essential.
We now turn to the second principle. In standard discrete choice models, scale is irrelevant under linear specifications, and identification is typically achieved by normalizing the variance of the error term. This property, however, does not extend naturally to a Weitzman model. The key reason is that the distribution of $\varepsilon_{ij}$ influences not only the scale of purchase values but also the magnitude of reservation values through $m_\varepsilon(c_{ij})$, which is generally nonlinear in $c_{ij}$. Consequently, $c_{ij}$ and the distribution of $\varepsilon_{ij}$ must be identified jointly based on observed actions. In the absence of exclusive search cost shifters, identifying the distributional parameters of $\varepsilon_{ij}$ requires two conditions. First, $m_\varepsilon(\cdot)$ must be homogeneous of degree one; otherwise, the estimated intercept for search costs becomes sensitive to scale normalization. Second, if $c_{ij}$ is stochastic, its distribution must satisfy specific conditions to ensure that the associated parameters are scale invariant. Moreover, while satisfying these conditions may enable identification in theory, heteroskedasticity often complicates estimation in practice.
We illustrate this point with a stylized specification from kim2010online, in which the consumer has a pre-search taste for each product before inspections. The taste is denoted by $\xi_{ij}$ and affects both reservation and purchase values. The specification is stated as:
Here, $\xi^u_{ij} = \xi_{ij}$ captures the unobserved stochasticity in reservation values, while the standard deviation of $\xi^z_{ij} = m_\varepsilon(c_0)$ is assumed to be zero. Based on this specification, we decompose the probability function in the stacked vectorized form as follows:
We focus on whether $\sigma_\varepsilon$ and $\sigma_\xi$ can be identified without additional assumptions. In this setup, $\xi_{ij}$ is a shared component that enters both the reservation and purchase values. Rescaling $\xi_{ij}$ does not affect the relative scale of the preference parameters. Hence, the central problem lies in the search cost parameter, as well as the absolute or relative magnitudes of $\sigma_\varepsilon$ and $\sigma_\xi$.
We first note that normalization based on $\sigma_\xi$ is not generally applicable, as $m_\varepsilon(c_0)$ may not be a homogeneous function of degree one. kim2010online provides a sufficient condition under which this normalization is valid: when $\varepsilon_{ij}$ is independently normally distributed, $m_\varepsilon(c_0)$ can be expressed in the form $m_\varepsilon(c_0) = \sigma_\varepsilon \cdot m(c_0/\sigma_\varepsilon)$, where the function $m(\cdot)$ is irrelevant to the distribution of $\varepsilon_{ij}$. Hence, we can normalize the model as follows:
For search costs, since they enter the probability function only through the intercept term given by $m(c_0/\sigma_\varepsilon)$, identification of $c_0$ can be achieved if the ratio $\sigma_\varepsilon/\sigma_\xi$ can be identified from heteroskedasticity. Although this ratio is, in principle, directly identifiable, empirical work documents substantial practical difficulties in doing so jiang2021consumer, chung2025simulated. The source of this difficulty is that $\xi_{ij}$, as unobserved heterogeneity, induces correlation between the purchase value and the reservation value for the same product. When such a correlation is present, stable identification typically requires additional exclusion restrictions, namely, observable variables that affect only purchase values or only reservation values; otherwise, identification may become fragile keane1992note. However, these exclusion restrictions are often difficult to justify from an economics perspective or infeasible due to data limitations.\footnote{yavorsky2021consumer provides a successful example of applying exclusion restrictions. They use distance to automobile dealers as an extra variable that affects only reservation values, thereby achieving identification of $\sigma_\varepsilon$.} In light of this reality, existing empirical studies commonly impose the assumption $\sigma_\varepsilon/\sigma_\xi = 1$ to obtain robust estimation performance, at the cost of making search cost estimates dependent on this assumption.\footnote{A key implication is that if the identification of the search cost relies on distributional assumptions, the resulting estimates should not be directly monetized ursu2024sequential, compiani2024online, greminger2024heterogeneous.} Appendix (ref) discusses the identification in a different specification chung2025simulated, which excludes $\xi_{ij}$ and introduces randomness into reservation values through a stochastic search cost $c_{ij}$. This specification removes the correlation between reservation values and purchase values, thereby yielding more stable identification.
The identification arguments discussed in this subsection apply broadly to sequential search settings beyond the Weitzman model. The principle that “only utility differences matter” requires the value indices of actions to contain conditionally independent randomness, typically implemented through univariate independent shocks. By contrast, the principle that “utility scales are arbitrary” does not always hold. Even when scale invariance is satisfied, the resulting heteroskedasticity may weaken identification in practice, making additional distributional assumptions a practical compromise. In later sections, we extend the analysis to extended models whose underlying processes still admit a ranking representation. We point out that the identification arguments also apply to the identification of extended models.
Rewriting the probability function as in Equation (ref) allows the use of established methods from ranking models to estimate a sequential search model. Among these methods, the GHK-style simulator is a natural choice for computing the simulated likelihood. This simulator has been used in prior studies jiang2021consumer, chung2025simulated to compute the likelihood function given by Equation (ref), whereas we apply it to compute Equation (ref). The following description provides a general guideline for the GHK-style simulator that applies not only to the Weitzman setting but also to all sequential search models discussed in the remainder of the paper.
Taking the model with specifications in Equations (ref) to (ref) as an example, we describe the implementation procedure, and show its application to the example in Figure (ref) in Figure (ref):
We assess the performance of our proposed simulator through a Monte Carlo experiment, comparing it with the GHK-style simulator presented in ursu2024sequential, a simplified version of the one applied in jiang2021consumer. For clarity, we refer to the existing simulator as the policy-based GHK simulator and to our proposed method as the rank-based GHK simulator.\footnote{We do not compare the rank-based GHK simulator with other methods, such as the Crude or Kernel-smoothed Frequency Simulators, because prior studies have shown that the policy-based GHK method outperforms these approaches. We acknowledge the contributions of ursu2024sequential and chung2025simulated, who provide extensive simulation-based comparisons between policy-based GHK and alternative methods. More recent, targeted approaches, such as importance sampling morozov2021estimation and Bayesian MCMC methods morozov2023measuring, onzo2025bayesian, could, in principle, also benefit from the simplified likelihood function derived from the ranking representation. Given the limited use of these methods in conventional ranking models, we do not pursue this discussion further in this paper.}
To ensure comparability, we adopt the benchmark specification from ursu2024sequential, constructing the policy-based GHK likelihood strictly following their publicly available MATLAB code, with modifications only to the number of products and the product attribute details.
Here, the purchase value of product $j$ consists of a series of attributes, $x^s_j$, and a consumer-specific price, $p_j$. All preference parameters are assumed to be homogeneous across consumers.\footnote{In theory, preference heterogeneity is parametrically identifiable in cross-sectional settings, while nonparametric identification is generally considered infeasible morozov2023measuring. In practice, the preference heterogeneity is difficult to identify even parametrically because of $\zeta_{ij}$. Appendix (ref) provides a detailed investigation. } Search costs are set to be constant. As discussed in Section (ref), while it is theoretically possible to estimate $\sigma_\varepsilon$, doing so without cost shifters is empirically challenging. We therefore fix $\sigma_\varepsilon = 1$. To ensure comparability independent of the number of simulation draws, we fix the number of groups of draws at 500. The results are reported in Table (ref).
We compare the estimation performance of the two simulators across two market sizes. The first corresponds to a minimal market or experimental environment with 8 products. The purchase value of each product is specified as a linear function of three binary attributes and price, encompassing all possible attribute combinations. The product with attributes $[0,0,0]$ is used as the normalized base alternative. The second corresponds to a typical medium-to-large market with 32 products. The product attributes are given by five binary attributes and price, and the product with attributes $[0, 0, 0, 0, 0]$ serves as the base alternative. In both settings, the search cost is held constant. The results are summarized in Columns (1) to (4) of Table (ref).
We find that in the small market (with an average of 1.4 products inspected), the two simulators yield very similar estimates for both search costs and attributes preference parameters, with negligible differences in RMSE.\footnote{RMSE is calculated following ursu2024sequential as $\sqrt{1/N_\theta \times \sum [\hat{\theta - \theta^{true}}]^2}$, where the summation covers all parameters, including attribute preferences and the search cost.} In the larger market (with an average of 3.3 products inspected), differences become pronounced: policy-based GHK performs better in estimating preference parameters, while rank-based GHK produces better estimates of search costs. Overall, rank-based GHK achieves a lower RMSE and slightly outperforms policy-based GHK.
Beyond estimation performances, the primary advantage of rank-based GHK over policy-based GHK lies in its simplicity and computational efficiency, which directly address the most demanding limitations of the latter.
First, in terms of implementation simplicity, the rank-based GHK is considerably more streamlined than policy-based GHK, requiring less than 40% of the latter's coding effort. This simplification arises because policy-based GHK must separately construct likelihood functions for different types of observed sequences, depending on whether the purchased product is the outside option, the last inspected, or an earlier inspected alternative. While in rank-based GHK, such distinctions only determine whether the distributional restriction applies when sampling $y_i$.
Second, rank-based GHK is computationally more efficient. In a market with 32 products, it requires less than 60% of the computation time of policy-based GHK. This efficiency stems from the lower simulation dimensionality: policy-based GHK typically draws unobserved components for all unselected purchase actions, requiring up to $J + |\mathcal{M}_i|$ values to be simulated per sequence. By contrast, rank-based GHK makes draws only for the values of selected actions, with a total of $J + 1$ values to be simulated.\footnote{The minimum simulation dimensionality is $J$. Under the current specification, $\zeta_{i1}$ is additionally drawn to compute the probability associated with $u_{i1}$, which may be unnecessary under alternative specifications.} On average, computing the simulated likelihood for a sequence requires 33.7 draws under policy-based GHK, but only 4.3 draws under rank-based GHK. For a fair comparison, we match the total number of draws of the two methods, with the results reported in Column (5) of Table (ref). When the total number of draws is held constant, rank-based GHK produces preference parameter estimates nearly identical to those from policy-based GHK while delivering substantially more accurate estimates of the outside option value and the search cost, with an RMSE approximately half that of the policy-based GHK.
In conclusion, compared with policy-based GHK, rank-based GHK achieves higher estimation accuracy while substantially reducing computational and implementation costs. Moreover, it fully retains the advantages of policy-based GHK over the Crude Frequency Simulator and the Kernel-smoothed Frequency Simulator, two other commonly used empirical methods.\footnote{The Crude Frequency Simulator is mainly suitable for short search sequences in small markets, as it requires a large number of draws to avoid zero-probability issues. The Kernel-smoothed Frequency Simulator alleviates this problem by introducing external smoothing, but its performance depends critically on the choice of smoothing parameters, which lack theoretical guidance and often require calibration through simulation, thereby increasing implementation complexity and limiting comparability across models. In contrast, the policy-based GHK method avoids these issues, with its primary drawback being implementation complexity ursu2024sequential.}
Another method for estimating ranking models is the exploded logit approach beggs1981assessing, chapman1982exploiting, which compiani2024online adapts to the Weitzman model through a “double logit” framework. Essentially, this approach also follows the probability function described in Equation (ref). By imposing a Gumbel distribution on both the randomness in reservation and purchase values \footnote{Technically, this can be achieved through a Gumbel-preserving distribution for random search costsmoraga2023consumer, which ensures that when the post-inspection taste shock $\varepsilon_{ij}$ follows a Gumbel distribution, the inspection propensity $\xi^z_{ij}$ derived from Equation (ref) also follows a Gumbel distribution.}, the approach delivers a closed-form solution for the likelihood function. Its main advantage lies in the strong identification performance induced by the Gumbel assumption. Its limitation, however, is that conditional probabilities in later steps still depend on previously revealed purchase values. Thus, sequential dependence is addressed but not eliminated. As a result, implementation requires the consumer's complete action sequence to be directly observed or, when unobserved, enumerated. We show in the next section and in Appendix (ref) that our strategy does not rely on such path-level observability and is more flexible for extended models in practice.
We now highlight the other main advantage of the ranking representation: its broad empirical applicability. Since Lemma (ref) applies to any branching project that satisfies Independence and Invariance, any sequential search process that can be regarded as a branching project satisfying these assumptions can be fully captured by an equivalent ranking, rendering it static and free of sequence dependence. Therefore, the ranking representation is particularly useful for handling many variants of sequential search models that would otherwise be difficult to analyze.
In this section, we first provide a broad definition of sequential search processes and, through several examples, illustrate that many commonly studied settings beyond the classical Weitzman-style can also be accommodated within this framework. We then focus on two representative examples to demonstrate how they admit a ranking representation and can be estimated by directly applying the rank-based GHK method. Finally, we characterize the application boundary of the ranking representation: its existence requires only an identified topological ordering among feasible actions. That is, the representation is well-defined provided that the set of feasible actions forms a unique directed acyclic graph up to transitive equivalence.
A broad definition of the sequential search process describes how a decision-maker gradually performs actions across multiple alternatives and ultimately selects one. The utility of each alternative may be fully known at the outset or may need to be revealed through a sequence of actions with immediate payoffs. For each alternative, the associated actions must be conducted in a fixed order, with subsequent actions becoming feasible only after the preceding ones are completed. Selecting an action may reveal partial utility information about one or more alternatives or may simply serve as a prerequisite for subsequent actions. The decision process unfolds over multiple stages, with the decision-maker selecting one feasible action at each stage. An alternative becomes eligible for eventual selection only once its utility is fully revealed. The process terminates upon making a final choice.
Under this definition, the Weitzman model constitutes a special case of sequential search models. We provide several additional examples to illustrate the range of encompassed models.
Despite differences in their specific settings, these examples share a key feature: they can all be mapped into multi-stage discrete choice processes over actions structured as a branching project. As in previous sections, we focus on sequential search processes that satisfy the Independence and Invariance assumptions. Under these conditions, the optimal strategies conform to the Gittins index policy keller2003branching, and Lemma (ref) applies, providing a unified foundation for the ranking representation and its associated empirical strategy.
In many empirical works using the Weitzman model, the inability to observe the complete search process poses a significant challenge. Under optimal rules, sequence dependence implies that missing information about the early search process distorts the conditional probabilities of subsequent decisions. Hence, empirical works typically require full observation of the entire search process. Otherwise, researchers must impute missing actions through assumptions or enumeration ursu2018power, compiani2024online, chung2025simulated, or reconstruct optimality in some less data-intensive settings based on revised optimal rules, such as the Eventual Purchase Theorem armstrong2017ordered, choi2018consumer, kleinberg2016descending. The former approach may introduce bias through restrictive assumptions or entail substantial computational costs from full enumeration, whereas the latter leads to a tough trade-off between retaining useful information that falls outside the revised optimal rules and the model's tractability.
Due to partial observability, insufficient information prevents the construction of a branching project corresponding to the original search process. This can be categorized into two cases.
The first case arises when some selected actions or their order are unobserved in some steps, while information on other steps remains fully observed. This allows us to directly recover the ranking relations implied in the underlying branching project. For example, in Figure (ref), suppose the order in which products 2 and 3 are inspected is unobserved, while it is known that product 1 is inspected in the first step and product 4 in the fourth step. This implies that inspecting products 2 and 3 both rank above inspecting product 4. Accordingly, in the branching project, these two steps can be treated as additional restrictions relative to inspecting product 4. The resulting branching project and the corresponding ranking representation are illustrated in Figure (ref).
The second case arises when missing information in some steps prevents the identification of feasible action sets in other steps. Hence, although the selected actions in the remaining steps are observed, the researcher cannot determine which actions are dominated by these selections, and therefore cannot construct the ranking representation. To address this issue, we construct a new branching project whose fully observed action sequence corresponds to the original partially observed sequence. The following lemma formalizes how the observed information is consistently mapped into this reconstructed process.
Composite actions offer an effective solution to partial observability. If a consumer inspects an alternative $j$ ($I_j$), then for any other alternative $k$, two scenarios arise. In the first, purchasing $k$ ($P_k$) is infeasible because its parent action, inspecting $k$ ($I_k$), has not been selected, implying that $I_j$ dominates $I_k$. In the second, $P_k$ is feasible but not selected, implying that $I_j$ dominates $P_k$. A composite action combining $I_k$ and $P_k$ unifies these cases, enabling comparison when $k$’s associated action is undetermined. Crucially, if the original process with individual actions satisfies Independence and Invariance, the process with composite actions also satisfies these assumptions. Therefore, Lemma (ref) remains applicable, and the Gittins index policy retains its optimality. Composite actions can additionally be combined with other individual actions.
Lemma (ref) can be flexibly applied to transform a partially observed sequential search process into a new branching project. As an example, consider the process illustrated in Figure (ref), but only observe that Product 3 is purchased. This implies that the observable action sequence is $\{I_3, P_3\}$. Denoting the composite action for inspecting and purchasing the same product $j$ as $E_j$, we transform this sequence into the branching project as in the left panel of Figure (ref).
For other cases involving partially observed action sequences, Appendix (ref) provides an algorithm for implementing this transformation with Lemma (ref). Based on the transformation, the next theorem delivers the ranking representation of the reconstructed sequential search process.
We formally define the preventive action as follows:
The definition of the preventive action guarantees that no feasible but unselected action dominates any observed selected action. This principle applies to both individual and composite actions. In a Weitzman-style sequential search process, if the purchased product is inspected before the final step, only the purchase step is forward-inaccessible; if the product is inspected at the final step, then both the final inspection and the purchase steps are forward-inaccessible. Accordingly, the definition of the preventive action in Theorem (ref) is consistent with that in Theorem (ref): as illustrated in Figure (ref), both Step 1 and Step 2 are forward inaccessible, and thus $\bar{a}$ is the lower-ranked between $I_J$ and $P_h$, and $y_i = \min\{u_{ih}, z_{ih}\}$, as in Proposition (ref).
The proof of Theorem (ref) is provided in Appendix (ref). The theorem establishes that, even without complete observation of a Weitzman-style sequential search process, a reconstruction process based on a partially observed action sequence still admits a ranking representation. This implication is illustrated in Figure (ref): applying Theorem (ref) to the branching project in the left panel yields the ranking representation in the right panel. Notice that Theorem (ref) classifies all observed feasible actions into three categories: selected actions in forward-accessible steps, selected actions in forward-inaccessible steps, and unselected actions. Incorporating all these categories ensures that the ranking representation entails no loss of information relative to the reconstructed sequential search process.
It can be seen that Theorem (ref) is a special case of Theorem (ref) under full observability. Moreover, existing propositions in the literature concerning various forms of partial observation can be directly derived from Lemma (ref) and Theorem (ref). We illustrate with two examples. First, consider the case where no search information is observed and only the purchase is known:
Proposition (ref) shows that Theorem (ref) subsumes the famous Eventual Purchase Theorem, a result widely used for allowing researchers to infer demand directly from a discrete choice model without requiring detailed search process data.\footnote{For example, moraga2023consumer derives a closed-form solution for consumers’ final purchase in a Weitzman model following Proposition (ref), under a Gumbel-preserving distribution for random search costs, embeds it within a BLP style framework, and applies it to the Dutch car market.}
Unobservable inspection order is a common limitation in empirical studies, also present in widely used sources such as Expedia hotel booking data ursu2018power, compiani2024online, greminger2024heterogeneous, kaye2024personalization. Based on Theorem (ref), the process can be immediately reformulated to yield three sets of inequalities that fully characterize optimality in this setting.\footnote{These inequalities provide an alternative formulation of Proposition 2.1 in jolivet2019consumer and Appendix F in greminger2024heterogeneous.}
For the remainder of this section, we conduct Monte Carlo simulations to evaluate the model’s estimation performance under varying degrees of partial observability using the rank-based GHK simulator. Specifically, we raise five illustrative scenarios of partial observability:
In all these scenarios, we do not rely on revisions to the optimal rules, external assumptions, or full enumerations. Instead, we apply Lemma (ref) and Theorem (ref) to establish a ranking representation for these partially observed search processes, and carry out estimation using modified rank-based GHK simulators. Appendix (ref) provides the implementation details for each case.\footnote{In Scenarios 3 and 5, the ranking representation can only be established for part of the search process, in which case the rank-based GHK method is applied locally and combined with the crude frequency method. }
The model specification used in the assessment is as follows:
We simulate 50 datasets, each consisting of 2,000 consumers and 8 products, without an outside option. Estimation relies exclusively on partial information from each dataset. The purchase value of each product is modeled as a linear function of price and three binary attributes, with the eight products covering all possible combinations of these attributes. The product with attributes $[0,0,0]$ is taken as the base alternative for location normalization. Table (ref) reports the estimation results. The first five columns present estimates for five partial-observability scenarios, while the final column reports estimates based on full observability. In Scenario 5, the observed action sequences cover six of eight products, forming a proper subset of the full choice set.
Table (ref) shows that linear preference parameters are well estimated across all partial observability scenarios. However, when only final purchase data are observed, the estimation of search costs deteriorates substantially, with both the standard deviation and RMSE much higher than in other cases. This is not surprising, as in this case the model collapses to a standard discrete-choice setting, in which identification of search costs relies only on variation in irregularly-distributed inspection propensities, making it weak and sensitive to sampling variation. As information on inspections increases, the precision of search cost estimates improves markedly.
These results highlight two core advantages of our empirical strategy in handling partial observability. First, we allow partially observed information to be fully exploited for estimation in a unified and parsimonious manner. Our strategy does not rely on additional or revised optimal rules, nor does it require extra assumptions or data processing tailored to a specific empirical setting. Instead, once partially observed action sequences are viewed as realizations of a constructed branching project, whose ranking representation can be directly used for implementation. Second, we allow partial observations to be used independently for estimation. When datasets are excessively large, the approach provides substantial flexibility in data usage. For example, when the main objective is to estimate consumer preferences, researchers can rely solely on the set of inspected products (Scenario 2). When a strategy of interest affects only the early stages of search, the analysis can be restricted to those stages (Scenario 4).
Note that correctly identifying the observability of actions is crucial for estimation. For any unselected action, the researcher must determine whether it was unobserved or observed but not selected. When this distinction is unavailable, separate likelihood contributions must be specified for both cases and summed, weighted by their observability probabilities.
In Appendix (ref), we apply the approaches above to the Expedia dataset to examine whether consumers reveal consistent preferences within the search process and in purchase decisions. Following the specification in ursu2018power, our results show systematic differences between preference estimates based on the searched set data and searched set and purchase data, suggesting that treating search and purchase decisions as arising from the same preference environment may be misspecified. In particular, consumers place greater weight on hotel star ratings at the purchase stage, while review scores, location scores, and chain affiliation tend to matter more at the search stage. These findings suggest that search and purchase decisions may reflect different preference tradeoffs, with practical implications for recommendation system design.
In increasingly rich clickstream data from online platforms, researchers can observe a wide range of consumer behaviors related to information acquisition. For example, consumers may scroll or navigate to the next page to discover additional products, open supplementary tabs to view detailed specifications, proceed to the checkout page to review shipping and tax costs, or decide to return a product based on the post-purchase experience. These actions suggest that information acquisition in the search process may occur not in a single step but gradually through multiple stages involving different types of actions. Characterizing and understanding the general sequential information-acquisition process in a sequential search model is a natural approach for studying the effects of marketing strategies or policy interventions.
However, incorporating richer types of information-acquisition actions into sequential search, as represented by optimal policies, poses substantial empirical challenges. Accommodating additional stages and action types often requires re-deriving the optimal rules, which is technically demanding and increases estimation complexity. In practice, researchers therefore often rely on partial information from the data, trading informational richness for tractability.
This subsection extends the ranking representation and the associated empirical strategy to sequential search with multi-stage information acquisition. The theory is straightforward: whenever the process can be expressed as a branching project that satisfies Independence and Invariance, (i) Lemma (ref) continues to apply, and (ii) the Gittins index policy remains optimal. The following theorem establishes the ranking representation for the extension case.
The proof of Theorem (ref) is provided in Appendix (ref). The theorem extends the main result from a structural perspective. Even when the assumptions on information acquisition stages and action types are relaxed, an optimal sequential search process that satisfies Independence and Invariance still admits a ranking representation. Theorem (ref) therefore arises as a special case of Theorem (ref), corresponding to a setting with two action types, a two-stage structure, and a common conditional preventive action for all unselected actions.
Theorem (ref) further broadens the applicability of the ranking representation. We use Example 5 from Section (ref), the sequential search with discovery model greminger2022optimal, to illustrate how the ranking representation can be applied to such settings.\footnote{Another example in Section (ref) is the two-stage sequential search model gibbard2022model, a brief discussion of which is offered in Appendix (ref). Note that some multi-stage information acquisition models, such as the search-purchase-return model in ibragimov2024clicks, assume that once a purchase is made, all other feasible actions are eliminated, leaving only the decision of whether to return the product, which violates the Invariance assumption. Such settings are typically addressed by re-deriving action values that incorporate the possibility of returns.} In this model, a product must be discovered before it can be inspected or purchased. While the consumer is initially aware of only a subset of products, they can expand this set via discovery actions through different channels like product lists or promotion pages. Within each channel, a subsequent discovery becomes feasible only after the preceding one is completed. Hence, products outside the initial feasible set must undergo one or more discovery steps before they become feasible for inspection.
To extend the ranking representation to discovery actions, these actions must also admit well-defined and comparable values. This requires that discovery actions satisfy the Independence and Invariance assumptions. Independence ensures that the expected payoff of any discovery action is unaffected by inspection outcomes or other discoveries, so its value is predetermined and does not involve belief updating. Invariance ensures that once a discovery becomes feasible, it remains feasible until selected, which guarantees a stable feasible set. In addition, greminger2022optimal imposes a weak monotonicity condition to ensure traceability. Along the same route, the value of any later discovery cannot exceed that of earlier ones. This prevents a discovery from being selected solely to access more valuable future discoveries.
Under this condition, similar to Equation (ref), greminger2022optimal defines the Gittins index for the $t$-th discovery on route $r$, referred to as the discovery value. Let
which is unknown, while its cumulative distribution function $G_{irt}(w)$ can be derived. The discovery value $q_{irt}$ is then defined implicitly as the solution to
where $c_{irt}^{dis}$ denotes the discovery cost of the $t$-th discovery on route $r$ for consumer $i$.
We illustrate how a sequential search process with discovery can be represented as a ranking through a simple example. Suppose the consumer initially knows only two products, 1 and 2, but can select $D_\mathrm{I}$ to discover product $3$ through a unique discovery route, which also generates a subsequent discovery opportunity $D_\mathrm{II}$. In this example, the consumer takes four sequential actions: inspecting product $1$ ($I_1$), performing the first discovery ($D_\mathrm{I}$), inspecting product $3$ ($I_3$), and purchasing product $3$ ($P_3$).
The branching project and ranking representations of this example process are shown in Figure (ref). The mapping between them is established by Theorem (ref). Among the four steps, the second, third, and fourth are forward-inaccessible, and the values of three conditional preventive actions can be computed from the values of selected actions $D_\mathrm{I}$, $I_3$, and $P_3$ in these steps. Among the unselected actions, $I_2$ and $P_1$ are feasible in all three forward-inaccessible steps, whereas $D_\mathrm{II}$ is infeasible in the second step (when $D_\mathrm{I}$ is selected).
We apply a modified rank-based GHK simulator for estimation based on the ranking representation. Its performance is demonstrated via a Monte Carlo study with the following specifications:
where $c^{ins}_{ijr}$ denotes the search cost for product $j$ on route $r$, $n_r$ denotes the number of products revealed per discovery on route $r$. Notice that we assume that $c^{dis}_{irt}$ is stochastic and follows a given distribution, mirroring the arguments in Section (ref): variation in discovery costs provides an independent source of randomness for discovery values. We further assume that the variances of the pre- and post-inspection taste shocks are equal to 1 to strengthen identification.
We perform the Monte Carlo simulation using 100 simulated datasets. Each dataset consists of 2000 consumers searching in a market of 1000 products. These products are split into two distinct routes: Route 1 (600 products) and Route 2 (400 products), where the latter offers lower average prices at the cost of reduced attribute variation. Each consumer begins with an initial set of 1 product plus an outside option and is randomly assigned 14 products to discover across the two routes. Throughout the process, each discovery action reveals $n_r = 2$ new products, unless only one remains undiscovered along the route. Importantly, consumers operate without knowing the total product count, such that they always believe that a discovery yields 2 products. As shown in Table (ref), the results confirm the effectiveness of our modified simulator.
The sequential search with discovery model has been empirically applied in zhang2024product. Building on the optimal rules proposed in greminger2022optimal, the paper implements estimation using a Kernel-smoothed Frequency Simulator. However, with a more complex search environment and a wider range of action types, implementation becomes considerably more difficult, and performance can be much weaker than that in the Weitzman model. By contrast, the rank-based GHK simulator is not subject to these limitations. Its implementation continues to follow the general guideline outlined in Section (ref), and its estimation performance shows no evident deterioration relative to that in the Weitzman model.
Notice that a multi-stage sequential search process can be partially observed. For example, greminger2024heterogeneous provides a set of implications characterizing the optimality of a sequential search with discovery process when the inspection order is unobserved, and constructs the likelihood function based on these implications. Appendix (ref) shows that these implications follow directly from a combination of Theorem (ref) and (ref) proposed in this paper.
The examples in the preceding sections demonstrate that the ranking representation remains valid across a broad class of sequential search settings. However, the necessary conditions for its applicability must be formally established: given an observed action sequence, under what constraints can the underlying sequential search process be represented as a ranking?
The validity of the ranking representation hinges on structural conditions that go beyond the assumptions of Independence and Invariance. As in Section (ref), we require that the search process can be decomposed into a sequence of discrete choices. Moreover, this sequence of discrete choices must satisfy the structural properties of a branching project.
The first condition requires that at each step, both the selected action and the alternative set are well-defined. A typical violation in which the feasible action set is unobserved occurs when the market choice set is unknown, preventing the researcher from determining the feasible alternatives against which the selected action is compared. Failure to observe the selected action occurs when an unobserved action is selected within the sequence. For example, if the data indicate only that a product is the second to be inspected, while the identity of the first inspected product remains unknown. In all scenarios discussed in Section (ref), partial observability prevents the original search process from being decomposed into a sequence of discrete choices. As a result, Lemma (ref) is required to construct composite actions, thereby restoring representability in the form of a branching project.
When the requirements for identifying the corresponding discrete choice sequence are satisfied, whether the sequence admits a branching project depends on the presence of sufficient and consistent dominance relations revealed in these choices. Drawing on concepts from graph theory, we establish the following theorem:
The proof of Theorem (ref) is stated in Appendix (ref). It rests on the following fact: a topological sort exists only when the underlying graph is a DAG bang2008digraphs, which allows all feasible actions to be represented as a ranking. When the discrete choices fail to form a DAG, cycles arise. For example, if we observe a consumer inspecting product A twice before and after inspecting a different product B, this implies a cycle between inspecting products A and B. This violates the structural requirement of a branching project that selected actions are removed from the feasible action set, and therefore no valid branching project can be established. If the discrete choices correspond to multiple DAGs, this suggests that the observed information is insufficient to determine the feasible action set or to establish relations among actions; thus, the ranking representation cannot be identified.
Theorem (ref) generalizes the ranking representation of sequential search. Under the assumptions of Invariance and Independence, it extends the representation from specific settings to a broader framework. Its primary structural requirement is that the feasible action set forms a unique directed acyclic graph, ensuring the existence and consistency of a topological ranking. From an empirical perspective, this property allows researchers to bypass ad hoc derivations of optimal policies for varying environments and instead construct likelihood functions directly from the ranking representation, providing a unified empirical approach across diverse market settings.
Theorem (ref) also helps identify potential redundant conditions that may arise in implementing sequential search models. Although the ranking representation in Theorem (ref) contains no redundant conditions, the representations constructed for extended models in Theorems (ref) and (ref) may still introduce redundant constraints. In such cases, one can construct the corresponding directed acyclic graph based on Theorem (ref) and apply transitive reduction to eliminate these redundancies, thereby avoiding the additional complexity induced by redundant constraints when implementing the model, in the same spirit as simplifying the optimal rules in Proposition (ref).
Finally, whenever a ranking representation applies under Theorem (ref), the rank-based GHK simulator remains a natural tool for estimation, though its applicability is not universal. Appendix (ref) discusses the scope and limitations of the simulator in greater detail. Yet, note that the GHK-style simulator is not the only method for performing the estimation. Generally speaking, once a ranking representation is established, estimation can draw on the full set of methods developed for ranking models. Improving computational strategies, particularly for large markets and long action sequences, remains an important direction for future research, for which the ranking representation provides a streamlined empirical foundation.\footnote{Recent progress includes wei2025pre, who develops a neural network-based approach for estimating sequential search models under a given specification.}
Before concluding, we briefly discuss violations of the fundamental assumptions of Independence and Invariance, which clarify both the implications and boundaries of optimal sequential search.
Although the ranking representation established in this paper is primarily intended for empirical application, its economic interpretation should not be overlooked. The central insight is that, given all action values and under Independence and Invariance, a consumer’s optimal sequential search can be equivalently represented as a ranking over all feasible actions. Put differently, an optimal sequential search can be viewed as a discrete choice at the level of action rankings, in which consumers choose among all possible rankings according to their preferences and search costs. Under this interpretation, the sequential search setting serves as a framework that extends the standard discrete choice problem from a single final purchase to a ranking of actions determined by their Gittins indices. The realized search process then serves only to reveal which category of partial ranking the consumer’s selections belong to. From the researcher’s perspective, once optimality is imposed, the realized search path is fully determined by ex ante draws, regardless of the complexity of the underlying model specification: it neither generates nor alters any source of randomness in the environment. Therefore, even without simplification, as in Proposition (ref), the sequential search problem can still be interpreted as a fully static discrete-choice problem defined over the space of possible rankings.
Therefore, under these assumptions, optimal sequential search is fundamentally static, and dynamics emerge only when these assumptions fail. Violations of Independence or Invariance can be taken as introducing dynamics into an otherwise static discrete choice problem through endogenous and exogenous channels, respectively.
First, relaxing Invariance introduces dynamics driven by exogenous variation. In this case, action values evolve over time or in response to other observable state variables, in ways that can be parameterized and empirically identified, leading to changes in consumers' rankings of actions across sessions. In a simple case in which such changes are not anticipated by consumers, the problem remains quasi-static: the demand structure within each session can still be characterized as an optimal sequential search, while cross-session differences reflect unanticipated ranking shifts induced by exogenous factors.\footnote{Closely related settings have been studied in discrete choice setups. For example, conlon2013demand considers unanticipated stockouts across sessions that generate sudden variations in observed choice sets.} Under this interpretation, researchers can characterize each session separately using the optimal sequential search framework, while identifying the parameters governing exogenous transitions from variation in rankings across sessions.\footnote{klein2024Do apply this method to study preference discovery in a multi-session Weitzman setting.}
In more complex environments, consumers may anticipate exogenous changes and adjust their search behavior accordingly. Search decisions are then made under forward-looking expectations. For example, consumers may understand that current search actions affect the future status of certain alternatives. The setting is then a dynamic decision problem with a given state-transition structure. For tractability, researchers may first estimate the transition matrix using frequency-based methods or Markov estimation, and then estimate the remaining structural parameters.\footnote{ursu2023search proposes a setting in which pausing search releases search fatigue and lowers subsequent search costs, with the search cost reduction effect being parameterized and estimated; elberg2019dynamic develops a belief-updating framework in repeated purchase settings, where beliefs evolve according to an empirically estimated transition matrix that is then used to identify search parameters; gardete2024multiattribute constructs a decision tree over attribute spaces, enabling joint estimation of belief-driven value transitions and search parameters.} In these settings, the Gittins index strategy is generally no longer optimal, and approximate index rules lin2015learning, compiani2024online are often required to quantify non-purchase actions.
Finally, relaxing Independence implies that the outcome of a search action affects the rankings of all other alternatives, which evolve through endogenous state transitions driven by realized search outcomes, with the transition structure itself also depending on prior stochastic realizations. A key example of Independence failure is cross-product learning. In this case, forward-looking optimal strategies must jointly account for current payoffs, the evolution of state variables, and the endogenous impact of current outcomes on future transitions. Given the complexity of the resulting intertemporal objective, empirical implementations often rely on approximate solutions to optimal strategies.\footnote{An example is ursu2020search, which departs from the sequential search framework of weitzman1979optimal and applies the framework of chick2012sequential.}
This paper establishes a theoretical equivalence between the optimal sequential search process and a partial ranking over feasible actions throughout the search process under the two fundamental assumptions of Independence and Invariance. Exploiting this equivalence, we strip out the sequential dependence issue and yield two empirical advantages. First, we develop a simple empirical strategy for the baseline Weitzman model, which delivers lower computational burden, improved performance, and reduced implementation complexity. Second, we show that the ranking representation and its associated empirical strategy extend to a broader class of sequential search settings, thereby allowing the approach to be applied to more realistic settings, such as partially observed search data or multi-stage information acquisition processes.
The implications of our findings are relevant to two primary research communities. For applied economists, this paper bridges the gap between optimal sequential search and ranking models, providing a rigorous microfoundation for incorporating granular search data into demand analysis. This perspective can be used to address a range of empirical challenges in demand estimation. For example, existing studies goeree2008limited often model demand under limited consideration as a two-stage process involving consideration set formation and purchase, whereas these two stages can be directly unified into a single ranking problem under the sequential search setting. The identification problem arising from zero market shares can be alleviated by exploiting non-zero search observations. Moreover, search sequences can serve as auxiliary data that enriches the identification of product complementarities and substitutabilities.
For marketing researchers, in response to the call in honka2024consumer, this paper provides a robust and flexible toolkit for analyzing consumer behavior in data-rich environments. As online platforms generate increasingly granular clickstream data, our methodology offers an efficient way to integrate such high-frequency observations into structural analysis. This not only deepens our understanding of how consumers interact with search intermediaries and retail interfaces but also provides an actionable framework for evaluating the effectiveness of marketing strategies deployed throughout the consumer search journey.
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