A Ranking Representation of Optimal Sequential Search

Tinghan Zhang

arxiv: 2501.07514 · v4 · submitted 2025-01-13 · 💰 econ.EM

A Ranking Representation of Optimal Sequential Search

Tinghan Zhang This is my paper

Pith reviewed 2026-05-23 05:45 UTC · model grok-4.3

classification 💰 econ.EM

keywords sequential searchoptimal searchranking representationconsumer searchempirical strategyinformation acquisitionsearch models

0 comments

The pith

Under independence and invariance, optimal sequential search holds exactly when a consistent ranking over all feasible actions exists.

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

The paper proves an equivalence: a sequential search process is optimal if and only if the feasible actions can be ordered by a fixed ranking that applies at every step. This equivalence rests on standard assumptions and replaces the usual requirement to solve for optimal policies that depend on prior outcomes. The result yields a direct empirical method that avoids repeated simulations of full search paths. It covers both fully observed sequences and cases with partial observation or staged information collection. Researchers gain a simpler route to estimate search costs and predict choices from observed action orders.

Core claim

Under the assumptions of Independence and Invariance, a sequential search process is optimal if and only if a corresponding ranking over all feasible actions throughout the process holds. This ranking representation converts the optimality condition into a static ordering that can be used directly for estimation and simulation.

What carries the argument

The ranking representation of optimal sequential search, which equates optimality to the existence of one consistent ordering of every feasible action at every decision point.

If this is right

The approach reduces simulation requirements while improving accuracy and computational efficiency for the classic search model.
It supplies a unified empirical strategy that handles partially observed action sequences.
It extends without change to multi-stage information acquisition problems such as discovery.
The same ranking device applies across a broad class of sequential search settings.

Where Pith is reading between the lines

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

Observed sequences could be used to recover the implied ranking directly and then test whether that ranking predicts later choices.
Data patterns that violate any possible ranking would indicate that independence or invariance fails in practice.
The representation might link sequential search to static ranking models used in other choice settings.
The method could be checked by comparing predicted search orders against new experimental data where true optimality is known.

Load-bearing premise

Independence and Invariance must hold for the optimality condition to be equivalent to the existence of a ranking.

What would settle it

An observed search sequence that is optimal yet cannot be generated by any single ranking of actions, or a sequence generated by a ranking that is nevertheless suboptimal, when the independence and invariance conditions are satisfied.

read the original abstract

Sequential search models provide a powerful framework for studying consumer search using rich data that records the sequence of consumer actions taken during the search process. In existing empirical applications, their implementation often builds on optimal policies, in which later decisions depend on outcomes from earlier actions that are often fully observed by researchers. Therefore, implementation is largely restricted by computation burden and limited model flexibility. This paper establishes a theoretical equivalence showing that, under common and mild assumptions of Independence and Invariance, a sequential search process is optimal if and only if a corresponding ranking over all feasible actions throughout the process holds, thereby introducing a ranking representation of optimal sequential search. This representation enables a novel, simple, and unified empirical strategy for implementing sequential search models. For the classic \cite{weitzman1979optimal} model, the proposed approach reduces simulation requirements while improving accuracy, computational efficiency, and ease of implementation. We further show that the same strategy extends to a broad class of sequential search settings, including partially observed action sequences and multi-stage information acquisition, such as discovery. Overall, the results enhance both the tractability and the empirical applicability of sequential search models.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper offers a ranking representation equivalent to optimal sequential search under independence and invariance, which could simplify empirical work if the math checks out.

read the letter

The core claim is that under independence and invariance, a sequential search process is optimal if and only if a ranking over all feasible actions holds throughout. This is framed as a new characterization distinct from the usual optimal policy setup in Weitzman 1979. The paper shows how this representation cuts simulation requirements and improves accuracy and speed for the classic model, then extends the same logic to partially observed sequences and multi-stage discovery. Those extensions are the part that actually broadens applicability to richer sequence data in empirical settings. The approach is presented as a unified, simpler strategy for implementation, which directly targets the computation burden that has limited these models so far. The assumptions are labeled common and mild, and the result is an if-and-only-if equivalence rather than an approximation. That structure is what makes the computational gains follow once the representation is accepted. The main soft spot is that the abstract gives no derivation steps or verification of how the assumptions produce the exact equivalence, so it is impossible to judge whether hidden restrictions appear in the proof or whether the ranking is straightforward to recover in applied cases. Independence in particular can be fragile once actions are correlated through unobserved factors. The citation pattern is straightforward and centers on the Weitzman baseline without obvious circularity. This paper is for empirical industrial organization researchers who already work with sequential search models and need better tractability for estimation with sequence data. A reader focused on computational methods in structural models would get the most out of the claimed efficiency gains. It deserves a serious referee to examine the proof and test the practical improvements.

Referee Report

0 major / 3 minor

Summary. The paper claims that under the assumptions of Independence and Invariance, optimal sequential search is equivalent to the existence of a ranking representation over feasible actions at each stage. This equivalence yields a simpler, unified empirical implementation strategy that reduces simulation needs and improves accuracy for the Weitzman (1979) model while extending to settings with partial observation and multi-stage discovery.

Significance. If the equivalence holds, the ranking representation offers a computationally lighter and more flexible approach to estimating sequential search models, directly addressing the implementation barriers noted in the abstract. The claimed reductions in simulation requirements and extensions to broader classes of models represent a practical advance for empirical work in consumer search.

minor comments (3)

[Abstract] Abstract: the phrase 'common and mild assumptions of Independence and Invariance' is used without a one-sentence definition or reference to their formal statement; adding this would improve accessibility for readers unfamiliar with the exact formulation.
[§2 or §3] The manuscript would benefit from an explicit statement (perhaps in §2 or §3) of how the ranking is constructed from the value functions under the stated assumptions, even if the full proof appears later.
[Empirical section on Weitzman model] Table or figure comparing simulation time and accuracy for the Weitzman benchmark versus the proposed ranking method would strengthen the computational-advantage claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the paper, the accurate summary of its contributions, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity; theoretical equivalence under external assumptions

full rationale

The paper derives an if-and-only-if equivalence between optimality of sequential search and existence of a ranking representation, conditioned explicitly on the maintained assumptions of Independence and Invariance. These assumptions are standard and external; the central claim does not reduce by construction to fitted parameters, self-citations, or redefinitions of its own inputs. No load-bearing steps match the enumerated circularity patterns, and the result is self-contained against the stated conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Independence and Invariance assumptions, which are presented as standard domain assumptions but not derived or tested within the paper.

axioms (1)

domain assumption Independence and Invariance assumptions
Described as common and mild assumptions under which the equivalence holds.

pith-pipeline@v0.9.0 · 5715 in / 1201 out tokens · 37022 ms · 2026-05-23T05:45:39.292690+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

[1]

Sincevi j′ >v i j, it must also hold thatv i j′ >y i j′ becausey i j′ ≤v i j

If this step is forward-inaccessible, thenv i j′ is never selected later. Sincevi j′ >v i j, it must also hold thatv i j′ >y i j′ becausey i j′ ≤v i j. This contradicts condition (ii) in the theorem

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[2]

If condition (i) holds, thenv ik <v i j <v i j′, which again contradicts condition (ii), as shown in the first case

If this step is forward-accessible andv i j′ is not selected later, then by the Invariance as- sumption, there exists a forward-inaccessible step afterjwhere some actionkis selected, vi j′ is feasible, but not selected. If condition (i) holds, thenv ik <v i j <v i j′, which again contradicts condition (ii), as shown in the first case

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[3]

In all three cases, we arrive at a contradiction

If this step is forward-accessible andv i j′ is selected later, then sincev i j <v i j′, there must be at least one intermediate step between the selection ofjandj ′ that violates condition (i). In all three cases, we arrive at a contradiction. Hence, Theorem 3 is proven. C The Search Propensity in an Additive Specification Without loss of generality, con...

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[4]

Only the final purchase is observed

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[5]

Only the set of inspected products and the final purchase are observed

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[6]

Chung et al

Only the first inspection and the final purchase are observed; 3The exponential distribution is a common example of this closure under scaling property. Chung et al. (2025) compared its estimation performance against the log-normal distribution, which does not share the property, and found the former to be more stable. Other examples include uniform distr...

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[7]

Only the search process over a subset of products is observed (excluding purchase); We adopt the specification defined by Equations (12) to (13), imposing the normalization σζ =1 and excluding the outside option. To maintain clarity, we illustrate the discussion using an example where a consumer’s actual search sequence is given byMi ={A,B,C,D,E}, the set...

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[10]

Drawζ iA,ζ iC,ζ iD andζ iE randomly to determinez d iA,z d iC,z d iD andz d iE

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[11]

For the other draws, assignp d i,a =1

For those draws withz d iA >w d iB, computep d i,a =Pr(u iA <w d iB |z d iA). For the other draws, assignp d i,a =1. Do this also for productsDandE. Computep d i =p d i,a ·p d i,c ·p d i,d ·p d i,e

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[12]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i . Take the average across draws to obtain the simulated likelihood. E.2 Scenario 2 In this case, all actions in the search process are observable, including inspections of all products and purchases among the inspected products, while only the inspection order is unobserved. Since the purchased p...

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[13]

Draw the heterogeneity in preferences and determineβ d i

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[15]

Computep d i1 =Pr(z iA >w d iB)·Pr(z iC >w d iB)·Pr(z iD >w d iB)

Drawζ iA,ζ iC andζ iD conditional onz iA >w d iB,z iC >w d iB andz iD >w d iB to determinez d iA,z d iC andz d iD. Computep d i1 =Pr(z iA >w d iB)·Pr(z iC >w d iB)·Pr(z iD >w d iB)

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[16]

Computep d i2 =Pr(u iA <w d iB |z d iA)·Pr(u iC <w d iB |z d iC)·Pr(u iD <w d iB |z d iD)·Pr(z iE <w d iB). A13

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[17]

larger than

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2. Take the average across draws to obtain the simulated likelihood. E.3 Scenario 3 In this case, the observable actions include the inspection and purchase of the first inspected product, as well as the inspection and purchase of the purchased product, each associated with its correspondi...

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[19]

Drawζ iB andε iB randomly to determinez d iB,u d iB andw d iB

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[20]

Computep d i1 =Pr(z iA >z d iB)

Drawζ iA conditional onz iA >z d iB to determinez d iA. Computep d i1 =Pr(z iA >z d iB)

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[21]

Drawζ iC,ζ iD andζ iE randomly to obtainz d iC,z d iD andz d iE. A14

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[22]

For the other draws, assignp d i2,c =1

For those draws withz d iC >w d iB, computep d i2,c =Pr(u iC <w d iB |z d iC). For the other draws, assignp d i2,c =1. Do this also for productsDandE. Computep d i2 =p d i2,c ·p d i2,d ·p d i2,e

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[23]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2. Take the average across draws to obtain the simulated likelihood. For the case where the purchased product is the first inspected product, the implementation procedure follows Scenario 1. E.4 Scenario 4 We assume that in this case, information regarding productCis missing; that is, it i...

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[24]

Make draws on the heterogeneity in preferences and determineβ d i

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[25]

Computew i =max{w d iC}

Drawζ iC andε iC randomly to determinew d iC. Computew i =max{w d iC}. If there are more products being unobserved, draw the effective value of them all and calculate their maximum

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[26]

Computep d i1 =Pr(z iD >w d i )

Drawζ iD conditional onz iD >w d iC to determinez d iD. Computep d i1 =Pr(z iD >w d i )

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[27]

Computep d i2 =Pr(z iA >z d iB)·Pr(z iB >z d iD)

Drawζ iB andζ iA sequentially conditional onz iB >z d iD andz iA >z d iB to determinez d iA and zd iB. Computep d i2 =Pr(z iA >z d iB)·Pr(z iB >z d iD)

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[28]

Compute pd i3 =Pr(u iA <z d iD |z d iA)·Pr(u iB <z d iD |z d iB)

Drawε iA,ε iB conditional onu iA <z d iD andu iB <z d iD to determineu d iA andu d iA. Compute pd i3 =Pr(u iA <z d iD |z d iA)·Pr(u iB <z d iD |z d iB)

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[29]

Computey d i =min{z d iD,max{u d iA,u d iB,w d iC,u d iD}}

Drawε iD randomly to obtainu d iD. Computey d i =min{z d iD,max{u d iA,u d iB,w d iC,u d iD}}

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[30]

Computep d i4 =Pr(z iE <y d i )

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[31]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2 ·p d i3 ·p d i4. Take the average across draws to obtain the simulated likelihood. The implementation of Scenario 4 is substantially more complex than in the first three sce- narios. In Scenarios 1–3, the observed action sequence corresponds to a chain within a partially ordered set. For...

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[32]

If not, assignp d i,1,0 =p d i,2,0 =1, skip Steps 2 to 3

Check ifJ(0)>J(1). If not, assignp d i,1,0 =p d i,2,0 =1, skip Steps 2 to 3

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[33]

Assignp d i,1,0 =1

Drawζ u i,J(0) to determinez d i,J(0). Assignp d i,1,0 =1. A18

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[34]

Computep d i,2,0 = ∏J(1)+1≤j≤J(0)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

Sequentially drawζ u i,J(0)−1,ζ u i,J(0)−2,· · ·,ζ u i,J(1)+1 to determinez d i,J(0)−1,· · ·,z d i,J(1)+1 con- ditional onz i,j >z d i,j+1 . Computep d i,2,0 = ∏J(1)+1≤j≤J(0)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 ). So far, we have accomplished the simulation of the ranking conditions for the last segment, similar to the implementation procedure for the...

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[35]

It corresponds to the value of discovering more alternatives through the route observed at the end of Segment t, orr(t)

Drawc dis i,r(t),t to determineq i,r(t),t, which is the discovery value realized in Segmentt. It corresponds to the value of discovering more alternatives through the route observed at the end of Segment t, orr(t). Several conditions need to be satisfied: •q i,r(t),t is larger than the reservation values of products discovered before or in Seg- menttwhile...

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[36]

If not, assignp d i,1,t =p d i,2,t =1, skip Steps 6 to 7

Check ifJ(t)>J(t+1). If not, assignp d i,1,t =p d i,2,t =1, skip Steps 6 to 7

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[37]

Drawζ u i,J(t) to determinez d i,J(t) conditional onz i,J(t) >q d i,r(t),t, calculatep d i,1,t =Pr(z i,J(t) ≥ qd i,r(t),t |qi,r(t),t =q d i,r(t),t )

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[38]

Computep d i,2,t = ∏J(t)+1≤j≤J(t)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

Sequentially drawζ u i,J(t)−1 ,ζ u i,J(t)−2 ,· · ·,ζ u i,J(t)+1 to determinez d i,J(t)−1 ,· · ·,z d i,J(t)+1 condi- tional onz i,j >z d i,j+1 . Computep d i,2,t = ∏J(t)+1≤j≤J(t)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

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[39]

So far, we have finished the simulation of the observed ranking conditions in the overall search and product discovery sequence

Repeat Steps 4 to 7 until exhausting the sequence up to the last SegmentT. So far, we have finished the simulation of the observed ranking conditions in the overall search and product discovery sequence

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[40]

IfJ(t+1)<h≤J(t), drawε ih conditional onu ih <min 1≤t′≤t{qd i,r(t′),t′}to determineu d ih, calculatep d i,3,t =Pr(u ih <min 1≤t′≤t{qd i,r(t′),t′}|qd i,r(1),1,q d i,r(2),2,· · ·,q d i,r(t),t ); else, do not drawε ih and assignp d i,3,t =1

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[41]

core value

Compute the sub-core values: fort=0,y d i0 =min{z d i,J(0),u d ih}; fort>0,y d it =q d i,r(t),t. So far, we have finished the simulation of the “core value” in each segment within the search and product discovery sequence

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[42]

For any router ′ that has not been discovered throughout the search process, set ¯qd i,r′ = +∞

Compute ¯qd ir for each router, which is equal to the drawn discovery value onrwith the smallestt. For any router ′ that has not been discovered throughout the search process, set ¯qd i,r′ = +∞. A19

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[43]

Computep d i,4 = ∏1≤r′≤NR Pr(qi,r′,0 <min{y d i0,min 1≤r′′≤NR,r′′̸=r′{¯qd i,r′′}}|yd i0,¯qd i1,· · ·¯qd i,NR) to account for the probabilities of the unselected discovery actions given the sub-core values of segments in which these actions are available

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[44]

Computep d i,5 = ∏0≤h′≤J(t),h ′̸=h Pr(uih′ <min {t:J(t)≥h ′}{yd it})to account for the probabil- ities of the unselected purchase actions given the sub-core values of segments in which these actions are available

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[45]

So far, we have finished computing the probabilities of the unselected actions of the ranking conditions, which include unrealized discovery, reservation, and purchase values

Computep d i,6 similar top d i,5 for those products discovered but not inspected corresponding to the sub-core values in the segments since their being discovered, to account for the probabilities of unselected inspection actions since they are available. So far, we have finished computing the probabilities of the unselected actions of the ranking conditi...

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[46]

Take the average across draws to obtain the simulated likelihood

Take products ofp d i,0,t,p d i,1,t,p d i,2,t,p d i,3,t,p d i,4,p d i,5 andp d i,6 as the simulated likelihood con- tribution of the draw. Take the average across draws to obtain the simulated likelihood. Hence, we obtain the simulated likelihood of the overall model. H Two-Stage Sequential Search Model This section introduces the PR representation and th...

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[47]

Randomly draw the Gittins indices for the actions selected in all forward-inaccessible steps: sampleG d c,B,G d a,B, andG d b,B randomly

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[48]

Calculate the probability that the Gittins index of each actionℓsmaller than the correspondingy iℓ: pd i1 =Pr(G a,A <y d i |y d i )·Pr(G b,C <y d i |y d i )

Calculatey iℓ for all unselected actions:y d iℓ =y d i =min{G d c,B,G d a,BGd b,B}. Calculate the probability that the Gittins index of each actionℓsmaller than the correspondingy iℓ: pd i1 =Pr(G a,A <y d i |y d i )·Pr(G b,C <y d i |y d i )

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[49]

Recursively draw actions in the reverse sequence, conditioning each draw on the Gittins A21 index being greater than that of the previously drawn action, except for consideration actions that immediately follow their corresponding browsing, and stop at the second browse: sampleG d b,C conditional onG b,C >G d b,B, calculatep d i2 =Pr(G b,C >G d b,B | Gd b,B)

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[50]

Calculate the conditional probability that the first browsing is selected:p d i3 =Pr(G b,A > Gd b,C | Gd b,C)

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[51]

Calculate the conditional probability that all immediate considerings are selected:p d i4 = Pr(Gc,A >G d b,C | Gd b,C)

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[52]

I Search and Product Discovery Model with Incomplete Data We consider the case of search and product discovery processes with incomplete information

Finally, the product of all probabilities above is taken as the simulated likelihood contri- bution for the sequence:L d i =p d i1 ·p d i2 ·p d i3 ·p d i4. I Search and Product Discovery Model with Incomplete Data We consider the case of search and product discovery processes with incomplete information. The setting we discuss here arises from search impr...

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[53]

Draw the heterogeneities in preferences and determineβ d i

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[54]

Compute ˜w2,d ih

Drawu d ih andz d ih randomly. Compute ˜w2,d ih

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[55]

Calculateq d ih and ˜w3,d ih

Draw the discovery value for the last discovered product ˜Jconditional on thatq i ˜J >˜w2,d ih . Calculateq d ih and ˜w3,d ih . Calculatep d i1 =Pr(q i ˜J >˜w2,d ih )

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[56]

Make draws on these reservation values if it is necessary to calculate the conditional probabilities of purchase values

Calculatep d i2 =Pr(z i j >˜w3,d ih )for allj∈ S 1 andp d i3 =Pr(z i j >˜w2,d ih )for allj∈ S 2. Make draws on these reservation values if it is necessary to calculate the conditional probabilities of purchase values

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[57]

Calculate the conditional probabilities of the following: •p d i4 = ∏ j∈S1 Pr(ui j <˜w3,d ih ); •p d i5 = ∏k∈ ¯S1 Pr(zik <˜w3,d ih ); •p d i6 = ∏ j∈S2 Pr(ui j <˜w2,d ih ); •p d i7 = ∏k∈ ¯S2 Pr(zik <˜w2,d ih )

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[58]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2 ·p d i3 ·p d i4 ·p d i5 ·p d i6 ·p d i7. Take the average across draws to obtain the simulated likelihood. If an outside option is available and not purchased, its purchase valueu i0 must be smaller than ˜w3,d ih , and the corresponding conditional probability can be computed together wi...

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[1] [1]

Sincevi j′ >v i j, it must also hold thatv i j′ >y i j′ becausey i j′ ≤v i j

If this step is forward-inaccessible, thenv i j′ is never selected later. Sincevi j′ >v i j, it must also hold thatv i j′ >y i j′ becausey i j′ ≤v i j. This contradicts condition (ii) in the theorem

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[2] [2]

If condition (i) holds, thenv ik <v i j <v i j′, which again contradicts condition (ii), as shown in the first case

If this step is forward-accessible andv i j′ is not selected later, then by the Invariance as- sumption, there exists a forward-inaccessible step afterjwhere some actionkis selected, vi j′ is feasible, but not selected. If condition (i) holds, thenv ik <v i j <v i j′, which again contradicts condition (ii), as shown in the first case

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[3] [3]

In all three cases, we arrive at a contradiction

If this step is forward-accessible andv i j′ is selected later, then sincev i j <v i j′, there must be at least one intermediate step between the selection ofjandj ′ that violates condition (i). In all three cases, we arrive at a contradiction. Hence, Theorem 3 is proven. C The Search Propensity in an Additive Specification Without loss of generality, con...

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[4] [4]

Only the final purchase is observed

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[5] [5]

Only the set of inspected products and the final purchase are observed

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[6] [6]

Chung et al

Only the first inspection and the final purchase are observed; 3The exponential distribution is a common example of this closure under scaling property. Chung et al. (2025) compared its estimation performance against the log-normal distribution, which does not share the property, and found the former to be more stable. Other examples include uniform distr...

work page 2025

[7] [7]

Only the search process over a subset of products is observed (excluding purchase); We adopt the specification defined by Equations (12) to (13), imposing the normalization σζ =1 and excluding the outside option. To maintain clarity, we illustrate the discussion using an example where a consumer’s actual search sequence is given byMi ={A,B,C,D,E}, the set...

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[8] [10]

Drawζ iA,ζ iC,ζ iD andζ iE randomly to determinez d iA,z d iC,z d iD andz d iE

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[9] [11]

For the other draws, assignp d i,a =1

For those draws withz d iA >w d iB, computep d i,a =Pr(u iA <w d iB |z d iA). For the other draws, assignp d i,a =1. Do this also for productsDandE. Computep d i =p d i,a ·p d i,c ·p d i,d ·p d i,e

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[10] [12]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i . Take the average across draws to obtain the simulated likelihood. E.2 Scenario 2 In this case, all actions in the search process are observable, including inspections of all products and purchases among the inspected products, while only the inspection order is unobserved. Since the purchased p...

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[11] [13]

Draw the heterogeneity in preferences and determineβ d i

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[12] [15]

Computep d i1 =Pr(z iA >w d iB)·Pr(z iC >w d iB)·Pr(z iD >w d iB)

Drawζ iA,ζ iC andζ iD conditional onz iA >w d iB,z iC >w d iB andz iD >w d iB to determinez d iA,z d iC andz d iD. Computep d i1 =Pr(z iA >w d iB)·Pr(z iC >w d iB)·Pr(z iD >w d iB)

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[13] [16]

Computep d i2 =Pr(u iA <w d iB |z d iA)·Pr(u iC <w d iB |z d iC)·Pr(u iD <w d iB |z d iD)·Pr(z iE <w d iB). A13

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[14] [17]

larger than

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2. Take the average across draws to obtain the simulated likelihood. E.3 Scenario 3 In this case, the observable actions include the inspection and purchase of the first inspected product, as well as the inspection and purchase of the purchased product, each associated with its correspondi...

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[15] [19]

Drawζ iB andε iB randomly to determinez d iB,u d iB andw d iB

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[16] [20]

Computep d i1 =Pr(z iA >z d iB)

Drawζ iA conditional onz iA >z d iB to determinez d iA. Computep d i1 =Pr(z iA >z d iB)

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[17] [21]

Drawζ iC,ζ iD andζ iE randomly to obtainz d iC,z d iD andz d iE. A14

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[18] [22]

For the other draws, assignp d i2,c =1

For those draws withz d iC >w d iB, computep d i2,c =Pr(u iC <w d iB |z d iC). For the other draws, assignp d i2,c =1. Do this also for productsDandE. Computep d i2 =p d i2,c ·p d i2,d ·p d i2,e

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[19] [23]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2. Take the average across draws to obtain the simulated likelihood. For the case where the purchased product is the first inspected product, the implementation procedure follows Scenario 1. E.4 Scenario 4 We assume that in this case, information regarding productCis missing; that is, it i...

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[20] [24]

Make draws on the heterogeneity in preferences and determineβ d i

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[21] [25]

Computew i =max{w d iC}

Drawζ iC andε iC randomly to determinew d iC. Computew i =max{w d iC}. If there are more products being unobserved, draw the effective value of them all and calculate their maximum

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[22] [26]

Computep d i1 =Pr(z iD >w d i )

Drawζ iD conditional onz iD >w d iC to determinez d iD. Computep d i1 =Pr(z iD >w d i )

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[23] [27]

Computep d i2 =Pr(z iA >z d iB)·Pr(z iB >z d iD)

Drawζ iB andζ iA sequentially conditional onz iB >z d iD andz iA >z d iB to determinez d iA and zd iB. Computep d i2 =Pr(z iA >z d iB)·Pr(z iB >z d iD)

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[24] [28]

Compute pd i3 =Pr(u iA <z d iD |z d iA)·Pr(u iB <z d iD |z d iB)

Drawε iA,ε iB conditional onu iA <z d iD andu iB <z d iD to determineu d iA andu d iA. Compute pd i3 =Pr(u iA <z d iD |z d iA)·Pr(u iB <z d iD |z d iB)

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[25] [29]

Computey d i =min{z d iD,max{u d iA,u d iB,w d iC,u d iD}}

Drawε iD randomly to obtainu d iD. Computey d i =min{z d iD,max{u d iA,u d iB,w d iC,u d iD}}

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[26] [30]

Computep d i4 =Pr(z iE <y d i )

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[27] [31]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2 ·p d i3 ·p d i4. Take the average across draws to obtain the simulated likelihood. The implementation of Scenario 4 is substantially more complex than in the first three sce- narios. In Scenarios 1–3, the observed action sequence corresponds to a chain within a partially ordered set. For...

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[28] [32]

If not, assignp d i,1,0 =p d i,2,0 =1, skip Steps 2 to 3

Check ifJ(0)>J(1). If not, assignp d i,1,0 =p d i,2,0 =1, skip Steps 2 to 3

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[29] [33]

Assignp d i,1,0 =1

Drawζ u i,J(0) to determinez d i,J(0). Assignp d i,1,0 =1. A18

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[30] [34]

Computep d i,2,0 = ∏J(1)+1≤j≤J(0)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

Sequentially drawζ u i,J(0)−1,ζ u i,J(0)−2,· · ·,ζ u i,J(1)+1 to determinez d i,J(0)−1,· · ·,z d i,J(1)+1 con- ditional onz i,j >z d i,j+1 . Computep d i,2,0 = ∏J(1)+1≤j≤J(0)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 ). So far, we have accomplished the simulation of the ranking conditions for the last segment, similar to the implementation procedure for the...

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[31] [35]

It corresponds to the value of discovering more alternatives through the route observed at the end of Segment t, orr(t)

Drawc dis i,r(t),t to determineq i,r(t),t, which is the discovery value realized in Segmentt. It corresponds to the value of discovering more alternatives through the route observed at the end of Segment t, orr(t). Several conditions need to be satisfied: •q i,r(t),t is larger than the reservation values of products discovered before or in Seg- menttwhile...

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[32] [36]

If not, assignp d i,1,t =p d i,2,t =1, skip Steps 6 to 7

Check ifJ(t)>J(t+1). If not, assignp d i,1,t =p d i,2,t =1, skip Steps 6 to 7

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[33] [37]

Drawζ u i,J(t) to determinez d i,J(t) conditional onz i,J(t) >q d i,r(t),t, calculatep d i,1,t =Pr(z i,J(t) ≥ qd i,r(t),t |qi,r(t),t =q d i,r(t),t )

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[34] [38]

Computep d i,2,t = ∏J(t)+1≤j≤J(t)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

Sequentially drawζ u i,J(t)−1 ,ζ u i,J(t)−2 ,· · ·,ζ u i,J(t)+1 to determinez d i,J(t)−1 ,· · ·,z d i,J(t)+1 condi- tional onz i,j >z d i,j+1 . Computep d i,2,t = ∏J(t)+1≤j≤J(t)−1 Pr(zi,j ≥z d i,j+1 |zi,j+1 =z d i,j+1 )

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[35] [39]

So far, we have finished the simulation of the observed ranking conditions in the overall search and product discovery sequence

Repeat Steps 4 to 7 until exhausting the sequence up to the last SegmentT. So far, we have finished the simulation of the observed ranking conditions in the overall search and product discovery sequence

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[36] [40]

IfJ(t+1)<h≤J(t), drawε ih conditional onu ih <min 1≤t′≤t{qd i,r(t′),t′}to determineu d ih, calculatep d i,3,t =Pr(u ih <min 1≤t′≤t{qd i,r(t′),t′}|qd i,r(1),1,q d i,r(2),2,· · ·,q d i,r(t),t ); else, do not drawε ih and assignp d i,3,t =1

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[37] [41]

core value

Compute the sub-core values: fort=0,y d i0 =min{z d i,J(0),u d ih}; fort>0,y d it =q d i,r(t),t. So far, we have finished the simulation of the “core value” in each segment within the search and product discovery sequence

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[38] [42]

For any router ′ that has not been discovered throughout the search process, set ¯qd i,r′ = +∞

Compute ¯qd ir for each router, which is equal to the drawn discovery value onrwith the smallestt. For any router ′ that has not been discovered throughout the search process, set ¯qd i,r′ = +∞. A19

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[39] [43]

Computep d i,4 = ∏1≤r′≤NR Pr(qi,r′,0 <min{y d i0,min 1≤r′′≤NR,r′′̸=r′{¯qd i,r′′}}|yd i0,¯qd i1,· · ·¯qd i,NR) to account for the probabilities of the unselected discovery actions given the sub-core values of segments in which these actions are available

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[40] [44]

Computep d i,5 = ∏0≤h′≤J(t),h ′̸=h Pr(uih′ <min {t:J(t)≥h ′}{yd it})to account for the probabil- ities of the unselected purchase actions given the sub-core values of segments in which these actions are available

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[41] [45]

So far, we have finished computing the probabilities of the unselected actions of the ranking conditions, which include unrealized discovery, reservation, and purchase values

Computep d i,6 similar top d i,5 for those products discovered but not inspected corresponding to the sub-core values in the segments since their being discovered, to account for the probabilities of unselected inspection actions since they are available. So far, we have finished computing the probabilities of the unselected actions of the ranking conditi...

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[42] [46]

Take the average across draws to obtain the simulated likelihood

Take products ofp d i,0,t,p d i,1,t,p d i,2,t,p d i,3,t,p d i,4,p d i,5 andp d i,6 as the simulated likelihood con- tribution of the draw. Take the average across draws to obtain the simulated likelihood. Hence, we obtain the simulated likelihood of the overall model. H Two-Stage Sequential Search Model This section introduces the PR representation and th...

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[43] [47]

Randomly draw the Gittins indices for the actions selected in all forward-inaccessible steps: sampleG d c,B,G d a,B, andG d b,B randomly

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[44] [48]

Calculate the probability that the Gittins index of each actionℓsmaller than the correspondingy iℓ: pd i1 =Pr(G a,A <y d i |y d i )·Pr(G b,C <y d i |y d i )

Calculatey iℓ for all unselected actions:y d iℓ =y d i =min{G d c,B,G d a,BGd b,B}. Calculate the probability that the Gittins index of each actionℓsmaller than the correspondingy iℓ: pd i1 =Pr(G a,A <y d i |y d i )·Pr(G b,C <y d i |y d i )

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[45] [49]

Recursively draw actions in the reverse sequence, conditioning each draw on the Gittins A21 index being greater than that of the previously drawn action, except for consideration actions that immediately follow their corresponding browsing, and stop at the second browse: sampleG d b,C conditional onG b,C >G d b,B, calculatep d i2 =Pr(G b,C >G d b,B | Gd b,B)

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[46] [50]

Calculate the conditional probability that the first browsing is selected:p d i3 =Pr(G b,A > Gd b,C | Gd b,C)

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[47] [51]

Calculate the conditional probability that all immediate considerings are selected:p d i4 = Pr(Gc,A >G d b,C | Gd b,C)

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[48] [52]

I Search and Product Discovery Model with Incomplete Data We consider the case of search and product discovery processes with incomplete information

Finally, the product of all probabilities above is taken as the simulated likelihood contri- bution for the sequence:L d i =p d i1 ·p d i2 ·p d i3 ·p d i4. I Search and Product Discovery Model with Incomplete Data We consider the case of search and product discovery processes with incomplete information. The setting we discuss here arises from search impr...

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[49] [53]

Draw the heterogeneities in preferences and determineβ d i

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[50] [54]

Compute ˜w2,d ih

Drawu d ih andz d ih randomly. Compute ˜w2,d ih

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[51] [55]

Calculateq d ih and ˜w3,d ih

Draw the discovery value for the last discovered product ˜Jconditional on thatq i ˜J >˜w2,d ih . Calculateq d ih and ˜w3,d ih . Calculatep d i1 =Pr(q i ˜J >˜w2,d ih )

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[52] [56]

Make draws on these reservation values if it is necessary to calculate the conditional probabilities of purchase values

Calculatep d i2 =Pr(z i j >˜w3,d ih )for allj∈ S 1 andp d i3 =Pr(z i j >˜w2,d ih )for allj∈ S 2. Make draws on these reservation values if it is necessary to calculate the conditional probabilities of purchase values

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[53] [57]

Calculate the conditional probabilities of the following: •p d i4 = ∏ j∈S1 Pr(ui j <˜w3,d ih ); •p d i5 = ∏k∈ ¯S1 Pr(zik <˜w3,d ih ); •p d i6 = ∏ j∈S2 Pr(ui j <˜w2,d ih ); •p d i7 = ∏k∈ ¯S2 Pr(zik <˜w2,d ih )

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[54] [58]

Take the average across draws to obtain the simulated likelihood

Compute the likelihood contribution of the drawL d i =p d i1 ·p d i2 ·p d i3 ·p d i4 ·p d i5 ·p d i6 ·p d i7. Take the average across draws to obtain the simulated likelihood. If an outside option is available and not purchased, its purchase valueu i0 must be smaller than ˜w3,d ih , and the corresponding conditional probability can be computed together wi...

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