Expert-Mediated Search

Meenal ChhabraDept. of Computer Science, Rensselaer Polytechnic Institute, Troy NY

Sanmay DasDept. of Computer Science, Rensselaer Polytechnic Institute, Troy NY

David SarneDept. of Computer Science, Bar Ilan University, Ramat Gan, Israel

Increasingly in both traditional, and especially Internet-based marketplaces, information is becoming a

traded commodity. We consider the impact of the presence of information-brokers, or experts, on markets

in which consumers receive noisy signals of the true values of opportunities; an example of such a market

is an Internet marketplace for used cars. An expert offers to disambiguate noisy signals, providing more

information, at a specified price, about the true value of an opportunity. In this paper, we model such

markets using the classic economic framework of one-sided search. We use our model to characterize the

optimal search strategy for buyers, which has a double-reservation value form. This characterization allows

us to derive the optimal pricing strategy for a monopolist expert who moves first by setting the price of her

services. We show that a market designer or regulator can improve social welfare by subsidizing the purchase

of expert services, and that the optimal level of subsidy for an additive measure of social utility is the level

that forces the buyer to exactly fully internalize the cost of provision of expert services. Finally, we extend our

analysis to cases where the expert sells bundles of services, and find that bundling can be pareto-improving

for the consumer and the expert; further, the key features of electronic marketplaces, namely lower search

costs for consumers and lower costs of production of expert services, both favor bundling.

1. Introduction

In one-sided search agents face a stream of opportunities that arise sequentially, and the process

terminates when the agent picks one of those opportunities (McCall 1970, McMillan and Rothschild

1994, Grosfeld-Nir et al. 2009). A classic example is a consumer looking to buy a used car. She

will typically investigate cars one at a time until deciding upon one she wants. Similar settings

can be found in job-search, house search, technology R&D and other applications (Lippman and

McCall 1976, Mortensen and Pissarides 1999, Bental and Peled 2002, Terwiesch and Loch 2004).

Sequential search theory has also been applied extensively in mathematical biology to model the

process of mate search (Janetos 1980, Wiegmann et al. 2010). Search is becoming increasingly

important in modern electronic marketplaces, both because of the lowering of search costs, as well

as the ability of consumers to turn to price search intermediaries, often artificial agents, to find

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the best prices or values for items they are interested in acquiring (Bakos 1997, Brynjolfsson and

Smith 2000, Haynes and Thompson 2008).

The decision making complexity in one-sided search arises from the fact that there is a cost

incurred in finding out the true value of any opportunity encountered (Bakos 1997, Kephart et al.

2000). For example, there is a cost to arranging a meeting to test drive a car you are considering

purchasing. The searcher needs to trade off the potential benefit of continuing to search and seeing

a possibly more valuable opportunity with the costs incurred in doing so. The optimal stopping

rule for such search problems has been widely studied, and is often a reservation strategy, where

the searcher should terminate search once she encounters an opportunity which has a value above

a certain reservation value or threshold (Weitzman 1979, McMillan and Rothschild 1994). Most

models assume that upon encountering an opportunity the searcher obtains its exact true value.

However, in many realistic settings, search is inherently noisy and searchers may only obtain a noisy

signal of the true value. For example, the drivetrain of a used car may not be in good condition,

even if the body of the car looks terrific. The relaxation of the assumption of perfect values not

only changes the optimal strategy for a searcher, it also leads to a niche in the marketplace for

new knowledge-brokers. The knowledge brokers, or experts, are service providers whose main role

is to inform consumers or searchers about the values of opportunities.

An expert offers the searcher the option to obtain a more precise estimate of the value of an

opportunity in question, in exchange for the payment of a fee (which covers the cost of providing

the service as well as the profit of the expert). To continue with the used-car example, when the

agent is intrigued by a particular car and wants to learn more about it, she could take the car

to a mechanic who could investigate the car in more detail, or make a repeat visit to see the car,

possibly bringing an experienced friend for a more thorough examination. In essence, the searcher

obtains more accurate information for an additional cost (either monetary or equivalent).

Sequential-search markets with experts are particularly interesting in the context of modern

electronic marketplaces for several reasons. First, there is now a proliferation of not just sellers, as

mentioned above, but competing marketplaces that attempt to match buyers and sellers (ranging

from highly organized marketplaces like autotrader.com to bulletin boards like craigslist.com),

and design choices affect how these marketplaces perform and serve the needs of customers. Second,

the transition to electronic marketplaces has greatly lowered search costs (the monetary, time,

and opportunity costs of examining new possibilities), which significantly affects market dynamics.

Third, this transition has led to the emergence of new kinds of experts; for example, independent

agencies like Carfax now monitor the recorded history of transactions, repairs, claims, etc. on cars,

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and can, to a certain extent, replace traditional experts like mechanics, performing similar services

at a much lower marginal cost of production of an expert report. Fourth, many decisions, including

pricing decisions and decisions on when to terminate search, can be taken by automated agents

operating in the best interests of different users.

In this paper we investigate optimal search and market design in environments where searchers

observe noisy signals and can obtain (i.e., query the expert for) the actual values for a fee. There

has been some previous work on search with noisy signals in the theoretical biology literature, in

the context of mate search (Wiegmann et al. 2010). However, this existing work only considers

restricted strategy spaces (an opportunity cannot be accepted by the searcher until its exact true

value is known; the option of accepting an uncertain offer without paying an additional revelation

cost is ignored), and also models the cost of revelation of actual values as fixed, rather than as

set by a strategic, self-interested agent. In contrast, we formulate the problem as a Stackelberg

game, where the expert makes the first move by setting the terms for its services (e.g., price) and

the searcher responds. By characterizing optimal strategies for searchers, we can then solve for

the price that maximizes the expert’s profit. We can use this analysis to understand the effects of

different market interventions, for example, subsidizing the purchase of expert services.

In the real world, experts may also sell their services in bundles, or as contracts over periods

of time. Motivated by this observation, we extend our framework so that we can compute the

optimal bundle size and price for an expert to offer, given different features of the marketplace.

There is a significant literature on bundling of goods and services ranging from daily necessities at

supermarkets to digital information goods (Hanson and Martin 1990, Bakos and Brynjolfsson 1999,

2000). However, research has typically focused on the bundling of complementary and substitute

goods. Many studies have looked at profitability and discounts offered in bundling services that

use common infrastructure, like phone, cable and internet service, or complementary services like

flights, hotels and rental cars. In the context of expert-mediated search, bundling has a somewhat

different meaning. In such markets bundling means that, for a predefined fee, the searcher can use

the expert services some fixed number of times. Such schemes are common in physical and virtual

markets. For example, as of this writing, Carfax offers a bundle of five reports for $44.99 (alongside

the option to buy a single report for $34.99).

Contributions: We consider a model of one-sided search where agents have access to noisy signals

and the option of consulting an expert. Our main contributions are as follows: (1) We show that

under a standard stochastic dominance assumption on the signal structure (that higher signals are

“good news” (Milgrom 1981)), the optimal search rule is characterized by a “double reservation

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value” strategy, wherein the searcher rejects all signals below a certain threshold, resuming search,

and accepts all signals above another threshold, thus terminating search, without querying the

expert; the agent queries the expert for all signals that are between the two thresholds. (2) We show

that in the presence of a monopolist expert, social welfare can be improved by subsidizing access to

expert services. We prove that, in our model, social welfare is maximized when the expert’s service

is subsidized to a level where its cost to the searcher is exactly the marginal cost the expert incurs

in producing an expert report (in the case of many digital/information services this cost is small).

In many electronic markets there exist both natural monopolies in information as well as special

relationships between marketplaces and experts (consider, for example, AutoTrader and Carfax).

Our result suggests that marketplaces may be able to compete more effectively for consumers by

improving their overall experience using a subsidization mechanism; the marketplace could then

leverage an increased consumer base in several ways including for advertising revenue or through

charging transaction fees. (3) We characterize optimal searcher and expert behavior in settings

where the expert offers service bundles. Surprisingly, we find that not only can bundling be rational

for the expert, it can even be pareto-improving for the expert and the searcher. We also show that

the key features of electronic marketplaces, namely lower search costs for consumers and lower

costs of production of expert services, both favor bundling.

Along with the general theory, we illustrate our results throughout in a specific, plausible distri-

butional model, in which the true value is always bounded by the signal value, and the probability

monotonically decays as the discrepancy between the two increases.

2. The Model

The standard one-sided search problem (McMillan and Rothschild 1994) considers an agent or

searcher facing an infinite stream of opportunities from which she needs to choose one. While the

specific value v of each future opportunity is unknown to the searcher, she is acquainted with the

(stationary) probability distribution function from which opportunities are drawn, denoted fv(x)

(x ∈ R). The searcher can learn the value of an opportunity for a cost cs (either monetary or in

terms of resources that need to be consumed for this purpose) and her goal is to maximize her total

utility, defined as the value of the opportunity eventually picked minus the overall cost incurred

during the search. Having no a priori information about any specific opportunity, the searcher

reviews the opportunities she encounters sequentially and sets her strategy as the mapping from

the value received to the set {terminate, resume}, i.e., a stopping rule; the rule specifies when to

terminate and when to resume search, based on the opportunities encountered.

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One can relax the standard assumption that the searcher receives the exact true value of an

opportunity. Instead, we assume that the searcher receives, at cost cs, a noisy signal s, correlated

with the true value according to a known probability density function fs(s|v). In addition, the

searcher may query and obtain from a third party (the expert) the true value v of an opportunity,

for which signal s was received, by paying an additional fee ce. In this case, the searcher’s strategy is

the mapping from a signal received to the set {terminate, resume, query}. The goal of the searcher

is to maximize the total utility received i.e., the expected value of the opportunity eventually picked

minus the expected cost of search and expert fees paid along the way.

We can generalize the model above further by allowing experts to offer their services in packages

(“bundles”). In this case, for a cost cke , the searcher obtains, upon purchasing a bundle, the right

to use the expert’s service k times along its search path with no additional cost. The searcher does

not necessarily need to make use of all the k queries, and similarly, if she uses them all she can

purchase additional bundles as required. The goal of the searcher is to find an optimal strategy as

before (mapping from a signal received to {terminate, resume, query}). When choosing to query

the expert the agent needs to pay a cost cke if she has not purchased a bundle yet or if she has

already used the expert’s services k times since the last bundle was purchased.

The first question that arises in the different model variants is how to characterize the optimal

strategy for the searcher, given the query cost ce set by the expert. A second question is how the

expert sets her service fee ce or bundling terms (k, cke) in order to maximize her (per-searcher)

expected profit, denoted πe. In this paper we consider a monopolist provider of expert services.

The searcher’s optimal strategy is directly influenced by the service terms, and therefore implicitly

determines the expected number of times the services of the expert are required, and thus the

expert’s revenue. The problem can be thought of as a Stackelberg game where the expert is the first

mover, and wants to maximize her profits with respect to the service terms she sets to searchers.

We assume that the provider of expert services has already performed the “startup work” nec-

essary, and only pays a marginal cost de per query. The expert’s per-searcher profit πe is thus a

function of the expected number of queries purchased by the searcher, denoted ηce , the cost of the

service ce and its cost of producing the service de. Formally, πe = (ce − de)ηce . When the expert

offers her services in bundles, we need to distinguish between the expected number of bundles

purchased by the searcher, denoted ηb, and the expected number of queries used by the searcher,

denoted ηcke (ηcke ≤ ηbk). The expert’s expected profit in this case is given by: πe = ckeηb− ηckede.

Assuming exogeneity of opportunities, social welfare, denoted W , is a function of the expected

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utility of searchers and the expected profit of the expert.1 We assume that social welfare is additive.

Since the whole process scales up linearly in the number of searchers, we can simply consider the

interactions involving a single searcher and the expert. The social welfare is then the sum of the

expected total utility to the searcher and the expected profit of the expert.

3. Optimal Policies

In this section, we derive optimal strategies for searchers and for monopolist experts. We begin

by analyzing environments without bundling. This serves as an important special case and also

allows us to build intuition that we can then extend to solve the more general case where experts

sell bundles of services. We first characterize the searcher’s optimal strategy and her expected use

of the expert’s services, given the fee ce set by the expert. From this we can derive the expert’s

expected profit as a function of the fee she sets, enabling maximization of the expert’s revenue. In

the last part of the section, we extend the analysis to bundles of arbitrary size.

3.1. The searcher’s optimal strategy

One-Sided Search: The optimal search strategy for cases where the actual value of an opportunity

can be obtained at cost cs can be found in the extensive literature of search theory (McMillan

and Rothschild 1994, McCall 1970). In this case, the searcher follows a reservation-value rule: she

reviews opportunities sequentially (in random order) and terminates the search once a value greater

than a reservation value x∗ is revealed, where the reservation value x∗ satisfies:

cs =

∫ ∞y=x∗

(y−x∗)fv(y)dy (1)

Intuitively, x∗ is the value where the searcher is precisely indifferent: the expected marginal utility

from continuing search and obtaining the value of the next opportunity exactly equals the cost

of obtaining that additional value. The reservation property of the optimal strategy derives from

the stationarity of the problem – since the searcher is not limited by the number of opportunities

it can explore, resuming search places her at the same position as at the beginning of the search

(McMillan and Rothschild 1994). Consequently, a searcher that follows a reservation value strategy

will never decide to accept an opportunity she has once rejected, and the optimal search strategy is

the same whether or not recall is permitted. The expected gain to the searcher from following the

optimal strategy in this case is x∗. The expected number of search iterations is simply the inverse of

the success probability, 11−Fv(x∗) , since this becomes a Bernoulli sampling process, as opportunities

arise independently at each iteration.

1 We abstract away from modeling the existence of “sellers” of opportunities, instead viewing them as exogenous, orelse as being offered at some “fair price” by the seller.

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One-Sided Search with Noisy Signals: When the searcher receives a noisy signal rather than the

actual value of an opportunity, there is no guarantee that the optimal strategy is reservation-value

based as it is in the case where values obtained are certain. The stationarity of the problem,

however, still holds, and an opportunity that has been rejected will never be recalled. In the absence

of any restriction over fs(s|v), the optimal strategy is based on a set S of signal-value intervals for

which the searcher terminates the search. The expected utility of the search in this case, denoted

V (S), is given by:

V (S) =−cs +V (S)

∫s/∈S

fs(s)ds+

∫s∈S

fs(s)E[v|s]ds (2)

The fact that the optimal strategy may not be reservation-value based in this case is because

there may be no systematic correlation between the signal and the true value of the opportunity.

Nevertheless, in most real-life cases, there is a natural correlation between signals and true values.

In particular, a fairly weak and commonly used restriction on the conditional distribution of the

true value given the signal goes a long way towards allowing us to recapture a simple space of

optimal strategies. This is the restriction that higher signal values are “good news” in the sense

that when s1 > s2, the conditional distribution of v given s1 first-order stochastically dominates

that of v given s2 (Wright 1986, Milgrom 1981). The condition requires that given two signals s1

and s2 where s1 > s2, the probability that the actual value is greater than any particular value v

is greater for the case where the searcher received signal s1. Formally:

Definition 1. Higher signals are good news (HSGN) assumption: If s1 > s2, then,

∀y,Fv(y|s1)≤ Fv(y|s2), where Fv(y|s) is the cumulative distribution function (cdf) of values given

the signal.

Corollary 1. For noisy environments satisfying the HSGN assumption, if s1 > s2, then,

E[v|s1]≥E[v|s2].

Proof: This is a standard result that is the core of the definition of first-order stochastic dominance.

It can be derived easily through integration by parts:

E[v|s1]−E[v|s2] =

∫ ∞y=−∞

y(fv(y|s1)− fv(y|s2))dy=[y(Fv(y|s1)−Fv(y|s2))

]∞y=−∞

−∫ ∞y=−∞

(Fv(y|s1)−Fv(y|s2))dy=

∫ ∞y=−∞

(Fv(y|s2)−Fv(y|s1))dy > 0 �

Where fv(y|s) and FV (y|s) are the density and cumulative distribution functions, respectively, of

the true value conditional on the signal received. This enables us to prove the following theorem.

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Theorem 1. For any probability distribution function fv(v|s) satisfying the HSGN assumption:

(a) the optimal search strategy is a reservation-value rule, where the reservation value, t∗, satisfies:

cs =

∫ ∞s=t∗

(E[v|s]−E[v|t∗]

)fs(s)ds (3)

(b) the expected utility of using the optimal search strategy satisfies: V (t∗) =E[v|t∗]

Proof: This proof uses standard techniques from dynamic programming and search theory; we

defer the details to Appendix A. Note that the expected number of search iterations is 11−Fs(t∗) ,

since this is again a Bernoulli sampling process.

The Expert Option: The introduction of an expert extends the number of decision alternatives

available to the searcher. When receiving a noisy signal of the true value, she can choose to (1)

reject the offer without querying the expert, paying search cost cs to reveal the signal for the next

offer; (2) query the expert to obtain the true value, paying a cost ce, and then decide whether to

resume search or not, after the true value v is revealed; or (3) accept the offer without querying

the expert, receiving the (unknown) true value of the offer.

As in the no-expert case, a solution for a general density function fv(v|s) dictates an optimal

strategy with a complex structure of the form of (S′, S′′, V ), where: (a) S′ is a set of signal intervals

for which the searcher should resume her search without querying the expert; (b) S′′ is a set of

signal intervals for which the searcher should terminate her search without querying the expert

(and pick the opportunity associated with this signal); and (c) for any signal that is not in S′ or

S′′ the searcher should query the expert, and terminate the search if the value obtained is above

a threshold V , and resume otherwise. V is the expected utility from resuming the search and is

given by the following modification of Equation 2:

V (S′, S′′, V ) =− cs +V (S′, S′′, V )

∫s∈S′

fs(s)ds+

∫s∈S′′

fs(s)E[v|s]ds

+

∫s 6∈{S′,S′′}

fs(s)(V (S′, S′′, V )Fv(V |s) +

∫ ∞y=V

yfv(y|s)dy− ce)ds (4)

The first element after the search cost on the right hand side applies to resuming search, in which

case the searcher continues with an expected utility V (S′, S′′, V ). The second element applies to

terminating search without querying the expert, in which case the expected utility is E[v|s]. The

remaining term applies to querying the expert; based on the value revealed by the expert, the

search is either: (a) resumed, with an expected utility V (S′, S′′, V ); or (b) terminated, with utility

equal to the revealed value. We can show that under the HSGN assumption, each of the sets S′

and S′′ actually contains a single interval of signals, as illustrated in Figure 1.

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Figure 1 Characterization of the optimal strategy for noisy search with an expert under HSGN assumption. The

searcher queries the expert if s∈ [tl, tu] and accepts the offer if the worth is greater than the expected

utility of resuming the search V . The searcher rejects and resumes search if s < tl and accepts and

terminates search if s > tu, both without querying the expert.

Theorem 2. For fv(y|s) satisfying the HSGN assumption (Definition 1), the optimal search

strategy can be described by the tuple (tl, tu, V ), where: (a) tl is a signal threshold below which the

search should be resumed; (b) tu is a signal threshold above which the search should be terminated

and the current opportunity picked; and (c) the expert should be queried given any signal tl < s< tu

and the opportunity should be accepted (and search terminated) if the value obtained from the expert

is above the expected utility of resuming the search, V , otherwise search should resume (see Figure

1). The values tl, tu and V can be calculated from solving the set of Equations 5-7:

V =− cs +V Fs(tl) +

∫ ∞s=tu

fs(s)E[v|s]ds+

∫ tu

s=tl

fs(s)

(V Fv(V |s) +

∫ ∞y=V

yfv(y|s)dy− ce)ds

(5)

ce =

∫ ∞y=V

(y−V )fv(y|tl)dy (6)

ce =

∫ V

y=−∞(V − y)fv(y|tu)dy (7)

Proof: See Appendix A.

The equations that characterize the searcher’s optimal strategy can also be derived from her

indifference conditions. The cost of purchasing the expert’s services must equal two different things:

(1) The expected increase in utility from consulting the expert when you would otherwise reject and

resume search (the condition for tl). tl is the point at which a searcher is indifferent between either

resuming the search or querying the expert: V =∫∞y=V

yfv(y|tl)dy + V Fv(V |tl)− ce, which trans-

forms into Equation 6. (2) The expected increase in utility from consulting the expert when you

would otherwise accept the offer, terminating search (the condition for tu). The searcher is indiffer-

ent between either buying without querying the expert or purchasing a query:∫∞y=V

yfv(y|tu)dy+

V Fv(V |tu)− ce =∫∞y=−∞ yfv(y|tu)dy, which transforms into Equation 7.

There is also a reasonable degenerate case where tl = tu(= t). This happens when the cost of

querying is so high that it never makes sense to engage the expert’s services. In this case, a direct

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indifference constraint exists at the threshold t, where accepting the offer yields the same expected

utility as continuing search, so V = E[v|t]. This can be solved in combination with Equation 3,

since there are now only two relevant variables.

3.2. Profit Maximization From the Expert’s Perspective

A monopolist expert moves first in the implied Stackelberg game by setting her query cost ce. The

searcher responds by following her optimal search strategy described above. Therefore, the expert

can solve for the searcher’s behavior, given knowledge of the search cost cs and the signal and value

distributions. The optimal fee from the expert’s perspective is thus the one that maximizes her

overall profit, defined as the product of the expected number of times her services are used by the

searcher, and the profit she makes per query.

Expected number of queries: The search strategy (tl, tu, V ) defines how many times the

expert’s services are used. Let Pj denote the probability that the searcher queries the expert exactly

j times along its search, until terminating, then Pj = Aj

(1−B)j(DA

+ C1−B ) where

A=Pr(tl ≤ s≤ tu ∧ v < V ) =

∫ y=tu

y=tl

∫ x=V

x=−∞fv(x|y)dxfs(y)dy (8)

B =Pr(s < tl) = Fs(tl) (9)

C =Pr(s > tu) = 1−Fs(tu) (10)

D=Pr(tl ≤ s≤ tu ∧ v≥ V ) =

∫ y=tu

y=tl

∫ x=∞

x=V

fv(x|y)dxfs(y)dy (11)

Since on each search iteration along the search the searcher queries the expert if the signal is

between tl and tu, the expected overall number of queries made is given by:

ηce = Pr(Searcher queries an expert) ∗ ηs

The value of ηs can be calculated by ηs = 1C+D

(see Appendix B.1) and the probability that the

searcher queries the expert at any search instance is given by A+D. Therefore:

ηce =A+D

C +D(12)

Expected profit of the expert: As formulated in Section 2, the expected profit of the expert

is: πe = E(Profit) = (ce− de)ηce . The expert can maximize the above expression with respect to ce

(ηce decreases as ce increases) to find the profit maximizing price to charge searchers.

3.3. Bundling

When having the option to offer bundles of queries, the expert needs to set both the bundle size

and its price. As in the a la carte model above, we begin by characterizing the optimal strategy of

the searcher given the expert’s choice of the bundle terms.

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Figure 2 MDP representation of the searcher’s problem when the expert offers bundles of her services (state

variable is the number of queries the searcher has remaining).

3.3.1. Characterizing Optimal Search Since the searcher cannot recall previous opportu-

nities, her state depends only on the number of remaining pre-paid queries, denoted γ. The system

can thus be modeled as a Markov Decision Process with k search states (γ = 0,1,2, . . . , k− 1) and

one termination state, as illustrated in Figure 2. Let Vγ denote the expected value-to-go of following

the optimal search strategy starting from state γ. The process begins when the searcher is in state

γ = 0. In this state, upon receiving a signal s, the searcher can either: (a) reject the current oppor-

tunity and continue search, starting from the same state γ; (b) accept the current opportunity

and terminate search; (c) query the expert for the true value of the current opportunity, incurring

a cost cke , and, based on the value received, either accept the current opportunity (terminating

the search) or reject the current opportunity and continue search from state k− 1. When in state

γ > 0, the searcher has the same options when receiving a signal s, except that under the choice of

querying the expert the agent does incur any cost, and if, based on the value received, she decides

to continue the search, it is resumed from state γ− 1.

While terminating without querying the expert is a legitimate option in any states γ > 0, we can

show that it is never preferred.

Lemma 1. When in state γ > 0, querying the expert dominates terminating search without query-

ing the expert.

F or any strategy that terminates search upon obtaining a signal s when in state γ > 0, consider

instead a modification of that strategy which queries the expert, and terminates only if the true

value, revealed by the expert, is greater than Vγ−1. This new strategy dominates, since if the true

value is less than Vγ−1 the searcher is better off resuming search, and she pays no marginal cost to

obtain the expert’s service in this instance. �

For any signal s received in state γ > 0, the expected utility if querying the expert, denoted by

M(s,Vγ−1), is given by: M(s,Vγ−1) =∫∞−∞max(x,Vγ−1)fv(x|s)dx. Therefore, the optimal strategy

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is to query the expert if M(s,Vγ−1)>Vγ . The expected utility of using the optimal strategy, when

starting from state γ > 0, is thus given by:

Vγ>0 =−cs +Vγ

∫M(s,Vγ−1)<Vγ

fs(s)ds+

∫M(s,Vγ−1)>Vγ

fs(s)M(s,Vγ−1)ds (13)

Similarly, when in state γ = 0, the expected utility when obtaining a signal s is: (a) E[v|s] if

terminating the search without querying; (b) V0 if resuming the search without querying the expert;

and (c) M(s,Vk−1) − cke on querying the expert. For any signal s, the choice which yields the

maximum among the three should be made. Let Λ = max(E[v|s], V0,M(s,Vk−1)− cke) and let ζ1,

ζ2 and ζ3 define the sets of signal support such that Λ equals V0, E[v|s], and M(s,Vk−1) − cke ,

respectively. The expected utility is then:

V0 =−cs +V0

∫ζ1

fs(s)ds+

∫ζ2

E[v|s]fs(s)ds+

∫ζ3

fs(s)(M(s,Vk−1)− cke

)ds (14)

The optimal search strategy can be obtained by solving the set of k − 1 equation instances of

Equation 13 (for 1 ≤ γ < k) in addition to Equation 14. An important property of the optimal

strategy is given in the following Lemma.

Lemma 2. When using the optimal strategy, Vγ ≤ Vγ+1 ∀ 0≤ γ < k− 1.

S uppose Vγ > Vγ+1. A searcher starting from state γ + 1 can follow the optimal strategy as

if starting from γ. In this case it will end up with the same value v while the accumulated cost

will be at worst equal and possibly less (in the case where search terminates after the last bundle

purchase) than the accumulated cost when starting the search from state γ; therefore the original

strategy could not have been optimal. �

The HSGN case: While the structure of the optimal strategy depends strongly on fv(y|s), for

some cases, such as under the HSGN assumption discussed above, the optimal strategy can have

a simple representation in the form of reservation values.

Theorem 3. For fv(y|s) satisfying the HSGN assumption the optimal strategy for a searcher

can be described as (see Figure 3):

(1) a tuple (tl, tu, Vk−1), corresponding to state γ = 0, such that for any signal s: (a) the search

should resume if s < tl; (b) the opportunity should be accepted if s > tu; and (c) the expert should

be queried if tl < s< tu and the opportunity accepted (and search terminated) if the value obtained

from the expert is above the expected utility of resuming the search, Vk−1, otherwise search should

resume; and

(2) a set of (k − 1) tuples (tγ , Vγ−1) corresponding to states γ ∈ 1,2 . . . , k − 1 such that: (a) the

13

(a) For γ = 0 (b) For γ > 0Figure 3 Characterization of the optimal strategy for noisy search with an expert offering bundles. For state 0

(left figure), the searcher queries the expert if s∈ [tl, tu] and terminates search if the worth is greater

than the utility of resuming search Vk−1. The searcher resumes search if s < tl and terminates without

querying the expert if s > tu. However, for any state γ > 0 (right figure), the searcher rejects the

opportunity (terminating search) if s < tγ and otherwise queries the expert before deciding.

search should resume if s < tγ; and (b) the expert should be queried if s > tγ and the opportunity

accepted if the value obtained from the expert is above the expected utility of resuming the search,

Vγ−1, otherwise search should resume.

Proof: See Appendix A.

Based on Theorem 3, we can construct the appropriate modifications of Equations 13 and 14 for

the HSGN case:

Vγ>0 = VγFs(tγ) +

∫ ∞s=tγ

fs(s)(∫ ∞

y=Vγ−1

yfv(y|s)dy+Vγ−1Fv(Vγ−1|s))ds− cs (15)

V0 = V0Fs(tl) +

∫ ∞s=tu

fs(s)E[v|s]ds+

∫ tu

s=tl

fs(s)(Vk−1Fv(Vk−1|s) +

∫ ∞y=Vk−1

yfv(y|s)dy− cke)ds− cs (16)

The values (tl, tu) and tγ ∀γ > 0 to be used in the optimal strategy are obtained by setting the

derivative of Equation 16 w.r.t tl and tu (separately) and that of Equation 15 w.r.t tγ ∀γ > 0 to

zero, resulting in (after integration by parts):

cke =

∫ ∞y=Vk−1

(y−Vk−1)fv(y|tl)dy (17)

cke =

∫ Vk−1

y=−∞(Vk−1− y)fv(x|tu)dy (18)

Vγ = Vγ−1Fv(Vγ−1|tγ) +

∫ ∞y=Vγ−1

yfv(y|tγ)dy (19)

Similar to the case of buying the expert’s services one at a time, Equations 17-19 can intuitively

be interpreted as describing the indifference values of the searcher. Equation 17 characterizes the

intuition that, at s= tl, the searcher is indifferent between either resuming the search or querying

the expert. Equation 18 means that, at s= tu, the searcher is indifferent between either terminating

the search or querying the expert. Finally, s= tγ in Equation 19 is the signal where the expected

14

gain from rejecting the opportunity with which it is associated is equal to the expected gain received

by deciding after querying the expert.

Based on Equations 15-19 we can construct a set of 2k + 1 equations from which the optimal

search strategy can be extracted. These are Equations 16, 17, and 18 for state γ = 0 and the 2k−2

equations resulting from Equations 15 and 19 for γ = 1,2 · · ·k− 1. We can solve the above system

of equations to calculate the equilibrium of the Stackelberg game. The searcher’s utility in this

case is given by V0 because the searcher starts from state 0, i.e., with no queries in hand.

3.3.2. The Expert’s Perspective We now turn to finding the expert’s profit as a function

of the bundle characteristics (k, cke) she sets. In the case of bundles of size one, we showed how to

compute this by computing the expected number of queries purchased when a particular price is

set. We will again need to compute this, but now based on both bundle characteristics k and cke . As

discussed in Section 2, we need to account for the fact that not all queries in a bundle may end up

being used, since the searcher may terminate search in some state γ 6= 0. Thus the expert’s profit

is πe = ckeηb− ηckede, where ηb is the expected number of bundles purchased and ηcke is the expected

number of queries used (de is again the marginal cost to the expert of producing an additional

report). Given a way of computing ηb and ηcke , the profit-maximizing bundle characteristics can be

determined.

Expected number of bundles purchased (ηb): This can be computed as the expected number of

times the searcher queries the expert when in state γ = 0 (either transitioning to state γ = k− 1

after, or terminating the search) (see Figure 2). Details are in Appendix B.2.

Expected number of queries used (ηcke ): The searcher may terminate search before exhausting all

the remaining queries purchased. This situation is unique to settings where the buyer (searcher

in our case) is buying the right to use a service rather than actually receiving the service (or a

product) upon purchase. For an expert offering a bundle (k, cke), the question of how many queries

purchased are actually used is important when de > 0. Details on how this can be computed are in

Appendix B.2.

A Subscription Model: Infinite Bundle Sizes A specific interesting case to look at is whether the

expert would ever want to sell an unlimited supply of services at a fixed price. This is equivalent to

an infinite-sized bundle. In this case the model reduces to two states. The searcher starts in state

γ = 0 and continues in this state until she either terminates search or transitions to state γ =∞.

In the latter case the searcher can keep querying the expert for any reasonable opportunity until

she finally finds a sufficiently good opportunity and terminates search (the existence of a search

cost ensures that this querying process does not go on forever). Being in state γ =∞ is equivalent

15

to being in the world of perfect signals. The optimal reservation value when in state γ =∞ can

thus be extracted from (e.g., (McMillan and Rothschild 1994)):

V∞ =−cs +V∞Fv(V∞) +

∫ ∞y=V∞

yfv(y)dy (20)

For state γ = 0, we can use appropriate modifications of Equations 16-18, replacing Vk−1 with V∞

(realizing that the searcher transitions to state γ =∞). The optimal strategy can be extracted

from the solution of this set of four equations.

4. Market Design

In this section, we describe possible uses of the theory described above to improve the design of

markets in which such search takes place. The prospect of designing or significantly influencing these

markets is not remote. Consider the design of a large scale Internet website like AutoTrader.com.

The listings for cars that users see are signals, and users may be unsure of a car’s true worth.

AutoTrader can partner with a provider of reports like Carfax, to make it easy for users to look up a

car’s worth. In order to be general, let us refer to AutoTrader as “the market” (or in some instances

as “the market designer”) and Carfax as the expert. The market wants to attract customers to it,

rather than to rival markets. The best way of doing this is to provide customers with a high value

shopping experience. The expert wishes to maximize its profits. Since the market and the expert

both have significant power, it is reasonable to imagine them coming up with different models of

the kinds of relationships they may have.

We focus here on one particular kind of market intervention, namely subsidization of expert

queries. In order to provide customers with the highest value shopping experience, the market

may choose to subsidize the cost of expert services. We consider a market designer who makes

payments to the expert in exchange for the expert providing services to users at a lower cost. A

typical problem with subsidization is that it often decreases social welfare because the true cost

of whatever is being subsidized is hidden from the consumer, leading to overconsumption of the

resource. In this instance, however, the natural existence of many monopolies in expert services,

combined with the existence of search frictions, make it quite possible that subsidies will in fact

increase social welfare.

We first define social welfare and then describe the effects of subsidization on social welfare. For

simplicity of exposition, and because it captures all the relevant intuition, we focus this discussion

on a setting without bundling. It is also worth noting that while we are thinking about private

markets here, this entire discussion is equally relevant to a big player like the government as market

designer or regulator, and independent providers of expert services.

16

4.1. Subsidization Model

Suppose a monopolist provider of expert services maximizes her profits by setting the query cost to

c∗e, yielding an expected profit πe = (c∗e − de)ηc∗e (this discussion is again on a per-consumer basis).

The market designer can step in and negotiate a reduction of the fee ce charged by the expert

to a value c′e, for the benefit of the searchers. In return for the expert’s agreement, the market

designer can offer a per-consumer payment β to the expert, which fully compensates the expert for

the decreased revenue, leaving her total profit unchanged. Since c′e < c∗e, ηc′e > ηc∗e (the consumer

queries more often because she has to pay less). The compensation for a requested decrease in the

expert’s fee from c∗e to c′e is thus β = (c∗e−de)ηc∗e − (c

′e−de)ηc′e . The overall welfare per agent in this

case increases by Vc′e − Vc∗e , where Vc′e and Vc∗e are the expected utilities of searchers according to

Equation 5-7, when the expert uses a fee c′e and c∗e respectively, at a cost β to the market designer.

The social welfare is given by the sum of utilities of all parties involved. Thus far, we have just

considered two: the searcher and the expert (of course this generalizes to multiple searchers as well,

since each search process would be independent). Then:

W = Vc∗e +πe (21)

where πe is the expert’s maximum profit, c∗e is the fee that maximizes the expert’s profit and V

is the searcher’s overall utility. When the market designer enters the picture by subsidizing expert

queries, the social welfare must also take the subsidy into account.

Since the expert is fully compensated for her loss due to the decrease in her fee, the change in

the overall social welfare is Vc′e −Vc∗e −β. Under the new pricing scheme c′e, and given the subsidy

β, the social welfare is given by W′= Vc′e +πe−β.

4.2. Welfare-Maximizing Subsidies

The first important question that arises is what level the market designer should set the subsidy

at in order to maximize social welfare. Here we prove that, in the model of subsidization described

above, social welfare is maximized at the point where the searcher pays exactly de per query, thus

fully internalizing the cost to the expert of producing the service. If the searcher had to pay less, it

would lead to inefficient overconsumption of expert services, whereas if she had to pay more, the

expected decline in the utility she receives from participating in the search process would outweigh

the savings to the market designer or government from having to pay less subsidy.

Theorem 4. Suppose the market designer pays the expert a flat subsidy β (per-customer) in

exchange for the expert reducing her per-query fee from c∗e to c′e such that β = (c∗e − de)ηc∗e − (c

′e−

17

de)ηc′e where de is the marginal cost of producing the service and ηce is the expected number of

queries to the expert when the expert fee is set to ce. Then social welfare, defined as the sum of a

searcher’s expected utility and the expert’s expected (maximum) profit, minus the subsidy payed to

the expert by the market designer, W′= Vc′e +πe−β, is maximized when β is set such that c

′e = de.

Proof: The searcher acts to maximize her utility. For convenience, we mildly abuse notation and

assume that E[ω], ηs() and ηce() are functions defined by the searcher’s optimal behavior from

solving the Bellman equation. Here E[ω] is the expected value of the final opportunity that is taken

(which is different from the expected utility of the searcher since it does not factor in costs), ηs()

is the expected number of opportunities examined by the searcher, analogous to ηs, and ηce(),

analogous to ηce , the expected number of times the expert is queried (although tangential to this

proof, details on computing these quantities can be found in Section 3.2 and Appendix B.1). Then

W =E[ω]− ηs(ce)cs− ηcece + ηce(ce− de) =E[ω]− ηs(ce)cs− ηcede (22)

We claim that social welfare is maximized when ce = de. We show this by appealing to the

optimal solution of a single searcher’s decision problem. The functions E[ω], ηs() and ηce() in this

case are such that the expression in Equation 23 is maximized.

E[ω]− ηs(de)cs− ηcede (23)

Suppose there exists some ce′ 6= de that maximizes the social utility (Equation 22). Let us now

define an alternative strategy for the single searcher above. The searcher pretends that the cost of

querying the expert is ce′ and follows the strategy given by that belief. Then the expected utility of

the searcher can be found by assuming the searcher pays cost ce′ but receives a “kickback” of ce′−deevery time. The expected utility of the searcher is then (E[ω]− ηs(ce′)cs− ηcece′) + ηce(ce′ − de).

By the Bellman Optimality Principle, this cannot be greater than Equation 23. Therefore, such a

ce′ 6= de does not exist. In addition, note that this is actually just the right hand side of Equation

22. Therefore, it must be the case that Equation 22 is maximized at ce = de. �

We demonstrate the empirical effects of subsidization in Section 5 for some natural distributions.

Our empirical results confirm the theory developed above and also provide some intuition for the

quantitative size of the effect. Another interesting implication of this theorem applies to the special

case when de = 0. We can think of this case as “digital services,” analogous to digital goods like

music MP3s – producing an extra one of these has zero marginal cost. Similarly, producing an extra

electronic history of a car, like a Carfax report, assuming it exists in the Carfax database, can be

considered to have zero marginal cost. In this case, there is no societal cost to higher utilization of

18

the expert’s services, so subsidy is welfare improving right up to making the service free. These are

the cases where it could make sense for the market designer or government to take over offering the

service themselves, and making it free, potentially leveraging the increased welfare of consumers

by attracting more consumers to their market, or increasing their fees.

5. A Specific Example

Equilibrium in expert-mediated search derives from a complex set of dynamics in the system. The

number of parameters affecting the equilibrium is substantial: the distribution of values, the corre-

lation between signals and values, search frictions, the cost of querying the expert, and (in the case

of bundling) bundle size, all affect the overall outcome of the process. Uncovering phenomenological

properties of the model is therefore difficult and restricted using a static analysis. Instead, we turn

to a specific example to outline some interesting effects of expert mediation in this domain. The

example is based on a particular, plausible distribution of signals and values. This case illustrates

the general structure of the solutions of the model and demonstrates how interventions by the

market designer can increase social welfare.

We consider a case where the signal is an upper bound on the true value. Going back to the

used car example, sellers and dealers offering cars for sale usually make cosmetic improvements to

the cars in question, and proceed to advertise them in the most appealing manner possible, hiding

defects using temporary fixes. Specifically, we assume signals s are uniformly distributed on [0,1],

and the conditional density of true values is linear on [0, s]. Thus

fs(s) =

{1 if 0< s< 1

0 otherwisefv(y|s) =

{2ys2

for 0≤ y≤ s0 otherwise

We can use these distributions to solve the systems of equations as described in detail in Section

3. Solving these leads to several interesting insights on the effects of expert-mediation and bundling

of services in such settings.

5.1. Expert Mediation

A solution to the expert-mediated problem for the setting described above can be obtained by

substituting the specific distribution in Equations 5 through 7 and simplifying:

V =−cs +V tl− ce(tu− tl) +V 3(tu− tl)

3tutl+

1− t2l3

(from Equation 5)

ce =2tl3

+V 3

3t2l−V (from Equation 6)

ce =

∫ V

0

(V − y)fv(y|tu)dy=V 3

3t2u(from Equation 7)

19

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.4

0.5

0.6

0.7

0.8

0.9

1

cs −>

Sig

na

l −

>

Signal s vs. Search cost cs for c

e = 0.05

tl

tu

Query Expert

Accept and terminate

Reject and resume

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

cs −>

V −

>

Agent utility V vs. Search cost cs

ce = 0.03

ce = 0.06

ce =0.12

No Expert

(a) Signal threshold vs cs (b) Variation of agent utility w.r.t cs and ceFigure 4 Effect of cs on the signal thresholds (tl,tu) and agent utility V .

We can find feasible solutions of this system for different parameter values, as long as the condition

tl < tu holds. Otherwise, when ce is high enough that querying never makes sense, a single threshold

serves as the optimal strategy, as in the case with no expert. In the latter case, we obtain the

optimal reservation value to be used by the searcher from Equation 3, yielding t∗ = 1−√

3cs.

The other thing to note here is that Equation 7 above is for the case when the support on signal

s is unbounded. When there is an upper limit on s, i.e., s ≤m for some m (as is the case here,

where signals are bounded in [0,1]), once tu reaches m (we never buy without querying), Equation

7 does not hold. Now the system rejects if the signal is below tl or queries if it is above.

Figure 4(a) illustrates how the reservation values tl and tu change as a function of cs for ce = 0.05.

The vertical axis is the interval of signals. As can be seen from the graph, in this specific case for

very small search costs (cs), the searcher never terminates search without querying the expert.2

Due to the low search cost the searcher is better off only querying the expert when a high signal

is received. The expert option is preferred over accepting without querying the expert for those

high signals, because if a low value is received from the expert then the cost of finding a new

opportunity with a high signal is low. As the search cost cs increases, there is some behavior that

is not immediately intuitive. The reservation values tl and tu become closer to each other until

coinciding at cs = 0.08, at which point the expert is never queried anymore. The reason for this

is that the overall utility of continuing search goes down significantly as cs increases, therefore

the cost of querying the expert becomes a more significant fraction of the total cost, making it

comparatively less desirable. This is a good example of the additional complexity of analyzing

a system with an expert, because in the static sense the cost of consulting the expert does not

2 When search costs are 0 the problem is ill-defined. The first point on the graph shows an extremely low, but non-zerosearch cost. In this case tu = 1 and tl is almost 1, but not exactly, and the expert is again always queried.

20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.02

0.04

0.06

0.08

0.1

0.12

Cost per query(ce)−>

πe−

>

Profit of Expert vs Cost per query for cs=0.01

d

e=0

de=0.025

de=0.05

Figure 5 Expert’s profit as a function of ce and de for cs = 0.01.

Without Subsidy With Subsidyde ce ηcece csηs Worth V W β ce ηcece csηs Worth V W0 0.06 0.10 0.08 0.7472 0.5653 0.6656 0.1003 0 0 0.1036 0.7928 0.6893 0.6893

0.025 0.06 0.10 0.08 0.7472 0.5653 0.6238 0.0585 0.025 0.05 0.09 0.7712 0.6295 0.62950.05 0.061 0.09 0.08 0.7365 0.5637 0.5806 0.0169 0.05 0.09 0.08 0.7536 0.5824 0.5824

Table 1 The different components of social welfare with and without subsidy for cs = 0.01. “Worth” is the

expected value of the opportunity eventually picked. Initially the decrease in query cost contributes

more to the increase in social welfare, but as de increases, this contribution becomes less significant.

Note that the first two columns in the case without subsidy are similar because the profit-maximizing

ce is the same and the searcher’s cost depends only on value of selected ce, not de.

change, so the fact that the expert should be consulted less and less frequently is counter-intuitive.

Figure 4(b) illustrates the change in the searcher’s welfare as a function of the search cost, cs, for

different values of the service fee, ce, charged by the expert. As expected, the searcher’s welfare

is better with the expert option than without, and the smaller the fee charged by the expert, the

better the searcher’s welfare.

Expected number of queries: We can find the expected number of queries in this case by using

our knowledge of the uniform distribution and the noise distribution in Equations 8-11, yielding

A= V 2( 1tl− 1

tu); B = tl; C = (1− tu); D= tu− tl−V 2( 1

tl− 1

tu)

which give the final expressions:

ηce =tltu(tu− tl)

tut2l − tlv2− tutl + tuv2; ηs =

1

1− tl− v2( 1tl− 1

tu)

The monopolist expert’s optimal strategy: Using the above derivations, it is now easy to calculate

numerically the value of c∗e that maximizes πe, trading off decreasing number of queries η and

increasing revenue per query ce. Figure 5 shows examples of the graph of expected profit for the

expert as a function of the expert’s fee, ce, for different values of de, the marginal cost to the expert

of producing an extra “expert report”, for cs = 0.01.

21

0 0.02 0.04 0.06 0.08 0.1 0.120

0.005

0.01

0.015

0.02

0.025

Subsidy(β) −>

Incr

ea

se in

so

cia

l we

lfare

−>

Increase in Social welfare vs Subsidyfor c

s = 0.01 and d

e = 0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−14

−12

−10

−8

−6

−4

−2

0

2

4

6x 10

−3

Increase in Social welfare vs Subsidyfor c

s = 0.01 and d

e = 0.025

Incr

ea

se in

so

cia

l we

lfare

−>

Subsidy (β) −>0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Subsidy (β) −>

Incr

ea

se in

so

cia

l we

lfare

−>

Increase in Social welfare vs Subsidyfor c

s = 0.01 and d

e = 0.05

for de = 0 for de = 0.025 for de = 0.05Figure 6 Increase in social welfare vs subsidy. When there is no marginal cost (de = 0), it is better for the market

designer to make the service available for free but when there is some marginal cost involved, then

increase in social welfare is concave, peaking at marginal cost.

5.2. Subsidizing the expert

As discussed above, the market designer or the government can guarantee the reduction of the

expert’s charge from c∗e to c′e, keeping πe constant, by paying a per-consumer subsidy β to the

expert. Figure 6 shows the improvement in social welfare, denoted δ(W ), as a function of the

subsidy paid to the expert, β, for various de values (where cs = 0.01). From the graphs, we do

indeed find that subsidization can lead to substantial increases in social welfare, even when there

is a significant marginal cost of producing an expert report. While this could be from a reduction

in search and query costs or an increase in the expected value of the opportunity finally taken,

the data in Table 1 indicates that the latter explanation is the dominant factor in this case. Also,

in accordance with Theorem 4, social welfare is maximized at the point where the searcher pays

exactly de per query, thus fully internalizing the cost to the expert of producing the extra report.

5.3. The Option for Service Bundling

We illustrate the optimal strategies for the searcher and the expert in the presence of the option for

service bundling using the same environment, leading to several insights on the effects of bundling.

Surprisingly, we find that forced bundling can be pareto improving. It can lead to improvement

not only in the profit of the monopolist expert, which is to be expected, but simultaneously to

improvement in the expected utility of the searcher from engaging in the search process. Figure

7 shows the effect and the intuition. The figure depicts the expected utility to the buyer and the

expert’s expected net profit (alongside the bundle and per-query costs) as a function of the bundle

size3 for cs = 0.01 and different de values. When the marginal cost of producing an expert report is

zero (as in the “digital services” case: an extra Carfax report can be produced essentially for free),

3 For each bundle size, the appropriate optimal cke is used.

22

0 5 10 15 20 25 30

0.565

0.57

0.575

0.58

0.585

0.59

Bundle size(k)−>

Searc

her

Utilit

y−

>

Expert Profit and Searcher Utility vs. Bundle size

Searcher Utility

0.1

0.101

0.102

0.103

0.104

0.105

0.106

0.107

0.108

0.109

0.11

Expert Profit

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

Bundle Size(k)−>

Cost−

>

Bundle Cost and Cost per query vs. Bundle Size

Bundle Cost ce

k

Cost per query ce

k/k

cs = 0.01 and de = 0 cs = 0.01 and de = 0

0 5 10 15 20 25 30

0.565

0.57

0.575

0.58

0.585

Bundle Size(k)−>

Searc

her

Utility

−>

Expert Profit and Searcher Utility vs. Bundle Size

0.076

0.078

0.08

0.082

0.084

0.086

Expert

pro

fit−

>

Searcher Utility

Expert Profit

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

Bundle Size(k)−>

Cost−

>

Bundle Cost and Cost per query vs. Bundle Size

Bundle Cost ce

k

Cost per query ce

k/k

cs = 0.01 and de = 0.01 cs = 0.01 and de = 0.01Figure 7 Effect of bundling on the price charged by the expert and its impact on the searcher’s utility and the

expert’s profit. The searcher’s utility increases as the bundle size increases. While the overall bundle

price, cke , increases with the increase in bundle size, the cost per query decreases rapidly.

there is no significant added cost to the expert. The search overall is becoming more efficient, and

the expert and the buyer can split the additional gain. On the other hand, for non-zero marginal

cost of producing an extra expert report, the benefit to the expert of selling higher bundle sizes

rapidly declines after bundle sizes of two and three, because there is a real cost incurred in producing

the additional reports that the searcher may demand.

Another interesting observation concerns the correlation between bundling and search cost (the

cost of seeing each opportunity initially, cs) faced by the user. In a world of high search costs,

users do not expect to keep searching for more than a few opportunities, so they are unlikely to be

willing to purchase a bundle of high size. Interestingly, when the search costs become very low, we

find that it can be optimal for the expert to sell an unlimited subscription to her services for a fixed

fee. Figure 8 shows an example of this phenomena in the zero marginal cost scenario. A search

cost cs = 0.001 yields a setting where it is optimal for the expert to offer infinite-size bundles, but

increasing the search cost slightly to 0.003 leads to high (but far from infinite) bundle sizes being

preferred. Looking back at Figure 7 we observe that when cs = 0.01, the optimal bundle size is

actually very small (in fact it is around 2-4 for several examples).

23

0 5 10 15 20 25 300.15

0.16

0.17

0.18

0.19

0.2

0.21

Bundle Size(k)−>

Expert

Pro

fit

Expert Profit vs. Bundle Size

optimal π vs k

optimal π for k=∞

5 10 15 20 25 300.16

0.161

0.162

0.163

0.164

0.165

0.166

Bundle Size(k)−>

Expert

Pro

fit−

>

Expert Profit vs. Bundle Size

optimal π vs k

optimal π for k=∞

cs = 0.001 and de = 0 cs = 0.003 and de = 0Figure 8 Expert’s profit for different bundle sizes for two different search costs when de = 0. When cs = 0.001,

selling infinite size bundles (offering an unlimited use subscription) is most profitable.

Both these observations, that increasing either the marginal cost of producing expert services

and/or the cost incurred by the buyer in searching lead to smaller bundle sizes being preferred,

correspond well to the real world. Unlimited subscription models are usually available in online

services, where vanishing marginal costs and low search costs dominate (like autocheck.com),

whereas traditional marketplaces have smaller bundle sizes. The second result, that increasing

search costs leads to smaller bundle sizes being preferred, provides support for the theoretical

result of Geng et al. (2005), who say that, in the setting of information goods, if consumers’

values for future goods do not decrease too quickly, then bundling is (approximately) optimal for

a monopolist, whereas if the values for future goods do decrease quickly bundling may not be

optimal. In the presence of high search costs, consumers are unlikely to want to use too many

queries to the experts, so the marginal utility of an extra query is rapidly decreasing compared to

the case where there are lower search costs.

6. Related Work

There has been relatively little explicit focus on noisy signals in the one-sided search literature. A

notable exception is the recent work of Wiegmann et al. (2010) on “multi-attribute” mate search:

in their model, it is assumed that males arrive stochastically (equivalent to opportunities in our

model), and picky females have to decide which male to mate with (the male always accepts). In the

multi-attribute scenario, they consider the existence of several “hidden” attributes, which can be

revealed on the payment of additional cost if the female chooses to do so. There are two significant

differences between this model and ours: first, Wiegmann et al assume that the only options

available to the searcher are to reject an offer or to pay the cost of revealing the next attribute –

they do not allow the searcher to accept the offer and terminate search until all hidden attributes

24

are revealed. Second, the cost of revealing additional attributes is exogenous, as is appropriate for

the application to biology, rather than a cost that is determined by the strategic behavior of an

expert who serves as an information broker.

More commonly, relaxation of the perfect signals assumption is found in models of two-sided

search (Burdett and Wright 1998), including marriage or dating markets (Das and Kamenica 2005)

and markets with interviewing (Lee and Schwarz 2009). The standard assumption in these cases

is usually that agents can get better signals through repeat interactions (e.g., cohabitation and

repeated interviews (Sahib and Gu 2002) . The literature has not to this point focused on the

decision problems faced by self-interested knowledge brokers, or how their presence affects the

market. Mediators in such models usually take the form of matchmakers rather than information

brokers (Bloch and Ryder 2000).

The literature on market interventions such as subsidies is, of course, enormous, so we limit our

discussion to interventions in markets where search plays an important role. The two-sided search

literature has considered the impact of search frictions on labor markets (Mortensen and Pissarides

1999, Diamond 1982). One classic regulatory intervention in these models is the introduction of a

minimum wage, which can be shown to be welfare increasing in many contexts (Ho 2000), but we

are unaware of any work on subsidizing providers of expert knowledge, as we discuss here.

The third major related strand of literature is on bundling. There is a large literature in economics

that demonstrates the extra profits monopolists can extract by bundling goods, and considers the

differences between complementary and substitute goods (Adams and Yellen 1976, Schmalensee

1984, inter alia). More relevant to our work is the relatively recent literature on bundling of infor-

mation goods, which typically have low or zero marginal cost of production (Bakos and Brynjolfsson

1999, 2000, Geng et al. 2005). While Bakos and Brynjolfsson (1999) initiated much of this line of

literature, Geng et al. (2005) provide a more recent model with several updates, so we compare

our qualitative results in Section 5 with their predictions. While much of this literature focuses on

deriving general conditions under which bundling may make sense, our work is quite different in

that we focus on bundling of expert services within the context of sequential search. Interestingly,

we are able to extend the intuitions from the theory of product bundling of Geng et al. (2005) to

the search domain, and our results are also in line with observations from the real world.

7. Discussion and Conclusions

This paper introduces a search model in which agents receive noisy signals of the true value of

an object, and can pay a (monopolist) expert to reveal more information. We show that, for a

25

natural and general class of distributions, the searcher’s optimal strategy is a “double reservation”

strategy, where she maintains two thresholds, an upper and a lower one. When she receives a signal

below her lower threshold, she rejects it immediately. Similarly, when she receives a signal above

her upper threshold, she accepts it immediately. Only when the signal is between the thresholds

does she consult the expert, determining whether to continue searching or accept the offer based

on the information revealed by the expert.

In many modern markets, there may be a special relationship between the marketplace itself and

the expert. There is then scope for the market designer (or, alternatively, a regulatory authority)

to subsidize the cost of querying the expert. This has the potential to improve social welfare if

the benefit to the searcher of having less friction in the process could potentially more than offset

the cost to the authority. The possible downside would be that if the authority provided too much

subsidy, this could lead to inefficient overconsumption of costly (to produce) expert services. We

show that subsidies can in fact be welfare-enhancing, and, in fact, prove that in our model social

welfare is maximized when the searcher has to pay exactly the marginal production cost of expert

services. Both the model and our results are significant for designers of markets in which consumers

will search and the need for expert services will arise naturally (like an Internet marketplace for

used cars); by enhancing social welfare, the market designer can take market share away from

competitors, or perhaps charge higher commissions, because it is offering a better marketplace for

consumers.

We also extend our analysis to cases where the expert can sell bundles of her services. Surprisingly,

we find that, not only can bundling increase expert profits, it can simultaneously benefit searchers

(who gain in expected utility of the search process). The intuition is that, by purchasing a bundle,

the searcher is less constrained by the marginal cost of expert services and can exploit the search

process to find a better opportunity. We find that larger bundles are preferred in situations with low

search costs and close-to-zero marginal costs of production of additional units of expert services.

Our results support existing theories of bundling of information goods in non-search settings;

bundling is discouraged if the search cost is high, even if the marginal cost is zero, because buyers

will typically not want to sample many opportunities, so the marginal benefit to them of extra

expert reports that are essentially “free” is minimized. It is also notable that these characteristics

(low search costs and marginal costs of production of services) correspond well to modern electronic

marketplaces; our model thus helps explain the increasing prevalence of bundling and subscription

models in these markets.

26

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Appendix A: Proofs

A.1. Proof of Theorem 1

The proof is based on showing that, if according to the optimal search strategy the searcher should resume

her search given a signal s, then she must necessarily also do so given any other signal s′ < s. Let V denote

the expected benefit to the searcher if resuming the search if signal s is obtained. Since the optimal strategy

given signal s is to resume search, we know V >E[v|s]. Given the HSGN assumption, E[v|s]≥E[v|s′] holds

for s′ < s. Therefore, V >E[v|s′], proving that the optimal strategy is reservation-value. Then, the expected

utility of the searcher when using reservation signal t is given by:

V (t) =−cs +V (t)

∫ t

s=−∞fs(s)ds+

∫ ∞s=t

E[v|s]fs(s)ds (24)

where Fs(s) is the cumulative distribution function of the signal s. Taking the first derivative with respect

to t on both sides and then setting it to 0 at t∗, we get:

dV (t)

dt=dV (t)

dtFs(t) +V (t)fs(t)−E[v|t]fs(t)

V (t∗) =E[v|t∗] fs(t∗) 6= 0

28

The condition V (t∗) = E[v|t∗] implies that the reservation value t∗ is the signal for which the searcher’s

utility of resuming search is equal to the expected value of the opportunity associated with that signal.

To verify that t∗ is a maximum, we calculate the second derivative.

d2V

dt2= Fs(t)

d2V

dt2+ fs(t)

dV

dt+d(V (t)fs(t))

dt− d(E[v|t]fs(t))

dtd2V

dt2|t∗ =

1

1−Fs(t∗)

(d((V (t)−E[v|t])fs(t))

dt|t∗)

=− fs(t∗)

1−Fs(t∗)

(dE[v|t]dt

|t∗)

Given the HSGN assumption, E[v|t] is an increasing function in t. Therefore dE[v|t]dt

> 0, this implies that the

value of the second derivative evaluated at t∗ is less than zero, which confirms that t∗ is indeed a maximum.

Finally substituting V (t∗) =E[V |t∗] in Equation 24 we obtain Equation 3 as follows:

V (t∗) =−cs +V (t∗)

∫ t∗

s=−∞fs(s)ds+

∫ ∞s=t∗

E[v|s]fs(s)ds

E[v|t∗] =−cs +E[v|t∗]∫ t∗

s=−∞fs(s)ds+

∫ ∞s=t∗

E[v|s]fs(s)ds

cs =

∫ ∞s=t∗

(E[v|s]−E[v|t∗])fs(s)ds

The value t∗ can be calculated using the above equation. �

A.2. Proof of Theorem 2

The proof extends the methodology used for proving Theorem 1. We first show that if, according to the

optimal search strategy the searcher should resume her search given a signal s, then she must also do so

given any other signal s′ < s. Then, we show that if, according to the optimal search strategy the searcher

should terminate her search given a signal s, then she must also necessarily do so given any other signal

s′′ > s. Again, we use V to denote the expected benefit to the searcher if resuming search.

If the optimal strategy given signal s is to resume search then the following two inequalities should hold,

describing the superiority of resuming search over terminating search (Equation 25) and querying the expert

(Equation 26):

V >E[v|s] (25)

V > V Fv(V |s) +

∫ ∞y=V

yfv(y|s)dy− ce (26)

Given the HSGN assumption and since s′ < s, Equation 25 holds also for s′. Similarly, notice that:

V > V Fv(V |s) +

∫ ∞y=V

yfv(y|s)dy− ce = V Fv(V |s) +

∫ ∞y=V

y dFV (y|s)− ce

= limT→∞

[V Fv(V |s) +

[yFv(y|s)

]Ty=V−∫ T

y=V

Fv(y|s)dy− ce]

= V +

∫ ∞y=V

(1−Fv(y|s))dy− ce

>V +

∫ ∞y=V

(1−Fv(y|s′))dy− ce = V Fv(V |s′) +

∫ ∞y=V

yfv(y|s′)dy− ce

and therefore Equation 26 also holds for s′ < s.

The proof for s′′ > s is exactly equivalent: the expected utility of accepting the current opportunity can

be shown to dominate both resuming the search and querying the expert. The optimal strategy can thus

29

be described by the tuple (tl, tu, V ) as stated above. Taking the derivative of Equation 5 (the main Bellman

Equation) w.r.t. tl and equating to zero, we obtain a unique tl which maximizes the expected utility.

∂V

∂tl=V fs(tl) +Fs(tl)

∂V

∂tl− fs(tl)V Fv(V |tl) +

∫ tu

tl

fs(s)∂V Fv(V |s)

∂tlds

− fs(tl)∫ ∞V

yfv(y|tl)dy−∫ tu

tl

fs(s)V fv(V |s)∂V

∂tlds+ cefs(tl)

=fs(tl)

(V (1−Fv(V |tl))−

∫ ∞V

yfv(y|tl)dy+ ce

)+∂V

∂tl

(Fs(tl) +

∫ tu

tl

fs(s)Fv(V |s)ds)

=fs(tl)

(∫ ∞V

(V − y)fv(y|tl)dy+ ce

)+∂V

∂tl

(Fs(tl) +

∫ tu

tl

fs(s)Fv(V |s)ds)

(27)

Substituting ∂V∂tl

= 0 in Equation 27 we get Equation 6: ce =∫∞V

(y−V )fv(y|tl)dy

To confirm that this is a maximum, we compute the second derivative:

∂2V

∂tl2 = fs(tl)

∂

∂tl

(∫ ∞V

(V − y)fv(y|tl)dy)

+ f ′s(tl)

(∫ ∞V

(V − y)fv(y|tl)dy+ ce

)+

∂2V

∂t2l

(Fs(tl) +

∫ tu

tl

fs(s)Fv(V |s)ds)

+∂V

∂tl

∂

∂tl

(Fs(tl) +

∫ tu

tl

fs(s)Fv(V |s)ds)

(28)

Substituting ∂V∂tl

= 0 and ce =∫∞V

(y − V )fv(y|tl)dy in Equation 28, the second and fourth terms vanish,

leaving:

∂2V

∂t2l= fs(tl)

∂

∂tl

(∫ ∞V

(V − y)fv(y|tl)dy)

+∂2V

∂t2l

(Fs(tl) +

∫ tu

tl

fs(s)Fv(V |s)ds)

∂2V

∂tl2

(1−Fs(tl)−

∫ tu

tl

fs(s)Fv(V |s)ds)

= fs(tl)∂

∂tl

(∫ ∞V

(V − y)fv(y|tl)dy)

∂2V

∂tl2

(∫ ∞tl

fs(s)ds−∫ tu

tl

fs(s)Fv(V |s)ds)

= fs(tl)∂

∂tl

(∫ ∞V

(V − y)fv(y|tl)dy)

∂2V

∂tl2

(∫ tu

tl

fs(s)(1−Fv(V |s))ds+

∫ ∞tu

fs(s)ds

)= fs(tl)

∂

∂tl

(∫ ∞V

(V − y)fv(y|tl)dy)

∂2V

∂tl2 =

−fs(tl)(∫ tutlfs(s)(1−Fv(V |s))ds+

∫∞tufs(s)ds)

∂

∂tl

(∫ ∞V

(y−V )fv(y|tl)dy)

The term on the right hand side outside the derivative is clearly negative. Now we establish that the derivative

of∫∞V

(y−V )fv(y|tl)dy w.r.t tl is positive, completing the proof.

∂

∂tl

(∫ ∞V

(y−V )fv(y|tl)dy)

=∂

∂tl

([(y−V )fv(y|tl)]∞V −

∫ ∞V

Fv(y|tl)dy)

=∂

∂tl

(∫ ∞V

(1−Fv(y|tl))dy)

= (Fv(V |tl)− 1)∂V

∂tl+

∫ ∞V

∂

∂tl(−Fv(y|tl)dy

Since ∂V∂tl

is 0, we need only consider the second term on the right hand side. By the HSGN assumption,

FV (y|tl) is a decreasing function in tl, therefore, the derivative is negative, and the term inside the integral

is positive, meaning the above term is positive, which completes the proof, showing that Equation 6 is a

condition for the maximum. The proof for Equation 7 is exactly similar. We can solve Equations 5 - 7 (three

equations in three variables) simultaneously to calculate V , tl, and tu. �

30

A.3. Proof of Theorem 3

F or Part (1), the proof is a simple extension of Theorem 2 for cases where queries are sold only one at a

time. We first show that if it is optimal for the searcher to resume search given a signal s, then it must also

be optimal for her to do so given any other signal s′ < s. Then, we show that if it is optimal for the searcher

to terminate search given a signal s, then it must also necessarily be optimal for her to do so given any other

signal s′′ > s.

If the optimal strategy given signal s is to resume search then V0 > max(E[v|s],M(s,Vk−1)− cke). From

the HSGN assumption, E[v|s]>E[v|s′] (since s′ < s). Similarly:

M(s,Vk−1) =

∫ ∞y=Vk−1

yfv(y|s)dy+Vk−1Fv(Vk−1|s)

=[yFv(y|s)

]∞y=Vk−1

−∫ ∞y=Vk−1

Fv(y|s)dy+Vk−1Fv(Vk−1|s) =

∫ ∞y=Vk−1

(1−Fv(y|s))dy+Vk−1

therefore, based on the HSGN assumption, M(s,Vk−1) > M(s′, Vk−1), and consequently V0 >

max(E[v|s′],M(s′, Vk−1)− cke), proving 1(a). The proof for s′′ > s is similar: the expected cost of accepting

the current opportunity can be shown to dominate both resuming the search and querying the expert, prov-

ing 1(b) and thus (together with 1(a)) also 1(c). The optimal strategy can thus be described by the tuple

(tl, tu, Vk−1) as stated in the theorem.

For Part (2), given Lemma 1, we only need to prove that if, according to the optimal search strategy the

searcher should resume her search given a signal s, then she must also do so given any other signal s′ < s.

The proof is similar to the one above, ignoring the option to terminate the search without querying the

expert, and substituting Vk−1 with Vγ−1 whenever applicable. �

Appendix B: Computations of Key Variables

B.1. A la carte Case:

Expected number of opportunities examined: Using the same notations from Section 3.2, the probability

of terminating the search at any search iteration is given by C +D, and these are independent Bernoulli

draws at each search iteration. Therefore the expected number of search iterations is ηs = 1/(C +D).

Expected value of the opportunity received: Instead of calculating the expected value of the opportunity

the searcher eventually ends up with, we can derive the value from the expected utility of the search:

E(ω) = V + ηscs + ηcece.

B.2. Expert offers bundles

Expected value of an opportunity received: This quantity helps us analyze how much a searcher loses

in search cost and query cost. It is also interesting to note how the search cost and query cost affects

the expected value of an opportunity received. Let ω be the random variable representing the value of an

opportunity received, then Ei(ω) represents the expected value of the opportunity if the search terminates

in state i and Pi(ω) represents the probability of terminating at that state.

Ei(ω) =

{E[v|v > Vi−1, s > ti] i > 0E[v|v>Vk−1,tl<s<tu]Pr(v>Vk−1∧tl<s<tu)+E[v|s>tu]Pr(s>tu)

Pr(v>Vk−1∧tl<s<tu)+Pr(s>tu)i= 0

31

Pi(ω) =

{∑j=∞j=0 P j

cycleP0(T )(∏h=k−1

h=i+1 Ph(T ))Pi(Term) i > 0∑j=∞

j=0 P jcycle(1−P0(T )) i= 0

E(ω) = E(Expected value of opportunity) =

i=k−1∑i=0

Ei(ω)Pi(ω)

B.2.1. Computing Expected Number of Bundles Purchased (ηb) and Expected Number of

Queries used (ηcke ) In order to calculate ηb and ηcke , we define and compute the following probabilities:

Represents General formulationHSGN formulation

Pγ→γ−1 Pr (querying and resuming,∫M(s,Vγ−1)>Vγ

fs(s)Fv(Vγ−1|s)dsmoving from state γ > 0 to γ− 1) Pr(s> tγ and v<Vγ−1)

Pγ→γ Pr (resuming without querying,∫M(s,Vγ−1)<Vγ

fs(s)ds

staying in state γ > 0) Pr(s< tγ)Pγ→ter Pr (querying and then

∫M(s,Vγ−1)>Vγ

fs(s)(1−Fv(Vγ−1|s))dsterminating from state γ) Pr(s≥ tγ and v≥Vγ−1)

P0→k−1 Pr (querying and resuming,∫ζ3fs(s)Fv(Vk−1|s)ds

moving from state γ = 0 to k− 1) Pr(tl ≤ s≤ tu and v<Vγ−1)P0→0 Pr (resuming without querying

∫ζ1fs(s)ds

when γ = 0) Pr(s< tl)P¬query0→ter Pr (terminating without querying

∫ζ2fs(s)ds

when γ = 0) Pr(s> tu)P query0→ter Pr (querying and terminating

∫ζ3fs(s)(1−Fv(Vγ−1|s))ds

when γ = 0) Pr(tl ≤ s≤ tu and v≥Vγ−1)

Notice that Pγ→γ−1 + Pγ→γ + Pγ→ter = 1, and similarly P0→k−1 + P0→0 + P¬query0→ter + P query0→ter = 1. Let Pγ(T )

denote the probability of eventually transitioning, when in state γ, to state γ−1 (for γ = 0 it is the probability

of eventually transitioning to state k−1). Let Pcycle be the probability of starting at a given state and getting

back to it after going through all other states (excluding search termination state). The values of Pγ(T ) and

Pcycle can be calculated as:

Pγ(T ) =

∞∑j=0

(Pγ→γ)jPγ→γ−1 =Pγ→γ−1

1−Pγ→γ; Pcycle =

k−1∏i=0

Pi(T )

Now let Pγ(Term), denote the probability of terminating the search in current state γ without transitioning

to state γ−1 (or k−1 if currently the searcher’s state is γ = 0), and P0(Buy∧Term) denote the probability

of eventually purchasing the bundle and then terminating the search when starting from state γ = 0. These

probabilities are given by:

Pγ(Term) = 1−Pγ(T ) =Pγ→ter

1−Pγ→γ

P0(Buy∧Term) =P query0→ter

1−P0→0

; P0(¬ Buy∧Term) =P¬query0→ter

1−P0→0

Expected number of bundles purchased (ηb): In order for the searcher to purchase the jth bundle, it first

needs to purchase j−1 bundles and use them fully. The probability for this is (Pcycle)j−1. Then, the searcher

needs to purchase an additional bundle and eventually terminate without further bundle purchases. There

are three possible scenarios by which the latter occurs: (1) the searcher purchases the bundle and terminates

32

the search in state γ = 0 (with probability P0(Buy ∧ Term)); (2) the searcher purchases the bundle and

terminates the search in some state γ > 0 (with probability P0(T )−Pcycle); and (3) the searcher purchases

the bundle, fully consumes it (i.e., returns to state γ = 0) however terminates search without further bundle

purchases (with probability PcycleP0(¬ Buy∧Term)). Therefore, the probability that the searcher purchases

exactly j bundles throughout its search is given by:

Pr(purchase j bundles) = (Pcycle)j−1(P0(Buy∧Term) +P0(T )−Pcycle +PcycleP0(¬ Buy∧Term)

)Since P0(T ) = 1−P0(¬ Buy∧Term) +P0(Buy∧Term) we obtain:

Pr(purchase j bundles) = P j−1cycle(P0(Buy∧Term) +P0(T ))(1−Pcycle)

Using the above, the expected number of bundles purchased, ηb, is given by:

ηb =

∞∑j=1

jPr(purchase j bundles) =

∞∑j=1

j(Pcycle)j−1(P0(Buy∧Term) +P0(T ))(1−Pcycle)

=

∞∑j=1

j

(k−1∏i=0

Pi(T )

)j−1(P0(Buy ∧ Term) +P0(T ))(1− (

k−1∏i=0

Pi(T )))

Expected number of queries used (ηcke ): In order to calculate the expected number of queries used, ηcke ,

we first calculate the probability that exactly m queries are eventually used, denoted Pm(Q). If m<k then

the only way to exhaust m queries is by transitioning from state γ = 0 to state k −m+ 1 and eventually

terminate the search without transitioning to the next state. For the case of m= k, there are two scenarios

according to which the k queries are exhausted. In both scenarios the searcher transitions to state γ = 1,

however while in the first the search terminates without transitioning to γ = 0 in the second the search

terminates after transitioning to γ = 0 however without purchasing a new bundle. Therefore:

Pm(Q) =

P0(T )(∏j=k−1

j=k−m+2Pj(T ))Pk−m+1(Term) m<k

P0(T )(∏j=k−1

j=2 Pj(T ))P1(Term) +PcycleP0(¬Buy∧Term) m= k

For m > k, we represent m as jk + i where i = m%k and j = bmkc. Here, j represents the number of full

cycles completed (when all queries in a bundle are used) and i represents the number of queries used prior

to terminating before finishing the j+ 1st round. This cyclic nature gives us the following recurrence:

Pm=jk+i(Q) = PcycleP((j−1)k+i)(Q) = · · ·= (Pcycle)jPi(Q)

Therefore the expected number of queries is given by:

ηcke =

∞∑m=0

mPm(Q) =

∞∑j=0

k∑i=1

(jk+ i)(Pcycle)jPi(Q) =

∞∑j=0

k∑i=1

(jk+ i)

(k−1∏l=0

Pl(T )

)jPi(Q) (29)

Expected number of opportunities examined: We know the searcher’s utility, V0, is the value of the

opportunity accepted minus the total search and query cost incurred along the process (V0 = E(ω)− ηscs−

ηbcke). We can use this relation to calculate the value of ηs as follows:

ηs =E(ω)− ηbcke −V0

cs