Competitive Information Disclosure in Search Markets⇤
Simon Board† and Jay Lu‡
Current Version: March 28, 2016
First Version: May 2015
Abstract
Buyers often search across sellers to learn which product best fits their needs. We study how
sellers manage these search incentives through their disclosure policies (e.g. product trials, reviews and recommendations), and ask how competition affects information provision. If sellers
can observe the beliefs of buyers then we show there is an equilibrium in which sellers provide the “monopoly level” of information, and derive conditions under which this equilibrium is
unique. In contrast, if buyers’ beliefs are private, then there is an equilibrium in which sellers
provide full-information as search costs vanish. Anonymity thus plays an important role in the
understanding of how information markets work.
1
Introduction
Starting with Stigler (1961), there has been a large literature examining the incentives of buyers to
search for better prices. This paper studies the incentives for buyers to search for better information,
and asks how sellers manage these incentives through their disclosure policies. We consider a setting
in which all sellers offer the same products and choose what information to release. Buyers then
engage in costly random search across sellers, trying to learn which product they prefer.
There are many environments that share these features. When shopping online, customers
typically face a vast array of products, so websites provide consumers with information to help
them filter the options. For example, when searching for a hotel in Los Angeles, a customer will
⇤
We are grateful to John Asker, Alessandro Bonatti, Emir Kamenica, Moritz Meyer-ter-Vehn, Marek Pycia and
Asher Wolinsky for comments, as well as seminar audiences at Berkeley, ESSET (Gerzensee), SED (Warsaw), SITE,
UNSW, UT Austin, UT Sydney, U Queensland, Munich, Bonn, World Congress (Montreal), Harvard-MIT, Microsoft,
Duke-UNC, UCSD, Chicago, NEGT (USC), UCR, ASU, Miami, Columbia/Duke/MIT/Northwestern IO Conference
and WUSTL. Keywords: Information Disclosure, Advertising, Learning, Search, Targeting, Cookies. JEL: D40, D83,
L13.
†
Department of Economics, UCLA. http://www.econ.ucla.edu/sboard
‡
Department of Economics, UCLA. http://www.econ.ucla.edu/jay
1
typically go to a booking website such as Expedia, and look at recommendations, reviews and photos.
Expedia chooses the information it shows customers in order to steer them towards more expensive
hotels, but also recognizes that customers can leave the site and obtain more information from other
websites such as Orbitz or Priceline. The model is also applicable to the market for information
brokers, where intermediaries try to help customers make decisions in complex environments. For
example, when investing their savings, individuals often seek out advice from a financial advisor.
The advisor prefers that investors purchase mutual funds with larger commissions, but also wants
to provide enough information to prevent them from searching for another advisor.
In this paper, we ask whether competition forces sellers to provide all pertinent information,
and show that the answer depends on the sellers’ information about buyers. We first suppose that
sellers can observe buyers’ beliefs; this may be because websites can track buyers via their cookies,
or because naive customers fully describe their preferences to financial advisors. We then show
that, in a broad range of environments, there is a unique equilibrium in which all sellers choose
the monopoly disclosure policy. That is, sellers manipulate their customers to purchase the most
profitable, rather than the most suitable product. In contrast, when buyers’ beliefs are private,
there is an equilibrium in which sellers fully reveal all their information as search costs vanish.
In the model, we consider a prospective buyer (“he”) who samples from sellers in order to learn
which product best suits his needs. Each seller (“she”) sells the same set of products and, when
sampled, chooses how much information to disclose in order to encourage the buyer to purchase. A
product is defined by the utility of the agent in each state, and the profit to the seller. We model the
seller’s information disclosure as a distribution of signal realizations that may increase or decrease
the buyer’s assessment of the product, as in Kamenica and Gentzkow (2011). After updating his
belief, the buyer chooses to buy a product, exit, or pay a search cost c and randomly sample another
seller. We study how much information sellers disclose and how this depends on sellers’ information
about buyers.
In Section 2, we suppose that buyers’ beliefs are public. First, we show that the monopoly
disclosure policy is always an equilibrium. Intuitively, if all sellers use the same disclosure policy no
buyer has an incentive to continue to search and, since the policy maximizes profits, no seller has
the incentive to defect. This monopoly policy may be very undesirable for customers: if there is a
single product and the buyer is skeptical, preferring not to buy in the absence of information, then
the monopoly policy leaves the buyer with zero utility.
Next, we derive conditions under which the monopoly policy is the unique equilibrium, as is the
case if there is a single product. Intuitively, the search cost allows one seller to provide a little less
information than the market, iteratively pushing the equilibrium towards the monopoly outcome.
However, in other examples, e.g. two vertically differentiated products, non-monopolistic equilibria
may also exist. Intuitively, a seller would like to provide a lot less information than the market,
2
but the search cost only allows for local deviations which may not exist. We show that if every
maximal (i.e. seller obtains her monopoly profit) belief is state-maximal (i.e. highest profit over
all possible states), then these local deviations are enough to prove that the monopoly policy is the
unique equilibrium. In addition to the single product case, this also covers cases when there are
two horizontally differentiated products or many “niche” products, that is, a state for each product
where that product is preferred by the buyer.
In Section 3, we suppose that buyers’ beliefs are private. That is, a seller knows a buyer’s initial
prior but does not know his current belief nor his past search behavior; the seller then chooses a
signal that is independent of past signals. In equilibrium, the buyer purchases (or exits) in the
first period. However, the possibility that the buyer can continue to search and mimic a new buyer
forces sellers to provide more information than when beliefs are public. First, we show that as
the search cost vanishes, there is a sequence of equilibrium buyer payoffs that converges to that of
full-information. Intuitively, if all other sellers provide (almost) full-information and search costs
are small, then any one seller must match this and also provide (almost) full-information.
Next, we derive conditions under which full-information is the unique limit equilibrium. As
search costs become small, any equilibrium has the property that signals are “partitional”, meaning
that the buyer either learns everything or nothing about whether an event has occurred. We show
that if every maximal conditional belief is state-constant (i.e. same profit over all possible states),
then full-information is the only limit equilibrium.
Comparing the private and public cases, we find that anonymity is a powerful force to ensure
sellers compete against one another. When beliefs are public, sellers can discriminate between new
and old buyers. Since old buyers are already informed, it is optimal for a seller to provide them with
less information, which lowers their outside option and undermines competition. When beliefs are
private, sellers cannot discriminate and the option of going elsewhere forces sellers to provide all the
pertinent information in the limit. This finding is important for applications. Our results provide
a theoretical foundation for the notion that tracking consumers can make them “exploitable”,1 and
imply that financial advisors can implicitly collude to guide naive consumers towards expensive
mutual funds, while sophisticated customers receive more informative advice.
Finally, in Section 4, we explore the robustness of these results. First, we show that if sellers
can perfectly coordinate their signals, then equilibria are monopolistic, even if beliefs are private.
Intuitively, a coordinated signal is informative to new buyers but not to old buyers, allowing the
sellers to discriminate between new and old buyers. Second, we consider a case in between public and
private beliefs where a seller can observe whether a buyer approached previous sellers, but cannot
observe the outcome of the signals. As in the public beliefs case, a seller is able to discriminate
1
For example, see “Facebook Tries to Explain Its Privacy Settings but Advertising Still Rules,” New York Times
(13th November 2014) or “Google to Pay $17 Million to Settle Privacy Case,” New York Times (18th November
2013).
3
between new and old buyers. Consequently, the monopoly policy is always an equilibrium and, in
our leading example, it is the only equilibrium. Third, we suppose buyers can pay a cost to prevent
sellers observing their beliefs (e.g. by deleting cookies). We show that when such a buyer becomes
anonymous, this exerts a positive externality on other buyers, meaning that buyers invest too little
in individual privacy measures. Moreover, no matter how small the cost of becoming anonymous,
we show that there is no equilibrium where all buyers do so, providing a rationale for regulating
tracking programs.
The paper contributes to the literature on search by allowing the seller to choose information
disclosure policies. In the benchmark model, when sellers choose prices, Diamond (1971) shows
that all sellers charge the monopoly price. Intuitively, the search cost generates a holdup problem,
allowing any one seller to raise her price slightly above the price set by others.2 We show that
when sellers choose disclosure policies, the analysis depends on what sellers observe about the
buyers: When beliefs are public, the logic is analogous to Diamond, while the case of private beliefs
contrasts sharply. In comparison to the Diamond model, when a seller provides information to a
visiting buyer, this changes his beliefs and changes how he acts at subsequent sellers.
The paper also contributes to a growing literature on information disclosure with competition
based on the Kamenica-Gentzkow framework.3 Gentzkow and Kamenica (2012) study senders who
release “coordinated” signals, characterizing the equilibrium signals and showing that additional
senders always increase the amount of information. Li and Norman (2014) generalize this result to
mixtures of simultaneous and sequential senders, and also show that the result can fail with independent signals. Hoffmann, Inderst, and Ottaviani (2014) suppose heterogeneous sellers simultaneously
release information to win over a customer, and show that competition increases information. In all
of these models a buyer receives the market information and then chooses his action; in our model,
the buyer must pay a search cost to acquire more information. This gives rise to the public vs.
private dichotomy that is the centerpiece of our analysis.4,5
Following, Crawford and Sobel (1982) we assume payoffs to the parties are exogenous. In some
applications, prices may be beyond the seller’s control (e.g. commission rates for Expedia or a
2
This result continues to hold if sellers make multiple offers to each buyer (Board and Pycia (2014)). However, it
breaks down if sellers sell heterogeneous products (Wolinsky (1986)), multiple products (Zhou (2014); Rhodes (2014)),
or if buyers pre-commit to a number of searches (Burdett and Judd (1983)).
3
See also Rayo and Segal (2010) and Aumann and Maschler (1995).
4
There are other related papers. Gentzkow and Kamenica (2015) consider a very general model and ask when
competition yields more information than collusion. Forand (2013) studies a model of information revelation with
directed search. Au (2015) studies dynamic information disclosure by a monopolistic seller.
5
In addition, there are a variety of papers concerning the revelation of information when the senders are informed.
One literature considers verifiable information (“persuasion games”). For example, Milgrom and Roberts (1986) finds
conditions under which competition leads to full revelation. More recently, Bhattacharya and Mukherjee (2013)
consider a model where senders are stochastically informed, characterizing the receiver’s preferences over the biases
of experts. A second literature supposes information is unverifiable (“cheap talk”). Here, Krishna and Morgan (2001)
consider sequential communication, while Battaglini (2002) analyzes a simultaneous game, both finding conditions
under which full revelation is possible when the senders have opposed preferences.
4
financial advisor), or there may simply be no transfers (e.g. within an organization). With other
applications, the seller may choose aggregate prices when faced with heterogeneous buyers, and then
individually target product recommendations. Anderson and Renault (2006) study a monopolist
selling a single good who can choose information and prices, and show that the seller should reveal
whether the buyer’s value exceeds the cost, and charge a price equal to the buyer’s expected value;
this implements the efficient allocation and fully extracts from the buyer. With more goods, such
as two vertically differentiated goods, matters become much more complicated, and a monopolist
must trade off the efficiency of the allocation and the rents acquired by the buyer.6
2
Public Beliefs
In this section we suppose buyers’ beliefs are public. We first describe the model and consider a
simple single-product example. Our first main result is that monopoly is always an equilibrium.
Our second main result is to derive conditions under which monopoly is the unique equilibrium.
2.1
Model
Overview. Time is discrete, and there is mass one of sellers and buyers. Each seller offers the
same set of products, and each buyer searches for one good. When a buyer approaches a seller, the
seller provides information, and the buyer chooses whether to buy, exit or continue searching.
Information. A buyer is initially uncertain about a finite set of states relevant for his decision
that we call S. Let
S be the set of beliefs about S, and assume without loss that the initial
belief places positive probability on every state. When a buyer approaches a seller, she observes
the buyer’s prior p 2
S, which may incorporate information from other sellers, and chooses a
distribution of posteriors such that the average posterior equals the prior. Formally, a disclosure
policy K is a Markov kernel where for each prior p, K (p) is a distribution of posterior beliefs
R
satisfying S q K (p, dq) = p. Let K̄ denote the full-information disclosure policy such that every
posterior is degenerate on some state, q = 1s . In other words, K̄ reveals all the information to the
buyer.
Shopping. After the buyer sees the seller’s signal, he updates his belief to form a posterior q 2
S
and chooses whether to (1) buy a product from the seller, (2) exit the market or (3) pay a search
cost c > 0 and pick a new seller at random. Each seller offers the same set of products {0, 1, . . . , I}.
Product i
1 delivers utility ui (s) in state s 2 S and profit ⇡i > 0 to the seller. We assume that
6
The seller can again achieve first-best if she can either make the price contingent on the state, or can use an
up-front fee (e.g. Eső and Szentes (2007)).
5
different products yield different profits, that is, ⇡i 6= ⇡j for all i 6= j. Product 0 is the exit option
normalized so that u0 is the zero vector and ⇡0 = 0.
If a buyer with belief q 2
S decides to stop shopping, then let i⇤ (q) = argmaxi q · ui denote his
optimal product, assuming ties are resolved in the seller’s favor.7 We can define the indirect utility
of the buyer by u (q) := ui⇤ (q) , and the profit of the seller by ⇡ (q) := ⇡i⇤ (q) . If a buyer decides to
continue shopping, then he pays the search cost, arrives at the new seller and the seller observes
the buyer’s new prior. The game then proceeds as above.
Seller’s strategy. Sellers choose disclosure policies to maximize profit. All sellers have identical
optimization problems, with the buyer’s belief being the only payoff relevant state variable. We
thus look for an equilibrium in which sellers use symmetric stationary disclosure policies.8 If all
other sellers choose a policy K, then the continuation value of a buyer with belief q equals
Z
Vc (K, q) := c +
max {r · u (r) , Vc (K, r)} K (q, dr)
(1)
S
A buyer then purchases or exits when their belief falls in a stopping set Qc (K) ⇢
Qc (K) = {q 2
Given a stopping set Q ⇢
S | q · u (q)
S, given by
(2)
Vc (K, q) }
S, a seller chooses a disclosure policy to maximize her profit. A
disclosure policy K̃ is optimal given Q iff for all possible disclosure policies L and priors p
Z
Z
⇡ (q) K̃ (p, dq)
⇡ (q) L (p, dq) .
Q
Q
A monopoly policy is a disclosure policy that is optimal given Q =
S; this corresponds to the
optimal disclosure policy in Kamenica and Gentzkow (2011).
Equilibrium. In a symmetric stationary equilibrium, each seller uses a disclosure policy K that is
optimal given the stopping set Qc (K), where the latter is derived from K following equation (2).
In this case, we call K an equilibrium.
Discussion. We assume that, after a buyer visits a seller, he observes the disclosure policy K.9
For some sellers, the policy is quite explicit, e.g. Expedia showing a certain number of reviews,
7
This ensures the seller’s profit function is upper semi-continuous, which is necessary for her best-response to be
well-defined. For the same reason, we also assume that a buyer purchases when indifferent between stopping and
continuing.
8
That is, the disclosure policy depends neither on calendar time nor the identify of the seller. This makes the
analysis simpler, but the assumption is not required for our leading example (see footnote 10). The assumption of
stationarity is without loss if sellers cannot observe the order in which a buyer approaches them.
9
If the buyer observes the disclosure policy before visiting, then the buyer can direct their search, leading
to Bertrand competition and full-information revelation. While some companies develop reputations for revealing
information, it is generally hard to describe a disclosure policy to a customer before they have visited (e.g. when
compared to describing a price).
6
Figure 1: Single Product (Public Beliefs)
(b) Non-Equilibrium Policy
(a) Monopoly Equilibrium Policy
Buyer
Buyer
∫ r⋅u( r) K (q , dr ) = q⋅u(q)
∫ r⋅u( r ) K (q , dr )
V c ( K , q)
V c ( K , q)
0
0
1
1
2
1
2
1
b
b(1−c)
Seller
Seller
π (q)
π (q)
∫Q π (r ) K (q , dr)
0
c>0
q⋅u( q)
c>0
Q
1
2
∫Q π (r ) K (q , dr)
0
1
Q
1
2
b
Q
1
Figure 1(a) shows that when there is a single product (Example 1), the monopoly policy is an equilibrium. The
top panel shows the buyer’s payoffs: the bold line is the payoff from stopping q · u (q), the shaded line is the expected
R
payoff S r · u (r) K (q, dr) under policy K, and the dotted line is the buyer’s value function Vc (K, q). The bottom
panel shows the seller’s payoffs: the bold line is the profit function ⇡ (q) and the shaded line is the expected profit
R
⇡ (r) K (q, dr) under K. Figure 1(b) shows that a more informative policy cannot be an equilibrium. This is
Q
because the seller can raise profits by providing a bit less information as demonstrated by the the dashed line.
or a car dealership offering product trials. With others, the disclosure policy can be learned from
the seller, e.g. looking at a variety of products at Amazon, or asking a financial advisor about
different investments. As in Kamenica and Gentzkow (2011), we assume a seller does not change
the disclosure policy depending on the state. This could be because she does not know the state (e.g.
the buyer’s idiosynchratic preferences), or because she is worried about being caught (e.g. hiding
bad reviews for a hotel). In this context, the “public beliefs” case corresponds to a theoretical ideal
that reflects the seller ability to infer a buyer’s belief via their previous shopping or click behavior
(e.g. Expedia) or through conversation (e.g. financial advisors). As we show in Section 4, the results
of this section broadly carry over to the “private” case if sellers can coordinate their signals, and
to the “semi-public” case in which a seller sees whether a buyer visited other sellers previously, but
does not observe the outcome of these signals.
7
2.2
Motivating Example
Example 1 (Single product). A seller has a single product to sell, there are two states {L, H},
and the buyer wishes to buy the product if state H is more likely. For example, a consumer considers
booking a resort in Big Bear, but does not know whether he will enjoy the trip. Suppose that payoffs
to the buyer are u (L) =
1 and u (H) = 1, so the buyer prefers to buy if his belief in the H state
p = Pr (H) is greater than 12 . A sale generates profits ⇡ = 1 for the seller. The resulting monopoly
policy provides just enough information to persuade the buyer to buy. In other words, if we let
denote the degenerate belief at posterior r 2
8
< p
K ⇤ (p) =
:(1 2p)
r
S, then the monopoly policy is given by
if p 2
{0}
+ 2p { 1 }
2
⇥1
⇤
,
1
2
⇥
if p 2 0, 12
This is illustrated in Figure 1(a). If the buyer starts with a prior p
1
2,
then the seller provides no
information and the buyer buys; if p < 12 , then the seller sends the buyer to posteriors 0 and
1
2
and
the buyer buys with probability 2p. For shorthand, we will sometimes call the latter the “perfect
bad news” policy and use the condensed notation p ! 0, 12 .
The monopoly policy is an equilibrium of the game. If all sellers choose such a policy a buyer
learns nothing from a second seller after leaving a first and, given the search cost, buys immediately.
Since the buyer purchases and the seller makes monopoly profits, she has no incentive to deviate.
More surprising, the monopoly policy is the the unique equilibrium of the game. To gain some
intuition, fix b > 12 , and suppose that the seller provides a perfect bad news signal p ! {0, b}, as
h
i
b
illustrated in Figure 1(b). This induces a stopping set Q = 0, 2b 1 c [ [b(1 c), 1], as shown in the
lower panel. Now, consider a seller who deviates and uses the policy p ! 0, max
1
2 , b (1
c)
.
Since there is a strictly positive search cost, a buyer who receives this signal will not subsequently
search and this deviation strictly raises the seller’s profits. As we prove below, this logic implies
that whenever sellers provide more information than the monopoly policy, there is a profitable
deviation.10
2.3
4
Monopoly is an Equilibrium
We now return to the general model, with multiple states and multiple products. First, we show
that any equilibrium policy will always send buyers inside their stopping set, meaning that buyers
only search once in equilibrium. Second, we show that the monopoly policy is always an equilibrium.
Finally, we show that the set of equilibria gets larger as the search cost rises.
10
In this example, one can show a stronger result: the monopoly disclosure policy is the only rationalizable
equilibrium. If we start with an stopping set Q0 = {0} [ {1} then, no matter what other sellers do, a buyer purchases
from i if his belief falls within Q1 = [0, c] [ [1 c, 1]. Iterating n
log(2)/ log(1 c) rounds, then Qn [ 12 , 1] and
the monopoly policy is the only undominated strategy remaining.
8
Lemma 1. If K is an equilibrium, then K (p, Qc (K)) = 1 for all p 2
function is
Vc (K, p) =
c+
Z
S
S. Thus, the buyer’s value
(3)
q · u (q) K (p, dq)
Proof. Let K be an equilibrium and Q := Qc (K) be the stopping set. We will prove K (p, Q) = 1
by contradiction. Suppose K (p) puts strictly positive weight outside Q. Consider the following
deviation: provide full-information to all buyers who would have ended up outside Q under K. Call
this new disclosure policy L (it is straightforward to check that L is a well-defined disclosure policy).
The seller’s profit from using L is
Z
Z
Z
⇡ (q) L (p, dq) =
⇡ (q) K (p, dq) +
Q
Q
S\Q
✓Z
◆
⇡ (r) K̄ (q, dr) K (p, dq)
Q
For any q that is not in Q, the buyer pays the search cost c > 0 in order to acquire some information.
Hence, after being given full-information, he must purchase with some strictly positive probability
(else he should have just stopped at q). Since all products have positive profits, this means the seller
R
strictly benefits by providing full-information at q, that is Q ⇡ (r) K̄ (q, dr) > 0. By assumption,
K puts strictly positive weight on beliefs outside Q, so the expected profit from the new policy
L is strictly larger than the profit from the original policy K contradicting the fact that K is an
equilibrium. Hence, K must put all its weight in the stopping set Q. Since the buyer stops with
probability one, this means the value function (1) is given by (3).
Lemma 1 says that a seller will only send a buyer to beliefs that induce him to stop. This does
not mean that the buyer will not get more information if he were to search again; it is just that
given the search cost, he would not benefit from this additional information so he might as well stop
after the first search.
We now wish to show that monopoly is always an equilibrium. To do this, we need a preliminary
result. Given a disclosure policy K, let AK ⇢
is provided, i.e. K (p) =
p
S be the set of absorbing beliefs where no information
for all p 2 AK . We say a disclosure policy is one-shot if it only takes
the buyer to absorbing beliefs, that is, K (p, AK ) = 1 for all p 2
S. This means the buyer gets no
more information if he were to search again under K.
Lemma 2. Given any equilibrium, there exists a one-shot equilibrium policy with the same profits.
Proof. See Appendix A.1.
To see why, suppose the seller were to send buyer p to non-absorbing beliefs. If the second
round of information strictly increased profits then the seller should incorporate this in the first
round of information. The seller must therefore be indifferent about providing the second round of
information, allowing us to ignore it and define a new monopoly policy which is one-shot.
9
We now present our first main result: monopoly is always an equilibrium. Note that this also
implies the existence of an equilibrium.
Theorem 1. A monopoly policy is an equilibrium.
Proof. Let K be some monopoly policy which exists by the theorem of the maximum. Note that
we can set c̄ sufficiently high such that Qc̄ (K) =
there exists a one-shot policy
K⇤
S so K is also an equilibrium. By Lemma 2,
with the same profits as K, so K ⇤ is a one-shot monopoly policy.
Now, for any search cost c, buyers receive no information at absorbing beliefs so they stop, i.e.
AK ⇤ ⇢ Qc (K ⇤ ). Since K ⇤ is one-shot, K ⇤ (p, AK ⇤ ) = 1 and the seller can achieve her monopoly
profit. That is,
Z
⇤
⇡ (q) K (p, dq) =
Qc (K ⇤ )
Z
⇤
⇡ (q) K (p, dq) =
AK ⇤
Z
⇡ (q) K ⇤ (p, dq)
S
Hence the seller has no incentive to deviate and K ⇤ is an equilibrium.
Intuitively, if all sellers use a one-shot monopoly policy, a buyer only receives one round of
information and so purchases from the first seller. Since all sellers are making their monopoly
profits, none has an incentive to defect.
The next result shows that the set of equilibria grows as search costs shrink.
Proposition 1. Given any equilibrium for c > 0, there is an equilibrium for all c0 < c with the
same profits.
Proof. Given an equilibrium for c > 0, Lemma 2 implies there is a one-shot equilibrium policy K
with the same profits. We show that K is also an equilibrium for any c0 < c. Let Qc and Qc0 denote
the stopping sets under K for search costs c and c0 respectively. First, observe that the buyer will
stop at any absorbing belief, so AK ⇢ Qc0 . Since K is one-shot, a seller’s profits from policy K are
unaffected by the reduction in cost. Moreover, the reduction in cost means that buyers’ continuation
values actually increase so the stopping set shrinks, that is, Qc0 ⇢ Qc . Since the seller would only
want to send the buyer to beliefs in the stopping set, the scope for deviations is smaller. We now
put this together. Since K is optimal given Qc , for any disclosure policy L,
Z
Z
Z
Z
⇡ (q) K (p, dq) =
⇡ (q) K (p, dq)
⇡ (q) L (p, dq)
⇡ (q) L (p, dq)
Q c0
Qc
Qc
Q c0
Thus, K is optimal given Qc0 and is an equilibrium under c0 as well. Moreover, K yields the same
profits under c0 as under c.
Intuitively, a decrease in costs does not change the profit from the previous equilibrium policy
but does make it harder to deviate. Hence the policy remains an equilibrium.
10
2.4
Uniqueness of the Monopoly Equilibrium
Theorem 1 establishes that monopoly is always an equilibrium. In this section we first provide an
example where there is a non-monopoly equilibrium. We then return to the general case, and derive
conditions such that monopoly that is the unique equilibrium.
Example 2 (Vertical differentiation). Suppose there are two vertically differentiated products and two states L and H, with p = Pr (H). The first product is a cheap, low-quality hotel yielding utility u1 =
; the second is an expensive, high-quality hotel, yielding utility u2 = ( 1, 1).
⇥
⇥
Thus, the buyer prefers no hotel if p 2 0, 13 , prefers the cheap hotel if p 2 13 , 23 and prefers the
⇥
⇤
expensive hotel if p 2 23 , 1 . Assume the expensive hotel also has a higher profit for the seller,
1 2
3, 3
⇡1 = 1 and ⇡2 = 23 . The monopoly policy here is as follows:
8
>
>
(1 3p) {0} + 3p { 1 }
>
>
3
<
K (p) = (2 3p) 1 + (3p 1) 2
{3}
{3}
>
>
>
>
:
p
⇥
if p 2 0, 13
⇥
if p 2 13 , 23
⇥
⇤
if p 2 23 , 1
By Theorem 1, the monopoly policy is an equilibrium; this is illustrated in Figure 2(a). One can
interpret this policy as the seller recommending the high-quality hotel for high beliefs, recommending
either the high- or low-quality hotel for middling beliefs, and recommending either the low-quality
hotel or no hotel for low beliefs.
When c > 0 is small enough, there is also a second equilibrium where sellers provide more
information,
K (p) =
8
<2
:
3p
{0}
2
+
3p
2
{ 23 }
p
⇥
if p 2 0, 23
⇥
⇤
if p 2 23 , 1
as illustrated in Figure 2(b). This gives rise to the stopping set Q = [0, "c ] [
"c > 0 and "¯c > 0. Hence a seller’s profits for q 2 Q are
⇡ (q) = 1[ 2
3
"¯c , 23 )
+
⇥2
3
⇤
"¯c , 1 for some
3
·1 2
2 [ 3 ,1]
This policy differs from the monopoly policy by never inducing the buyer to purchase the low-quality
hotel. As shown in Figure 2(b), the presence of the search cost allows the seller to provide a little
less information than her competitors; however, there is no local deviation that is profitable. If one
seller induces a buyer to have a posterior q = 13 , as under the monopoly policy, then the buyer would
refuse to buy, correctly anticipating that other sellers will provide significantly more information.4
We now provide a sufficient condition for monopoly to be the unique equilibrium. Let ⇧⇤ (p) be
11
Figure 2: Vertical and Horizontal Differentiation (Public Beliefs)
(a) Vertical Differentiation: Monopoly
(b) Vertical Differentiation: Non-monopoly
Buyer
Buyer
(c) Horizontal Differentiation: Monopoly
Buyer
c>0
c>0
c>0
0
2
3
1
3
1
ɛc
Seller
0
0
1
3
̄ɛ c
1
2
3
Seller
1
3
Q
2
3
1
0
0
1
3
1
2
3
Seller
Q
2
3
1
3
Q
1
0
Q
1
3
2
3
Q
Figures 2(a) and (b) show the two equilibria that arise in the vertical differentiated example. Figure 2(c) shows
the unique equilibrium that arises in the horizontally differentiated example; this coincides with the monopoly policy.
For a detailed description of the different lines, see the caption for Figure 1.
the monopoly profit of the seller as a function of the buyer’s belief p. We say a belief p is maximal
if it yields the maximal (monopoly) profit, that is, ⇡ (p) = ⇧⇤ (p). In other words, a monopolist can
achieve her monopoly profit by providing no information. A belief p is state-maximal if it yields the
maximal state-wise profit, that is
⇡ (p) = max ⇡ (1s )
s2Sp
where Sp ⇢ S is the support of p. We first present a preliminary lemma.
Lemma 3. If all maximal beliefs are state-maximal, then they are in any equilibrium stopping set.
Proof. See Appendix A.2.
To illustrate what it means for maximal beliefs to be state-maximal, consider Figure 3(a) where
there are three products {1, 2, 3} with ⇡1 > ⇡2 > ⇡3 . Hi denotes the set of beliefs where the buyer
favors product i and note that si 2 Hi for all i 2 {1, 2, 3}. Thus, the most profitable product is
favored in state s1 , the second most profitable in state s2 , and the third most profitable in state
s3 . The shaded dark region represents the set of state-maximal beliefs. Given an interior belief p
that is not state-maximal, the monopoly policy can be decomposed in two steps. First, it releases a
perfect bad news signal about s1 , taking the prior p ! {q1 , q23 } where q1 just persuades the buyer
to buy the first product and q23 is on the edge between s2 and s3 . Second, the policy releases a
12
1
Figure 3: Monopoly Policy (Public Beliefs)
(a) Maximal
Certainty-maximal
(b) Maximal
Certainty-maximal
s1
s1
H1
H1
q1
q1
p
p
H2
H3
s3
H2
q 23
q2
s2
s3
s2
Figure 3(a) shows the monopoly policy when all maximal beliefs are also state-maximal. In this
case, the state-maximal beliefs are in the stopping set of any equilibrium. Figure 3(b) shows how
complications can arise when not all maximal beliefs are state-maximal. Here, p is maximal but not
state-maximal as there is a more profitable product in the support of p. In this case, we can find a
non-monopoly equilibrium where p is not in the stopping set.
perfect bad news signal about s2 such that q23 ! {q2 , 1s3 }, where q2 just persuades the buyer to
purchase good u2 .11 Note that in this example, all maximal beliefs are state-maximal. Figure 3(b)
shows an example where not all maximal beliefs are state-maximal. In particular, it is optimal for a
monopolist to provide no information at p so p is maximal. However, p is clearly not state-maximal
as product 2 is not the most profitable product in the support of p.
We now provide some intuition for Lemma 3. We argue that if all maximal beliefs are statemaximal, then all state-maximal beliefs must be in the stopping set of any equilibrium. Consider
Figure 3(a) again and suppose there is a state-maximal belief in H1 that is not in the stopping set
Qc (K) of some equilibrium K used by all sellers. Note that a buyer with a belief on the boundary
of Qc (K) \ H1 gets no information since he is already purchasing the most profitable product. By
a continuity argument, we can find a belief close enough to the boundary but not in the stopping
set such that the buyer gets very little information. Since the search cost is strictly positive, this
small amount of information is not enough to incentivize the buyer to shop, so the buyer stops at
that belief yielding a contradiction.
Once we have established that all maximal beliefs are also in the stopping set of any equilibrium,
then in any equilibrium, any seller can always convince the buyer to stop and purchase at any
maximal belief. As a result, the seller can keep monopoly profits.
11
If p is state-maximal, then there is some freedom about the monopoly policy but since the buyer’s decision is
the same (he buys the product), such differences are payoff irrelevant.
13
Theorem 2. If all maximal beliefs are state-maximal, then all equilibria are monopoly.
Proof. Let K be an equilibrium policy and ⇧ be the corresponding profit function. Let K ⇤ be a
one-shot monopoly equilibrium as guaranteed by Theorem 1. Since K ⇤ is one shot, it only sends the
buyer inside its absorbing set AK ⇤ . Since K ⇤ provides no information at every q 2 AK ⇤ , it means
that q must be maximal. By the premise and Lemma 3, every q 2 AK ⇤ must also be in the stopping
set Qc (K). Letting ⇧⇤ denote the monopoly profit, we have
Z
Z
Z
⇤
⇧ (p) = max
⇡ (q) L (p, dq)
⇡ (q) K (p, dq) =
L
Qc (K)
Qc (K)
⇡ (q) K ⇤ (p, dq) = ⇧⇤ (p)
AK ⇤
Hence, K must be a monopoly policy as well.
The basic intuition for Theorem 2 is as follows. Given that other sellers are using some equilibrium policy, a seller can always deviate by providing a very informative policy and sending the
buyer to state-maximal beliefs. If all maximal beliefs are also state-maximal, then we can find such
a deviation where the seller obtains her monopoly profit. This is because Lemma 3 ensures that
all maximal beliefs are in the stopping set of any equilibrium. Hence, the seller can send buyers to
these maximal beliefs and keep her monopoly profit. As a result, all equilibria are monopoly.
An important step in the above argument is the iterative logic we used in Lemma 3 to establish
why state-maximal beliefs must be in the stopping set of any equilibrium. These local deviation
arguments are analogous to those from Example 1. Such arguments do not apply in Example 2
as the maximal belief where the cheap hotel is recommended is not state-maximal. Figure 3(b)
shows a three-dimensional extension of Example 2 and illustrates the problem whenever there is
a “middle” product. In this case, for sufficiently high ⇡2 , the monopoly policy involves giving the
buyer no information at p. As in Example 2, there is another equilibrium: if all others sellers “skip
over” product 2 and send the buyer p ! {q1 , 1s3 }, then it is optimal for any seller to ignore product
2 as well. Such non-monopoly equilibria cannot arise if maximal beliefs are also state-maximal.
How general is the sufficient condition for Theorem 2? The following corollary illustrates a series
of examples where maximal beliefs are all state-maximal.
Corollary 1. All equilibria are monopoly if:
(a) There is one product.
(b) There are two “horizontally differentiated” products in that u1 (s)
(c) Products are “niche” in that p · ui
0 implies u2 (s)  0.
0 implies p · uj  0 for any j 6= i.
Proof. See Appendix A.3.
Corollary 1 provides sufficient conditions for uniqueness. Part (a) considers the case of a single
good, extending Example 1 to any finite number of states. Part (b) considers two “horizontally
14
differentiated” goods where, in each state, buyers only want one or other good (although for intermediate beliefs they may want both). This case is illustrated in Figure 2(c) and means that no
product is “stuck in the middle”, as in Example 2. Finally part (c) considers a large number of
“niche” goods such that buyers only want one good at any belief. This case is illustrated in Figure
3(a) and again means that no product is “stuck in the middle”.
It is notable that our condition for Theorem 2 only involves the monopoly policy. Since the
uniqueness result depends on finding a series of local deviations from any potential equilibrium to
monopoly, a full description of necessary and sufficient conditions on the products would be very
cumbersome. Nevertheless, our sufficient condition is also necessary in the following sense: whenever
there are maximal beliefs in the support of all monopoly policies that are not state-maximal, then
we can find a search cost sufficiently small such that a non-monopoly equilibrium exists.
3
Private Beliefs
In this section, we suppose buyers’ beliefs are private. As a result, sellers cannot provide different
information to buyers with different beliefs. We first describe the model and demonstrate how
private beliefs affect the equilibrium in the single product example. Our first main result is that
in the limit, as search costs become small, there is an equilibrium that is payoff-equivalent to fullinformation. In general, there may be multiple equilibria and we characterize these limit equilibria.
We then derive conditions under which the full-information is the unique limit equilibrium.
3.1
Model
Overview. Time is discrete. There is unit mass of sellers, and a unit mass of buyers arriving each
period. As before, each seller offers the same set of products, and each buyer searches for one good.
When a buyer approaches a seller, she sees neither the buyer’s current belief nor his past search
behavior. The seller then chooses to disclose an independent signal, and the buyer chooses whether
to buy, exit or pay the search cost c and continue.
Information. A seller knows the buyers’ initial prior p which lies in the interior of
S, however
she does not know whether he has learned information from other sellers. Each seller chooses a
signal structure on some space ⇥ that is independent of all previous signals received by the buyer.12
Any potential buyer with prior q can also observe the signal and learn about the state. Let r (✓)
be the posterior of a buyer with prior p after observing a signal ✓ 2 ⇥, and let rq (✓) denote the
posterior of a buyer with prior q who observes the same signal. The posteriors of the two buyers
12
Since we are interested in large markets where coordination might be difficult, we assume signals are independent.
We discuss this in Section 4.1.
15
are related as follows
rq (✓) =
where
q
satisfying
:
S !
q
(r (✓))
S is a mapping from the posteriors of buyer p to the posteriors of buyer q
[
q (r)] (s) = P
r(s)
q (s) p(s)
s
0
(4)
0
r(s )
q (s0 ) p(s
0)
Intuitively, for any two buyers, the proportionality of their likelihood ratios for any two states
remains constant when we update via Bayes’ rule. That is, if buyer q starts off twice as optimistic
as buyer p, then buyer q will remain twice as optimistic as p after any signal realization.13
Since this mapping
is independent of the signal structure, sellers can simply choose a signal
R
policy µ which is a distribution of posteriors for a buyer with prior p that satisfies S q µ (dq) = p.
q
Let µ̄ denote the full-information signal policy which has full support on all degenerate posteriors.
In other words, µ̄ reveals all the information to the buyer.
Seller’s strategy. Sellers choose signal policies to maximize profit. As in the public case, we look
for an equilibrium in which sellers use symmetric stationary signal policies. Note that since a buyer
with prior q has a different distribution over states than a buyer with prior p, he also has a different
distribution over signals as well. Let µq denote the signal distribution from the perspective of buyer
q and note that trivially, µp = µ. If all other sellers choose a policy µ, then the continuation value
of a buyer with belief q equals
Vc (µ, q) :=
c+
Z
S
max {
q
(r) · u (
q
(r)) , Vc (µ,
q
(r))} µq (dr)
The stopping set of the buyer is then given by
Qc (µ) := {q 2
S |q · u (q)
Vc (µ, q) } .
(5)
At any time, there may be a variety of generations of buyers in the market. Nevertheless, it is
always a best-response for a seller to provide enough information such that new buyers purchase
with probability one.14 We thus consider equilibria where there are only new buyers in the market.
Given a stopping set Q ⇢
S, a seller chooses a signal policy to maximize her profit. A signal
13
For a formal derivation, see Lemma 4 in Appendix B.1.
The reasoning is as follows. Suppose some buyers are in the market for multiple generations so at any one time,
there is a non-degenerate measure of beliefs in the market. Suppose all sellers use policy µ and in the steady state,
each seller sells to a unit measure of buyers. Since c > 0, any buyer who does not stop initially after the first search will
eventually stop in some finite time in the future. Consider a policy µ̃ that sends all buyers to their eventual stopping
posterior immediately. By using µ̃, the seller still sells to a unit measure of buyers and obtains the same profits but µ̃
may induce some old buyers to purchase, strictly increasing her profits (as is the case in Example 1). Hence, inducing
new buyers to purchase immediately is always a best-response. Note that this restriction affects neither Theorem 3
(which is an existence result) nor Theorem 4 (as buyer payoffs converge in the limit for all generations of buyers).
14
16
policy µ̃ is optimal given Q iff for all possible signal policies ⌫
Z
Z
⇡ (q) µ̃ (dq)
⇡ (q) ⌫ (dq) .
Q
Q
Equilibrium. In a symmetric stationary equilibrium, each seller uses a signal policy µ that is
optimal given the stopping set Qc (µ), where the latter is derived from µ following equation (5). In
this case, we call µ an equilibrium. We also consider what happens to equilibria when search costs
vanish. A limit equilibrium is a sequence of decreasing search costs cn ! 0 and associated equilibria
µn such that the buyer value functions Vcn (µn , q) converge.
3.2
Motivating Example
Example 1 (cont). As before there are two states {L, H} and the buyer’s initial prior is p = Pr (H).
First, suppose that p < 12 , so that the monopoly policy provides a some amount of information. We
claim that as c ! 0, the only equilibrium policy is full-information µ̄. To see this, first observe that
the value function is convex and increasing in the posterior q. Since information has no value at
the boundaries, the stopping set is of the form Qc (µ) = [0, a] [ [b, 1], where a 
1
2
 b. As a result,
the seller’s optimal policy is a perfect bad news signal p ! {0, b}, as shown in Figure 4(a).
If the buyer with prior p receives a signal p ! {0, b}, then a buyer with prior b will have
posteriors p ! {0,
b (b)}.
Using the the expression for
b pb
b (b)
=
b (0)
=0
b +(1 b) 11
b
p
b
from equation (4), we get
with probability
b
p
with probability 1
b
(b)
b
b
(b)
b
By the definition of the stopping set Qc (µ), the buyer with belief b must be indifferent between
stopping and searching again to obtain additional signals, that is
2b
1=
b
(2
b (b)
b2
[1 + p
c (1
b (b)
1)
c
Rearranging, this becomes
p)] b + p = 0.
(6)
Now, for small enough search cost c, only the larger root b = b⇤ (c) of this quadratic exists when
p < 12 . Hence, the unique equilibrium is that sellers provide a signal p ! {0, b⇤ (c)} which induces
an stopping set Qc (µ) = [0, a⇤ (c)] [ [b⇤ (c) , 1]. Moreover, as c ! 0, the value of searching increases
and b⇤ (c) ! 1, as shown in Figure 4(b). This means that the seller provides full-information.
Intuitively, since the monopoly policy provides some information then, as the cost vanishes, a buyer
can obtain a large number signals at low cost and become almost fully-informed. Hence the current
17
Figure 4: Single Product (Private Beliefs)
(b) Full-information Limit Equilibrium
(a) Non-monopoly Equilibrium
Buyer
Buyer
c→0
c>0
W ( μ , q)
V c ( μ , q)
q⋅u(q)
0
a (c)
p
1
2
b (c)
1
0
Seller
p
1
2
b (c)→ 1
Seller
π (q)
∫Q π (r ) μ(dr )
0
Q
p
1
2
Q
1
0
Q
p
1
2
Q
1
Figure 4(a) shows when there is a single product (Example 1) with private beliefs and the buyer has
prior p < 12 , there exists a non-monopoly equilibrium: a perfect bad news policy p ! {0, b⇤ (c)} that
induces the stopping set Qc (µ) = [0, a⇤ (c)] [ [b⇤ (c) , 1]. The top panel shows the buyer’s payoffs: the
bold line is the payoff from stopping q · u (q), the shaded line is the expected payoff
R
W (µ, q) := S q (r) · u ( q (r)) µq (dr) under policy µ, and the dotted line is the buyer’s value
function Vc (µ, q). The bottom panel shows the seller’s payoffs: the bold line is the profit function
R
⇡ (q) and the shaded line is the expected profit Q ⇡ (r) µ (dr) under µ. Figure 4(b) shows that when
the search cost vanishes, c ! 0, this equilibrium converges to full-information, b⇤ (c) ! 1.
seller must provide a lot of information in order to beat this outside option and make a sale.
When p
1
2,
a monopolist would provide no information and there are two limit equilibria as
search costs vanish: full-information and no information. First observe that no information is an
equilibrium for any search cost; if sellers provide no information, the buyer will not search and,
since “no information” is the monopoly policy, no seller will defect. There are two further equilibria
⇤
⇥
which correspond to the two roots of the quadratic (6), denoted by {b⇤L (c) , b⇤H (c)} ⇢ 12 , 1 . As
c ! 0, b⇤L (c) ! p, which corresponds to no information, whereas b⇤H (c) ! 1, which corresponds to
full-information.
4
This example shows how there are equilibria that are much more informative than the monopoly
policy. When beliefs are public, a buyer who receives information from one seller obtains no useful
information if he were to go back to the market. However, when beliefs are private, the buyer can
reject the first offer, pretend to be uninformed and receive more information from another seller in
18
the market. The possibility of taking this outside option allows competition to function.
3.3
Full-Information is a Limit Equilibrium
We now return to the general model, with multiple states and multiple goods. We show that we
can always find a sequence of decreasing search costs c ! 0 and associated equilibria such that
the buyer obtains his full-information payoff in the limit. In other words, a full-information limit
equilibrium always exists.
Theorem 3. Full-information is a limit equilibrium.
Proof. See Appendix B.2.
The basic intuition is as follows. As costs become small, a buyer either learns everything or
nothing about a state. Although a buyer only visits one seller on the equilibrium path, he always
has the option of visiting c
1
2
1
sellers for a total search cost of c 2 . As c approaches zero, so does the
total search cost. Hence, if each seller provides some information about a state, then the buyer will
be fully-informed in the limit. This means that if all sellers provide information about all states
then a single seller must match them and also provide full-information information about those
states. Note that Theorem 3 says that the limit equilibrium policy must be payoff-equivalent to the
full-information buyer payoff, but the seller may not provide all information. This occurs because
a buyer who makes the same decision in two different states will not pay a search cost to separate
them.
We now provide an overview of the proof to show how we construct such a sequence of equilibria.
Fix a search cost c > 0 and suppose all other sellers use policy µ. The other sellers’ policies generate
a value function Vc (µ, ·) for the buyer and a corresponding stopping set Qc (µ). Faced with this
stopping set, the current seller has a set of optimal signal policies which we denote by 'c (µ).
In other words, µ ◆ 'c (µ) is the best-response correspondence which describes the best-response
policies given that all other sellers use policy µ.
We first need to show that given a small search cost c, there exists an equilibrium µ 2 'c (µ).
Note that the set of all signal policies is a convex compact space. If the best-response correspondence
'c is also nonempty, convex valued and has a closed graph, then a direct application of the KakutaniFan-Glicksberg fixed point theorem implies that an equilibrium exists. Unfortunately existence is
an issue as 'c may not have a closed graph. To see this, let us return to the vertical differentiation
case in Example 2 with initial prior p = 13 . Consider a sequence of policies µn and optimal policies
⇥
⇤ ⇥
⇤
⌫n 2 'c (µn ) such that (µn , ⌫n ) ! (µ, ⌫) and Qc (µn ) = 0, 13 "n [ 23 , 1 , where "n ! 0. Note
that for each "n > 0, the buyer will never purchase the low-quality hotel, so the seller’s optimal
⇥
⇤ ⇥
⇤
policy ⌫n is a perfect bad news signal p ! 0, 23 . However, in the limit as Qc (µn ) ! 0, 13 [ 23 , 1
19
the optimal policy ⌫ ⇤ 2 'c (µ) provides no information. Hence ⌫ 62 'c (µ) and the correspondence
does not have a closed graph.
Although the fixed point theorem may not be directly applicable, we show that it is still possible
to construct a sequence of equilibria that yields the full-information payoff to the buyer in the limit
as search costs vanish. The construction works as follows. Let D" be the set of beliefs such that
the buyer’s stopping payoff is "-away from that of full-information and let M" be the set of signal
policies with support in D" . For example, in Figure 5(a) the highlighted D" regions are all close to
the vertices corresponding to beliefs where the buyer obtains his full-information payoff. When " is
sufficiently small, we can find a small search cost c such that the mapping 'c restricted to signals
in M" does have a closed graph (see sketch below). We can thus apply our fixed point theorem to
show that for every sufficiently small " > 0, there exists an equilibrium µ 2 M" . Now, as " ! 0 and
c ! 0, the buyer’s equilibrium value function Vc (µ, ·) converges to that of full-information. Hence,
full-information is always a limit equilibrium.
We now provide a sketch of why 'c has a closed graph when signals are in M" for sufficiently
small ". Consider a sequence of policies µn 2 M" and best-responses ⌫n 2 'c (µn ) such that
(µn , ⌫n ) ! (µ, ⌫). Let ⌫ ⇤ 2 'c (µ) be an optimal policy given µ. Consider Figure 5(a) again and
note that we can find " small enough such that whenever ⌫ ⇤ recommends some product i, that is
⌫ ⇤ (Hi ) > 0, then we can find some state such that 1s 2 Hi as well. In Figure 5(a), this occurs
whenever " is small enough such that the D" regions never intersect H4 . Since buyer value functions
are convex and continuous, whenever ⌫ ⇤ puts weight on some belief q i 2 Qc (µn ) \ Hi we can find
some qni 2 Qc (µn ) \ Hi along the line segment from q i to 1s such that qni ! q i . Hence, we can define
a sequence of policies ⌫n⇤ with weights on qni such that ⌫n⇤ ! ⌫ ⇤ . Since ⌫n⇤ has support in Qc (µn ),
the profits of ⌫n⇤ converges to that of ⌫ ⇤ in the limit. Since ⌫n is weakly more profitable than ⌫n⇤ for
every µn , it follows that ⌫ = limn ⌫n is weakly more profitable than ⌫ ⇤ = limn ⌫n⇤ . This establishes
that ⌫ 2 'c (µ), as required.
3.4
Uniqueness of the Full-Information Limit Equilibrium
Theorem 3 establishes that there always exists a limit equilibrium that is payoff equivalent to
full-information. In some situations however, there are multiple limit equilibria, some of which
provide less than full-information. In this section, we characterize these limit equilibria as revealing
partitional information about the state space, and provide a sufficient condition for when the fullinformation limit equilibrium is unique.
20
Figure 5: Limit Equilibria (Private Beliefs)
(a) Full-information Limit Equilibrium
(b) Monopoly-optimal Partition
s1
p{s }
s1
1
H1
Dɛ
p
H4
Dɛ
H2
H3
H1
s2
s3
s3
p{s , s }
2
(c) Not Monopoly-optimal Partition
p ' {s }
s2
3
(d) Monopoly-optimal Partition not always Limit Equilibrium
s1
s1
1
r
q
H2
p'
p
H1
H1
s3
p ' {s , s }
2
s3
s2
p{s , s }
2
3
s2
3
Figure 5(a) shows the construction of a full-information limit equilibrium in the proof of Theorem 3.
Figure 5(b) shows a limit equilibrium where the buyer only learns about s1 . Note that every event is
maximal since the buyer is purchases the most profitable reachable product at every conditional belief.
Figure 5(c) shows the case with a different prior p0 where learning about s1 is not a limit equilibrium
as p{s2 ,s3 } is not maximal. Figure 5(d) shows that not every partition containing maximal events
corresponds to a limit equilibrium. Even though all other sellers only provide information about s1
and every event is maximal, each seller has an incentive to deviate and provide some information
about s2 versus s3 (i.e. sending the buyer to posterior r instead of q).
We first present an example of a limit equilibrium that is not payoff equivalent to full-information.
Suppose there are three states S = {s1 , s2 , s3 } and a single product as in Figure 5(b). Suppose
that all other sellers provide information about s1 only, so a buyer never receives any information
that allows him to distinguish between s2 or s3 . The buyer’s posterior beliefs thus lie on the line
connecting s1 with the conditional belief p{s2 ,s3 } , i.e., the projection of p on the event {s2 , s3 }.
From the current seller’s perspective, a buyer near the belief p{s2 ,s3 } purchases her product and the
21
seller has no incentive to provide more information. This means that if all other sellers provide no
information about s2 versus s3 , the current seller also has no incentive to provide such information.
As search costs go to zero, buyers know for sure whether s1 is true or not, but they learn nothing
about s2 versus s3 . This partial-information policy is a limit equilibrium and gives the buyer a
strictly lower payoff than full-information. Of course, Theorem 3 implies that full-information is
also a limit equilibrium, so this example exhibits multiple limit equilibria.
On the other hand, consider Figure 5(c) where the buyer has prior p0 and does not purchase the
product at the conditional belief p{s2 ,s3 } . In this case, the partial-information partition is no longer
a limit equilibrium. The reason is that for beliefs close to p{s2 ,s3 } , sellers have an incentive to provide
information about s2 versus s3 . Once sellers provide a little information about s2 versus s3 , then
as the search cost falls, a buyer can visit many sellers and fully learn about this dimension. Hence,
a single seller must match the market and the unique limit equilibrium is that of full-information.
We now formally characterize limit equilibria. Let E be some partition of the state space S. For
any event E 2 E and belief p 2
given the event E, that is,
S such that p (E) > 0, let pE 2
pE (s) =
p (s)
p (E)
for all s 2 E, and pE (s) = 0 otherwise. Geometrically, if
the event E, then pE is the projection of p onto
S be the conditional belief of p
E is the sub-simplex corresponding to
E.
Given any signal policy µ, let Eµ be the limit partition under µ, that is, the information that
is revealed in the limit after infinite iid draws of the signal under µ. Formally, Eµ is the smallest
set of events that cover S and all conditional posteriors rE are constant µ-a.s. for all E 2 Eµ . Note
that this implies that rE = pE µ-a.s.. Also define µE as the limit signal policy corresponding to
revealing Eµ to the buyer, that is,
µE :=
X
p (E)
pE .
E2Eµ
This is best illustrated with some examples. In the case of no information release, Eµ is the
trivial partition {S} and µE = µ =
p.
information will be released (e.g. p >
If µ provides no information, then even in the limit, no
in Example 1). In the case of full-information release, Eµ is
1
2
the finest partition {{s1 } , {s2 } , {s3 }} and µE = µ̄. Since all other sellers are already providing fullinformation, providing full-information is a limit equilibrium (e.g. µ that has support in D" in Figure
5(a)). For the intermediate case, consider Figure 5(b) where µ only provides information about s1 so
Eµ = {{s1 } , {s2 [ s3 }}. As search costs become arbitrarily small, a buyer learns everything about
all events in Eµ but nothing more than Eµ .
We now formalize these ideas. Let W (µE , q) denote the payoff from a buyer with prior q who
learns about the partition Eµ .15 A limit equilibrium is partitional if the sequence of equilibria
15
Formally, this means W (µE , q) :=
R
S
(
q
(r) · u (
q
(r))) (µE )q (dr).
22
payoffs Vcn (µn , q) converges to W (µE , q) where µ is a limit of µn . An event E ⇢ S is maximal if
the conditional belief pE is maximal.
Proposition 2. All limit equilibria are partitional and all events in the partition are maximal.
Proof. See Appendix B.3.
The intuition behind this result is that if sellers reveal a little information about state s versus
s0
then, as the search cost vanishes, a buyer can visit a large number of sellers and perfectly
distinguish them. In order to make a sale, any one seller must therefore match the market, meaning
that any limit information structure must be partitional. Now, suppose the market induces partition
Eµ = {{s1 } , {s2 , s3 }}, as in Figure 5(b). It must be the case that, for a buyer with belief close to
the conditional belief p{s2 ,s3 } , the seller does not want to release any extra information about s2
versus s3 . In other words, the event {s2 , s3 } is maximal. Suppose this is not the case as in Figure
5(c) where the seller has an incentive to release more information about s2 versus s3 . This induces
the buyer to keep searching in order to acquire more information which contradicts the fact that Eµ
is the limit partition. Thus, any event in the limit partition Eµ must be maximal.
Even though every limit equilibrium corresponds to limit partition where all events are maximal,
it is not true that any partition that satisfies this property must be a limit partition. For example,
consider Figure 5(d) in which there are two products and consider the partition E = {{s1 } , {s2 , s3 }}.
In the limit, as the buyer learns either s1 or {s2 , s3 } perfectly, the seller has no incentive to provide
information about s2 versus s3 since every event in E is maximal. However, away from the limit,
the seller wishes to provide information about s2 versus s3 . In particular, the seller would like to
send the buyer to posterior r rather than q. Hence, there is no equilibrium that corresponds to E
despite the fact that every event in E is maximal.
We now provide a sufficient condition for when full-information is the unique limit equilibrium.
We say a belief p is state-constant if it yields the same seller profit at all possible states in its
support, that is, ⇡ (p) = ⇡ (1s ) for all s 2 Sp . Since all products have unique profits, this means
that once the buyer has reached posterior p, there is only one possible product the buyer could
choose. Finally, an event E state-constant if pE is state-constant.
Theorem 4. If all maximal events are state-constant, then all limit equilibria are full-information.
Proof. From Proposition 2, all limit equilibria are partitional where all events in the partition are
maximal. Let E be this limit partition and note that since all maximal events are also state-constant,
the buyer attains his full-information payoff at each pE for all E 2 E. As a result, all limit equilibria
must be full-information.
To illustrate this result, note that the examples in Figures 3(a) and 5(c) both satisfy this
sufficient condition. In Figure 3(a), the only maximal events are degenerate, which are trivially
23
state-constant. In Figure 5(c), the maximal event {s1 , s3 } is not degenerate but it is state-constant
as conditional on {s1 , s3 }, the buyer will only exit. Thus, all maximal events in the example are
state-constant. Note that whenever there is some event that is not maximal, sellers will always have
an incentive to release more information even in the limit. This competition drives the market to
provide full-information in the limit. With two states, the analysis is even simpler. This extends
the analysis of Example 1 to multiple goods.
Corollary 2. Suppose there are two states. If a monopolist provides some information, full-information
is the unique limit equilibrium. Otherwise, both full and no information are limit equilibria.
Proof. Follows from Proposition 2 and Theorem 4.
Taken in combination, Theorems 2 and 4 imply that when maximal events are state-constant
and search costs go to zero, buyers receive a higher payoff under private beliefs than under public
beliefs. When beliefs are public, buyers only attain the monopoly payoff, even when search costs are
arbitrarily small. The ability to discriminate between new and old buyers allows sellers to implicitly
collude. When beliefs are private, buyers can attain their full-information payoff in the limit. Even
though buyers purchase from the first seller on-path, the option to mimic an uninformed buyer
and receive more information forces sellers to compete against each other and provide much more
information to buyers.
4
Extensions
In this section we first explore the robustness of the results. In Section 4.1 we consider the case
of private beliefs, but allow sellers to coordinate their signals. In Section 4.2 we suppose a seller
sees whom a buyer has visited in the past, but does not see the realizations of the signals. In both
cases, the seller can discriminate between new and old buyers, implying that the forces are broadly
similar to the public belief model. In Section 4.3 we suppose a buyer can become anonymous at a
small cost and choose between the public and private cases.
4.1
Private Beliefs and Coordination
The contrast between our results on monopoly and full-information equilibria in Sections 2 and 3
respectively can be interpreted as whether sellers have the ability to coordinate, that is, whether
sellers can use signals such that a first seller’s signal is completely (Blackwell) sufficient for that of
a second seller. When beliefs are public, the second seller can always provide no information as a
means of coordinating and we obtain monopoly equilibria. When beliefs are private and signals are
independent across sellers, sellers face difficulty coordinating, competition kicks in and we obtain
full-information limit equilibria.
24
How necessary is independence for breaking coordination and delivering full-information? Clearly,
if each seller can see the previous seller’s signal perfectly, then sellers can successfully coordinate
and monopoly will again be an equilibrium as in Section 2. In particular, suppose there is a public
randomization device that sellers use for signaling. In this case, even if beliefs are private, monopoly
is still an equilibrium. This is because each seller knows the policies of previous sellers, the realizations of the public randomization device and thus the belief of the agent. We are thus in the world
of Section 2 and the proof of Theorem 1 applies.
On the other hand, full independence is unnecessary for full-information in the limit; as long
as there is some amount of imperfect coordination where the first seller’s signal is not completely
sufficient for that of a second seller, our results on the full-information limit equilibrium holds. The
key property is whether a buyer learns all relevant information as the number of signals grows large.
There are a couple of ways to interpret imperfect coordination. First, one can think of the degree
of coordination as reflecting the nature of the technology generating the signals. For example, when
hotel websites shows reviews written by individuals then full coordination seems very unlikely.
However, when the hotels show pictures of the hotels, these photos are often provided by the hotel
themselves and thus serve as a coordination device.
Second, the degree of coordination also depends on how similar the sellers are to one another. In
the model, all sellers are identical. This means that if we introduce a public randomization device
x, there is an equilibrium where sellers use it to enforce the monopoly, and another equilibrium
where sellers ignore the coordination device and choose independent signals (after all, if all other
sellers are ignoring it, then seller i may as well ignore it). Both of these equilibria can be purified,
depending on the degree of similarity of the sellers. To see this, let’s return to the single-product
example:
Example 1 (cont). As before, the underlying state is binary s 2 {L, H}, and suppose the prior
is p =
1
3.
To justify the coordinated equilibria, suppose that some buyers are more profitable
than others, and this is common across sellers. That is, suppose we perturb the profits so that
⇡ = 1 + x, where (s, x) are i.i.d., and the common profit shock x ⇠ U [0, 1] is unknown by buyers.
The monopoly policy is then to provide two signals, with seller sending “low” to buyers with s = L
and x < 1/2, and “high” to buyers with either s = H or s = L and x > 1/2. Since sellers use the
state x to coordinate their signals, monopoly is the unique equilibrium.
To justify the independent equilibrium, we can perturb the profits by i.i.d. profit shocks. That
is, suppose the profit when seller n serves buyer m is given by ⇡ = 1+xn,m , where (s, xn,m ) are i.i.d.,
and the idiosyncratic profit shock xn,m ⇠ U [0, 1] is unknown by buyers. As before, the monopoly
policy is to partition the utility space into the same two signals, however any two sellers will provide
independent information about s. Hence, the unique equilibrium provides more information than
monopoly, and converges to full-information as c ! 0.
25
4
4.2
Semi-Public Beliefs
In Sections 2–3 we considered two extreme cases wherein a seller either sees the belief of the buyer,
or she knows nothing about the buyer’s history. Instead, suppose that a seller sees which other
sellers a buyer has previously visited, but that she does not know the outcome of the signal. To
be more precise, suppose that each seller chooses information independently. The nth seller that
a buyer approaches then chooses a signal policy µn as a function of previous disclosure policies
(µ1 , . . . , µn
1 ).
We first observe that the monopoly policy is a sequential equilibrium.
Proposition 3. If sellers observe the number of past sellers visited and their policies, then the
monopoly policy is a sequential equilibrium.
Proof sketch. We consider the following strategies: Seller 1 provides the monopoly level of information, and the buyer accepts this offer; if the buyer rejects, sellers n
2 believe the posterior is
drawn from the support of the monopoly posteriors and provide no information. If seller 1 deviates,
the subsequent sellers will see the deviation and may provide more information; the buyer can then
accept seller 1 or reject, whichever gives them higher utility.
To show this is a sequential equilibrium, we have to verify no player wishes to defect. Seller
1 obtains monopoly profits, so cannot be better off. The buyer gains nothing by rejecting, since
subsequent sellers provide no information. Now, suppose a buyer approaches seller n
2, and
previous sellers followed the equilibrium. She believes the buyer’s belief q is in the support of the
monopoly policy, and that subsequent sellers provide no information; hence providing no information
is optimal for her. This is also the limit of mixed strategies: if the buyer approaches seller 2, then
she can see who trembled and, if it is the buyer, will optimally provide no information.
The underlying intuition is that when a buyer approaches seller 2, she knows that seller 1 provides
the optimal amount of information. Given that the two sellers are in essentially the same position,
they have the same stopping sets and it is optimal for seller 2 to provide no more information.
The next question is whether there are any other equilibria. In Example 1, the monopoly policy
is the unique equilibrium.
Example 1 (cont). Since the buyer’s belief is private information, his value function is convex
and each seller n faces an stopping set of the form [0, ↵n ] [ [
n , 1].
initial prior p and thus provides a perfect bad news signal p ! {0,
seller 2, then she knows the buyer’s belief q lies in {0,
1 }.
Seller 1 knows the buyer has
1 }.
If the buyer approaches
If q = 0, then seller 2 cannot affect the
buyer’s posterior; her optimal policy is thus to assume the buyer received a positive signal from the
first seller and use a perfect bad news policy
1
! {0,
2 }.
Notice, that seller 2 would use exactly
the same policy if she could observe the belief of the buyer! The same logic applies to subsequent
sellers, so the equilibria of the game coincide with the game with public beliefs. Given that the
26
monopoly policy is the only rationalizable outcome (see footnote 10), it is also the equilibrium in
this case with intermediate observability.
4.3
4
The Choice of Anonymity
In Sections 2–3 we examined two polar cases, with private and public beliefs. In the applications,
we think that the degree of anonymity reflects the ease of tracking customers online via cookies, or
the sophistication of customers when dealing with financial advisors. In the former case, customers
can affect their anonymity by deleting cookies, and activating the “do not track” mode on their
browser. Motivated by this, we suppose that buyers can pay a small cost
to become anonymous.
We first argue that buyers who become anonymous exert a positive externality on other buyers. We
then observe that the free-riding is so severe that there is no equilibrium in which all buyers choose
to become anonymous.
There is a unit mass of buyers and sellers, and all buyers start with prior p. At time t = 0,
a buyer can choose to pay cost
in order to become anonymous. After this decision, buyers can
be described by two dimensions: their belief q when meeting a seller and their type {A, T }, which
describes whether they are anonymous (A) or tracked (T ). When revealing information, a seller
must treat an anonymous buyer (A, q) in the same way as a new tracked buyer (T, p). The seller
can, however, give different signals to old tracked buyers.
One can understand much of the economics by considering a simple 2 ⇥ 2 game where buyers
choose whether or not to be anonymous. The following table provides an illustration showing the
pure strategy payoffs in Example 1 with initial prior p =
1
3
and c = 0.05.
Other Buyers
Anonymous
Buyer m
Anonymous
Tracked
0.30
Tracked
0.15
0.30
0
First, suppose all other buyers are tracked, so we are in the model from Section 2. If buyer m is
also tracked then he receives a single “monopoly” signal from a seller and immediately buys since he
knows a second seller will refuse to provide more information. In Example 1, this means the buyer
receives 0 utility. If buyer m becomes anonymous then he can get the initial “monopoly” signal,
delete the seller’s cookie and then receive “monopoly” signals from subsequent sellers, paying cost c
for each search. In Example 1, this yields expected utility 0.15
. Next, suppose all other buyers
are anonymous, so we are in the model from Section 3. If buyer m is tracked then he receives a
single “competitive” signal that is more informative than the monopoly signal. Since he is tracked,
27
no sellers provide any further information to him. If buyer m becomes anonymous he has the option
to receive a sequence of “competitive” signals. However, on the equilibrium path, the buyer will
purchase at the first opportunity.
The first point to observe is that when other buyers become anonymous, this exerts a positive
externality on buyer m. Intuitively, when faced with lots of anonymous buyers, competition is more
effective and sellers provide more information in equilibrium. This benefits any buyer, independent
of whether they are anonymous or tracked.
Second, observe that no matter how small the cost , there is no equilibrium where everyone
is anonymous. When all other buyers are anonymous then sellers provide enough information to
prevent the searching; hence buyer m has no incentive to become anonymous himself. To illustrate,
consider the above table. If
> 0.15 there is a unique equilibrium in which everyone is tracked. If
2 (0, 0.15) then there is no pure strategy equilibrium.16
5
Conclusion
This paper studied a market where buyers search for better information about a product and
sellers choose how much information to disclose. When beliefs are public, the monopoly policy is an
equilibrium; when maximal beliefs are state-wise maximal, it is the only equilibrium (e.g. if products
are horizontally differentiated or niche). When beliefs are private, full-information revelation is a
limit equilibrium; when maximal events are state-constant, it is the only equilibrium (e.g. if products
are niche and the buyer only purchases if he’s quite certain). More generally, our paper provides
the first attempt to study information disclosure in search markets and, in doing so, shows how the
seller’s information about the buyer is crucial in determining the equilibrium outcome.
Our results highlight the role that anonymity can play in fostering competition between information providers. This is important because it illustrates how tracking technology may undermine the
competitive mechanism and support collusive equilibria. It also illustrates how financial advisors
can take advantage of naive investors. More broadly, regulators should be wary of any technology
that enables sellers to discriminate between new and old buyers and thereby discourage searching,
including methods of correlating disclosure policies.
The current model considers markets with a large number of sellers. We expect our results
will generalize. Intuitively, so long as a seller thinks a leaving buyer will return with less than
100% probability, she will pick posteriors within the stopping set; any one seller then affects the
stopping set, but the main insights are unaffected. The current model also has the feature that
16
If we wish to characterize the mixed strategy equilibrium played by the buyers, we have to move beyond the
simple 2 ⇥ 2 game. Unfortunately, this is difficult to analyze since anonymous buyers will search multiple times
before buying, meaning that a single seller faces a heterogeneous population of buyers. However, one can then show
that, in the most informative equilibrium, an increase in the proportion of anonymous buyers raises the amount of
information disclosed.
28
consumers only search once on the equilibrium path. This would no longer be the case if buyers
had sufficiently heterogeneous priors (e.g. Kolotilin et al. (2015)), or had match specific tastes (e.g.
Wolinsky (1986)). Nevertheless, we expect our broad insights would carry over so long as it is
optimal to provide less information to a buyer who returns to the market after they have rejected
a seller’s initial offer.17
17
For example, if the priors are uniformly distributed over a small interval, e.g. [p
29
", p + "].
References
Aliprantis, C., and K. Border (2006): Infinite Dimensional Analysis. Springer.
Anderson, S. P., and R. Renault (2006): “Advertising Content,” American Economic Review,
96(1), 93–113.
Au, P. H. (2015): “Dynamic Information Disclosure,” RAND Journal of Economics, 46(4), 791–823.
Aumann, R. J., and M. Maschler (1995): Repeated games with incomplete information. MIT
press.
Battaglini, M. (2002): “Multiple referrals and multidimensional cheap talk,” Econometrica, 70(4),
1379–1401.
Bhattacharya, S., and A. Mukherjee (2013): “Strategic Information Revelation when Experts
Compete to Influence,” RAND Journal of Economics, 44(3), 522–544.
Board, S., and M. Pycia (2014): “Outside options and the failure of the Coase conjecture,” The
American Economic Review, 104(2), 656–671.
Burdett, K., and K. L. Judd (1983): “Equilibrium price dispersion,” Econometrica: Journal of
the Econometric Society, pp. 955–969.
Crawford, V. P., and J. Sobel (1982): “Strategic information transmission,” Econometrica,
50(6), 1431–1451.
Diamond, P. A. (1971): “A Model of Price Adjustment,” Journal of Economic Theory, 3(2),
156–168.
Eső, P., and B. Szentes (2007): “Optimal information disclosure in auctions and the handicap
auction,” The Review of Economic Studies, 74(3), 705–731.
Forand, J. G. (2013): “Competing through Information Provision,” International Journal of Industrial Organization, 31(5), 438–451.
Gentzkow, M., and E. Kamenica (2012): “Competition in Persuasion,” Discussion paper, University of Chicago.
(2015): “Information Environments and the Impact of Competition on Information Provision,” Discussion paper, University of Chicago.
Hoffmann, F., R. Inderst, and M. Ottaviani (2014): “Persuasion through Selective Disclosure:
Implications for Marketing, Campaigning, and Privacy Regulation,” Discussion paper, Bocconi.
Kamenica, E., and M. Gentzkow (2011): “Bayesian Persuasion,” American Economic Review,
101(6), 2590–2615.
30
Kolotilin, A., M. Li, T. Mylovanov, and A. Zapechelnyuk (2015): “Persuasion of a Privately Informed Receiver,” Working paper.
Krishna, V., and J. Morgan (2001): “A Model of Expertise,” Quarterly Journal of Economics,
116(2), 747–775.
Li, F., and P. Norman (2014): “On Bayesian Persuasion with Multiple Senders,” Discussion
paper, UNC.
Milgrom, P., and J. Roberts (1986): “Relying on the Information of Interested Parties,” The
RAND Journal of Economics, pp. 18–32.
Rayo, L., and I. Segal (2010): “Optimal Information Disclosure,” Journal of Political Economy,
118(5), 949–987.
Rhodes, A. (2014): “Multiproduct Retailing,” Review of Economic Studies, forthcoming.
Stigler, G. J. (1961): “The Economics of Information,” Journal of Political Economy, 69(3),
213–225.
Wolinsky, A. (1986): “True Monopolistic Competition as a Result of Imperfect Information,”
Quarterly Journal of Economics, 101(3), 493–511.
Zhou, J. (2014): “Multiproduct search and the joint search effect,” The American Economic Review,
104(9), 2918–2939.
31
A
Proofs for Section 2
A.1
Proof of Lemma 2
Let K be any equilibrium with stopping set Qc (K). By Lemma 1, we know that K (p, Qc (K)) = 1
for all p 2
S. We first show that under K, buyers will only end up at stopping beliefs where
the seller is indifferent to providing no information. Formally, this means that if we denote the
indifference set by
R :=
(
q 2 Qc (K)
Z
⇡ (r) K (q, dr) = ⇡ (q)
Qc (K)
)
,
then K (p, R) = 1. Suppose otherwise, that is, K (p, Qc (K) \R) > 0 for some p 2
S. Consider
the deviation of providing a second round of information to all buyers who would have ended up
outside R under K. Call this new policy L; it is straightforward to check that it is a disclosure
policy. The seller’s profit from using L is
✓Z
◆
Z
Z
Z
⇡ (q) L (p, dq) =
⇡ (q) K (p, dq) +
⇡ (r) K (q, dr) K (p, dq)
Qc (K)
R
Qc (K)\R
S
Z
Z
Z
>
⇡ (q) K (p, dq) +
⇡ (q) K (p, dq) =
⇡ (q) K (p, dq)
R
Qc (K)\R
Qc (K)
where the strict inequality follows from the definition of R and our premise that K (p, Qc (K) \R) >
0. Since this contradicts the fact that K is an equilibrium, it must be that K (p, R) = 1.
We now construct a one-shot policy as follows. Let K̃ be the policy that gives no information
when the seller is indifferent but otherwise follow K, that is,
8
<K (p) if p 2
6 R
K̃ (p) :=
:
if p 2 R
p
We now show that K̃ is one-shot. First note that every belief p 2 R is absorbing under K̃, that
is, R ⇢ AK̃ . Hence all we need to show is that K̃ only sends buyers to beliefs in R. Note that if
p 62 R, then K̃ = K so K̃ (p, R) = K (p, R) = 1. On the other hand, if p 2 R, then K̃ (p) =
p
so
K̃ (p, R) = 1 again. This means that K̃ only sends buyers to beliefs in R, and since all beliefs in R
are absorbing, K̃ must be one-shot.
We now verify that K̃ is an equilibrium as well. First, we claim that both K and K̃ have
the same stopping sets, that is, Qc (K̃) = Qc (K). Since K̃ provides less information than K,
by Blackwell ordering, it follows that Vc (K̃, p)  Vc (K, p) so Qc (K) ⇢ Qc (K̃). Now, suppose
p 2 Qc (K̃)\Qc (K). Since p 62 Qc (K), p cannot be in R so K (p) = K̃ (p) by the definition of
K̃. Since buyers immediately stop under both K and K̃, their value functions under both policies
must satisfy (3). Hence, Vc (K̃, p) = Vc (K, p) so p 2 Qc (K) yielding a contradiction. Thus
32
Qc (K) = Qc (K̃). Next, we show that there are no profitable deviations from K̃. For any disclosure
policy L,
Z
⇡ (q) K̃ (p, dq) =
Qc (K̃)
Z
Z
⇡ (q) K (p, dq)
Qc (K)
⇡ (q) L (p, dq) =
Qc (K)
Z
⇡ (q) L (p, dq) (7)
Qc (K̃)
Here, the first equality comes from the definition of K̃ and the fact that the stopping sets are the
same. The second inequality follows from the fact that K is an equilibrium and the final equality
again comes from the equivalence of the stopping sets. Thus, K̃ is also an equilibrium. Note that
from the first equality of equation (7), K̃ yields the same profit as K.
A.2
Proof of Lemma 3
Suppose all maximal beliefs are state-maximal. Consider some equilibrium K and let
Z
⇧ (p) :=
⇡ (q) K (p, dq)
Qc (K)
be the corresponding equilibrium profit. Fix some p that is state-maximal so
⇡i = ⇡ (p) = max
⇡ (1s0 ) = ⇡ (1s )
0
s 2Sp
where ⇡i and s 2 Sp are the corresponding profit and state. We will show that p is also a stopping
belief under K, that is, p 2 Qc (K).
Let Hi be the set of beliefs where product i is chosen under no information. Consider beliefs
between p and the degenerate belief at s, that is,
p := p + (1
for
) 1s
2 [0, 1]. Note that since ⇧ is concave and upper semicontinuous, ⇧ (p ) as a function of
2 [0, 1] is also concave and upper semicontinuous. This implies that ⇧ (p ) as a function of
is
continuous on [0, 1].
By our assumption, maximal beliefs are all state-maximal so ⇡i is the highest attainable profit
in
Sp . Note that the convexity of Hi implies that p 2 Hi as both p 2 Hi and 1s 2 Hi . Since ⇡i
is the highest attainable profit in
Sp , ⇡i is also the monopoly profit at p , that is ⇧⇤ (p ) = ⇡i .
This implies that
⇧ (p )  ⇧⇤ (p ) = ⇡i
Finally, define
¯ := inf { 2 [0, 1] | ⇧ (p ) = ⇡i }
and p̄ := p ¯ . Note that by the continuity of ⇧ (p ), ⇧ (p̄) = ⇡i .
We will show that ¯ = 0. Suppose otherwise so we can find a increasing sequence
33
n
% ¯ . Let
pn := p
n
so by continuity
lim ⇧ (pn ) = ⇧ (p̄) = ⇡i
n
Since ⇡i is the highest attainable profit, it must be that limn K (pn , Hi ) = 1. Note that pn cannot
be in the stopping set Qc (K). This is because if pn is in the stopping set, then the seller can achieve
profit ⇡i by providing no information, that is ⇧ (pn ) = ⇡ (pn ) = ⇡i However, this contradicts the
fact that n < ¯ so pn 62 Qc (K). Hence, it must be that
pn · ui < Vc (K, pn ) =
c+
Z
S
(q · u(q)) K (pn , dq)
where the last the expression follows from equation (3). Rearranging terms yield
✓Z
◆
Z
pn · u i
q K (pn , dq) · ui < c +
(q · u(q)) K (pn , dq)
Hi
S\Hi
Note that both the left expression and the second term in the right expression vanish as K (pn , Hi ) !
1. Since c > 0, we can find some < ¯ such that the above inequality is not satisfied or p 2 Qc (K).
This implies ⇧ (p ) = ⇡i again contradicting the fact that
n
< ¯ . Thus, ¯ = 0 so ⇧ (p) = ⇧ (p̄) =
⇡i .
Since ⇡i is the most profitable product in
Sp and ⇧ (p) = ⇡i , it must be that K (p, Hi ) = 1.
Hence, the information under K (p) has no effect on the buyer’s purchasing behavior so the buyer
might as well stop. Formally, this means that
Z
Vc (K, p) = c +
(q · ui ) K (p, dq) =
Hi
c + p · ui < p · ui
so p 2 Qc (K). Since maximal beliefs are all state-maximal, they must be in the stopping set Qc (K)
as well.
A.3
Proof of Corollary 1
We will show that in all three cases, maximal beliefs are state-maximal and then apply Theorem 2.
Let ⇧⇤ denote the monopoly profit function. Note that part (a) follows trivially from either (b) or
(c). For part (b), consider two “horizontally differentiated” products with ⇡1 > ⇡2 . We will prove
this by contradiction. Suppose there is a maximal belief p that is not state-maximal, so
⇧⇤ (p) = ⇡2 < ⇡1 = ⇡ (1s )
for some s 2 Sp . Since 1s 2 H1 , u1 (s)
0 and “horizontal differentiation” implies that 1s · u2 =
u2 (s)  0. Let p0 be the conditional belief knowing that s is not true, so
p = p (s) 1s + (1
34
p (s)) p0
Note that 1s · u2  0 and p · u2
product at p0 so ⇡ (p0 )
0 imply that p0 · u2
0. Hence, the buyer chooses some profitable
⇡2 . This implies that the seller can earn a strictly higher payoff by
providing a perfect signal about state s contradicting that ⇧⇤ (p) = ⇡2 . Hence, all maximal beliefs
are state-maximal.
The argument for part (c) is analogous. We prove it again by contradiction. Suppose there is a
maximal belief p that is not state-maximal, so
⇧⇤ (p) = ⇡i < ⇡1 = ⇡ (1s )
where ⇡1 is the state-wise optimal product in Sp and s 2 Sp . Again, let p0 be the conditional belief
knowing that s is not true, so p = p (s) 1s + (1
1s · ui  0. On the hand, we know that p · ui
0 so
p (s)) p0 . By “nicheness”, 1s · u1
p0 · u
i
0 from the definition of
again, this means that p0 · uj  0 for all j 6= i. Hence, we know that ⇡ (p0 )
0 implies that
p0 .
By “nichness”
⇡i so by the same
argument as before, the seller can obtain a strictly higher payoff by providing a perfect signal about
s yielding a contradiction.
B
Proofs for Section 3
This section includes the proofs for Theorems 3 and 4. In subsection B.1, we first introduce some
preliminary notation and results that will be useful in subsections B.2 and B.3 when we prove the
theorems
B.1
Preliminaries
We first establish some useful preliminary notation and results. Recall that when sellers use a signal
policy µ, a buyer with prior p has posterior beliefs distributed according to µ. Since signals are
all independent, without loss of generality we can define the canonical signal space as the set of
posterior beliefs of buyer p. In other words, each signal realization corresponds to a posterior belief
r2
S of the buyer with prior p. Letting L (s, dr) be the conditional signal distribution, the joint
distribution of states and signals from the perspective of buyer p is given by Bayes’ rule as
(8)
p (s) L (s, dr) = µ (dr) r (s)
Now, a buyer with prior q has signal distribution µq and for each signal realization r, his posterior
belief is
q
(r). Our first lemma shows how to obtain the following expression for
in Section 3.1.
35
q
(r) that we saw
Lemma 4. If buyer q has posterior
(r) whenever buyer p has posterior r, then µ-a.s.
!
X
r (s)
r (s)
[ q (r)] (s)
q (s)
= q (s)
p (s)
p (s)
q
s2S
Proof. Since signals are all iid, the joint distribution of states and signals from the perspective of
buyer q is
q (s) L (s, dr) = µq (dr) [
q
(r)] (s)
Note this is because both buyer p and buyer q are faced with the same signal structure L. Combining
this with equation (8) yields the following
µq (dr) [
q
(r)] (s) = µ (dr) q (s)
r (s)
p (s)
(9)
which is well-defined since p has full support on S. If we sum both sides of equation (9) over all
states s 2 S, we get the following expression for µq (dr)
µq (dr) = µ (dr)
X
q (s)
s2S
r (s)
p (s)
(10)
Combining equations (9) and (10) yields the desired result.
Consider a buyer with prior q0 who searches t times when all sellers are using signal policy µ.
Let qt :=
qt
1
(rt ) be his posterior belief in period t given the t-period signal realization rt 2
S.
Assuming zero search costs, his payoff at the end of the t periods is given by
Z
t
W (µ, q0 ) : =
max (ui · qt ) µq0 (dr1 ) · · · µqt 2 (drt 1 ) µqt 1 (drt )
( S)t
i
When t = 1, we just set W := W 1 . We first show that W t has a simpler expression.
Lemma 5. For any signal policy µ, belief q and time period t
t
W (µ, q) =
Z
( S)t
max
i
X
s
r1 (s)
rt (s)
ui (s) q (s)
···
p (s)
p (s)
!
µ (dr1 ) · · · µ (drt )
Proof. Using the expression for µqt from equation (10) in the proof of Lemma 4, we obtain
!
Z
X
rt (s)
t
W (µ, q0 ) =
max (ui · qt )
qt 1 (s)
µq0 (dr1 ) · · · µqt 2 (drt 1 ) µ (drt )
p (s)
( S)t i
s
!!
Z
X
X
rt (s)
=
max
ui (s) qt (s)
qt 1 (s)
µq0 (dr1 ) · · · µqt 2 (drt 1 ) µ (drt )
p
(s)
( S)t i
s
s
36
Since qt :=
qt
1
(rt ), from Lemma 4, we have
W t (µ, q0 ) =
Z
( S)t
max
i
X
s
rt (s)
ui (s) qt 1 (s)
p (s)
!
µq0 (dr1 ) · · · µqt
2
(drt
1 ) µ (drt )
Repeating the above argument for each µq , we obtain the desired expression for W t
We now show that if the signal is informative about some event, then the buyer will eventually
learn it. In order to prove this, recall some of the notation from Section 3.4. Let pE denote the
conditional probability of an event E ⇢ S under the prior p. Given a signal policy µ, Eµ is the
smallest set of events that cover S and all conditional posteriors are equal to the conditional prior,
that is, rE = pE µ-a.s. for all E 2 Eµ . Recall that µE is the limit partitional signal policy given µ,
that is
µE :=
First, we confirm that Eµ is a partition of S.
X
p (E)
pE
E2Eµ
Lemma 6. Eµ is a well-defined partition of S.
Proof. Let Ēµ be the set of all events E ⇢ S such that rE = pE µ-a.s.. First, note that since
r{s} = 1s = p{s} for all s 2 S, Ēµ includes all the singleton states so it covers S trivially. We show
that if E, E 0 2 Ēµ are not disjoint, then E [ E 0 2 Ēµ . Let s⇤ 2 E \ E 0 so µ-a.s.
p(s⇤ )
p(E 0 )
=
r(s⇤ )
r(E 0 ) .
Hence, µ-a.s. for all s 2 E and
s0
2
p(s⇤ )
p(E)
=
r(s⇤ )
r(E)
and
E0,
p (s⇤ )
p (E)
p (s)
=
=
⇤
r (s )
r (E)
r (s)
p (E 0 )
p (s0 )
=
=
r (E 0 )
r (s0 )
This implies that µ-a.s.
p E [ E 0 = p (E) + p E 0 \E
X
X
p (E)
p (E 0 )
=
r (s)
+
r (s)
r (E)
r (E 0 )
0
s2E
=
Thus, for all s 2 E [ E 0 ,
p(s)
r(s)
=
p (s⇤ )
r (s⇤ )
p(E[E 0 )
r(E[E 0 )
s2E \E
r E [ E0
so E [ E 0 2 Ēµ . This means that whenever there are
non-disjoint events in Ēµ , their union is also in Ēµ , so the smallest set of such events Eµ ⇢ Ēµ is
well-defined and forms a partition of S.
We now show that the payoff of a buyer who searches indefinitely eventually converges to that
its limit partitional signal policy.
37
Lemma 7. For any signal policy µ, W t (µ, q) ! W (µE , q).
Proof. Fix a signal policy µ so Eµ is a well-defined partition by Lemma 6. Since
s 2 E 2 Eµ , we can rewrite
Wt
r(s)
p(s)
=
r(E)
p(E)
for all
from Lemma 5 as
!
X
r
(E)
t
W t (µ, q) =
max
···
ui (s) q (s) µ (dr1 ) · · · µ (drt )
t
i
p
(E)
p
(E)
( S)
s2E
E
!
Z
X r1 (E)
rt (E)
=
max
···
q (E) (ui · qE ) µ (dr1 ) · · · µ (drt )
p (E)
p (E)
( S)t i
Z
X r1 (E)
E
Defining yt (E) :=
r1 (E)
p(E)
t
W (µ, q) =
=
Z
t (E)
· · · rp(E)
q (E), we can rewrite this as
( S)t
X
max
q (E)
E
where µE (dr) :=
i
Z
✓P
( S)t
yt (E
E0 P
0 ) (u
yt
✓P
E0
max
i
i · qE 0 )
(E 0 )
E
r(E)
p(E) µ (dr).
0
◆ X
E
!
yt (E) µ (dr1 ) · · · µ (drt )
y (E 0 ) (ui · qE 0 )
Pt
0
E 0 yt (E )
◆
µE (dr1 ) · · · µE (drt )
Consider two events E, E 0 2 Eµ such that E 6= E 0 . We want to show that the ratio of yt (E 0 ) to
yt (E) goes to zero µE -a.s. as t ! 1. Now,
✓
◆
1
yt (E 0 )
1
r1 (E 0 )
rt (E 0 ) p (E)
p (E) q (E 0 )
log
= log
···
···
t
yt (E)
t
p (E 0 )
p (E 0 ) r1 (E)
rt (E) q (E)
✓
◆
✓
◆
0
X
0
1
rt (E ) p (E)
1
q (E 0 )
=
log
+
log
t 0
p (E 0 ) rt0 (E)
t
q (E)
t t
By the law of large numbers, we have
1
yt (E 0 )
log
!
t
yt (E)
Z
log
S
✓
r (E 0 ) p (E)
p (E 0 ) r (E)
Since E 6= E 0 , by the definition of Eµ , it must be that
µ-measure. This is because if
r(E 0 )
r(E)
r(E 0 )
r(E)
◆
µE (dr)
is different on some strictly positive
is constant µ-a.s., then for all s 2 E and s0 2 E 0 ,
r (s)
r (E)
r (E 0 )
r (s0 )
=
=
=
p (s)
p (E)
p (E 0 )
p (s0 )
so E [ E 0 2 Eµ which contradicts the fact that Eµ is the smallest such partition of S by definition.
By Jensen’s inequality,
✓
◆
✓Z
Z
r (E 0 ) p (E)
log
µE (dr) < log
p (E 0 ) r (E)
S
✓Z
= log
S
S
◆
r (E 0 ) p (E)
µE (dr)
p (E 0 ) r (E)
◆
✓R
r (E 0 )
µ (dr) = log
p (E 0 )
38
S
r (E 0 ) µ (dr)
p (E 0 )
◆
=0
r(E 0 )
r(E)
where the strict inequality is due to the fact that
0
)
converges to something strictly less than zero. Hence, it must be that log yytt(E
(E) !
for all
yt (E 0 )
1
t log yt (E)
0
)
or yytt(E
(E) ! 0
is non-constant µE -a.s.. Thus,
1
6= E. By dominated convergence, this implies that
P
✓
◆
Z
X
y (E 0 ) (ui · qE 0 )
t
E0 P
lim W (µ, q) =
q (E)
max lim
µE (dr1 ) · · · µE (drt )
0
t
t
( S)t i
E 0 y (E )
E
Z
X
=
q (E)
max (ui · qE ) µE (dr1 ) · · · µE (drt )
E0
=
E
i
( S)t
E
X
q (E) max (ui · qE )
i
From Lemma 5 and the definition of µE , we have
!
!
Z
X
X
X
r (s)
q (s)
W (µE , q) =
max
ui (s) q (s)
µE (dr) =
p (E) max
ui (s)
i
p (s)
p (E)
S i
s
E
s2E
✓
◆
X
X
q (E)
=
p (E) max
ui · qE =
q (E) max (ui · qE )
i
i
p (E)
E
E
so limt W t (µ, q) = W (µE , q).
B.2
Proof of Theorem 3
We now prove that there always exists a full-information limit equilibrium. We will be making use
of the preliminary results in the previous section which uses some of the notation from Section 3.4.
Given any " > 0, let D" be the set of beliefs such that the buyer’s stopping payoff is at least "-away
from his full-information payoff. In other words,
D" := {q 2
S | q · u (q)
W (µ̄, q)
"}
where µ̄ is the full-information policy. Let M" be the set of signal policies that only put weight in
D" , that is, µ (D" ) = 1 for all µ 2 M" . Denote ⇧c (µ, ⌫) as the profit of a seller for using policy ⌫ if
all other sellers are using policy µ, that is,
⇧c (µ, ⌫) :=
Z
⇡ (q) ⌫ (dq)
Qc (µ)
and define the best-response correspondence 'c : M" ◆ M" such that ⌫ 2 'c (µ)
⇧c (µ, ⌫)
⇧c µ, ⌫ 0
for any other signal policy ⌫ 0 .
The main idea is that we will construct a sequence of equilibrium payoffs that converges to
the buyer’s full-information payoff as search costs vanish. For each ", we will find some small
39
enough search cost c such that 'c (µ) is non-empty for all µ 2 M" . We will then use the Kakutani–Fan–Glicksberg (KFG) theorem to show that there exists an equilibrium policy µ 2 M" . Hence,
as " ! 0, we can find a sequence of decreasing search costs and corresponding equilibrium policies
µ 2 M" such that the buyer’s payoff converges to W (µ̄, p).
First, we show that the domain M" satisfies the usual conditions so that we can apply KFG.
Lemma 8. M" is a non-empty, compact and convex metric space.
Proof. To see why M" is non-empty, note that the full-information policy µ̄ puts weight only on
degenerate beliefs so µ̄ (D" ) = 1 and µ̄ 2 M" . The convexity of M" follows trivially. To see why
M" is compact, note that since
S is a compact metric space, the set of all signal policies is also
a compact metric space by Theorem 15.11 of Aliprantis and Border (2006). By Theorem 3.28
of Aliprantis and Border (2006), it is also sequentially compact. We now show that M" is also
sequentially compact. Consider the sequence µm 2 M" so it has a convergent subsequence µn ! µ.
Since µn are all signal policies, it must be that
Z
Z
q µ (dq) = lim
q µn (dq) = p
n
S
S
so µ is also a signal policy. Moreover, since µn (D" ) = 1 and D" is closed, by Theorem 15.3 of
Aliprantis and Border (2006),
µ (D" )
lim sup µn (D" ) = 1
n
so µ 2 M" . Thus, M" is sequentially compact and therefore compact.
Next, we show that 'c is convex-valued. The proof follows directly from the fact that expected
profits are linear in signal policies.
Lemma 9. 'c (µ) is convex for all µ 2 M" .
Proof. Let ⌫1 , ⌫2 2 'c (µ) and consider ⌫ := a⌫1 + (1
a signal policy and
⇧c (µ, ⌫) = a⇧c (µ, ⌫1 ) + (1
a) ⌫2 for some a 2 [0, 1]. Note that ⌫ is also
a) ⇧c (µ, ⌫2 )
⇧c µ, ⌫ 0
for any other signal policy ⌫ 0 . Hence, ⌫ 2 'c (µ) so 'c (µ) is convex.
As an intermediary step, we show that the buyer’s value function is jointly continuous in both
signal policies and posterior beliefs.
Lemma 10. The buyer value function Vc is continuous in (µ, q)
Proof. Fix some c > 0 and suppose all sellers are using the signal policy µ. If the buyer searches
once, then his payoff is
Vc1 (µ, q) :=
c + W (µ, q)
40
Using the expression for W from Lemma 5, we see that W is continuous by Corollary 15.7 of
Aliprantis and Border (2006). Hence, Vc1 is continuous and define a sequence of finite-period value
functions iteratively where
Vct+1 (µ, q) :=
c+
Z
Recall from equation (10), that
S
n
max max
i
µq (dr) = µ (dr)
t
q (r) · ui , Vc (µ,
X
s
and also note that
q
q (s)
o
(r))
µq (dr)
q
r (s)
p (s)
(r) is continuous in q. Hence, applying the same argument for Vc1 inductively,
we conclude that Vct is continuous. Since Vct converges uniformly to the value function Vc , the latter
is also continuous.
The next step is to show that the best-response correspondence 'c takes on non-empty values
so we can apply KFG. While this may not be true in general, we will choose " small enough
such that this holds. Set "¯ > 0 such that it is less than the minimal non-zero difference between
the buyer’s full and partitional information payoffs. In other words, choose "¯ such that whenever
W (µ̄, p) > W (µE , p) for any limit partitional policy µE , then
"¯ < W (µ̄, p)
W (µE , p)
Note that such an "¯ always exists as there are only a finite number of partitions of S.
We now show that for all "  "¯, we can always find a search cost c small enough such that 'c
takes on non-empty values on M" . The main idea is as follows. Recall that D" is the set of beliefs
such that the buyer’s payoff is "-close to his full-information payoff. Now, whenever µ only sends the
buyer to beliefs in D" for some "  "¯, then the definition of "¯ ensures that the buyer’s payoff under
µ approaches that of full-information as search costs vanish. Hence, the buyer will only stop if he
gets close to his full-information payoff so we can find a search cost c small enough such that the
stopping set Qc (µ) is completely contained in D" for all µ 2 M" . Since any optimal best response
signal policy ⌫ 2 'c (µ) will not send the buyer outside his stopping set, it must be that ⌫ also puts
full weight in D" so ⌫ 2 M" as well.
Lemma 11. If "  "¯, then there exists a c" > 0 such that for all c  c" , 'c (µ) is non-empty for all
µ 2 M" .
Proof. Let "  "¯. We first show that for every µ 2 M" , the value function Vc (µ, q) converges to
W (µ̄, q) uniformly as c ! 0. For each search cost c, suppose the buyer searches tc times where tc is
the largest integer less than c
1
2
1
so tc c  c 2 . Since the buyer can always do this, his value function
41
is at least his payoff from visiting tc sellers. In other words,
1
tc c + W tc (µ, q)
Vc (µ, q)
c 2 + W tc (µ, q)
As c ! 0, tc ! 1 so from Lemma 7
lim inf Vcn (µ, q)
W (µE , q)
n
We now show that W (µE , q) = W (µ̄, q). First note that since µ (D" ) = 1, the definition of D"
means that µ-a.s.
max r · ui
W (µ̄, r)
i
"
Note that the left-hand expression is convex in r so if we let r =
X
E2Eµ
r (E) max (rE · ui )
i
0
max @
i
X
E2Eµ
1
P
E2Eµ
r (E) rE A · ui
r (E) rE , then µ-a.s.
W (µ̄, r)
"
Since rE = pE µ-a.s. for all E 2 Eµ , rearranging we have that µ-a.s.
"
W (µ̄, r)
X
E2Eµ
r (E) max (pE · ui )
i
Note that the buyer’s full-information payoff W (µ̄, r) is linear in r. Hence, taking expectations
with respect to µ yields
"
W (µ̄, p)
X
E2Eµ
p (E) max (pE · ui ) = W (µ̄, p)
i
W (µE , p)
By the definition of "¯, this implies that W (µ̄, p) = W (µE , p). From Lemma 5, it is straightforward
to show that this implies W (µE , q) = W (µ̄, q).
We thus have lim inf n Vcn (µ, q)
W (µ̄, q). Since µ̄ is full-information, this implies that the
buyer’s value function converges to full-information as search costs vanish, that is, Vcn (µ, q) !
W (µ̄, q). We will use Dini’s theorem (Theorem 2.66 of Aliprantis and Border (2006)) to show that
this convergence is uniform on the domain M" ⇥
S. Note that M" ⇥
S is compact by Lemma
8 and Vc is continuous by Lemma 10. Since Vcn is decreasing monotonically, Vcn (µ, q) ! W (µ̄, q)
uniformly by Dini’s Theorem. Hence, we can find some c" > 0 such that
|Vc" (µ, q)
W (µ̄, q)| < " for all (µ, q) 2 M" ⇥
42
S
This implies that for all c  c"
Qc (µ) ⇢ Qc" (µ) = {q 2
⇢ {q 2
S | q · u (q)
S | q · u (q)
W (µ̄, q)
Vc" (µ, q) }
" } = D"
If ⌫ is a best-response policy to any µ 2 M" , then by the same argument as in Lemma 1, ⌫ must
have full support in the stopping set, that is, ⌫ (Qc (µ)) = 1. Hence,
⌫ (D" )
⌫ (Qc (µ)) = 1
so ⌫ 2 M" as desired.
Going forward, we assume " < "¯. The last step before we can apply KFG which is to show
that 'c has a closed graph. In general, this may not be true, but again we will find " small enough
such that this holds. Let µn ! µ, ⌫n 2 'c (µn ) and ⌫n ! ⌫. If we can show that ⌫ 2 'c (µ), then
'c has a closed graph. Let ⌫ ⇤ 2 'c (µ) be a best-response policy to µ. Recall that Hi is the set
of beliefs where ui is chosen under no information, assuming ties are resolved in the seller’s favor.
The first step is to find " small enough such that whenever ⌫ ⇤ recommends some product i, that is,
⌫ ⇤ (Hi ) > 0, then we can find some state s 2 S where 1s 2 Hi as well. For instance, in the vertical
differentiation example of Section 2.4, this means that the optimal policy never recommends the
low-quality hotel. In Figure 5(a) of Section 3, this means that D" is close enough to the vertices
such that the seller will never recommend product 4.
Let
u denote the smallest non-zero difference between buyer payoffs across all products at all
degenerate beliefs. In other words,
ui (s)| for all i, j and s 2 S where uj (s) 6= ui (s).
u  |uj (s)
Define "˜ > 0 such that for every ⇡j > ⇡i ,
"˜ <
✓
⇡j
⇡i
⇡j
◆
u
This is the smallest profitable profit deviation for the smallest difference in buyer payoffs. Note that
this is well-defined as the number of products and states are both finite. Set "⇤ := min {¯
", "˜}.
Lemma 12. Let "  "⇤ . Then ⌫ ⇤ (Hi ) > 0 implies 1s 2 Hi for some s 2 S.
Proof. We will prove this by contradiction. Suppose ⌫ ⇤ (Hi ) > 0 for some product i and 1s 62 Hi
for all s 2 S. Since "  "¯, ⌫ ⇤ 2 M" so we know that ⌫ ⇤ (Hi \ D" ) > 0. Now, consider a belief
q 2 Hi \ D" . We will show that providing full-information at q will yield a strictly higher profit for
the seller which contradicts the fact that ⌫ ⇤ is a best-response policy. Let Si ⇢ S denote the set
of states such that product i is buyer-optimal, that is, ui (s) = u (1s ) for all s 2 Si . Since 1s 62 Hi
for all s 2 S by assumption, this means that for every s 2 S, either s 62 Si or s 2 Si but there is
another buyer-optimal product that is more profitable for the seller, that is ⇡ (1s )
43
⇡j > ⇡i for
some product j. Hence, by providing full-information at q, the seller will get profit
X
X
q (s) ⇡ (1s )
q (s) ⇡ (1s ) +
s62Si
s2S
X
q (s) ⇡j
⇡j
X
q (s)
(11)
s2
s2Si
as all profits have positive profits. Since q 2 Hi \ D" , we know that
q · ui = q · u (q) W (µ̄, q) "
X
"
q (s) (u (1s ) ui (s))
s
=
X
q (s) (u (1s )
ui (s))
( u)
s62Si
This implies that
X
q (s)
s62Si
q (s) = 1
X
q (s)
1
s62Si
s2Si
Combining this with inequality (11), we get
X
X
q (s) ⇡ (1s )
s2S
⇣
⇡j 1
" ⌘
= ⇡j
u
"
u
"
⇡j
> ⇡i
u
where the last strict inequality follows from the definition of "˜. Since providing full-information at
q is more profitable for the seller, ⌫ ⇤ cannot be optimal yielding a contradiction.
The next step is to show that we can find a sequence of policies ⌫n⇤ that approximates the seller’s
optimal payoff with arbitrary precision.
Lemma 13. Let "  "⇤ . Then there exists a sequence of disclosure policies ⌫n⇤ such that
⇧c (µn , ⌫n⇤ ) ! ⇧c (µ, ⌫ ⇤ )
Proof. We are going to partition the stopping set Qc (µ) into different regions corresponding to the
different products. In other words, define Qi := Qc (µ) \ Hi so all the Qi together form a partition
of Qc (µ). For each product such that ⌫ ⇤ Qi > 0, let q i be the ⌫ ⇤ -average belief in Qi , that is,
Z
1
i
q := ⇤ i
q ⌫ ⇤ (dq)
⌫ (Q ) Qi
Note that q i 2 Qi since Qi is convex. Since ⌫ ⇤ is a signal policy, the prior p must be the ⌫ ⇤ -average
of the beliefs q i . Moreover, since each ⌫ ⇤ Qi is strictly positive, p must be in the interior of the
convex hull of all q i .
Note that if every q i is in every stopping set Qc (µn ), then a policy with supports on q i will
give the seller exactly the optimal profit ⇧c (µ, ⌫ ⇤ ) and we are done. Unfortunately, q i may not be
in Qc (µn ) which means that the limit profit from such a policy may be less than ⇧c (µ, ⌫ ⇤ ). To
44
overcome this, we will construct a sequence of beliefs qni 2 Qc (µn ) such that qni ! q i and the profits
from a sequence of policies with supports on qni converge to ⇧c (µ, ⌫ ⇤ ). We do this as follows. For
each product i and signal µn , define
i
n
i
n
= 1 if q i 2 Qin and
:=
c
c + Vc (µn , q i )
q i · ui
otherwise. Note that in the case of the latter, Vc µn , q i > q i · ui so
qni :=
i i
nq
i
n
+ 1
i
n
< 1. Finally, define
1si
where 1si 2 Hi as guaranteed by Lemma 12. Since Vc is continuous by Lemma 10, Vc µn , q i !
Vc µ, q i  q i · ui where the inequality follows from the fact that q i 2 Qc (µ). By the definition of
i,
n
this implies that
i
n
! 1 and qni ! q i . Since p is in the interior of the convex hull of the average
beliefs q i , for large enough n, p is also in the convex hull of the beliefs qni . Hence, for each n, we
can define a signal policy ⌫n⇤ with finite support on each qni such that ⌫n⇤
qni
Finally, since Vc (µn , ·) is convex,
Vc µn , qni 
Hence, each qni 2 Qc (µn ) so
⇧c (µn , ⌫n⇤ ) =
i
n Vc
µn , q i + 1
i
n
Vc (µn , 1si )
=
i
n Vc
µn , q i + 1
i
n
( c)
=
i
n
X
i
⇡i ⌫n⇤
! ⌫ ⇤ Qi .
q i · ui  q i · ui
qni
!
X
⇡i ⌫ ⇤ Qi = ⇧c (µ, ⌫ ⇤ )
i
as desired.
In this last step, we use the sequence of policies ⌫n⇤ that converges to the optimal policy ⌫ ⇤ to
show that the limit policy ⌫ is also optimal. This proves that 'c has a closed graph.
Lemma 14. If "  "⇤ , then 'c has a closed graph.
Proof. Let µn ! µ, ⌫n 2 'c (µn ) and ⌫n ! ⌫. We want to prove that ⌫ 2 'c (µ). First, we show that
⌫ (Qc (µ)) = 1. Note that ⌫n (Qc (µn )) = 1 for all n as otherwise the seller can profitably deviate
by providing information at any belief that is not stopping. In order to prove that ⌫ (Qc (µ)) = 1,
we will show that the stopping set mapping Qc is upper hemi-continuous and takes on non-empty,
closed values. We can then use Theorem 17.13 of Aliprantis and Border (2006) which says that
in this case, ⌫n (Qc (µn )) = 1 for all n implies that ⌫ (Qc (µ)) = 1. Note that the non-emptiness
and closedness of Qc are trivial (the latter follows from the continuity of the value function from
Lemma 10). For upper hemi-continuity, suppose (µm , qm ) ! (µ, q) where qm 2 Qc (µm ). Thus,
45
qm · u (qm )
Vc (µm , qm ) so by continuity again, q · u (q)
Vc (µ, q) implying q 2 Qc (µ). Hence, Qc
is upper hemi-continuous, so applying the theorem yields ⌫ (Qc (µ)) = 1.
R
Since ⌫ (Qc (µ)) = 1, the seller profits are ⇧c (µ, ⌫) = S ⇡ (q) ⌫ (dq). Since ⇡ is upper semicontinuous, the expected profit of the seller is also upper semicontinuous with respect to policies (see
Theorem 15.5 of Aliprantis and Border (2006)). Using ⌫n⇤ as defined from Lemma 13, we thus have
Z
Z
⇧c (µ, ⌫) =
⇡ (q) ⌫ (dq) lim supn
⇡ (q) ⌫n (dq)
S
S
= lim supn ⇧ (µn , ⌫n )
⇤
= ⇧c (µ, ⌫ )
lim ⇧c (µn , ⌫n⇤ )
n
⇧c (µ, ⌫)
In other words, since the payoffs from the optimal policies ⌫n must be higher than those of the
approximate policies ⌫n⇤ and the profits of the latter converge to that of the optimal policy ⌫ ⇤ , so
must the profits from ⌫n . Thus ⇧c (µ, ⌫) = ⇧c (µ, ⌫ ⇤ ) so ⌫ 2 'c (µ) as desired.
Finally, we put everything together. Consider a sequence of "n ! 0 such that "n < "⇤ and let cn
be such that 'cn is non-empty as guaranteed by Lemma 11. Combined with Lemmas 8, 9 and 14,
the preconditions of KFG (Corollary 17.55 of Aliprantis and Border (2006)) are satisfied so there
exists an equilibrium µn 2 M"n . Moreover, from Lemma 11, we know that |Vcn (µn , q)
W (µ̄, q)| <
"n ! 0 so the buyer’s value function converges to his full-information payoff in the limit. We have
thus constructed a full-information limit equilibrium as desired.
B.3
Proof of Proposition 2
Recall a belief q is maximal if ⇡ (q) = ⇧⇤ (q) where ⇧⇤ is the monopoly payoff. An event E is
maximal if pE is maximal. We first show that all limit equilibria are partitional. Consider a
sequence of decreasing search costs cn ! 0 and associated equilibria µn such that the buyer value
functions Vcn (µn , q) converges to some limit function V ⇤ (q). Let E be any limit partition, that is,
for any N > 0, there is some n > N where E = Eµn (note there may be multiple limit partitions).
We will show that the buyer’s value functions converges to the payoff from learning E. Define a
1
subsequence of equilibria µm where Eµm = E. If we let tm be the largest integer less than cm2 , then
the buyer’s value function is at least his payoff from visiting tm sellers, that is
Vcm (µm , q)
tm cm + W tm (µm , q)
1
2
cm
+ W tm (µm , q)
By Lemma 7, for all µm , W t (µm , q) ! W (µE , q) as t ! 1. Hence, limm Vcm (µm , q)
W (µE , q).
To see why the limit value function is weakly less than W (µE , q), note that since Eµm = E, µE is
more informative than µm . This implies that the value function under µm is less than that from
46
µE . Hence, if we let q2 :=
q
(r), then
Vcm (µm , q) =
cm +

cm +
=
cm +
<
Z
Z
S
max (q2 · u (q2 ) , Vcm (µm , q2 )) (µm )q (dr)
S
max (q2 · u (q2 ) , Vcm (µm , q2 )) (µE )q (dr)
X
E2E
X
q (E) max (qE · u (qE ) , Vcm (µm , qE ))
q (E) qE · u (qE ) = W (µE , q)
E2E
Thus, V ⇤ (q) = limm Vcm (µm , q) = W (µE , q). This proves that all limit equilibria are partitional.
We now show that every event in the partition E is maximal, that is ⇡ (pE ) = ⇧⇤ (pE ) for all
E 2 E. We prove this by contradiction. Suppose there exists some E ⇤ 2 E that is not maximal so
the monopoly policy K ⇤ provides non-trivial information at pE ⇤ . In other words,
Z
⇤
⇡ (pE ⇤ ) < ⇧ (pE ⇤ ) =
⇡ (q) K ⇤ (pE ⇤ , dq)
(12)
E⇤
Define the policy ⌫ that is essentially the same as the partitional information µE but provides
another round of information according to the monopoly policy at pE ⇤ . In other words,
X
⌫ :=
p (E)
pE
+ p (E ⇤ ) K ⇤ (pE ⇤ )
E2E\{E ⇤ }
Also, since the set of signal policies are compact, we can assume that µm converges to some signal
policy µ without loss of generality.
We first show that the seller profit from using policy ⌫ is at most his profit from using the limit
policy µ. Since Eµm = E, µm provides no information at all q 2
for all q 2
E and E 2 E,
Vcm (µm , q) =
Thus,
E and E 2 E. This implies that
cm + q · u (q) < q · u (q)
E ⇢ Qcm (µm ) and since ⌫ only has support in
that µm is an equilibrium, this implies that
Z
⇧cm (µm , µm )
E for all E 2 E, ⌫ (Qcm (µm )) = 1. Given
⇧cm (µm , ⌫)
Z
⇡ (q) ⌫ (dq)
⇡ (q) µm (dq)
S
S
By the upper semicontinuity of profit functions (Theorem 15.5 of Aliprantis and Border (2006)), we
have
Z
⇡ (q) µ (dq)
S
lim sup
m
Z
⇡ (q) µm (dq)
S
47
Z
⇡ (q) ⌫ (dq)
S
(13)
so the profit from µ is at least that from ⌫.
We now show that the profits from µ and µE are the same so inequality (13) cannot be true. This
is because ⌫ is the same as µE except at pE ⇤ where ⌫ yields a strictly higher profit (see inequality
(12)). First, since Vcm (µm , q) ! W (µE , q),
Qcm (µm ) = {q 2
S | q · u (q)
Vcm (µm , q) } ! {q 2
S | q · u (q) = W (µE , q) }
Also, since µm (Qcm (µm )) = 1 for all m, we can use Theorem 17.13 of Aliprantis and Border (2006)
as in the proof of Lemma 14 to obtain
µ ({r · u (r) = W (µE , r)}) = 1
(14)
Finally, since Eµm = E for all m, it must be that for all E 2 E,
µ ({rE = pE })
lim sup µn ({rE = pE }) = 1
n
(15)
Equations (14) and (15) imply that the profit from µ and µE are the same so we have a contradiction
as desired.
48