The Signaling Value of Product Simplicity∗
Sho Miyamoto†
December 14, 2013
Abstract
The value of a technology good is hard both for consumers to
assess at the time of purchase and for a firm to communicate credibly
to consumers. This paper demonstrates that product simplicity is
instrumental in signaling product value and persuading consumers
to purchase a product. I discuss two forms of product simplicity:
(i) simple product instructions and (ii) simple user interface. The
result that product simplicity has a signaling value holds whether or
not consumers are averse to dealing with complexities.
Keywords: signaling, persuasion, simplicity, technology goods
∗
An earlier version of this paper was titled “Signaling by Blurring.” I am especially grateful to
John Nachbar and Maher Said for their generous advice and support. I would also like to thank Filippo Balestrieri, Marcus Berliant, Johannes Hörner, Navin Kartik, David Levine, Shintaro Miura, David
Myatt, Tymofiy Mylovanov, Joel Sobel, and anonymous referees for helpful comments. Participants in
the WUSTL Theory Lunch, the 5th Workshop in Decisions, Games, and Logic, the 22nd International
Conference on Game Theory, and the 4th World Congress of the Game Theory Society provided valuable
feedback. I acknowledge financial support from Center for Research in Economics and Strategy (CRES),
in the Olin Business School, Washington University in St. Louis.
†
Department of Economics, Washington University in St. Louis, [email protected].
1
1
Introduction
The value of a technology good is hard for consumers to assess at the time
of purchase. This is because consumers lack detailed knowledge of how the
technology actually works, before they use it on a frequent basis. If a firm
has a product with high value, how can the firm communicate this private
information credibly and persuade more consumers to purchase the product?
This is the question I address in this paper.
I show that product simplicity is instrumental in communicating product
value and persuading consumers to purchase a product. It is optimal for the
firm to use product simplicity in order to provide a credible signal that its
product has a high value for consumers. I construct two models on the same
framework of signaling games between a firm and consumers, and discuss two
forms of product simplicity: (i) simple product instructions and (ii) simple
user interface.
For example, Apple uses product simplicity as an important message in
product marketing and company branding. The company has consistently
downplayed user manuals in order to claim that its products are usable even
without detailed instructions. It also insists on simple user interface design
as a visible cue that its designers have a good understanding of what outputs
the users want. After Apple’s success in consumer electronics, an emphasis
on product simplicity as a way to communicate product value to consumers
seems to be a growing trend in the industry. One can also observe the same
trend in the industry of website designs, where the designers strive to simplify
websites to invite more visitors.
In Section 2, I provide a framework where a firm wants to sell a product
to consumers who have different outside options and are uncertain about
their utility from the product. This uncertain user utility depends on the
firm’s private type and its observable choice. At the time of purchase, consumers use the product’s observable characteristic to make inferences about
the firm’s type and its product value. Provided that the firm wants to persuade more consumers to purchase the product as well as cares about each
purchaser’s satisfaction from the product, this framework is a signaling game
where the firm’s costly signal is a choice that decreases the user’s utility from
the product.
2
As one such costly signal, I show in Section 3 that the firm may want to
provide simple product instructions. In the model here, the firm informs consumers about how to best use its product through the instructions. Providing
simple instructions is essentially introducing noise in information transmission about product usage. The firms can produce high value products if they
better understand user behaviors, and only those firms can provide simple
instructions and still maintain high purchaser satisfaction. Given this monotone structure, consumers expect high satisfaction from the product with
simple instructions.
The firm’s problem here is different from the disclosure decision of product attributes as in Lewis and Sappington [9], because more information
disclosure is better for all consumers in the case of instructions.1 We do
not observe the full disclosure result of Milgrom and Roberts [11], however,
since the firm’s private information is not verifiable. Also, the reason the
firm withholds valuable information is not because the firm prefers the other
party’s decision when learning is costly to other decisions, as in Ellison and
Ellison [7] and Perez-Richet and Prady [14], but because the firm wants to
signal another piece of private information.
In Section 4, I present a model of user interface design and show that the
firm can use simple user interface to signal product value. I use a conceptual framework for human-computer interaction discussed by Norman [15]
and model the user’s product experience as a hypothetical communication
between a firm and the user. The user starts with an intention or a psychological goal, communicates it to the product through inputs, and obtains
outputs that are the firm’s preprogramed response to the inputs. Designing
simple user interface makes it more difficult for the users to specify what
outputs they want. Here, the firms can produce high value products if they
better understand the users’ minds, and only those firms can use simple user
interface and signal high user satisfaction to consumers.
Two forms of product simplicity are compared in Section 5. In both forms,
product simplicity limits information exchange between a firm and the users.
Through the instructions, the firm informs the users about product usage.
1
The firm’s problem is not a mechanism design problem either, which Kamenica and
Gentzkow [8] and Rayo and Segal [17] have studied.
3
Through user interface, the users specify what outputs they want to see. The
direction of information flow is opposite, but both forms of product simplicity
introduce noise in information transmission. With simple instructions, the
users do not learn enough about product usage. With simple user interface,
the users cannot command a product to produce outputs as they desire.
The good type firms find it strategically advantageous to use this observable
noise to signal their types. An observation that noise in the communication
channel serves as a costly signal has not been discussed even in more general
environments of strategic information transmission.2
In Section 6, I examine the robustness of the results to an additional
assumption that consumers are averse to dealing with complex information.
Since product simplicity limits the amount of information that the users deal
with, consumers would value a moderate degree of simplicity even without
inferences about firm type. Here it is shown that the firm adds an extra
degree of simplicity to signal product value.
2
Product Uncertainty and Consumer Decisions
Here I provide a framework upon which I build in Sections 3 and 4. In this
section I focus on analyzing how a signaling structure arises when a firm tries
to persuade consumers to purchase its product. I defer the interpretation of
the model to the following sections.
2.1
A signaling game between a firm and consumers
A firm produces a product with marginal cost m and sells it to a unit mass
of consumers at competitive price p. If consumers purchase the product,
they get v − c(h, z), which depends on the firm’s type z ∈ [z, z̄] ≡ Z and
the firm’s observable choice h ∈ [h, h̄]. About the firm’s type z, consumers
have some prior belief represented by a distribution with positive density on
2
The idea that noise in the communication channel can improve information transmission was discussed by Myerson [13] and Blume et al. [4]. Blume and Board [3] study how
the Sender’s strategic choice of a message adds endogenous noise to the exogenous noise
of vague natural language.
4
Z. I discuss assumptions on c(h, z) at the end of this subsection, and the
following sections provide stories about z, h, and c(h, z).
If a consumer does not purchase the product, he or she gets 1 − w from
their outside option, where w ∈ [0, 1] is distributed according to G(w) that
is twice continuously differentiable with full support on [0, 1].
The firm’s profit per sale is p − m, but it also cares about the user’s
utility from the product v − c(h, z) because of concerns about the firm’s
future profits. For example, purchasers’ word of mouth may affect the sales
of the same product to a later group of consumers or purchasers may update
their brand image of the firm based on their satisfaction from the purchased
product. We model the firm’s gain from unit sale as the weighted average of
the profit per sale and the user’s utility, which is (1−α)(p−m)+α(v−c(h, z))
for a weight α ∈ (0, 1). If we let q denote the measure of purchasers, the firm
maximizes q [1 − ρc(h, z)], where I drop a constant term (1 − α)(p − m) + αv
α
and use ρ ≡ (1−α)(p−m)+αv
.
For ease of exposition, I normalize parameters so that v − p = 1 and
α ≤ (1 − α)(p − m) + αv. With this normalization, a consumer’s benefit from
purchase is 1 − c(h, z) and the firm’s objective is to maximize q [1 − ρc(h, z)],
where ρ takes a value from (0, 1].
The timing of the game is as follows: (1) the firm learns z and chooses
h; and (2) each consumer observes h and w before they decide whether to
purchase the product or to exercise their outside options.
Assumption 1 shows all the restrictions on c(h, z). Parts (i) and (iv)
are technical assumptions. (ii) says that the firm should choose higher h
to increase product value. (iii) says that the firm of lower z type produces
a product with higher value for any fixed choice of h. Given (v) and (vi),
no firm type would choose h as low as h and any firm type can find some
purchasers by choosing h̄.
Assumption 1. (i) c(h, z) is positive and twice continuously differentiable.
(ii) ch (h, z) < 0. (iii) cz (h, z) > 0. (iv) chz (h, z) ≤ 0. (v) c(h, z) > 1.
(vi) c(h̄, z̄) < 1.
We note that with Assumption 1, this game is a signaling game in which
consumers use the firm’s choice of h to make inferences about its type z. It
5
is costly for the firm to choose h below h̄, since it reduces the user’s utility
from the product and the firm cares about that. Assumption 1 imposes
a monotone structure on the problem so that lower z types are more able
to choose low choices of h and that lower z types induce higher q or more
consumer purchases.
2.2
Signaling equilibrium
I analyze a perfect Bayesian equilibrium of this game. Let ρ(z|h) denote consumers’ posterior belief about z given the observed choice of h. A consumer
purchases the product if
Z z̄
w>
c(h, z)γ(z|h) dz.
(1)
z
This inequality determines the measure of purchasers q̂(h) when the firm
chooses h.
Let η(z) be the firm’s pure-strategy choice of h given z and let q(z) =
q̂(η(z)) be the equilibrium measure of purchasers for type z. The firm’s
incentive compatibility condition is that for all z ∈ Z,
z ∈ arg max q(z̃) [1 − ρc (η(z̃), z)] .
z̃∈Z
(IC)
As is the case with a standard signaling game, this game has multiple
equilibria without further restrictions on consumers’ belief when they observe h chosen with zero probability in equilibrium. To obtain a sharp prediction, we refine off-equilibrium beliefs by condition D1, developed by Cho
and Kreps [5] and Banks and Sobel [1]. After the firm’s deviation to an
off-equilibrium signal, consumers believe that the signal should have been
chosen by the type that is most likely to deviate. This deviator type finds
the deviation preferable to its equilibrium strategy for the lowest measure of
purchasers after the deviation.3
3
Technically, our refinement is different from theirs because there are multiple Receiver
types in this model. They compare the set of mixed-strategy best responses to a deviation
for a single Receiver that make a Sender type willing to deviate. We instead compare the
set of profiles of pure-strategy best response for all Receiver types that make a Sender
6
Given any off-equilibrium signal h ∈
/ η(Z), we let
ξ(z|h) =
q(z) [1 − ρc(η(z), z)]
1 − ρc(h, z)
denote the measure of purchasers that makes type z indifferent between
choosing its equilibrium choice η(z) and deviating to the off-equilibrium
choice h. Condition D1 requires that for any off-equilibrium choice h ∈
/ η(Z),
0
0
γ(z |h) > 0 if and only if ξ(z |h) ≤ ξ(z|h) for all z ∈ Z such that ξ(z|h) > 0.
This refinement is motivated by the following forward induction reasoning. If, on observing an off-equilibrium signal, all consumers purchase the
product, many types would want to choose the signal.4 If instead less consumers purchase the product, fewer types would find the deviation profitable.
As the measure of purchasers decreases, the firm type that is most likely to
deviate will be identified. Consumers believe that the off-equlibrium signal
was chosen by this firm type.
With this belief refinement, Proposition 1 shows that a unique prediction
of the game is a separating equilibrium, where the firm that can produce a
product with higher value chooses lower h and does not maximize the user’s
utility from the product. The choice of low h signals that the firm is a good
type with low z, since a bad type with high z would not be able to satisfy
purchasers if it chooses low h. At equilibrium, consumers learn the value of
the product v − c(h, η −1 (h)) from the firm’s choice of h. As the firm chooses
lower h, consumers’ expected utility from the product increases because the
positive effect of learning that the firm is a better type more than offsets the
direct negative effect of lower h. Consequently, more consumers purchase a
product with low h.
Proposition 1. Under Assumption 1, a D1 equilibrium exists, is unique, and
is separating. At the equilibrium, the firm’s choice η(z) is strictly increasing
in its type z and η(z̄) = h̄. The user’s utility v − c(η(z), z) and the measure
of purchasers q(z) are strictly decreasing in the firm’s type z.
type willing to deviate. In light of forward induction reasoning, our refinement is still that
of condition D1. See Cho and Sobel [6] and Ramey [16] for details about condition D1 in
signaling games.
4
It follows that the intuitive criterion of Cho and Kreps [5] has no bite in this model.
7
3
Signaling through Product Instructions
In the first application, consumers do not know how to use a product as well
as how usable the product is made. A firm informs the users about product
usage through product instructions. By product instructions, I mean any
kind of medium through which consumers learn about product usage, such
as package instructions, user guides, and the instructions in advertisement.
In this section, I first show that the game can be mapped into the framework of the previous section. Then, I use the result of Proposition 1 to
discuss how the firm uses the instructions as a signal of product usability.
3.1
Uncertainty about product usage
Suppose that a consumer’s satisfaction from a product is v − (a − ω)2 , where
ω ∈ R represents the ideal way to use the product and a ∈ R how the
consumer actually uses it. Consumers initially have no information about
ω, endowed with a flat prior on the real line.5 As a producer, the firm has
information represented by a noisy signal x that is an unbiased estimator of
ω with variance z. The firm knows how much it knows about ω, which is
captured by z.
The reason the firm does not know how to best use its own product is
because this knowledge depends on understanding user behaviors. A firm
creates its product based on its mental model of how a hypothetical user
behaves with the product. How close this user image x is to the ideal image
ω depends on how much the firm studies the users. This study includes
collecting data about the users through consumer research and conducting
scientific research in such fields as human factors (ergonomics) and cognitive
science. Through its own decision on how many resources to spend on the
study, the firm is aware of how much it understands the users.
The firm discloses its knowledge x through product instructions. Consumers would learn x exactly if the firm discloses all information. Otherwise
5
Alternatively, we can use a circle with arc length measure as the space of ω and a. The
analysis is the same in this alternative model as long as the length of the circumference
is large enough. Otherwise, the unique D1 equilibrium will be separating only for high z
types and pooling among low z types.
8
they would learn not the exact point of x but something close to it. How close
consumers’ learning is to x depends on the amount of information provided in
the instructions, which is denoted by h ∈ [h, h̄]. Formally, consumers receive
a noisy signal s that is an unbiased estimator of x with variance b(h), where
b(h) is a strictly decreasing and twice continuously differentiable function
with b(h) > 1 and b(h̄) = 0.
When consumers decide whether or not to purchase a product, they examine product instructions to learn about it. At this moment, they can
identify h and infer b(h) through the detailedness and informativess of the
instructions.
We analyze a game where a firm chooses the detailedness of product
instructions, through which consumers learn about its usability represented
by the expected utility from the product. The firm has private information
with respect to x (its product usage) and z (how much it understands the
users), but the firm is essentially characterized by z because it does not lie
about x intentionally if it does not fully reveal it. In the game, the firm first
chooses h given its type z. Then consumers observe h and w and receive a
noisy signal s before they decide whether or not to purchase the product.
3.2
The signaling value of simple product instructions
When consumers observe h and receive a noisy signal s, their expected utility
from the product is
i
h
2
2
v − Eω,x,z (a − ω) |s, b(h) = v − Ex,z Eω [(a − ω) |x, z]s, b(h)
h
i
= v − Ex,z (a − x)2 + z s, b(h)
= v − (a − s)2 − b(h) − E [z|b(h)] .
To maximize this, consumers would choose a = s and use the product as
they understand from the instructions. Given the normalization, v − p = 1,
consumers purchase the product if
w > b(h) + E [z|b(h)] .
9
Anticipating a = s, the firm expects that purchasers will get
h
i
2
2
v − Eω,a (a − ω) |x, z, b(h) = v − Ea (a − x) + z x, z, b(h)
= v − b(h) − z.
The firm’s gain from unit sale is the weighted average of profit per sale and
purchaser satisfaction, which is (1 − α)(p − m) + α(v − b(h) − z). With the
α
normalization of ρ ≡ (1−α)(p−m)+αv
∈ (0, 1], the firm effectively maximizes
q [1 − ρ(b(h) + z)].
Now if we let c(h, z) = b(h) + z and assume that [z, z̄] ( (0, 1), we can see
that the game here fits the framework of Section 2 and that Assumption 1
is satisfied. We invoke Proposition 1 to get get the following corollary about
the simplicity of product instructions (the inverse of h) and the usability of
a product (the expected utility from the product).
Corollary 1. The firm that better understands the users provides simpler
product instructions. Consumers infer the usability of a product by observing how simple the instructions are written. The firm that provides simpler
instructions sells its product to more consumers, because at equilibrium, consumers know that the product with simpler instructions is more usable.
The result here helps explain Apple’s novel marketing strategy in the consumer electronics industry. All its products boast minimalist user guides that
make a sharp contrast to typical electronics’ manuals with lengthy product
details. Suppressing detailed product information provides a credible signal
that its products are actually easy to use, since otherwise a thick manual
would be indispensable to aid users. When Apple unveiled iMac in 1998,
it ran a commercial ad called “Simplicity Shootout” and demonstrated how
easy it is to set up the computer without a bulky setup manual.
4
Signaling through User Interface
For the second application, we consider the user interface of a technology
product. The user interface, often abbreviated as UI, is the space where
the users input information for outputs. With the framework of Section 2, I
discuss how the firm uses the interface design as a signal of product quality.
10
4.1
Uncertainty about user intention
The user starts with an intention or a psychological goal and commands the
product to provide outputs for this goal. The user interface of the product
collects information about the user’s intention through input information
and it automatically returns outputs given the input-output mapping that is
preprogrammed by the firm. Since the firm and the user can have different
mental models about the product technology, there generally exists a gap
between the user’s intention and the product’s outputs.
To capture this idea, we suppose that the user’s satisfaction from a product is v − (a − ω)2 , where ω ∈ R represents the user’s intention and a ∈ R
the product’s outputs. The user knows his or her intention ω but the firm
has no information about it without the user’s inputs. With a flat prior on
the real line, the firm updates its belief about ω based on input information
and it returns outputs a.
When a firm designs its product, it specifies how much information the
user inputs at the interface h ∈ [h, h̄]. The firm is uncertain about how the
user would communicate about ω through his or her choices of the inputs.
Thus the firm interprets the user’s future inputs as a noisy signal x = s + of their intention ω. The user’s description s, which is an unbiased estimator
of ω with variance b(h), becomes more precise with more inputs. We again
assume that b(h) is a strictly decreasing and twice continuously differentiable
function with b(h) > 1 and b(h̄) = 0. The noise term , which is independent
of s and has mean zero and variance z, captures miscommunication due to
the fact that the firm and the user hold different mental models about the
technology. The firm is aware of how much it knows about the users, which
means that it knows z.
When consumers decide whether or not to purchase a product, they want
to learn about the product quality. But, at this point, they can only learn h
through the design of user interface. They see how many pieces of information
the user interface collects from the user and thus how complicated the user
interface is made.
We analyze a game where the firm designs the user interface of its product,
through which consumers learn about the product’s quality. The firm with
private information about z chooses h. Consumers observe h and w and
11
decide whether or not to purchase the product. When they use the purchased
product, they discover their intention ω and input information for outputs
a. For the users, the outputs are random because they do not know how the
firm preprogrammed the input-output mapping.
4.2
The signaling value of simple user interface
At the stage of product design, the firm interprets the user’s future inputs
as a noisy signal x of their intention ω. Since x is an unbiased estimator of
ω with variance b(h) + z, the firm’s expectation of user satisfaction is
v − Eω (a − ω)2 |x, b(h) = v − (a − x)2 − b(h) − z.
To maximize this, the firm preprograms the input-output mapping to output
a = x for the signal x. Anticipating a = x, consumers purchase a product if
w > b(h) + E [z|b(h)] ,
where we used the normalization, v − p = 1.
The firm’s gain from unit sale is the weighted average of profit per sale
and purchaser satisfaction, which is (1 − α)(p − m) + α(v − b(h) − z). With
α
the normalization of ρ ≡ (1−α)(p−m)+αv
∈ (0, 1], the firm effectively maximizes
q [1 − ρ(b(h) + z)].
Here again if we let c(h, z) = b(h) + z and assume that [z, z̄] ( (0, 1), we
can see that the game fits the framework of Section 2 and that Assumption 1
is satisfied. We invoke Proposition 1 to get get the following corollary about
the simplicity of user interface (the inverse of h) and the product’s quality
(the expected utility from the product).
Corollary 2. The firm that better understands the users’ intention uses simpler user interface for its product. Consumers infer the quality of a product by
observing how simple the user interface is made. The firm that uses simpler
user interface sells its product to more consumers, because at equilibrium,
consumers know that the product with simpler user interface provides them
better outputs.
We can use this result to understand why Apple insists on simple user
interface design. In reference to the simplicity of user interface, Apple’s
12
designers say that they study the user’s desires well enough to be able to get
rid of the choices that are not essential, and that their products are better
because of their better understanding of the user’s mind.
A programmable thermostat by Nest Labs provides another example. Despite the fact that it is a programmable thermostat as its competitors, its
sales pitch emphasizes that “it programs itself” because it is a learning thermostat. The users do not need to program it for the ideal room temperature
and can “use it like [their] old thermostat.”6 The user interface looks like an
old nonprogrammable thermostat and the users are supposed to simply spin
a dial to turn it up and rotate it back to turn it down. Although more user
inputs are ideal for a better programming, the simple user interface plays a
role of signaling the quality of the new technology.
5
Discussions
In both applications, the user’s satisfaction from a product is represented
by v − (a − ω)2 . But they had different stories about a and ω. In the first
application, the firm has better information about ω and the user chooses a
given the firm’s noisy message about ω through product instructions. In the
second application, consumers know ω and the firm specifies outputs a for the
user’s inputs that are informative about ω. Although the firm’s choice of h
controls information transmission about ω in both applications, the direction
of information flow is opposite.
Despite these differences, both results are derived from Proposition 1 and
thus they rely on the same insight. In a setup where information transmission
improves the user’s utility, the firm limits this information transmission in
order to signal that consumers can get a high utility from the product. Limiting information flow works as a credible signal because only good type firms
can afford the inefficiency of limiting information transmission. In these applications, the good type firms know more about users and are able to satisfy
the users with less information flow.
6
“Life with
thermostat.
Nest
Thermostat”
at
https://nest.com/thermostat/life-with-nest-
13
6
Complexity aversion
We have seen how product simplicity in the forms of product instructions
and user interface can be used to signal the user’s utility from the product.
But there is an alternative explanation for why consumers value some degree
of product simplicity. It is because they derive negative utilities from dealing
with complexities.
For instance, engineers in the computer game industry were aware of
consumers’ complexity aversion as early as 1972. For Atari’s arcade video
game, Pong, the only instructions were: “Avoid missing ball for high score.”
One of the creators says that “[games] have to be simple, no instructions in
general. You should be able to be intuitively involved in the game without
having to read a 40 page manual.”7
Now suppose that the users incur cognitive cost δk(h) when they deal with
the amount of information h, where k(h) is positive and strictly increasing
in h and δ ≥ 0 represents the degree of complexity aversion. The consumer’s
expected utility from the product is now v − E [c(h, z)] − δk(h).
We add the following assumption so that the results of Section 2 and this
section are easy to compare. It guarantees that any firm type can find some
purchasers by choosing h∗ (δ), which is uniquely defined given the convexity
assumption (i).
Assumption 2. (i) c(h, ·) and k(h) are strictly convex in h. (ii) c (h∗ (δ), z̄)+
δk(h∗ ) < 1, where h∗ (δ) is a unique minimizer of c(h, z̄) + δk(h).
Proposition 2 shows that the main result still holds even if we take consumers’ complexity aversion into account. The only difference is that the
worst type now chooses h∗ (δ) instead of h̄ because the users dislike dealing
with complex information.
Proposition 2. Under Assumption 1 and 2, the presence of complexity
aversion does not change the qualitative result of Proposition 1, except that
η(z̄) = h∗ (δ).
7
“An Interview with Nolan Bushnell” at http://www.interviewsbyspencer.com/Bushnell 607.htm.
14
The firm may use simple product instructions or simple user interface for
two different reasons. First, consumers value a moderate degree of simplicity
because they do not want to spend cognitive resources reading long product
instructions or working with complex user interface. Second, the firm needs
to persuade consumers to purchase its product and wants to use these forms
of product simplicity as a credible signal that the product has a high value
for consumers.
7
Conclusion
In this paper, I presented a framework where a firm controls an observable
aspect of its technology product to signal the user’s utility from the product.
I used this framework to discuss the signaling value of simple instructions
and simple interfaces.
The optimal design problem of a technology good is an interesting topic to
examine to a further degree. The technology design is not just a visual style.
The designer makes a series of decisions long before the user see the product.
What color should I display here? Should this come on this page or next?
What changes does the user can make? These questions are sometimes hard
to answer, because the product designer and the user are different people.
An easy solution for the designer is to make it a preference for the user to
decide instead. As I discussed in this paper, this will not make a good design
in general. For one thing, it signals that the designer does not understand
the user well. For another, the user has to spend cognitive resources on the
decision.
How many decisions should the designer and the user each make? If the
number of the user’s decisions decrease with product simplicity, what is the
optimal degree of simplicity? Albert Einstein once said: “Everything should
be made as simple as possible, but not simpler.” Its exact meaning and
the underlying economic theory are nontrivial and challenging. It might be
fruitful to investigate these questions in the spirit of Battigalli and Maggi
[2]. I leave this for future research.
15
A
Appendix
Proofs of Propositions 1 and 2
Here I prove Proposition 1 and 2 together. Let U (h, z) = q̂(h) (1 − ρc(h, z))
and U ∗ (z) = U (η(z), z).
Lemma 1. U ∗ (z) is positive and continuous on Z and strictly decreasing
in z.
Proof. First, observe that q̂(h̄) > 0 and U (h̄, z) > 0 for all z ∈ Z, since if h =
R z̄
h∗ (δ) then z c(h∗ (δ), z)γ(z|h∗ (δ))dz+δk(h∗ (δ)) ≤ c(h∗ (δ), z̄)+δk(h∗ (δ)) < 1
for any belief γ(·|h∗ (δ)). Since the firm can always choose h = h∗ (δ), it must
be that for any z ∈ Z, q(z) > 0 and U ∗ (z) > 0. Next, if z < z 0 then (IC)
and cz (h, z) > 0 imply that U ∗ (z) ≥ U (η(z 0 ), z) > U ∗ (z 0 ). Last, suppose
that there exist ẑ ∈ Z and > 0 such that U ∗ (ẑ−) − U ∗ (ẑ+) = . From the
uniform continuity of U (h, z) in z, there exists δ > 0 such that if |z − z 0 | < δ
then |U (η(z), z) − U (η(z), z 0 )| < . But, for z < ẑ < z 0 , U (η(z), z) =
U ∗ (z) > U ∗ (ẑ) > U ∗ (z 0 ) ≥ U (η(z), z 0 ) and hence U (η(z), z) − U (η(z), z 0 ) >
, a contradiction.
Lemma 2. η(z) is weakly increasing in z and q(z) is weakly decreasing in z.
Proof. Let z < z 0 and suppose η(z) > η(z 0 ) and ρc(η(z), z 0 ) < 1. From (IC)
for z and z 0 ,
q(z) [1 − ρc (η(z), z)] ≥ q(z 0 ) [1 − ρc (η(z 0 ), z)]
q(z 0 ) [1 − ρc (η(z 0 ), z 0 )] ≥ q(z) [1 − ρc (η(z), z 0 )]
Since q(z) > 0 and U ∗ (z) > 0 for all z ∈ Z, we can rewrite these inequalities
as
1 − ρc (η(z 0 ), z)
q(z)
1 − ρc (η(z 0 ), z 0 )
≤
≤
.
(2)
1 − ρc (η(z), z)
q(z 0 )
1 − ρc (η(z), z 0 )
Define ζ(z̃) =
ζ 0 (z̃) =
1−ρc(η(z 0 ),z̃)
.
1−ρc(η(z),z̃)
Since
−ρcz (η(z 0 ), z̃) [1 − ρc (η(z), z̃)] + ρcz (η(z), z̃) [1 − ρc (η(z 0 ), z̃)]
< 0,
[1 − ρc (η(z), z̃)]2
we have ζ(z) > ζ(z 0 ), which contradicts the inequalities in (2). Therefore
q(z)
η(z) < η(z 0 ). Then, from (2) and (IC), we obtain q(z
0 ) ≥ ζ(z) ≥ 1.
16
Lemma 3. In a D1 equilibrium, η(·|h) must satisfy:
(i) if h ∈ [0, η(z)), then γ(z|h) > 0 iff z = z;
(ii) if h ∈ η(z̄), h̄ , then γ(z|h) > 0 iff z = z̄; and
(iii) if h ∈ (η(z̃−), η(z̃+)) for z̃ ∈ Z, then γ(z|h) > 0 iff z = z̃,
where for z̃ = z or z̄, we let η(z−) = η(z) and η(z̄−) = η(z̄).
Proof. Let z < z 0 and h ∈
/ η(Z). Define φ(h) =
φ0 (h) =
1−ρc(h,z)
1−ρc(h,z 0 )
and observe that
−ρch (h, z) [1 − ρc(h, z 0 )] + ρch (η(z), z̃) [1 − ρc(h, z)]
< 0,
[1 − ρc(h, z 0 )]2
since we have cz (h, z) > 0 and chz (h, z) ≤ 0. Consider the following cases.
(1) When h < η(z) < η(z 0 ), we have
q(z) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
ξ(z|h)
=
·
·
ξ(z 0 |h)
q(z 0 ) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
1 − ρc (η(z 0 ), z 0 ) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
≤
·
·
1 − ρc (η(z), z 0 ) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
φ (η(z))
< 1.
=
φ(h)
(∵ (2))
(2) When h < η(z) = η(z 0 ), we have q(z) = q(z 0 ) and
ξ(z|h)
q(z) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
=
·
·
ξ(z 0 |h)
q(z 0 ) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
φ (η(z))
=
< 1.
φ(h)
(3) When η(z) < η(z 0 ) < h, we have
ξ(z|h)
q(z) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
=
·
·
ξ(z 0 |h)
q(z 0 ) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
1 − ρc (η(z 0 ), z) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
≥
·
·
1 − ρc (η(z), z) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
φ (η(z 0 ))
=
> 1.
φ(h)
17
(∵ (2))
(4) When η(z) = η(z 0 ) < h, we have q(z) = q(z 0 ) and
ξ(z|h)
q(z) 1 − ρc (η(z), z) 1 − ρc (h, z 0 )
=
·
·
ξ(z 0 |h)
q(z 0 ) 1 − ρc (η(z 0 ), z 0 ) 1 − ρc (h, z)
φ (η(z 0 ))
> 1.
=
φ(h)
Since γ(z 0 |h) > 0 iff ξ(z 0 |h) ≤ ξ(z|h) for all z ∈ Z such that ξ(z|h) > 0,
the result (i) follows from (1) and (2), and (ii) from (3) and (4). To see (iii),
note that η(z) ≤ η(z̃−) for all z < z̃ and η(z̃+) ≤ η(z) for all z > z̃, which
implies that (iii) follows from the similar arguments to the ones for (i) and
(ii).
Lemma 4. η(z) is strictly increasing and continuous in z, and η(z̄) = h∗ (δ).
Proof. First, define T (z) = {z̃ ∈ Z : η(z̃) = η(z)}. From
(1), if h = η(z 0 ) for
R
0
c(h,z) dF (z)
some z 0 ∈ Z then q̂(h) = 1 − G [w̄(h)], where w̄(h) ≡ T (zR ) dF (z) + δk(h).
T (z)
Next, we note that since η is monotone, if it has discontinuities they are
jump discontinuities. To see that η is strictly increasing is to show |T (z)| = 1
for all z ∈ Z.
By way of contradiction, suppose |T (z 0 )| > 1 for some z 0 ∈ Z. By the
monotonicity of η, T (z 0 ) is an interval. We now check that this type z 0 has
an incentive to deviate in any possible case. Let z 00 = inf T (z 0 ). If z 00 = z,
then Lemma 3 (i) and (iii) imply that type z 0 would deviate to h = η(z 0 ) − for small > 0 and get perceived as type z. So suppose z 00 > z. If η is
discontinuous at z 00 , then Lemma 3 (iii) implies that type z 0 would deviate
to h = η(z 0 ) − for small > 0 and get perceived as type z 00 . If instead η
is continuous at z 00 , then type z 0 would deviate to η(ẑ) = η(z 0 ) − for small
> 0 and get perceived as type ẑ.
If η is discontinuous at some point, say z 0 , then η being strictly increasing
implies U ∗ (z 0 −) < U ∗ (z 0 +), which contradicts the continuity of U ∗ . So η is
continuous everywhere.
Finally, note that η(z̄) = h∗ (δ) as otherwise a deviation is possible for
z = z̄.
Lemma 5. A D1 equilibrium exists and is unique.
18
Proof. In a separating equilibrium, (1) implies that q(z) = 1−G [c (η(z), z) + δk(η(z))].
Then (IC) becomes
z ∈ arg max [1 − G [c (η(z̃), z̃) + δk(η(z̃))]] [1 − ρc (η(z̃), z)] .
z̃∈Z
As a monotonic function, η is differentiable almost everywhere. The necessary first order condition at points of differentiability gives
η 0 (z) =
Γ1 (η(z), z)
Γ2 (η(z), z)
(3)
Γ1 (η(z), z) = −cz (η(z), z)G0 [ŵ(η(z), z)] [1 − ρc(η(z), z)]
Γ2 (η(z), z) = [ch (η(z), z) + δk 0 (η(z))] G0 [ŵ(η(z), z)] [1 − ρc(η(z), z)]
+ ρch (η(z), z) [1 − G [ŵ(η(z), z)]] ,
where ŵ(η(z), z) = c(η(z), z) + δk(η(z)). Let f (z, η) denote the right hand
side of (3) with η(z) being replaced by η. We need to find a unique solution
η̃ to the terminal value problem
η 0 = f (z, η), η(z̄) = h∗ (δ).
n
o
Define D = (z, η) : z ≤ z ≤ z̄, ĥ(z) ≤ η ≤ h∗ (δ) , where ĥ(z) is defined
such that c(ĥ(z), z) + δk(ĥ(z)) = c(h∗ (δ), z̄) + δk(h∗ (δ)). Since f ∈ C(D) and
(z̄, h∗ (δ)) ∈ D, there exists a local solution η̃ on (z 0 , z̄] for some z 0 ∈ (z, z̄).
Since f (z, η) is Lipschitz continuous in η (as ∂f
∈ C(D)), the uniqueness
∂η
of solutions is guaranteed. It remains to show that the local solution η̃ is
continuable to [z, z̄], which requires Lemma 6.8
Lemma 6. Let D̃ be a nonempty connected set in the (z, η) domain and let
f be a bounded and continuous function on D̃. Suppose η̃ is a solution of
η 0 = f (z, η) on the interval (z̃, z̄]. Then (i) the left-hand limit of η̃ at z̃,
η̃(z̃+) exists, and (ii) if (z̃, η̃(z̃+)) ∈ D̃ then the solution η̃ is continuable to
the left past the point z = z̃.
To show η̃ is continuable to [z, z̄], it suffices to show that (z̃, η̃(z̃+)) ∈ D
for any z̃ ∈ (z, z̄). Note that 1 − ρc(η, z) > 1 − c(η, z) > 1 − c(η, z) −
8
This is Theorem 3.1 in Miller and Michel [12].
19
δk(η) ≥ 1 − c(h∗ (δ), z̄) − δk(h∗ (δ)) for any (z, η) ∈ D. Also, ŵ(η, z) ≤
c(h∗ (δ), z̄) + δk(h∗ (δ)) < 1 for any (z, η) ∈ D. From our model assumption,
G0 (w) is bounded away from 0. Thus, f (z, η) is bounded on D. Now I
claim that (z̃, η̃(z̃+)) ∈ D for any z̃ ∈ (z, z̄). To see this, note that for any
(z, η(z)) ∈ D,
η 0 (z) < −
cz (η(z), z)
= ĥ0 (z).
ch (η(z), z) + δk 0 (η(z))
Then we observe that for any z̃ ∈ (0, 1),
Z
z̄
η̃ 0 (z) dz
η̃(z̃+) = η̃(z̄) − lim
z↓z̃ z
Z z̄
> h(z̄) −
ĥ0 (z) dz
z̃
= ĥ(z̃).
To show that the necessary condition is sufficient, we need to prove that
the strict incentive compatibility condition is satisfied for any z ∈ Z.9 Define
V (z, z̃, h) = [1 − G [ŵ(h, z̃)]] [1 − ρc(h, z)]. We show that for any z ∈ Z,
{η(z)} = arg max V (z, η −1 (h), h).
(4)
h∈η(Z)
Note that (3) can be written as
η 0 (z) = −
V2 (z, z, η(z))
.
V3 (z, z, η(z))
(5)
Also, if 1 − ρc(h, z) > 0, then for any z̃ ∈ Z,
V2 (z, z̃, h) = −cz (h, z̃)G0 [ŵ(h, z̃)] [1 − ρc(h, z)] < 0.
Furthermore, for any z, z̃ ∈ Z,
∂ V3 (z, z̃, η(z̃))
ρ [1 − G(ŵ)] [ρch + chz G0 (ŵ)(1 − ρc)]
=
< 0.
∂z V2 (z, z̃, η(z̃))
cz [G0 (ŵ)(1 − ρc)]2
9
The technique found here is due to Mailath [10].
20
(6)
(7)
By way of contradiction, suppose that there exists z ∈ Z\{0} such that
(4) is not true. Let h0 be a maximizer of V such that h0 6= η(z). Note that
V2 (z, z̃, h0 ) 6= 0 for any z̃ ∈ Z, since 1 − ρc(h, z) > 0 by Lemma 1.
(i) Suppose that η −1 (h0 ) = z 0 ∈ intZ. By the first order condition for (4),
−1 0
0 dη
V2 (z, z , h )
+ V3 (z, z 0 , h0 ) = 0.
dh h=η(z0 )
By (5),
dη −1 + V3 (z 0 , z 0 , h0 ) = 0.
V2 (z , z , h )
dh h=η(z0 )
0
0
0
Combining these equations gives
V3 (z, z 0 , h0 )
V3 (z 0 , z 0 , h0 )
=
,
V2 (z, z 0 , h0 )
V2 (z 0 , z 0 , h0 )
which contradicts (7).
(ii) Suppose that η −1 (h0 ) = z. By the first order condition for z,
−1 0 dη
V2 (z, z, h )
+ V3 (z, z, h0 ) ≤ 0.
dh h=η(z)
By (5),
V3 (z, z, h0 ) V3 (z, z, h0 )
+
V2 (z, z, h ) −
≤ 0.
V2 (z, z, h0 ) V2 (z, z, h0 )
0
But (6) and (7) imply the opposite strict inequality, a contradiction.
(iii) Suppose that η −1 (h0 ) = z̄. By the first order condition for z,
−1 0 dη
V2 (z, z̄, h )
+ V3 (z, z̄, h0 ) ≥ 0.
dh h=η(z̄)
By (5),
V3 (z̄, z̄, h0 ) V3 (z, z̄, h0 )
V2 (z, z̄, h ) −
+
≥ 0.
V2 (z̄, z̄, h0 ) V2 (z, z̄, h0 )
0
But (6) and (7) imply the opposite strict inequality, a contradiction.
21
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