ISSN (Print) 0473-453X

Discussion Paper No. 1216

ISSN (Online) 2435-0982

PECUNIARY EMULATION
AND INVIDIOUS DISTINCTION:
SIGNALING UNDERBEHAVIORAL DIVERSITY

Junichiro Ishida
Wing Suen

October 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanPecuniary Emulation and Invidious Distinction:
Signaling under Behavioral Diversity
JUNICHIRO ISHIDA
Osaka University
WING SUEN
University of Hong Kong
October 17, 2023
Abstract. We introduce behavioral diversity to an otherwise standard signaling
model, in which a fraction of agents choose their signaling actions according to
an exogenous distribution. These behavioral agents provide opportunities forstrategic low-type agents to successfully emulate higher types in equilibrium,
which in turn reduces the cost for strategic high-type agents to separate fromlower types. Behavioral diversity thus improves the equilibrium payoffs toall types of strategic agents. The model also exhibits a convergence property
which is intuitively more appealing than the least-cost separating equilibriumof the standard setting.
Keywords. least-cost separating equilibrium; insensitivity to prior; behavioral
diversity; equilibrium emulation
JEL Classification. D82; Z13

1.

Introduction

From the musical film My Fair Lady to the television drama Inventing Anna, emulating
the ways of the upper class is often depicted as a means to improving one’s social image
and gaining acceptance into high society. In real life, conspicuous consumption is usedby elites to signal their superior status (Veblen, 1899), but is increasingly also adopted
by the aspiring middle class to emulate the elites. LVMH, the world’s leading purveyor ofluxury products, has experienced an almost fourfold increase in its revenue from 2010 to2022.1 Many of its luxury goods are no doubt bought by the truly affluent, but most of therevenue growth is probably driven by the merely well-to-do. Conspicuous consumption isoften viewed in economics through the lens of the standard Spence (1973) signaling model(in which perception about one’s type matters), or status signaling models such as Hopkins
and Kornienko (2004) and Hoppe et al. (2009) (in which perception about one’s relative
rank matters). The focus, however, is usually given to separating equilibria: despite the
incentive of lower types to mimic higher types, the higher types choose signaling actions
just costly enough to deter imitation. Successful emulation, in other words, is not an
equilibrium phenomenon in these models.
In reality, some low-type agents may have high gaming ability (witness the Elizas andAnnas of the real world) that allows them to pool with high types. Frankel and Kartik(2019) show that in a signaling model with multi-dimensional heterogeneity, signaling
actions are not fully revealing, and successful emulation can occur in equilibrium. Another possible deviation from the idealized signaling model is preference diversity. In his
critique of Veblen’s (1899) thesis, Mencken (1919) wrote, “Do I enjoy a decent bath because I know that John Smith cannot afford one—or because I delight in being clean? Do
I admire Beethoven’s Fifth Symphony because it is incomprehensible to Congressmen andMethodists—or because I genuinely love music?” If music lovers of high social status havepreferences that do not have reputation concerns as described by signaling models, thenthese agents will not move on to more difficult repertoire even when the masses flock to
Beethoven in emulation of the higher class. Finally, some of the higher-type agents may
be financially constrained. The landed aristocracy of the nineteenth century might havepreferred to choose even more conspicuous luxury items to differentiate themselves frombankers and industrialists, but many were not able to afford them.
1ros

Total revenue of the LVMH Group increased from 20 billion euors to 79 billion euduring this period.
Source:https://www.statista.com/statistics/245852/

total-revenue-of-the-lvmh-group-worldwide/.

1

In this paper, we capture these deviations due to varying signaling costs, preference heterogeneity, or financial constraints by positing that a fraction of the agents are behavioral
in the sense that their choice of signaling action is exogenously given by a possibly typedependent probability distribution. The remaining agents have standard single-crossing
preferences and may potentially gain from emulating the behavioral agents or emulatingother types of strategic agents. We use this simple model to explore how the possibilityof emulation affects the signaling choices and welfare of strategic agents and to elucidate
the underlying mechanism that leads to those changes.
Main results. Signaling models typically admit a plethora of equilibria. A number ofrefinement criteria have been proposed to narrow down the set of “reasonable equilibria.” Among those, the most widely adopted are the Intuitive Criterion and D1 of Cho andKreps (1987) as they provide a sharp characterization of equilibria: the least-cost separating equilibrium (LCSE) is the only possible form of equilibrium that survives those criteria.
This exclusive focus on LCSE, however, entails one undesirable feature that the equilibrium outcome is invariant to prior belief about the agent’s type. This feature precludes
us from discussing, for instance, the agent’s incentive to build reputation in an earlier
stage of a more dynamic setting. Moreover, insensitivity to the prior leads to an implausible implication as noted by Mailath et al. (1993): consider a two-type signaling modelwith almost complete information where there are very few low-type agents, say one ina million; even in this case, high-type agents are still forced to choose an action, which
is bounded away from their complete information optimum, to separate from low-typeagents that are almost non-existent. This argument suggests that there is a sharp discontinuity between complete information and near-complete information environments. Kreps
and Sobel (1994, p. 860) note that the lack of continuity is “the intuitively least appealing
aspect” of the refinement concepts.
In the model that admits behavioral diversity, we show that the equilibrium signalinglevel is decreasing in the agent’s prior reputation. As a consequence, all types benefit
from having a better prior reputation. The driving force behind this result is the supplyof emulation opportunities relative to the demand. An improvement in the agent’s priorreputation leads to more quality emulation opportunities, which directly benefit lower
types who are in need of emulation. In addition, this reduces the need for lower types tomimic higher types, thereby enabling those higher types to separate more easily at lower
signaling actions.

2This finding leads to two important implications. On the empirical side, suppose wewant to test the Veblen effect by examining the statistical relationship between conspicuous
consumption and wealth. Regressing the level of consumption just on the level of wealthis misspecified because conspicuous consumption depends also on the individual’s priorreputation, which could well be correlated with the level of wealth. If we just look at
the correlation between conspicuous consumption and wealth, we may end up findinga non-monotone relationship, because many truly wealthy individuals may have alreadyestablished their status (i.e., a high prior belief) and feel less compelling need to engage
in conspicuous consumption than the aspiring middle class who have no such status justyet.2
On the theoretical side, this finding indicates the possibility of bridging the gap between complete information and near-complete information environments. In Section 3,
we consider a two-type example in which the equilibrium signaling level of the high typedecreases and converges to the complete information optimum as the prior probability
that the agent’s type is high tends to 1. Section 4 shows that this convergence result holds
generally, for an arbitrary number of types and arbitrary decision rules of the behavioral
types. Moreover, the extent of behavioral diversity required for the convergence can bearbitrarily small—the condition that should be satisfied in virtually any signaling situation.
Our analysis thus suggests that the lack of convergence in the standard setup need not beviewed as a shortcoming of the refinement concepts, as it may well be a consequence ofthe theoretical simplification that excludes behavioral diversity from analysis.
Literature. In his celebrated treatise, Veblen (1899) emphasized two fundamental motives for conspicuous consumption: pecuniary emulation and invidious distinction. Pecuniary emulation refers to the act of a member of a lower class to meet the consumption
standards of a higher class to signal that he belongs to that class, whereas invidious distinction occurs when a member of a higher class consumes conspicuously to distinguishhimself from the lower classes. Between the two notions, there is indication that Veblen
saw pecuniary emulation as the more important force behind conspicuous consumption.
He noted that “[a]mong the motives which lead men to accumulate wealth, the primacy,
both in scope and intensity, therefore, continues to belong to this motive of pecuniary
emulation” (Veblen, 1899, p. 40) and devoted an entire chapter to the treatment of thisconcept in his book. Despite his emphasis, however, the analytical focus of the signaling lit2

A standard approach to account for this potential non-monotonicity is counter-signaling; see Feltovich
et al. (2002), Araujo et al. (2007), and Chen et al. (2022).

3

erature, ever since its inception by Spence (1973), has been primarily on the transmissionof information via the act of separation, which can be seen as a manifestation of invidious
distinction: in the standard setup, the agent’s behavior in equilibrium is driven solely by
the desire to distinguish himself but not by the desire to emulate others.3 In this paper,
we revisit Veblen’s old insight and explore the role of emulation that has hitherto received
scant attention in the literature.
Standard refinement criteria consistently select the LCSE in signaling models. Mailath
et al. (1993) are among the first to point out the rather disturbing implication of thisprediction, which promoted them to propose an alternative refinement criterion called
undefeated equilibrium. In this paper, instead of modifying the ways to refine equilibria, we
modify the model structure, by introducing potential behavioral diversity among agents ofthe same type. In our model, we posit that the agent’s signaling action may be determinedby exogenous stochastic shocks. This approach is related to the literature on noisy signaling(Matthews and Mirman, 1983; de Haan et al., 2011; Heinsalu, 2018), in which the agent’s
chosen action is subject to stochastic shocks. Also, closely related in this regard is Daleyand Green (2014) who consider a setting where the receiver can observe both the sender’saction and a type-dependent noise. A crucial difference is that in this strand of models,a shock occurs after the signaling choice is made, whereas in our model with behavioral
diversity, a shock occurs before the signaling choice.
There are several works that may fall under the category of “signaling under behavioral
diversity.” Frankel and Kartik (2019) and Ball (2022) consider a signaling model in which
agents have multidimensional attributes and may differ in gaming ability. Dilme and Li(2016) consider a dynamic signaling model with dropout risk. They study how the possibility of exogenous dropout affects the trading dynamics. Although we cast our model in astatic setting, their model can be seen as a special case of ours; as a consequence, the twomodels share some technical properties.4 The focus of our study is, however, different,3

In equilibrium, it is not possible for any type to emulate a higher type. The desire to emulate only
works behind the scenes to determine how far the higher type must go, but that itself does not impact his
own signaling choice.
4
Dilme and Li (2016) show that in the model with two types, the equilibrium signaling level decreases
and coincides with the complete information outcome as the prior belief becomes sufficiently high. Daleyand Green (2014) also obtain the convergence result under an extra condition called RC-informativeness,but their underlying reasoning is different: in their model, it is the lack of single crossing that prevents thehigh type from fully separating. This is manifested in the fact that pooling in their model occurs at the top
while pooling in ours occurs at the bottom.

4

as we aim to make a broader point that the possibility of emulation is a ubiquitous andintegral aspect of signaling in society.

2.

A signaling model with behavioral diversity

Consider a signaling model with an agent (sender) and a market (receiver). The agent
is characterized by his type i ∈ {1, . . . , I}. The agent’s type is his private information andcannot be directly observed by the market. We denote the prior belief by P := (p1 , . . . , p I ),where pi ∈ (0, 1) is the probability that the agent’s type is i. The prior belief is an importantprimitive of the model, which captures the agent’s prior reputation.
The model is otherwise standard, except that the agent’s action is determined exogenously with some positive probability. Specifically, with probability δ ∈ (0, 1), the agent is
a behavioral agent whose action is drawn from a set D := [0, d] of “exogenous actions.”
For each type, the distribution of exogenous actions is continuous on D with full support.
We use βi (a) to represent the density of type i choosing action a, which summarizes thePIdecision rules of a behavioral agent in a reduced-form way.5 Let q(a) := i=1 pi βi (a) anddefinePIτ(a) :=

i=1

pi βi (a)iq(a)∈ (1, I)as the corresponding reputation at action a among the behavioral agents. With the remaining probability 1 − δ, the agent is strategic and chooses signaling action a ∈ R+ tomaximize his payoff. We view δ as capturing the extent of behavioral diversity that may
originate, for example, from differences in gaming ability, differences in preferences, orthe presence of external factors that exogenously constrain the agent’s signaling choice.
The payoff to type i is given by Ui (a, t) where t ∈ [1, I] denotes the market’s belief(expectation) of the agent’s type. The payoff function is continuously differentiable, strictly
decreasing in a, and strictly increasing in t. It also satisfies the standard single-crossing
property. Let
M RSi (a, t) := −∂ Ui (a, t)/∂ a∂ Ui (a, t)/∂ t

5

All of our results hold for any arbitrary set D, although the argument becomes more involved when D is
finite and has probability masses. The simplest example of this situation is where the agent has no reputation
concerns and chooses the complete information optimum with probability δ, in which case D = {0}.

5

be the marginal rate of substitution at (a, t). The single-crossing property is equivalent to
M RSi 0 (a, t) > M RSi 00 (a, t) for all (a, t) and i 00 > i 0 .
Let µ(a) denote the market’s belief (expectation) of the agent’s type when his signaling
action is a. The strategy of each type i is denoted by πi , a probability distribution of actions
on R+ . Throughout the analysis, we focus on signaling equilibria defined as follows:
1. Given µ(·), a0 ∈ arg maxa Ui (a, µ(a)) if a0 ∈ supp(πi );2. µ(·) is consistent with {πi } and Bayes’ rule whenever applicable.
As usual, the set of signaling equilibria can be quite large. The D1 refinement requires
that, for any ai 0 ∈ supp(πi 0 ), ai 00 ∈ supp(πi 00 ), and any off-equilibrium action a0 , if for all t,
Ui 0 (a0 , t) ≥ Ui 0 (ai 0 , µ(ai 0 )) =⇒ Ui 00 (a0 , t) > Ui 00 (ai 00 , µ(ai 00 )),then the off-equilibrium belief associated with action a0 must assign zero probability to theagent being of type i 0 . In our setting, however, the standard D1 criterion is not directly
applicable because there are no off-path actions on D. We thus introduce an extended
version of the criterion. Consider a discretized space D n := {d1n , . . . , dnn } where d1n = 0 and
R dn
d jn = ( j − 1)d/n for j = 1, . . . , n, and let βin (d j ) := d nj+1 βi (a) da, so that a behavioral agentjof type i chooses d j ∈ D n with probability βin (d j ). We say that an equilibrium in our modelwith continuous D satisfies (the extended version of) D1 if it is the limit of equilibria withdiscrete D n that satisfy D1 as n goes to infinity. We refer to such equilibrium simply as D1
equilibrium hereafter. In effect, this restriction requires that for any a0 ∈ D which is not in
the support of the equilibrium strategy of any type, no agent has an incentive to deviate
to a0 even when the belief µ(a0 ) = τ(a0 ) is replaced by some other “off-equilibrium” beliefµ̂(a0 ) that satisfies the D1 criterion.6

3.

An example with two types

We may illustrate the main ideas by a simple example with two types. In this example, we
refer to type 1 as the low type and to type 2 as the high type, and let the prior probabilitiesof these two types be 1 − p and p respectively. Also assume for simplicity that the distribution of exogenous actions is type-independent and uniform on D, so that q(a) = 1/d andτ(a) = 1 + p for all a ∈ D.
6

When there are only two types, we may require equilibrium to satisfy an extended version of the Intuitive

Criterion defined in a similar manner, and all the arguments will continue to hold.

6

Let ui and ui be the payoff lower bound and upper bound, respectively, for type i. Since
the highest reputation that can be achieved is t = 2 and effort is costly, it is easy to see
ui := Ui (0, 2) for both types. Note also that the low type can always choose a = 0 andsecure at least t = 1, so that u1 = U1 (0, 1). Finally, define s such that U1 (s, 2) = U1 (0, 1),
which is the action chosen by the high type in the LCSE. Since the low type can secure atleast a payoff of U1 (0, 1), we have u2 = U2 (s, 2) as the payoff lower bound for the hightype. Note that u1 and u2 are the respective payoff that each type earns in the LCSE.
As we will detail in the general case, our model admits a unique D1 equilibrium. In
this equilibrium, the high type always chooses a least-cost separating action and gets the
reputation t = 2. This means that once we fix the high type’s payoff u2 ∈ Υ := [u2 , u2 ], we
can uniquely pin down his corresponding action, s2 (u2 ), from u2 = U2 (s2 (u2 ), 2). Definex2 (u2 ) := (s2 (u2 ), 2) as the low type’s reservation allocation. We say that an action a isbinding for the low type if choosing that action unilaterally gives him a higher payoff than
the reservation allocation. Formally, let t = φ1 (a; u2 ) represent the indifference curve oftype 1 that passes through x2 (u2 ). Then, a is binding if τ(a) > φ1 (a; u2 ) > 1.7 Define
B1 (u2 ) := {a ∈ D : τ(a) > φ1 (a; u2 ) > 1}
as the set of binding actions for the low type when the high type’s payoff is u2 .
In D1 equilibrium in which the high type’s payoff is u2 , the low type randomizes overa ∈ B1 (u2 ) with some density π1 (a; u2 ). Given this, all of the low type’s on-path actions
must give him the same payoff, i.e., µ(a) = φ1 (a; u2 ) for all a ∈ B1 (u2 ). Thus, for eachu2 ∈ Υ and a ∈ B1 (u2 ), the density π1 (a; u2 ) must satisfy Bayes’ rule and the indifference
condition:µ(a) =

δ(1 + p)(1/d) + (1 − δ)(1 − p)π1 (a; u2 )= φ1 (a; u2 ).
δ(1/d) + (1 − δ)(1 − p)π1 (a; u2 )Note that we have π1 (a; u2 ) = 0 for a ∈/ B1 (u2 ). Solving the above equation yields§ªδ(1 + p − φ1 (a; u2 ))π1 (a; u2 ) = max,0 .
(1 − δ)(1 − p)d(φ1 (a; u2 ) − 1)(1)

The equilibrium value of u2 is determined by the requirement that the low type’s mixed
strategy is a valid probability distribution. LetZΠ1 (u2 ) :=π1 (a; u2 ) da.

(2)a∈(0,d]7

It is possible to have τ(a) > φ1 (a; u2 ) = 1. In the general analysis, we treat this case separately because

the mixed strategy (given below) is not well defined at a such that φ1 (a; u2 ) = 1.

7

If Π1 (u2 ) ≤ 1, then the equilibrium payoff to the high type is u∗2 = u2 , and the low type’s
mixed strategy is given by the density π1 (·; u∗2 ), with an atom of mass 1 − Π1 (u∗2 ) at a = 0.8
If Π(u2 ) > 1, then the equilibrium payoff u∗2 satisfies Π1 (u∗2 ) = 1, and the low type’s mixed
strategy is given by the density π1 (·; u∗2 ). Observe that Π1 (u2 ) is strictly decreasing in u2
with Π1 (u2 ) = 0. Therefore, u∗2 exists and is unique.
In this environment, the low type can potentially choose one of the binding actions
to emulate the behavioral agents, or the separating action to mimic the high type. These
two choices are substitutes for the low type and must yield the same payoff. The costof mimicking the high type is given by the cost of the separating action s2 (u2 ), which
we call the “price of signaling.” The price of signaling is determined by the supply ofemulation opportunities relative to the demand, which is captured by (2). For a given u2 ,if Π1 (u2 ) > 1, emulation opportunities are in excess supply, and the price s2 (u2 ) must go
down; if Π1 (u2 ) < 1, they are in excess demand, and the price must go up.
The magnitude of the relative supply of emulation opportunities depends on the balance between the numerator and the denominator of π1 (·; u2 ) in (1). Suppose, for example, that the agent’s prior reputation p increases. Because a behavioral agent on average
has a higher type, emulating a behavioral agent leads to a higher inference about one’s
type. In a sense this effect represents an increase in the supply of emulation opportunities,
and is reflected in an increase in the numerator of π(·; u2 ). At the same time, a higher
p means that there are fewer low types who are attempting to emulate the behavioral
agents. The demand for emulation decreases, and this effect is reflected in a decrease in
the denominator of π(·; u2 ). Both effects raise the overall relative supply Π1 (u2 ), making
emulation a more attractive choice. In equilibrium, the price of signaling s2 (u2 ) has to fallto compensate for the difference. Since s2 (u2 ) is decreasing in u2 , this shows that u∗2 must
increase with p. Of course, a lower price of signaling benefits the low type, and u∗1 also
increases with p.
Since φ1 (a; u2 ) ≥ 1 for any u2 ∈ Υ , (1) shows that the supply of emulation opportunities goes to 0 in the limit as p approaches 0, which raises the price of signaling to the
maximum level. The equilibrium then converges to the LCSE where the maximum levelof signaling is required, and the payoffs to both types approach their lower bounds ui . At
It is possible to have Π1 (u2 ) converge to a finite number if the indifference curves are concave in a,
in which case there could be an equilibrium in which the low type chooses a = 0 with strictly positive
probability. This possibility can be ruled out and the equilibrium mixed strategies are always smooth if theindifference curves are weakly convex. See Online Appendix.
88

the other end, as p approaches 1, the demand for emulation goes to 0. The relative supply
thus diverges to infinity, which makes emulation almost free. The equilibrium converges
to the complete information outcome that requires no signaling effort from the high type,
and the payoffs to both types approach their upper bounds ui .

4.

General analysis

4.1.

Equilibrium existence and characterization

We now extend the analysis to incorporate an arbitrary number of types and an arbitrary
distribution of exogenous actions. We propose an algorithm to find the set of binding
actions and the corresponding mixed strategy recursively for each type, from the second
highest type (type I − 1) to the lowest (type 1). Since the essence of this procedure isalready described in the previous section, we only provide a brief overview of the algorithm
and relegate its details to Appendix A.
In any D1 equilibrium, the highest type must choose a least-cost separating action
and get reputation t = I. The algorithm thus starts with the highest type and his payoffu I ∈ Υ := [u I , u I ].9 Once we fix u I ∈ Υ , the fact that the highest type is getting reputationt = I allows us to uniquely pin down the fully separating action. Denote this action bys I (u I ). In the following, we use ai (u I ) to denote the lower bound of the support of typei’s strategy when the payoff to type I is fixed at u I . Consider the indifference curve oftype I − 1 that passes through the reference allocation x I−1 (u I ) := (a I (u I ), I), which wedenote by φ I−1 (·; u I ). We derive the set of binding actions and the corresponding mixed
strategy for type I − 1, in a manner similar to that described in the two-type example. We
repeat this process to obtain ai (u I ) recursively for each i. A necessary condition for finding
an equilibrium is that we reach the final round to determine a1 (u I ) for the lowest type.
An equilibrium is obtained if a1 (u I ) is well defined, and the mixed strategy of type 1 on[a1 (u I ), a2 (u I )) is a well-defined probability distribution.
According to this algorithm, each type i ≤ I − 1 adopts a possibly mixed strategy on[ai (u I ), ai+1 (u I )). For each u I ∈ Υ , we can identify one indifference curve for each typeon which the allocations of that type lie. We refer to the lower envelope of those indifference curves as the allocation path and denote it by Φ(·; u I ); see Figure 1 for a graphicalillustration. For brevity, if u∗I is the equilibrium payoff to type I, we let φi∗ (·) := φi (·; u∗I )9The upper bound is u I = U I (0, I). If we let s be the action chosen by type I in the LCSE, the lower boundis given by u I = U I (s, I).

9

3

2

1

Figure 1. The equilibrium allocation path (with three types) is the lower envelope of the two equilibrium indifference curves, φ1∗ (·) and φ2∗ (·), depicted by the thick blue line. The thin blue line represents
the LCSE, whose allocation path stays strictly below the equilibrium allocation path.

be the equilibrium indifference curve of type i and Φ∗ (·) := Φ(·; u∗I ) be the equilibrium∗
allocation path. In equilibrium, the action space is divided into I − 1 intervals, [ai∗ , ai+1)

for i = 1, . . . , I − 1, where ai∗ := ai (u∗I ). The equilibrium set of binding actions is:∗Bi∗ := {a ∈ [ai∗ , ai+1
) : τ(a) > φ ∗ (a) > i}.

Also define si∗ := si (u∗I ) such that Φ∗ (si∗ ) = i if it exists, which is the fully separating action
for type i.
Proposition 1. For any δ > 0 and {βi }, there exists a unique D1 equilibrium characterizedby Φ∗ (·). In the equilibrium,
1. Type I chooses a fully separating action s∗I with probability 1;
2. Type i = 1, . . . , I − 1 randomizes over Bi∗ ∪ {si∗ } if Bi∗ is nonempty, and chooses the fully
separating action si∗ with probability 1 if it is empty.
Proof. See Appendix A.

10

4.2.

Comparative statics

Fix some equilibrium Φ∗ (·). For a ∈ Bi∗ , when type i adopts an equilibrium mixed strategy
with density π∗i (·), the equilibrium strategy must satisfy
Φ (a) =∗

δq(a)τ(a) + (1 − δ)pi π∗i (a)iδq(a) + (1 − δ)pi π∗i (a).

∗Define an indicator function ι(·) such that ι(a) = i if a ∈ [ai∗ , ai+1). Let a := sup{a : τ(a) >Φ∗ (a)} be the largest binding action and S ∗ := {si∗ } be the set of fully separating actions
under Φ∗ (·). This formulation allows us to express the mixed strategy by
δq(a)(τ(a) − Φ∗ (a))∗π (a) = max,0 ,(1 − δ)pι(a) (Φ∗ (a) − ι(a))(3)

for a ∈ [0, a)\S ∗ ,10 which can be interpreted as the equilibrium relative supply of emulation
opportunities at action a.
Aside from the preferences, the primitives of the model are summarized by Θ :=(δ, P, {βi }). Now consider an alternative environment characterized by Θ̂ := (δ̂, P̂, {β̂i })and define¨π̂(a) = maxδ̂q̂(a)(τ̂(a) − Φ∗ (a))«,0 ,(1 − δ̂)p̂ι(a) (Φ∗ (a) − ι(a))

PI
PI
p̂
β̂
(a)i
/q̂(a).
where q̂(a) := i=1 p̂i β̂i (a) and τ̂(a) :=i=1 i i
Lemma 1. The equilibrium payoffs to all types increase weakly if π̂(a) ≥ π∗ (a) for all a ∈[0, a) \ S ∗ .
Proof. See Appendix B.
A change in parameters that changes π∗ (·) to π̂(·) increases the relative supply of emulation opportunities. Lemma 1 states that such a change increases the equilibrium payoffs
to all types.11 Propositions 2–3 below follow directly from this lemma.
Proposition 2. Suppose the type distribution changes from P to P̂, where (a) p̂i ≤ pi for
all i ≤ ι(a) (with at least one strict inequality); and (b) p̂i ≥ pi for all i > ι(a). Then, theequilibrium payoffs to all types are weakly higher under P̂ than under P.
10

As in the two-type example, we exclude the fully separating actions because the mixed strategy is notwell define at a ∈ S ∗ .
11
The result holds strictly if the indifference curves are weakly convex. See Online Appendix.

11

The change from P to P̂ described in Proposition 2 implies stochastic dominance. Thus
the supply of emulation opportunities (numerator of π∗ (·)) increases. Further, condition(a) of the proposition implies that the demand for emulation from types below ι(a) (denominator of π∗ (·)) decreases. This proposition broadly suggests that better prior reputation is beneficial for all types: lower types gain from emulating the behavioral agents,
which enables higher types to separate at lower costs.
Proposition 3. For any δ and δ̂ > δ, the equilibrium payoffs to all types are weakly higher
under δ̂ than under δ.
Proposition 3 concerns the impact of a change in the extent of behavioral diversity.
From (3), an increase in δ raises the relative supply π∗ (·) of emulation opportunities,
which benefits all types by Lemma 1. Note that this result implies that all types are weakly
better off compared to the LCSE which prevails when δ = 0. Although the standard analysis of signaling focuses on one particular dimension of reputation, there are typically manydifferent dimensions along which an agent can gain or lose his reputation. Which dimension to focus on clearly depends on the agent’s preferences about what should constitutean important quality of life. In a homogeneous society where people have similar values
and aspire to achieve similar goals, more resources must be expended for signaling. A
society with diverse values and aspirations would muddle information and make signaling
less informative, but that can lead to a Pareto improvement as it reduces the intensity of
wasteful signaling.
4.3.

Convergence

We now examine the convergence property as the underlying information asymmetry disappears. To this end, we pay attention to the largest binding action and show that it converges to 0 as the prior type distribution converges to a degenerate distribution at some
type j.
Proposition 4. For any " > 0, there is p" < 1 such that a < " for all p j > p" .
Proof. Observe that lim p j →1 τ(a) = j for all a ∈ D. Also, since lim p j →1 = ∞ for i 6= j, a∗jconverges to 0. If τ(a) converges from below in a neighborhood of a, a∗j ≥ a, and thiscompletes the proof. If τ(a) converges from above, type j randomizes over [a∗j , a∗j+1 ). As
τ(a) converges to j, a converges to 0

12

Given that lim p j →1 τ(a) = j for all a ∈ D, Proposition 4 implies that the allocations oftype j collapse to a single point (0, j), and so do the allocations of all types below j. Types
lower than j receive ui (0, j), which is higher than their respective complete information
payoff. For types above j, the allocations converge to the LCSE of the standard setting
where the lowest type is type j (rather than type 1). Their payoff is less than the complete
information payoff but is higher than their payoff from the LCSE of the standard setting.

5.

Conclusion

The signaling literature often focuses on separating equilibria where the agent successfully coveys his private information via costly action. This paper looks at the flip side ofsignaling, with emphasis on the possibility of emulation (pooling) in equilibrium, by introducing behavioral diversity into the standard setup. The possibility of emulation changes
the structure of incentives, not only for those who emulate higher types but also for those
who attempt to distinguish themselves from lower types. The extended model delivers
new insights into the role of external factors that are absent in the standard setup. We also
note that with the slightest degree of behavioral diversity, the model exhibits a convergenceproperty which is intuitively more appealing than the LCSE of the standard setting.

13

Appendix A: Proof of Proposition 1
Characterization. In any discretized game, we can apply the standard D1 refinement torule out any pooling at a ∈/ D n . We first argue that any D1 equilibrium in a discretizedgame must have the following properties:• The highest type chooses a fully separating action with probability 1;• The lowest action chosen by type i is on the equilibrium indifference curve of typei − 1.
To show the first claim, suppose type I adopts a mixed strategy. For some action a0 in the
support of this mixed strategy such that µ(a0 ) < I, type I must be pooled with lower types.
Type I must then have an incentive to deviate slightly upward from those actions because
D1 would assign probability 1 to type I (observe that no type can randomize continuously
over an interval in a discretized game). This shows that type I must adopt a pure strategy.
The action chosen by type I must be fully separating because D1 assigns probability 1 toany upward deviation, giving type I an incentive to deviate slightly above. To show thesecond claim, suppose that there is some type i whose lowest action is bounded away fromthe equilibrium indifference curve of type i − 1. Type i then has an incentive to deviateslightly below because D1 assigns probability 1 to such a deviation.
Algorithm. In each round k ≥ 1, we determine the allocations of type I − k. Let
x I−k (u I ) := (a I−k+1 (u I ), t I−k+1 (u I ))be the reservation allocation of type I − k, where ai (u I ) is the lowest action chosen by typei and t i (u I ) = µ(ai (u I )) is the corresponding reputation for that action. In round 1, foreach u I ∈ Υ , we obtain a I (u I ) = s I (u I ) such that U I (s I (u I ), I) = u I and t I (u I ) = I. Fork ≥ 2, the algorithm inherits a reservation allocation x I−k (u I ) from the previous round,
and we let φ I−k (·; u I ) represent the indifference curve of type I − k that passes throughthis reservation allocation. Define s I−k (u I ) such thatI − k = φ I−k (s I−k (u I ); u I ),if φ I−k (0; u I ) ≤ I − k, and let a I−k (u I ) = s I−k (u I ). If φ I−k (0; u I ) > I − k, s I−k (u I ) is not welldefined, and we let a I−k (u I ) = 0. Define the tentative set of binding actions for type I − kbyB̂ I−k (u I ) := {a ∈ (a I−k (u I ), a I−k+1 (u I )) : τ(a) > φ I−k (a; u I )}.
14

For each a ∈ B̂ I−k (u I ), find π I−k (·; u I ) such that
φ I−k (a; u I ) =δq(a)τ(a) + (1 − δ)p I−k π I−k (a; u I )(I − k).
δq(a) + (1 − δ)p I−k π I−k (a; u I )For a ∈
/ B I−k (u I ), we let π I−k (a; u I ) = 0. Solving the above equation yields§ªδq(a)(τ(a) − φ I−k (a; u I ))π I−k (a; u I ) = max,0 .
(1 − δ)p I−k (φ I−k (a; u I ) − I + k)Let
Π I−k (u I ) :=Z

π I−k (a; u I ) da.
a∈(a I−k ,a I−k+1 )There are three possibilities.
• If Π I−k (u I ) > 1, there must be some a I−k (u I ) ∈ B̂ I−k (u I ) such thatZ
π I−k (a; u I ) da = 1,a∈(a I−k (u I ),a I−k+1 (u I ))and we set t I−k (u I ) = φ I−k (a I−k (u I ); u I ).
• If Π I−k (u I ) ≤ 1 and s I−k (u I ) is well defined, we let type I − k choose s I−k (u I ) withprobability 1 − Π I−k (u I ) and set (a I−k (u I ), t I−k (u I )) = (s I−k (u I ), I − k).
• If Π I−k (u I ) ≤ 1 but s I−k (u I ) is not well defined, u I is too high, and we terminate thealgorithm.
Given a well-defined a I−k (u I ), define B I−k (u I ) := B̂ I−k (u I ) ∩ [a I−k (u I ), a I−k+1 (u I )) as the setof binding actions.
Suppose that u I is set in the right range, and the algorithm reaches the final round.
In round I − 1, given x1 (u I ), we derive s1 (u I ) if it is well defined. If s1 (u I ) > 0, u I is too
low, and we terminate the algorithm. A necessary condition for finding an equilibrium istherefore φ1 (0; u I ) ≥ 0, so that a1 = 0. Given this, we obtain B1 (u I ) and Π1 (u I ) as above.
There are now four possibilities.
• If Π1 (u I ) > 1, u I is too low, and we terminate the algorithm.
• If Π1 (u I ) = 1, we have an equilibrium.
• If Π1 (u I ) < 1 and s1 (u I ) = 0, we let type 1 choose s1 (u I ) = 0 with probability 1 −Π I−k (u I ) and we have an equilibrium.
15

• If Π I−k (u I ) < 1 but s I−k (u I ) is not well defined, u I is too high, and we terminate thealgorithm.
Existence. The characterization result suggests that the algorithm described above is the
only way to construct a D1 equilibrium, if any. The algorithm produces an allocation
path for each u I ∈ Υ . Observe that ai (·) is continuous and strictly decreasing at u0I if
ai (u0I ) > 0. First, by definition, a I (·) is continuous and strictly decreasing. Now consider
how a change in ai+1 (u I ) affects ai (u I ). Suppose there is an infinitesimal increase in u Ifrom u0I to u00I , which in turn implies a I (u0I ) < a I (u00I ). Then, φ I−1 (a; u0I ) < φ I−1 (a; u00I ) for
all a. If B I−1 (u0I ) is empty, we have a I−1 (u0I ) > a I−1 (u00I ), again by definition. If B I−1 (u0I ) isnonempty, π I−1 (a; u0I ) > π I−1 (a; u00I ) for all a ∈ B I−1 (u0I ). Since sup B I−1 (u0I ) > sup B I−1 (u00I ),we must haveZa∈(a I−1 (u0I ),sup B I−1 (u00I ))π I−1 (a; u00I )da < 1,which proves a I−1 (u0I ) > a I−1 (u00I ). The claim is established because this argument holds
for any i < I − 1 as long as ai (u0I ) > 0.
This fact suggests that the allocation path shifts up continuously as u I increases. We
thus need to find an allocation path that can satisfy the equilibrium condition by adjustingu I . Observe that for any δ > 0 and {βi }, we have a I−1 (u I ) > 1 and Π I−k (u I ) < 1 if u I isclose enough to u I . Therefore, u I is too high when it is close to the upper bound. At theother end, as u I gets close to u I , there is u0I ≥ u I such that φ1 (0; u0I ) = 1. By continuity,
there must be u00I > u0I such that φ2 (0; u00I ) = 2. For u I ∈ (u0I , u00I ), Π1 (u I ) is well defined andstrictly decreasing in u I . Note that limuI ↑u00I Π1 (u I ) = 0. If Π1 (u0I ) ≥ 1, we can find a uniqueu∗I such that Π1 (u∗I ) = 1 and this constitutes an equilibrium. If u0I = u I and Π1 (u I ) < 1, we

let type 1 choose a = 0 with probability 1 − Π1 (u I ), and this is a unique equilibrium.

Appendix B: Proof of Lemma 1
We first show that u∗I increases if π̂(a) ≥ π∗ (a) for all a ∈ [0, a) \ S ∗ . Suppose we run thealgorithm under the new set of parameters Θ̂ with initial value u∗I for type I. Let Φ̃(·) be
the corresponding allocation path and [ãi , ãi+1 ) be the corresponding partition. Define«¨δ̂q̂(a)(τ̂(a) − Φ̃(a)),0 ,π̃(a) = max(1 − δ̂)p̂ι̃(a) (Φ̃(a) − ι̃(a))
16where ι̃(a) = i if a ∈ [ãi , ãi+1 ). Also let i := ι(a). Because π̂(a) ≥ π∗ (a) for a ∈ [0, a) \ S ∗ ,we have ãi+1 ≥ a∗ , which also implies that ãi ≥ a∗ and Φ̃(a) ≤ Φ∗ (a) for a such thati+1i

ι(a) = ι̃(a) = i − 1. The last fact ensures that π̃(a) ≥ π∗ (a) for this interval of a. Together
with ãi ≥ a∗ , this implies ãi−1 ≥ a∗ . We can repeat this argument back to type 1 to showii−1that ã1 ≥ a1∗ . Thus, the equilibrium payoff to type I must increase weakly.
We now argue that the equilibrium payoffs to all types weakly increase. Use Φ̂∗ (·) todenote the equilibrium allocation path under Θ̂. Define ι̂(·) analogously and let«¨∗δ̂q̂(a)(τ̂(a)
−
Φ̂
(a))
,0 .
π̂∗ (a) = max(1 − δ̂)p̂ι̂(a) (Φ̂∗ (a) − ι̂(a))∗
Let î ≥ i be the new i and [âi∗ , âi+1) be the new equilibrium partition under Θ̂. Suppose

the equilibrium payoff to some type k is strictly lower under Θ̂. If there are multiple types
that are made worse off by the change in distribution, let k represent the lowest such type.
Note that k cannot be greater than î, because this would imply that the least-cost separating
solution starting from type k would be entirely below the original allocation path, violating
the previously established result that the equilibrium payoff to type I increases. Because
the equilibrium indifference curve of type k − 1 under Θ̂ is weakly above that under Θ,while the equilibrium indifference curve of type k under Θ̂ is below that under Θ, thesingle-crossing property requires that âk∗ < ak∗ . If k = 1, we must have âk∗ ≤ ak∗ . Therefore,in either case, âk∗ ≤ ak∗ . Furthermore, Φ̂∗ (a) ≤ Φ∗ (a) implies π̂∗ (a) > π∗ (a) for a such that∗∗
ι(a) = ι̂(a) = k. Together with âk∗ ≤ ak∗ , this implies âk+1< ak+1. This in turn implies

that Φ̂∗ (a) < Φ∗ (a) and therefore π̂∗ (a) > π∗ (a) for a such that ι(a) = ι̂(a) = k + 1.
∗∗∗∗
. Repeating this argument shows that< ak+2, this implies âk+2< ak+1
Together with âk+1Φ̂∗ (a) < Φ∗ (a) for a such that ι(a) = ι̂(a) = i. But a strictly lower equilibrium indifference
curve Φ̂∗ (a) for type i would imply a strictly lower equilibrium indifference curve for typeI, contradicting the result that the equilibrium payoff to type I must increase.

17

Online Appendix to
Pecuniary Emulation and Invidious Distinction:
Signaling under Behavioral Diversity
JUNICHIRO ISHIDAWING SUEN

October 17, 2023
In the baseline model, we cannot rule out the possibility that some type i may choose afully separating action (i.e., the action that gives him the reputation of his own type) withstrictly positive probability, even when the set of binding actions is nonempty and thisRtype adopts a mixed strategy. This is because Π I−k (a; u I ) = a∈(s (u ),aπ (a; u I ) da
) I−k
I−k

I

I−k+1

may converge to a number less than 1, even though lima↓sI−k (uI ) π I−k (a; u I ) = ∞. Below,
we show that this possibility can be ruled out, and the mixed strategies are always smoothwith no probability masses if the indifference curves are weakly convex.12 Moreover, under
this condition, Lemma 1 and Propositions 2 and 3 hold strictly.
Let φi (·) be the indifference curve of type i that passes through (si , i) for some si . Given
that si is binding (i.e., τ(si ) > φi (si )), it suffices to show that
Z si +"
lim
s↓si

s

1
da
φi (a) − idiverges for any " > 0. To show this, consider a linear function η(a − si ) + i that passes
through (si , i), where η is set to satisfy η" + i = φi (si +"). Observe that this linear function
always stays (weakly) above φi (a) for a ∈ [si , si + ") if φi (·) is weakly convex. This in turnimplies that for any " > 0,Z si +"
lim
s↓si

s

1
da ≥ lims↓si
φi (a) − i

Z si +"s1da = ∞.
η(a − s)This ensures that Π I−k (u I ) is always greater than 1 and hence a I−k (u I ) > s I−k (u I ), so thatS ∗ is always empty.
Lemma 1 states that the equilibrium payoffs to all types increase weakly if π̂(a) ≥ π∗ (a)for all a ∈ [0, a) \ S ∗ . This is because even if π̂(a) > π∗ (a) for some a, this increase in therelative supply of emulation opportunities may be entirely absorbed by the corresponding
12

If the reputation payoff is additively separable and linear in t, this is equivalent to assuming that thecost of signaling effort is weakly convex in a, as often assumed in applications.

18

decrease in the probability of choosing the fully separating action, in which case the equilibrium allocation path would not shift up. Under the assumption that the indifference
curves are weakly convex, this can be restated as follows.
Lemma 1.A. The equilibrium payoffs to all types increase strictly if π̂(a) ≥ π∗ (a) for alla ∈ [0, a) with strict inequality for some a ∈ [0, a).
Given this result, both Propositions 2 and 3 can also be restated accordingly, where theequilibrium payoffs are strictly higher under Θ̂ in each instance.

19References
Araujo, Aloisio, Daniel Gottlieb, and Humberto Moreira, “A Model of Mixed Signals
with Applications to Countersignalling,” RAND Journal of Economics, 2007, 38 (4),1020–1043.
Ball, Ian, “Scoring Strategic Agents,” 2022. Working Paper, MIT.
Chen, Chia-Hui, Junichiro Ishida, and Wing Suen, “Signaling under Double-Crossing
Preferences,” Econometrica, 2022, 90 (3), 1225–1260.
Cho, In-Koo and David M. Kreps, “Signaling Games and Stable Equilibria,” Quarterly
Journal of Economics, 1987, 102 (2), 179–221.
Daley, Brendan and Brett Green, “Market Signaling with Grades,” Journal of Economic
Theory, 2014, 151, 114–145.
de Haan, Thomas, Theo Offerman, and Randolph Sloof, “Noisy Signaling: Theory andExperiment,” Games and Economic Behavior, 2011, 73 (2), 402–428.
Dilme, Francesc and Fei Li, “Dynamic Signaling with Dropout Risk,” American Economic
Journal: Microeconomics, 2016, 8 (1), 57–82.
Feltovich, Nick, Richmond Harbaugh, and Ted To, “Too Cool for School? Signalling andCountersignalling,” RAND Journal of Economics, 2002, 33 (4), 630–649.
Frankel, Alex and Navin Kartik, “Muddled Information,” Journal of Political Economy,
2019, 127 (4), 1739–1776.
Heinsalu, Sander, “Dynamic Noisy Signaling,” American Economic Journal: Microeconomics, 2018, 10 (2), 225–249.
Hopkins, Ed and Tatiana Kornienko, “Running to Keep in the Same Place: Consumer
Choice as a Game of Status,” American Economic Review, 2004, 94 (4), 1085–1107.
Hoppe, Heidrun C., Benny Moldovanu, and Aner Sela, “The Theory of Assortative
Matching Based on Costly Signals,” Review of Economic Studies, 2009, 76 (1), 253–281.
Kreps, David M. and Joel Sobel, “Signalling,” in Robert J. Aumann and Sergiu Hart, eds.,
Handbook of Game Theory, Vol. II, Amsterdam: Elsevier, 1994, chapter 25, pp. 849–867.
20

Mailath, George J., Masahiro Okuno-Fujiwara, and Andrew Postlewaite, “Belief-Based
Refinements in Signalling Games,” Journal of Economic Theory, 1993, 60 (2), 241–276.
Matthews, Steven and Leonard J. Mirman, “Equilibrium Limit Pricing: The Effects ofPrivate Information and Stochastic Demand,” Econometrica, 1983, 51 (4), 981–996.
Mencken, Henry Louis, Prejudices, First Series, New York: Knopf, 1919.
Spence, Michael, “Job Market Signaling,” Quarterly Journal of Economics, 1973, 87 (3),355–374.
Veblen, Thorstein, The Theory of the Leisure Class, New York: Macmillan, 1899. Mentor
edition, The New American Library.

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