ISSN (Print) 0473-453X

Discussion Paper No. 1103

ISSN (Online) 2435-0982

Chia-Hui Chen
Junichiro Ishida
Wing Suen

Secondly Revised October 2021
Revised December 2020
October 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Signaling under Double-Crossing Preferences∗
CHIA-HUI CHEN
Kyoto University

JUNICHIRO ISHIDA
Osaka University

WING SUEN
University of Hong Kong
October 13, 2021

Abstract. This paper provides a general analysis of signaling under doublecrossing preferences with a continuum of types. There are natural economic
environments where the indifference curves of two types cross twice, such thatthe celebrated single-crossing property fails to hold. Equilibrium exhibits athreshold type below which types choose actions that are fully revealing andabove which they pool in a pairwise fashion, with a gap separating the actions chosen by these two sets of types. The resulting signaling action is quasiconcave in type. We also provide an algorithm to establish equilibrium existence by construction.
Keywords. single-crossing property; countersignaling; local incentive compatibility; global incentive compatibility; pairwise-pooling
JEL Classification. D82; I21∗

We are grateful for the helpful comments provided by Daniel Gottlieb, Michihiro Kandori, Navin Kartik,
the referees, the editor, and conference and seminar participants at the 4th Spain-Japan Meeting on Economic Theory, CUHK-HKU-HKUST Theory Seminar, the Econometric Society European Meeting, Keio University, the University of Tokyo, and Concordia University. The first author acknowledges financial support
from JSPS KAKENHI 17H13699. The second author acknowledges financial support from JSPS KAKENHI(15H05728, 18H03640, 19H01468, and 20H05631), and the Joint Usage/Research Center for Behavioral
Economics at ISER and Osaka University for its International Joint Research Promotion Program.

1.

Introduction

The single-crossing property, also known as the Spence-Mirrlees condition, has remained
a cornerstone of signaling (Spence, 1973) and screening (Mirrlees, 1971) models since
their inception. In the context of the classic education signaling model of Spence (1973),
the single-crossing property states that an indifference curve of a higher type (in the spaceof education level and wages) crosses that of a lower type once and only once. This assumption captures the idea that the marginal cost of education is less expensive for moreable workers. As a result, they find it profitable to signal their ability through investing in
education while less able workers do not choose to mimic, thus making it possible to separate the two types by observing their educational choices. The assumption also plays a keyrole in simplifying incentive compatibility constraints to make the analysis tractable. The
validity of this assumption, however, is not necessarily evident, as requiring single crossing
to hold globally (for all signaling and reputation levels) imposes strong restrictions on the
structure of preferences. There is no guarantee that the insights gained from the class ofmodels characterized by the single-crossing property can be extended straightforwardly toa model with a wider scope.
The possibility that the single-crossing property may fail to hold in some environments
has been acknowledged in the literature (Mailath, 1987; Hörner, 2008), and there aresporadic and independent attempts to examine this situation in the analysis of signaling(Feltovich et al., 2002; Kolev and Prusa, 2002; Araujo et al., 2007; Daley and Green, 2014;
Bobtcheff and Levy, 2017; Frankel and Kartik, 2019; Chen et al., 2021; Degan and Li,2021).1 Much of this literature considers a small number of discrete types or some specificpayoff functions. In this paper, we provide an analysis of a standard signaling model with acontinuum of types, except that the usual single-crossing property is replaced by a doublecrossing property—indifference curves of two types cross twice in the relevant space. The
paper makes four contributions.
First, we provide a general framework and identify the key preference features, captured by Assumptions 2 and 3 in Section 2, that lead to the double-crossing property. To
the best of our knowledge, this is the first general analysis of signaling that does not impose
1

There are also some attempts to relax the single-crossing property in screening models. See Smart(2000), Araujo and Moreira (2010) and Schottmüller (2015). Matthews and Moore (1987) introduce
double-crossing utility curves in a multidimensional screening problem, but their focus and formulation
are different from ours, which relies on double-crossing indifference curves.

1the single-crossing property. When single crossing does not hold, an obvious concern isthat local incentive compatibility does not guarantee global incentive compatibility. It is often thought that this fact makes any analysis under such an environment complicated andintractable, especially when there are many types. Our analysis provides a systematic wayof understanding double-crossing preferences and yields new insight into the relationshipbetween local incentive compatibility and global incentive compatibility.
Second, we use examples to show that there are many situations of economic interest that exhibit the double-crossing property, suggesting that the set of assumptions we
identify is not only technically convenient but also economically relevant and meaningful.
One factor that potentially breaks the single-crossing property is that gains from signaling
are typically bounded. Beyond some level the gains diminish as an agent invests morein signaling. Moreover, higher, more productive types may reach this point of diminishing returns at lower signaling levels than do lower types. Thus, the benefit-cost ratio of
signaling is initially greater for higher types than for lower types, but the comparison isreversed past some signaling level, resulting in the double-crossing property. We provide
several examples to capture this principle and show that the single-crossing property is not
as robust as generally believed, as it can be easily turned into the double-crossing property
with minor modifications of the underlying environment.
Third, we provide a characterization of equilibria in Section 4. Despite the potential
complication that arises from the lack of single crossing, equilibrium exhibits a remarkably
simple structure. We introduce Low types Separate and High types Pairwise-Pool (LSHPP)
equilibrium, and Theorem 1 establishes that any D1 equilibrium under the double-crossing
property is LSHPP. In such an equilibrium, there is a threshold type above which two distinct types (or two intervals of types) pair up to choose the same signaling action. The equilibrium signaling action is quasi-concave in type above the threshold. Below the threshold,types choose fully revealing actions. Our notion of LSHPP is a generalized version of Low
types Separate and High types Pool (LSHP) equilibrium introduced by Kartik (2009). An
important difference from Kartik’s (2009) model (and also from Bernheim (1994)) is that
there is no exogenous bound on the signaling space. Instead, “pairwise-pooling” is theresult of endogenous constraints induced by the double-crossing property.
Finally, in Section 5, we provide an algorithm to find an LSHPP equilibrium that works
for any continuous type distribution. Theorem 2 establishes the existence of an equilibrium via this construction. Pairwise-pooling is related to a phenomenon known as “countersignaling,” where low and high types pool by refraining from costly signaling while2intermediate types separate from those types by signaling (Feltovich et al., 2002; Araujo
et al., 2007; Chung and Eso, 2013). However, establishing a countersignaling equilibrium
is not straightforward, and our understanding of countersignaling is limited to specificcontexts. Our equilibrium construction generalizes the notion of countersignaling to thatof pairwise-pooling, clarifies the forms it can take, and enables us to establish its existence
under general conditions. We later discuss in detail how our framework extends the existing literature and sheds new light on this seemingly perverse yet pervasive phenomenon.

2.

Model

We consider a standard signaling model, except that the usual single-crossing property isreplaced by a double-crossing property, which we define more precisely below. An agent,
characterized by his type θ ∈ [θ , θ ], chooses a publicly observable action (signaling level)a ∈ R+ . The agent’s type is his private information and is distributed according to acontinuous function F (·) with full support. The payoff to an agent is u(a, t, θ ), where t is
the market’s perception of his type, or his “reputation”; i.e., t = E[θ | a]. We assume thatsignaling is costly and that the agent benefits from a higher reputation.2
Assumption 1. u : R+ × [θ , θ ]2 → R is twice continuously differentiable, strictly increasing
in t and strictly decreasing in a.
In the subsequent analysis, we make heavy use of the marginal rate of substitution
between signaling action a and reputation t, defined asm(a, t, θ ) := −ua (a, t, θ ).
u t (a, t, θ )It measures the increase in reputation that is needed to compensate for a higher signalinglevel. Signaling is relatively inexpensive when the marginal rate of substitution is low.
If we let t = φ(a, u, θ ) represent the indifference curve of type θ at utility level u in the(a, t)-space, then the marginal rate of substitution gives its slope. Specifically, φa (a, u, θ ) =m(a, φ(a, u, θ ), θ ).
Preferences satisfy the single-crossing property if whenever a lower type θ 00 is indifferent between a higher signaling action a2 to a lower signaling action a1 , a higher typeThe assumption that ua (·) < 0 is just made for the sake of expository clarity. All of our results holdeven if signaling is not always costly. See Appendix G for the general case. We maintain the assumption thatu t (·) > 0 throughout this work. Liu and Pei (2020) consider a signaling model in which the sender’s payoff
2from reputation is type-dependent and may not be monotone. We leave that extension for future research.

3θ 0 strictly prefers the higher action a2 . This is equivalent to requiring that m(a, t, θ 0 ) <m(a, t, θ 00 ) for any θ 0 > θ 00 and any (a, t), which implies that an indifference curve of ahigher type crosses that of a lower type once and from above. We often refer to this caseas the “standard setup.”
We relax the standard setup to allow for “double-crossing preferences.” Our focus is
to study situations in which the single-crossing property holds when the signaling level islow but fails when it is high.
Definition 1 (Double-crossing property). For any θ 0 > θ 00 , there exists a continuous function
D(·; θ 0 , θ 00 ) : [θ , θ ] → R+ such that the following hold:(a) if a < a0 ≤ D(t 0 ; θ 0 , θ 00 ), thenu(a, t, θ 00 ) ≤ u(a0 , t 0 , θ 00 ) =⇒ u(a, t, θ 0 ) < u(a0 , t 0 , θ 0 );(b) if a > a0 ≥ D(t 0 ; θ 0 , θ 00 ), thenu(a, t, θ 00 ) ≤ u(a0 , t 0 , θ 00 ) =⇒ u(a, t, θ 0 ) < u(a0 , t 0 , θ 0 ).
The locus of points {(a, t) : a = D(t; θ 0 , θ 00 )} is a “dividing line” that partitions the(a, t)-space into two regions. For signaling actions to the left of the dividing line, thestandard single-crossing property holds for types θ 0 and θ 00 . To the right of the dividing
line, the reverse single-crossing property holds: whenever the lower type θ 00 is indifferent
between a higher action a2 and a lower action a1 , the higher type θ 0 strictly prefers thelower action. The double-crossing property does not impose any specific restrictions on
the rankings between actions on opposite sides of the dividing line. It also does not requireD(t; θ 0 , θ 00 ) to be monotone in t.
Assumption 2. u(·) satisfies the double-crossing property.
For θ 0 > θ 00 and any (a, t), m(a, t, θ 0 ) − m(a, t, θ 00 ) is negative in the standard setup.
Assumption 2, on the other hand, implies that this difference is single-crossing from below
in a with the crossing point at a = D(t; θ 0 , θ 00 ). But the latter condition alone does not
imply Assumption 2.3 Since we impose no restriction on the shape of D(·; θ 0 , θ 00 ) other3

If utility is additively or multiplicatively separable in reputation t, then the single-crossing difference in
a is sufficient to ensure the double-crossing property.

4than that it is a continuous function of t, we could have a situation where the dividing line
crosses an indifference curve more than once. However, suppose an indifference curveφ(·, u0 , θ 00 ) of type θ 00 crosses D(·; θ 0 , θ 00 ) at a1 and a2 . Indifference curves of the higher
type θ 0 that cross φ(·; u0 , θ 00 ) must be steeper than the latter between a1 and a2 , which
implies that type θ 0 strictly prefers the lower action a1 to the higher action a2 . This would
violate Definition 1(a). Assumption 2 therefore implies that an indifference curve cancross a dividing line only once in the (a, t)-space. More specifically, if φ(·, u0 , θ 0 ) is theindifference curve of type θ 0 that passes through (a0 , t 0 ), then a0 ≤ D(t 0 ; θ 0 , θ 00 ) implies
a < D(φ(a, u0 , θ 0 ); θ 0 , θ 00 ) for a < a0 .
Formally, suppose that type θ 00 attains utility level u0 at (a0 , t 0 ). We require that thedifference in the marginal rate of substitution between two types is single-crossing frombelow along an indifference curve of one type (say, the lower type): for θ 0 > θ 00 ,(m(a, φ(a, u0 , θ 00 ), θ 0 ) − m(a, φ(a, u0 , θ 00 ), θ 00 )≤0

if a ≤ a0 ≤ D(t 0 ; θ 0 , θ 00 ),≥0

if a ≥ a0 ≥ D(t 0 ; θ 0 , θ 00 ),(1)with strict inequality except when a = a0 = D(t 0 ; θ 0 , θ 00 ). It is clear that Assumption 2 issatisfied if and only if there exists D(·; θ 0 , θ 00 ) such that (1) holds; thus (1) can be adopted
as an alternative definition of the double-crossing property. For completeness, we provide
a proof of this claim in Appendix D.
In Figure 1, we show indifference curves of types θ 0 and θ 00 in the (a, t)-space. To
the left of the dividing line D(·; θ 0 , θ 00 ), an indifference curve of the higher type θ 0 mustcross φ(·, u0 , θ 00 ) from above. To the right, it must cross φ(·, u0 , θ 00 ) from below. At theboundary, the indifference curves of the two types are tangent to each other, with thehigher type having indifference curves that are “more convex.” One implication is thatwhenever the lower-type θ 00 prefers the allocation (a0 , t 0 ) at the tangency point to someother allocation (a0 , t 0 ), the higher-type θ 0 would strictly prefer the former to the latter.
The relevance of this implication for incentive compatibility will become apparent later in
the analysis.
Assumptions 1 and 2 are sufficient for an analysis of signaling under double-crossing
preferences when there are only two types. To allow for a general analysis with multiple
types, we need to make assumptions about how the dividing line D(·; θ 0 , θ 00 ) shifts withrespect to θ 0 and θ 00 .4
4

With three types, for example, there would be three dividing lines (one for each pair of types) and six

5Figure 1. Double-crossing property. The indifference curve of a higher-type θ 0 crosses that of a lowertype θ 00 twice. Along the dividing line, higher types have more convex indifference curves.

Assumption 3. For any t, D(t; θ 0 , θ 00 ) is continuous and strictly decreases in θ 0 and in θ 00 .
The dividing line D(·; θ 0 , θ 00 ) is defined for θ 0 > θ 00 . We extend the domain of D toallow for θ 0 ≥ θ 00 by defining, for any t,D(t; θ , θ ) := 00lim − D(t; θ , θ 00 ) = lim
D(t; θ 0 , θ ).
0
+
θ →θ

θ →θ

Because D(·; θ 0 , θ 00 ) is monotone in θ 0 and θ 00 and is bounded, the limit is well defined.
Definition 2. (a, t) is in the SC-domain of type θ if it belongs to the set SC(θ ) := {(a, t) :a < D(t; θ , θ )}; and it is in the RSC-domain of type θ if it belongs to RSC(θ ) := {(a, t) :a > D(t; θ , θ )}.
Assumption 3 implies that the SC-domain shrinks with type (i.e., SC(θ 0 ) ⊂ SC(θ 00 ) forθ 0 > θ 00 ) and that the RSC-domain expands with type. If (a, t) is in the SC-domain oftype θ , then for any two types lower than θ , the higher type has a lower marginal rateof substitution at this point than the lower type. This follows because a < D(t; θ , θ ) <D(t; θ 0 , θ 00 ) for any θ ≥ θ 0 > θ 00 . If (a, t) is in the RSC-domain of type θ , then for any two
types higher than θ , the higher type has a higher marginal rate of substitution than thelower type. If (a, t) is on the boundary of the SC-domain and RSC-domain of type θ , thentype θ has a lower marginal rate of substitution than any other type. In other words,
a = D(t; θ , θ ) =⇒ θ = argmin m(a, t, θ 0 ).

(2)

θ0

possible rankings of these dividing lines for each value of t. Any analysis becomes unmanageable without
further restrictions as the number of types increases.

6

Assumption 3 is not easy to interpret in terms of preferences. The following result isuseful for relating it to the marginal rate of substitution.
Lemma 1. Suppose that preferences satisfy the double-crossing property. Then, Assumption
3 holds if and only if m(a, t, θ ) is strictly quasi-convex in θ .
Proof. For any given (a, t), pick an arbitrary pair of types θ 0 and θ 00 such that a = D(t; θ 0 , θ 00 ).
If no such pair exists, then m(a, t, ·) is strictly monotone and hence strictly quasi-convex.
By definition, this means that m(a, t, θ 0 ) = m(a, t, θ 00 ). Suppose that D(t; θ 0 , ·) is decreasing. For θ1 < θ 00 , a < D(t; θ 0 , θ1 ) implies m(a, t, θ 00 ) = m(a, t, θ 0 ) < m(a, t, θ1 ). Forθ2 ∈ (θ 00 , θ 0 ), a > D(t; θ 0 , θ2 ) implies m(a, t, θ2 ) < m(a, t, θ 0 ). If D(t; ·, θ 00 ) is decreasing,then for θ3 > θ 0 , a > D(t; θ3 , θ 00 ) implies m(a, t, θ 0 ) = m(a, t, θ 00 ) < m(a, t, θ3 ). Since
this holds for any arbitrary pair (θ 0 , θ 00 ), Assumption 3 implies that m(a, t, ·) is strictly
quasi-convex.
Conversely, suppose m(a, t, θ ) is quasi-convex. Take any (a, t) such that a = D(t; θ 0 , θ 00 ).
For θ1 < θ 00 , m(a, t, θ1 ) > m(a, t, θ 0 ) implies a < D(t; θ 0 , θ1 ). For θ2 ∈ (θ 00 , θ 0 ), m(a, t, θ2 ) <m(a, t, θ 0 ) implies a > D(t; θ 0 , θ2 ). This shows that D(t; θ 0 , ·) is decreasing. A similar argument establishes that D(t; ·, θ 00 ) is decreasing.
Given this result, an alternative way to state Definition 2 is that (a, t) belongs to the
SC-domain of type θ if m(a, t, ·) is locally decreasing at θ , and it belongs to the RSCdomain of type θ if m(a, t, ·) is locally increasing at θ . In the standard setup, the marginal
rate of substitution strictly decreases in type, reflecting the assumption that higher types
have lower signaling costs. The double-crossing property with Assumption 3 is relevant forsituations in which the marginal costs of signaling are the lowest for intermediate types.
Note, however, that the type with the minimum marginal signaling cost is not fixed. For
low signaling levels relatively higher types have the minimum signaling cost, and for highsignaling levels relatively lower types have the minimum signaling cost.

3.

An Example: Signaling with News

While our specification is a natural way to define double-crossing preferences, the assumptions we adopt impose economically meaningful restrictions on preferences, which may or
may not be reasonable depending on the context of the application. Specifically, Assumption 2 implies that indifference curves of higher types are more convex than those of lower
types. In the standard setup, the relevant issue is which type has steeper indifference7

curves. Under double-crossing preferences, the issue is of higher order: we need to determine how the rate of change in marginal rates of substitution is related to the agent type,for which there appears to be no a priori obvious specification.
To better motivate the modeling choices we make and to demonstrate the relevanceof our analysis, we first discuss a leading example of our model; more examples are provided in Section 6.3. The example builds on an insight that has been scrutinized in theliterature: the single-crossing property may fail in signaling models with additional information sources such as news or “grades” (Feltovich et al., 2002; Daley and Green, 2014).
For illustration, we use a very simple formulation of additional information. The literature
has developed more complicated models.
Consider an environment where there are two sources of information: a signaling action and a test outcome. The test outcome is binary, either pass or fail, and the agent
passes the test with probability β0 + βθ (where β > 0). If the agent passes the test, thenhe is promoted and earns λV . If he fails, he is fired, and his outside payoff depends on his
reputation. Let the outside payoff be λt < λV . The agent’s utility is
γa a2u(a, t, θ ) = (β0 + βθ )λV + [1 − (β0 + βθ )]λt −
+
,
θ
2


where the last term in parentheses represents the signaling cost, and γ > 0 is a cost parameter. The marginal rate of substitution ism(a, t, θ ) =γ + aθ
.
λθ [1 − (β0 + βθ )]For θ 0 > θ 00 , m(a, t, θ 0 ) − m(a, t, θ 00 ) is single-crossing from below in a. Since m(a, t, θ ) is
independent of t, this suffices for Assumption 2 to hold. Assumption 3 also holds becausem(a, t, θ ) is quasi-convex in θ .
In this class of models, the single-crossing property breaks down because higher types
have less incentive to engage in costly signaling as they know that their type will be partially revealed by exogenous news anyway. Because of this, the marginal gain from signaling is not necessarily higher for higher types. As Feltovich et al. (2002) illustrate, thistype of model often leads to a phenomenon known as “countersignaling,” in which higher
types refrain from costly signaling. We later show that the possibility of countersignaling
is a common feature of equilibrium under double-crossing preferences.

8

4.

Characterization

This section provides a characterization of signaling equilibria that survive the D1 criterion.
Let S : [θ , θ ] → R+ denote the sender’s strategy, and let µ : R+ → ∆[θ , θ ] be the belief
about agent type. Define T (θ 0 ) := Eµ(S(θ 0 )) [θ ] as the equilibrium reputation of type θ 0 .
Definition 3. A signaling equilibrium is a pair of strategy S(·) and belief µ(·) such that thefollowing hold:(a) given µ(·), S(θ ) ∈ arg maxa u(a, Eµ(a) [θ ], θ ) for each θ ∈ [θ , θ ]; and(b) µ(·) is consistent with S(·) and Bayes’ rule whenever applicable.
This definition is equivalent to perfect Bayesian equilibrium if we introduce a fictitious
player (“the market”) who chooses t after observing a to minimize the loss function, (t −θ )2 . Signaling models typically exhibit a plethora of equilibria because off-equilibrium
beliefs are not pinned down by Bayes’ rule. We introduce the standard D1 refinement(Cho and Kreps, 1987), which requires that, for any off-equilibrium action a, the beliefµ(a) satisfies the following restriction: if there are θ 0 and θ 00 such that, for all t,u(a, t, θ 0 ) ≥ u(S(θ 0 ), Eµ(a) [θ ], θ 0 ) =⇒ u(a, t, θ 00 ) > u(S(θ 00 ), Eµ(a) [θ ], θ 00 ),then θ 0 6∈ supp µ(a). This restriction eventually comes down to comparing the marginal
rates of substitution at the point of pooling: a slight upward deviation to an off-equilibrium
action is attributed to the type with the lowest marginal rate (whose signaling cost is
the lowest), while a slight downward deviation is attributed to the type with the highest
marginal rate.5 Throughout the analysis, we simply refer to a signaling equilibrium thatsatisfies the D1 refinement as an “equilibrium.”
In what follows, we use S(θ − ) and T (θ − ) to denote the left limit and S(θ + ) and T (θ + )to denote the right limit at θ . Let Q(a) := {θ : S(θ ) = a} denote the set of types whochoose a in equilibrium. We refer to Q(a) as a pooling set if it contains at least two types.
4.1.

Full separation

Consider a fully separating strategy s∗ (·) for some interval of types, where T (θ ) = θ in thisinterval. Incentive compatibility requires type θ to have no incentive to mimic adjacent5

A frequently used alternative is the Intuitive Criterion, but it does not pin down a unique outcome evenunder single-crossing preferences when there are more than two types. We adopt D1 because it predicts aunique outcome under single-crossing preferences and hence provides an ideal reference point.

9

types:u(s∗ (θ ), θ , θ ) ≥ u(s∗ (θ + ε), θ + ε, θ ).
In the limit, this condition can be written ass∗ 0 (θ ) =1.
m(s(θ ), θ , θ )(3)An equilibrium is fully separating if the entire type space [θ , θ ] is separating. In this case,the initial condition must satisfy s∗ (θ ) = argmaxa u(a, θ , θ ) = 0.
If indifference curves are single-crossing, the solution to the differential equation (3)with initial condition s∗ (θ ) = 0 constitutes a fully separating equilibrium (Mailath, 1987).
This solution is also known as the least cost separating equilibrium, or the “Riley outcome”(Riley, 1979).
In our model, there is a dividing line D(·; ·, ·) that separates the (a, t)-space into two
distinct domains. No fully separating solution can extend beyond the dividing line.
Proposition 1. There is no fully separating equilibrium if there exists θ 0 < θ such thats∗ (θ 0 ) = D(θ 0 ; θ 0 , θ 0 ).
Proof. Let θ1 be a type that is slightly above θ 0 such that s∗ (θ 0 ) = D(θ 0 ; θ 0 , θ 0 ). Recall from
(2) that at (s∗ (θ 0 ), θ 0 ), type θ 0 has the lowest marginal rate of substitution. Moreover,by the double-crossing property, the indifference curve of the higher type θ1 that passes
through (s∗ (θ 0 ), θ 0 ) stays strictly above that of type θ 0 for all a0 > s∗ (θ 0 ). Therefore, if
type θ 0 is indifferent between (a0 , t 0 ) and (s∗ (θ 0 ), θ 0 ), then type θ1 must strictly prefer(s∗ (θ 0 ), θ 0 ). This shows that s∗ (·) cannot extend beyond the dividing line. Given this, theonly remaining possibility is that s∗ (·) jumps at some θ ≤ θ 0 , but this is clearly not incentive
compatible because T (·) must be continuous at θ .
This result essentially follows from the fact that the equilibrium signaling level must
be weakly increasing in the SC-domain and weakly decreasing in the RSC-domain. The
reason why S(·) cannot decrease in the SC-domain is the same as in the standard setup
with single-crossing preferences. In the RSC-domain, if type θ 00 is indifferent between apair of actions, S(θ 00 ) > S(θ 0 ), then a higher-type θ 0 has a higher signaling cost than does
type θ 00 and must strictly prefer the lower action S(θ 0 ). Thus S(·) cannot increase in the
RSC-domain.
10If (s∗ (θ ), θ ) belongs to SC(θ ) for all θ , then the model reduces to the standard setup.
For double-crossing preferences to have any bite, therefore, we need to examine the situation where s∗ (·) hits the boundary before it reaches the highest-type θ . The remainder ofthe paper addresses this situation.
4.2.

Pooling equilibria under D1Under the D1 criterion, the standard setup predicts the least-cost separating equilibrium,
which is distribution-free. This is not the case for our model, where some form of poolingcan survive the D1 criterion. As a consequence, the distribution of types has a nontrivial
impact on the equilibrium allocation.
For any (a, t) and any set of types Q, letθmax (a, t; Q) := argmax m(a, t, θ ),
θ ∈Qθmin (a, t; Q) := argmin m(a, t, θ ).
θ ∈QConsider a pooling set Q(a) of types who choose (a, t) in equilibrium, with t = E[θ |θ ∈ Q(a)]. Suppose further that actions slightly above or below a are not chosen by any
type in equilibrium. Then, under D1, a slight upward deviation from (a, t) is attributed totype θmin (a, t; Q(a)) while a slight downward deviation is attributed to θmax (a, t; Q(a)). To
satisfy D1, the equilibrium reputation must be greater than these off-equilibrium beliefs:
t ≥ max {θmax (a, t; Q(a)), θmin (a, t; Q(a))} .

(4)

If m(a, t, θ ) is monotone in θ , then θmax (a, t; Q(a)) and θmin (a, t; Q(a)) must be at theextremal points of Q(a). Since t ∈ (min Q(a), max Q(a)), (4) cannot be satisfied for any
pooling set Q(a). This is why no pooling equilibrium can survive D1 in the standard setup.
Under double-crossing preferences, however, m(a, t, θ ) may not be monotone in θ , thereby
leaving some room for pooling equilibria.
4.3.

Low types separate and high types pairwise-pool

Below, we show that equilibrium under double-crossing preferences exhibits a particularform of pooling, which can be seen as a generalized version of LSHP equilibrium introduced by Kartik (2009).
Definition 4. A sender’s strategy is LSHPP (Low types Separate and High types Pairwise-Pool)
if there is some θ0 ∈ [θ , θ ] such that:11

(a) S(θ ) = s∗ (θ ) for θ ∈ [θ , θ0 ).
(b) S(θ ) is discontinuous only at θ = θ0 with an upward jump if θ0 < θ .
(c) There exist θ∗ ∈ (θ0 , θ ) and p : [θ0 , θ∗ ] → [θ∗ , θ ] such that the following hold forθ ∈ (θ0 , θ∗ ]: (i) S(·) is continuous and weakly increasing; and (ii) p(·) is continuousand strictly decreasing with p(θ0 ) = θ , p(θ∗ ) = θ∗ , and S(p(·)) = S(·).
An equilibrium is an LSHPP equilibrium if the sender’s strategy is LSHPP. Our notionof LSHPP equilibrium includes full separation (θ0 = θ ), full pooling (θ0 = θ and S(·)
is constant for θ ∈ [θ , θ ]), and LSHP equilibrium (θ0 ∈ (θ , θ ) and S(·) is constant forθ ∈ [θ0 , θ ]) as special cases. An important feature of the LSHPP strategy is that it canhave at most one “gap” (i.e., discontinuity) at θ0 .
Part (c) of the definition embodies the reason why we call it pairwise-pooling, wheretypes θ and p(θ ) are “pairwise matched” to choose the same action for θ ∈ [θ0 , θ∗ ). It
also implies that S(·) is quasi-concave above the gap (i.e., among types above θ0 ). The
quasi-concavity of S(·) with S(θ0 ) = S(θ ) suggests that for any action a ≥ S(θ0 ) chosenin equilibrium, Q(a) must be a pooling set (except possibly for a = maxθ S(θ ), where
Q(a) may be a singleton or a pooling set). See Figure 2 for an illustration. An LSHPP
equilibrium exhibits countersignaling whenever S(·) is not constant above the gap. In
Figure 2, the highest-type θ chooses a signaling action lower than that chosen by any othertype in (θ0 , θ ). The highest equilibrium signaling action is chosen by some intermediatetypes. This observation indicates that countersignaling that has been discussed in various
contexts is a consequence that pertains to double-crossing preferences.
The next statement is one of the main results of this paper.
Theorem 1. Any D1 equilibrium is LSHPP if Assumptions 1 to 3 are satisfied.
4.4.

A sketch of the proof

The proof of Theorem 1 is lengthy and is relegated to Appendix A. Here, we provide the key
steps and a heuristic argument to illustrate the underlying intuition of our characterization,with a particular focus on two important properties of an LSHPP equilibrium: continuity
and quasi-concavity.
Since the properties of the fully separating region are tightly pinned down by the differential equation (3) and the initial condition, it suffices to examine what pooling patterns
12Figure 2. LSHPP strategy. Below the gap, S(·) coincides with the least cost separating solution s∗ (·).
Above the gap, S(·) is quasi-concave. There is mass pooling at a1 and at a2 , and atomless pooling in
the neighborhood of a3 .

are feasible above the gap. The following result is useful for narrowing down the possible
forms of pooling.
Lemma 2. Suppose that there is an interval (θ 00 , θ 0 ) such that S(·) is continuous and strictly
monotone, and Q(S(θ )) is a pooling set for some θ in this interval. Then, there exists p(·)
such that for all θ ∈ (θ 00 , θ 0 ), (a) Q(S(θ )) = {θ } ∪ {p(θ )}; and (b) m(S(θ ), T (θ ), θ ) =m(S(θ ), T (θ ), p(θ )).
Proof. Suppose that there is pooling only at some points in the interval. Then, we canfind two arbitrarily close types θ1 and θ1 + ε in (θ 00 , θ 0 ) such that Q(S(θ1 )) is a pooling set while Q(S(θ1 + ε)) is a singleton (where ε may be positive or negative). Let(S(θ1 ), T (θ1 )) = (a1 , t 1 ). Since S(·) is monotone, type θ1 must pool with some other types
outside of (θ 00 , θ 0 ). To satisfy incentive compatibility, however, T (·) must be continuous on(θ 00 , θ 0 ). Since any type that can pool with type θ1 is bounded away from θ1 , type θ1 must
pool with both types above θ 0 and below θ 00 to maintain the continuity of T (·). However,if three or more types pool at the same action, by Lemma 1, we can find a type θ2 ∈ Q(a1 )such that m(a1 , t 1 , θ1 ) 6= m(a1 , t 1 , θ2 ). This is a contradiction because type θ2 must havean incentive to deviate to an action slightly above or slightly below. This shows that there
must be pooling over the entire interval. Given this, we can apply the same argument asabove to show that Q(S(θ )) contains exactly two types, θ and p(θ ), for all θ ∈ (θ 00 , θ 0 ).
The fact that m(S(θ ), T (θ ), θ ) = m(S(θ ), T (θ ), p(θ )) follows immediately.
13

This result suggests that two different types of pooling can emerge in equilibrium. First,
it is possible to have pooling in the usual sense, where a positive measure of types choose
the same action. We refer to this pattern of pooling as mass pooling. Lemma 2 shows that
there can be a different kind of pooling, which we call atomless pooling, where exactly
two types are paired together for each action level, and the pooling set Q(a) has measurezero. For example, in Figure 2, the pooling set Q(a3 ) contains exactly two types, and S(·)
is locally increasing at one of these types and locally decreasing at the other type. Under
atomless pooling, the marginal rate of substitution at (S(θ ), T (θ )) must be the same forthe paired types. One implication is clear: (S(θ ), T (θ )) belongs to the SC-domain of type
θ and to the RSC-domain of type p(θ ).
When there is mass pooling, the pooling set may be either connected or disconnected.
In Figure 2, Q(a1 ) is a connected pooling set while Q(a2 ) is disconnected. It is straightforward to handle connected pooling sets because they must be an interval. Disconnected
pooling sets are more complicated as they potentially admit infinitely many different forms.
Below, we provide an intuitive explanation for why S(·) must be continuous and quasiconcave. Establishing these properties is the key to the proof of Theorem 1.
Continuity. An important fact that leads to continuity is that it is generally infeasible
to jump from one pooling set to another under D1. To illustrate this point, suppose that
(S(θ ), T (θ )) = (a p , t p ) for θ ∈ [θ p , θ j ] ∪ [θ j , θ p ], and (S(θ ), T (θ )) = (a1 , t 1 ) for θ ∈(θ j , θ j ), where a1 > a p .6 It is relatively straightforward to establish that actions slightly
below a1 are not chosen in equilibrium. Then, to prevent downward deviation from a1
given an off-equilibrium belief that satisfies D1, we must have m(a1 , t 1 , θ j ) ≥ m(a1 , t 1 , θ j ).
However, under Assumption 2, if type θ j is indifferent between (a p , t p ) and (a1 , t 1 ), type
θ j must strictly prefer (a p , t p ) to (a1 , t 1 ), which is a contradiction.
There are many different pooling patterns in principle, but we can essentially apply
the same argument to show that it is not feasible to have any pooling in (θ j , θ j ) if S(·) is
discontinuous on [θ p , θ p ]. This would imply that if S(·) is discontinuous at θ j or at θ j , thenit must be fully separating on (θ j , θ j ). But if this is the case, it must be strictly increasing,and so (S(θ ), T (θ )) ∈ SC(θ ) for all θ ∈ (θ j , θ j ). Then, S(·) cannot jump down at θ j−−because (S(θ j ), T (θ j )) is in the SC-domain of type θ j , which is again a contradiction.
Quasi-concavity. Once the continuity of S(·) above the gap is established, it is easy toLemma 5 in the proof implies (a1 , t 1 ) ∈ RSC(θ j ). As such, it is not possible to have a1 < a p because
S(·) can jump up only in the SC-domain.
6

14

see why S(·) must be quasi-concave. As this result has some independent interest, we state
it as a separate proposition.
Proposition 2. Under Assumptions 2 and 3, if S(·) is continuous and incentive compatible
on any interval [θ 00 , θ 0 ], then S(·) must be quasi-concave on this interval.
Proof. Suppose that to the contrary, this function is not quasi-concave; i.e., there exist
θ1 < θ2 < θ3 on [θ 00 , θ 0 ] such that min{S(θ1 ), S(θ3 )} > S(θ2 ). If S(θ1 ) ≥ S(θ3 ) > S(θ2 ),then we can always find θ̃ ∈ (θ1 , θ2 ) such that S(θ3 ) > S(θ̃ ) > S(θ2 ) by the continuity ofS(·). Hence, without loss of generality, we assume S(θ3 ) > S(θ1 ) > S(θ2 ).
Since the lower-type θ1 prefers a higher action than type θ2 , Assumption 2 implies
S(θ1 ) > D(T (θ1 ); θ2 , θ1 ).
Moreover, since type θ1 prefers a lower action than the higher-type θ3 , we haveS(θ1 ) < D(T (θ1 ); θ3 , θ1 ).
These two inequalities imply D(T (θ1 ); ·, θ1 ) is increasing, which violates Assumption 3.
Proposition 2 relies only on continuity and incentive compatibility; it does not dependon other restrictions imposed by signaling models (such as D1 or the requirement that
T (θ 0 ) = Eµ(S(θ 0 )) [θ ]). This result therefore has general applicability for mechanism design.
We provide more discussion on this point in Section 6.1.
In our current context, the continuity and quasi-concavity of S(·) imply that poolingtakes the form of pairwise-pooling for types above some type θ0 , as described in Definition
4(c) of an LSHPP strategy. For types below θ0 , incentive compatibility in the SC-domain
requires S(·) to follow the least-cost separating solution s∗ (·), as described in part (a)of the definition. Finally, pairwise-pooling between types θ0 and θ (and possibility withother types higher than θ0 ) and full separation for types below θ0 imply that the difference
between T (θ0+ ) and T (θ0− ) must be positive. Because the utility function is continuous,
the discontinuity of T (·) accounts for the upward jump in S(·) at θ0 , required by part (b)of an LSHPP strategy.

5.

Existence

This section establishes the existence of an equilibrium by construction. To this end, weneed a technical assumption to ensure that the density function of types, denoted by f , iscontinuous and positive in the interior of the type space.
15

Assumption 4. F : [θ , θ ] → [0, 1] is continuously differentiable and strictly increasing.
5.1.

Equilibrium conditions

The equilibrium signaling pattern for θ < θ0 is pinned down by the least-cost separating
solution S(θ ) = s∗ (θ ) and T (θ ) = θ . Above the gap, there are three objects that need to
be determined. Let θ∗ ∈ argmaxθ ∈[θ0 ,θ ] S(θ ) denote the boundary type (to be made more
precise below). Let σ : [θ0 , θ∗ ] → R+ represent the signaling action taken by type θ , andτ : [θ0 , θ∗ ] → [θ0 , θ ] represent the reputation of type θ . Additionally, let the (decreasing)
function p : [θ0 , θ∗ ] → [θ∗ , θ ] represent the type that is paired with type θ in choosing thesame signaling action. Once we pin down these three functions, we can determine(S(θ ) = σ(θ ) and T (θ ) = τ(θ )if θ ∈ [θ0 , θ∗ ],S(p(θ )) = σ(θ ) and T (p(θ )) = τ(θ ) if p(θ ) ∈ (θ∗ , θ ].
These objects are defined in this way because any pooling action is chosen either by exactly
two types or by two intervals of types. When there is atomless pooling, σ(·) and τ(·) arestrictly increasing, and when there is mass pooling, σ(·) and τ(·) are locally flat.
In equilibrium, the following set of conditions must be satisfied.
Bayes’ rule. The equilibrium belief τ(·) must be consistent with the equilibrium strategies and Bayes’ rule on the path of play. Under atomless pooling, the pooling set has
measure zero. We require the “pointwise” belief to satisfy:
τ(θ ) =f (θ )f (p(θ )) |p0 (θ )|θ+p(θ ).
f (θ ) + f (p(θ )) |p0 (θ )|f (θ ) + f (p(θ )) |p0 (θ )|It is often more convenient to solve this for p0 (θ ) and write
p0 (θ ) =f (θ )
θ − τ(θ ).
f (p(θ )) p(θ ) − τ(θ )(5)Local incentive compatibility. In equilibrium, no type has an incentive to mimic adjacenttypes. The incentive constraint for separation isu(σ(θ ), τ(θ ), θ ) ≥ u(σ(θ + ε), τ(θ + ε), θ ),for θ ∈ [θ0 , θ∗ ). At the limit, we obtain
σ0 (θ ) =

τ0 (θ ).
m(σ(θ ), τ(θ ), θ )16(6)

Pairwise-pooling. When there is atomless pooling, local incentive compatibility mustbe satisfied for both θ and p(θ ). This boils down to the restriction (Lemma 2) that thetwo paired types must have the same marginal rate of substitution:m(σ(θ ), τ(θ ), p(θ )) − m(σ(θ ), τ(θ ), θ ) = 0.
For the ease of notation, we sometimes use m(·) to represent the marginal rate of substitution evaluated at (σ(θ ), τ(θ ), θ ) and m̂(·) to represent the value evaluated at (σ(θ ), τ(θ ), p(θ )).
Taking the derivative with respect to θ then gives[m̂a (·) − ma (·)] σ0 (·) + [m̂ t (·) − m t (·)] τ0 (·) = mθ (·) − m̂θ (·)p0 (·).
5.2.

(7)

Alternating between mass pooling and atomless pooling

The simplest form of an LSHPP equilibrium is the one with full pooling above the gap.
Despite its simple appearance, however, we cannot ensure that such an equilibrium always
exists.
Consider an equilibrium in which S(θ ) = s∗ (θ ) for θ ∈ [θ , θ0 ) and S(θ ) = a p > s∗ (θ0 )for θ ∈ [θ0 , θ ]. Let t p = E[θ | θ ≥ θ0 ] be the equilibrium reputation above the gap. In
equilibrium, type θ0 must be indifferent between (s∗ (θ0 ), θ0 ) and (a p , t p ), which pins downa unique a p for each θ0 . The problem is that this particular choice of a p may not satisfy
D1. Specifically, D1 requires that the marginal rate of substitution for this gap type must
be no smaller than that of type θ :m(a p , t p , θ0 ) ≥ m(a p , t p , θ );otherwise, a slight downward deviation from a p would be attributed to type θ . Furthermore, to prevent upward deviation from a p under D1 requires t p ≥ θmin (a p , t p ; Q(a p )).
There is a continuum of candidate gap types, but it is possible that none of them satisfies
these two restrictions. In this case, it is not feasible to construct an equilibrium with only
mass pooling.
Another possibility is to find an equilibrium with only atomless pooling above the gap.
However, this construction does not always work either, as it requires tight restrictions on
payoff and type distribution functions. Consider a pure atomless pooling equilibrium inwhich σ(·) is strictly increasing on [θ0 , θ∗ ]. At (σ(θ 0 ), τ(θ 0 )), types θ 0 and p(θ 0 ) must havethe same marginal rate of substitution. Moving along the trajectory, at (σ(θ 0 +ε), τ(θ 0 +ε)),17the indifference curve of type p(θ 0 ) must be steeper than that of type θ 0 according to the
double-crossing property. Algebraically, this means that[m̂a (·)σ0 (·) + m̂ t (·)τ0 (·)] − [ma (·)σ0 (·) + m t (·)τ0 (·)] > 0,
evaluated at θ 0 .7 Observe that the left-hand side of this inequality corresponds to the lefthand side of (7). Given this, to satisfy the condition for atomless pooling, the right-handside of (7) must also be strictly positive. In other words, the marginal rates of substitutionof types θ 0 + ε and p(θ 0 + ε) must change in such a way to make them tangent at (σ(θ 0 +ε), τ(θ 0 + ε)). A necessary condition to sustain atomless pooling is thusmθ (·) − m̂θ (·)p0 (·) > 0.

(8)

In condition (8), mθ (·) < 0 and m̂θ (·) > 0 because (σ(θ ), τ(θ )) is in the SC-domain oftype θ and in the RSC-domain of type p(θ ). This means that for a given ratio mθ (·)/m̂θ (·)(under some fixed preferences), p0 (·) must be sufficiently negative. Once a trajectory(σ(·), τ(·)) is fixed, however, p(·) and p0 (·) are uniquely pinned down from the type distribution, and there is hence no degree of freedom. From equation (5) for the pointwise
belief, with (σ(·), τ(·)) fixed, the absolute value of p0 (·) is proportional to f (·)/ f (p(·)).
This suggests that atomless pooling is more likely to emerge when the type distribution
has a thin tail, but the construction is not always guaranteed to succeed.
This discussion raises a concern about equilibrium existence when it is not feasible toconstruct either of the two simple forms of equilibrium. Below, we prove the existenceof an equilibrium by providing an explicit algorithm that always leads to a solution thatsatisfies the equilibrium conditions. The main idea behind the algorithm is to switch backand forth between mass pooling and atomless pooling along the equilibrium trajectory
whenever one type of pooling becomes infeasible.
5.3.

Algorithm and equilibrium existence

There are a number of ways to construct an equilibrium in our model. We focus on a version that seeks to produce atomless pooling wherever possible. The details of the algorithm
are provided in Appendix B. Here, we only provide a brief account of it.
7

A more precise algebraic argument is as follows. Substituting (6) into the left-hand side shows that it
has the same sign as m̂a (·) − ma (·) + m̂(·)(m̂ t (·) − m t (·)). Under atomless pooling, types θ and p(θ ) havethe same marginal rate of substitution at (σ(θ ), τ(θ )). Letting a = σ(θ ) and ε > 0, condition (1) impliesm(a + ε, φ(a + ε, u, θ ), p(θ )) > m(a + ε, φ(a + ε, u, θ ), u), θ ). Taking the limit gives m̂a (·) + m̂ t (·)φa (·) >ma (·) + m t (·)φa (·). The conclusion follows since m̂(·) = φa (·).

18

First, we pick a boundary type on the dividing line, i.e., some type θ∗ such that σ(θ∗ ) =D(θ∗ ; θ∗ , θ∗ ). This choice is motivated by the concern that if there is mass pooling in aneighborhood of σ(θ∗ ), the off-equilibrium belief associated with an upward deviation
does not exceed θ∗ .
Starting from this type, we solve the system of differential equations (5), (6), and (7)for θ ≤ θ∗ . If the solution (σ(θB ), τ(θB ), p(θB )) violates constraint (8) at some point θB ,then we switch to mass pooling by finding a pair (θ E , p(θ E )) such thatm(σ(θB ), τ(θB ), θ E ) − m(σ(θB ), τ(θB ), p(θ E )) = 0,E [θ | θ ∈ [θ E , θB ] ∪ [p(θB ), p(θ E )]] − τ(σB ) = 0,
and condition (8) holds for (θ E , p(θ E )). If a solution does not exist, then we simply setp(θ E ) = θ and set θ E to be the type that solves the second equation of the equations
system. If a pair (θ E , p(θ E )) that satisfies the two equations exists, then we switch back to
atomless pooling by solving the system of differential equations using the initial conditionfor θ E , and this process repeats. The iteration stops when θ reaches the point wherep(θ ) = θ . Such θ corresponds to θ0 .
For any θ∗ , following the above algorithm yields a well-defined θ0 (and a candidate
solution (σ(·), τ(·), p(·)) for θ ∈ [θ0 , θ∗ ]). We denote this mapping from θ∗ to θ0 as ζ :[θ , θ ] → [θ , θ ]. In any interior LSHPP equilibrium (i.e., θ0 ∈ (θ , θ )), there must be full
separation below the gap. To pin down an equilibrium for the entire type space, type θ0must be indifferent between (s∗ (θ0 ), θ0 ) and (σ(θ0 ), τ(θ0 )). For θ0 = ζ(θ∗ ), define∆u (θ∗ ) := u(s∗ (θ0 ), θ0 , θ0 ) − u(σ(θ0 ), τ(θ0 ), θ0 ).
Equilibrium requires ∆u (θ∗ ) ≤ 0, with strict inequality only if θ0 = θ .
Theorem 2. An LSHPP equilibrium exists if Assumptions 1 to 4 are satisfied.
In the proof of Theorem 2 (Appendix C), we show that the mapping ∆u (·) is continuous, and that there exists θ∗ such that either ∆u (θ∗ ) = 0, or ∆u (θ∗ ) < 0 and ζ(θ∗ ) = θ .
By construction, the candidate solution obtained satisfies all the local incentive compatibility constraints. In the proof, we show that local incentive compatibility implies global
incentive compatibility. Since this part of the argument is of independent interest, we will
provide more discussion on this point in Section 6.1. Finally, we also show that given
the off-equilibrium beliefs that satisfy D1, no type has an incentive to deviate to an offequilibrium action.
19

5.4.

Comparing equilibria

We establish the existence of an equilibrium by construction, but there are typically otheralgorithms that can consistently find an LSHPP equilibrium. For example, in the algorithm
described in Section 5.3, we solve the system of differential equations until we reach apoint at which condition (8) is violated. If we switch from atomless pooling to mass poolingat an earlier point while (8) still holds, then this provides an alternative mapping from θ∗to θ0 that may also satisfy all the equilibrium restrictions. In other words, multiple LSHPP
equilibria can exist.
To further illustrate the properties of an equilibrium under double-crossing preferences,
we illustrate how the equilibrium varies with changes in some key parameters of the model
when we use the same algorithm. Although comparative statics is cumbersome when there
are multiple equilibria, this exercise still allows us to elucidate some general tendencies
and important insights.
Consider the signaling with news application of Section 3. We choose parameters so
that


a a2
+
,
u(a, t, θ ) = λ (θ + (1 − θ )t) −θ2

and let θ be uniformly distributed on [0.1, 0.5]. Let a p represent a pooling action in a fullpooling equilibrium, and let t p = E[θ ] = 0.3. Preventing downward deviation requiresm(a p , t p , 0.1) ≥ m(a p , t p , 0.5). Since the marginal rate of substitution does not depend ont p in this case, this requirement reduces to a p ≤ 8. Preventing upward deviation requires
θmin (a p , t p ; Q(a p ))) ≤ 0.3, which reduces to a p ≥ 40/9. Furthermore, u(a p , t p , 0.1) ≥u(0, 0.1, 0.1) for any a p ≤ 8 if λ ≥ 5600/9. We can conclude that for λ ≥ 5600/9, any
action a p ∈ [40/9, 8] can constitute part of a full pooling equilibrium.
We now turn to the case where there is some separation in equilibrium. In general,an equilibrium exhibits less separation and more pooling (i.e., θ0 decreases towards θ ) as
the returns to signaling λ become larger. As λ increases, higher types need to take evenhigher actions to separate because lower types now have more incentive to mimic. The
equilibrium action taken by higher types cannot be unbounded, however, because of the
double-crossing property: as the equilibrium action increases, it enters the RSC-domain
where it is more costly for higher types to choose higher actions. As we have seen, we
can always construct a fully pooling equilibrium when λ ≥ 5600/9, but the equilibriumoutcome exhibits less pooling when the returns to signaling fall below this threshold.

20

,t

,t

,t

Figure 3. Equilibrium actions for different values of λ and κ. Higher λ corresponds to larger returns
to signaling. The case of κ = −12.5 corresponds to a type distribution with thinner right tail than theuniform distribution. The red line shows the locus of (S(θ ), T (θ )) in the (a, t)-space above the gap.
The blue line plots a = S(θ ) against θ .

Figure 3 further illustrates this tendency: when λ increases from 50 to 100, the rangeof the fully separating region shrinks (i.e., θ0 decreases) and the equilibrium actions for all
types increase. This is different from the standard setup, where an increase in the returns
to signaling only stretches out equilibrium actions but yields no qualitative impact on theform of the equilibrium.
In the left and middle panels of Figure 3, S(·) is flat above the gap. This example
thus shows that countersignaling is not a necessary consequence of the double-crossing
property. To construct an equilibrium with atomless pooling, we manipulate the type distribution by letting f (θ ) = 2.5 + κ(θ − 0.3) for θ ∈ [0.1, 0.5]. Atomless pooling is more
likely to emerge as the slope parameter κ decreases. Figure 3 shows that for λ = 100,
the equilibrium is LSHP when κ = 0 (uniform distribution) but exhibits atomless poolingwhen κ = −12.5, as atomless pooling is more likely to emerge when the type distribution
has a thinner tail.

6.

Discussion

6.1.

Incentive compatibility under double-crossing preferences

Although we study double-crossing preferences in signaling models, the methods developed in this paper are useful for analyzing related environments that exhibit this class ofpreferences, especially in the context of mechanism design. To illustrate this point, we
now look at the problem from a different perspective and examine the set of “allocations,”21

(S(·), T (·)), that can be incentive compatible, without imposing any association between
S(·) and T (·). In the following discussion, we let φ ∗ (·; θ ) := φ(·, u(S(θ ), T (θ ), θ ), θ ) be
the indifference curve of type θ that passes through his equilibrium allocation (S(θ ), T (θ ))and refer to it as the equilibrium indifference curve of type θ .
One qualitative feature of an LSHPP equilibrium is that S(·) must be (weakly) quasiconcave. Proposition 2, which establishes quasi-concavity, requires only incentive compatibility and continuity, and its logic is very simple in light of our framework.8 To use
this result in an environment with double-crossing preferences, we only need to establish
continuity.9 In this paper, we exploit special features of the signaling model—namely, T (·)
is consistent with S(·) and Bayes’ rule on the equilibrium path and satisfies the D1 criterion off the equilibrium path—to prove continuity. In other potential applications suchas screening or mechanism design models with a continuum of types, the continuity ofS(·) often follows from incentive compatibility or from optimality, and the conclusion of
Proposition 2 remains valid in these different settings.
Another feature of our model is that we rely only on local incentive compatibility toconstruct an equilibrium and verify that incentive compatibility holds globally for all types.
In mechanism design under single-crossing preferences, it is well known that monotonicity
and local incentive compatibility imply global incentive compatibility (Maskin and Riley,
1984; Mailath, 1987), a result that greatly simplifies the analysis. Under double-crossing
preferences, an obvious concern is that ensuring incentive compatibility can become complicated and intractable: as Hörner (2008) remarks in an encyclopedic article on signaling
and screening, “Little is known about equilibria when single-crossing fails.” Below we
explain how our setup allows us to overcome this issue.
An allocation satisfies local IC if no type has an incentive to deviate locally. Formally,for each θ ∈ [θ , θ ], there is an ε > 0 such that u(S(θ ), T (θ ), θ ) ≥ u(S(θ 0 ), T (θ 0 ), θ )for all θ 0 ∈ (θ − ε, θ + ε). In general, under double-crossing preferences, this condition is insufficient to ensure global incentive compatibility. However, local IC together
with a pairwise-matching condition would be sufficient under Assumptions 2 and 3. Letθ∗ := sup{θ 0 : S(θ ) ≤ D(T (θ ); θ , θ ) for all θ ≤ θ 0 }. We say that an allocation satis8

Of course, our characterization result, Theorem 1, states much more than this, as it establishes pairwisepooling, and hence the proof necessarily becomes more involved.
9
It is not possible to rule out the possibility that S(θ1 ) ≥ S(θ3 ) > S(θ2 ) for θ1 < θ2 < θ3 by incentive
compatibility alone because Assumptions 2 and 3 do not impose enough restrictions on the rankings ofallocations across domains.

22

fies the pairwise-matching condition if for any θ 0 > θ∗ , there exists a θ 00 ≤ θ∗ such that
(S(θ 0 ), T (θ 0 )) = (S(θ 00 ), T (θ 00 )) and m(S(θ 0 ), T (θ 0 ), θ 0 ) = m(S(θ 00 ), T (θ 00 ), θ 00 ).
Proposition 3. Under Assumptions 2 and 3, an allocation that satisfies local IC and the
pairwise-matching condition is incentive compatible.
Proof. Because S(θ ) ≤ D(T (θ ); θ , θ ) for all θ ∈ [θ , θ∗ ], local IC implies that S(·) is weakly
increasing on this interval. We first show that incentive compatibility holds for any pair oftypes on this interval. Consider any two types θ1 < θ2 ≤ θ∗ . By Assumption 3,
S(θ2 ) ≤ D(T (θ2 ); θ2 , θ2 ) < D(T (θ2 ); θ2 , θ1 ).

(9)Assumption 2 requires that φ ∗ (·; θ2 ), the equilibrium indifference curve of type θ2 , cannot
cross D(·; θ2 , θ1 ) to the left of S(θ2 ), such that the single-crossing property holds along thisindifference curve:a < D(φ ∗ (a; θ2 ); θ2 , θ1 ) for a ≤ S(θ2 ).

(10)lower-type θ1 < θ2 always has a higher marginal rate of substitution than type θ2 .
We argue that any locally IC allocation must stay below φ ∗ (a; θ2 ) for a ∈ [S(θ1 ), S(θ2 )].
Suppose the opposite is true, and let T (θ 00 ) ≥ φ ∗ (S(θ 00 ); θ2 ) with T (θ ) < φ ∗ (S(θ ); θ2 )for all θ ∈ (θ 00 , θ2 ]. Local IC then implies that φ ∗ (·; θ 00 ) reaches φ ∗ (·; θ2 ) from above atsome a00 ∈ [S(θ 00 ), S(θ2 )). However, this is a contradiction because by (10), type θ 00 musthave a higher marginal rate of substitution at any point on φ ∗ (·; θ2 ). This shows thatT (θ1 ) < φ ∗ (S(θ1 ); θ2 ), and so type θ2 has no incentive to mimic type θ1 . Similarly, any
locally IC allocation must stay below φ ∗ (a; θ1 ) for a ∈ [S(θ1 ), S(θ2 )]. Suppose the opposite is true, and let T (θ 0 ) ≥ φ ∗ (S(θ 0 ); θ1 ) with T (θ ) < φ ∗ (S(θ ); θ1 ) for all θ ∈ [θ1 , θ 0 ).
Local IC implies that φ ∗ (·; θ1 ) reaches φ ∗ (·; θ 0 ) from above at some a0 ∈ (S(θ1 ), S(θ 0 )].
However, this is a contradiction because by (10), type θ1 must have a higher marginal rateof substitution at any point on φ ∗ (·; θ 0 ). This shows that T (θ2 ) < φ ∗ (S(θ2 ); θ1 ), and thus
type θ1 has no incentive to mimic type θ2 .
This argument shows that no type below θ∗ has an incentive to mimic any other typebelow θ∗ . By the pairwise-matching condition, for any allocation received by type θ 0 > θ∗ ,there is a type θ 00 ≤ θ∗ such that (S(θ 0 ), T (θ 0 )) = (S(θ 00 ), T (θ 00 )). Since any type below θ∗
prefers his own allocation to (S(θ 00 ), T (θ 00 )), he prefers his own allocation to (S(θ 0 ), T (θ 0 )).
Therefore global incentive compatibility holds for types below θ∗ .
Next consider a type θ 0 > θ∗ . Let θ 00 ≤ θ∗ be the type that receives the same allocation,(a p , t p ), as type θ 0 , with m(a p , t p , θ 0 ) = m(a p , t p , θ 00 ). Assumption 2 implies that φ ∗ (·; θ 0 ) is23

tangent to and is “more convex” than φ ∗ (·; θ 00 ), and hence must stay strictly above φ ∗ (·; θ 00 )for all a 6= a p . This means that whenever type θ 00 prefers his allocation (a p , t p ) to someallocation (a, t), type θ 0 must also prefer (a p , t p ) to (a, t). We have already establishedthat type θ 00 prefers (a p , t p ) to (S(θ ), T (θ )) for any θ ∈ [θ , θ ]. Therefore, global incentive
compatibility holds for types above θ∗ .
Our LSHPP equilibrium satisfies the pairwise-matching condition, and therefore global
incentive compatibility holds. Importantly, this condition is also satisfied in a variety ofother settings, such as the screening model of Araujo and Moreira (2010). This is becausewith continuous types, the graph of the allocation (S(·), T (·)) is often the lower envelopeof the indifference curves of the types choosing their respective allocations, thus forcing
two types that pool at the same allocation to have the same marginal rate of substitution.
Both the quasi-concavity result (Proposition 2) and global incentive compatibility result(Proposition 3) depend on Assumptions 2 and 3: Assumption 2 states that the SC-domainlies to the left of the dividing line and the RSC-domain lies to the right, and Assumption
3 states that the dividing line shifts to the left with type. This is obviously not the only
possible specification of double-crossing preferences. In principle, the SC-domain can lie
to the right of the dividing line or the dividing line can move to the right. This gives
rise to four specifications. For example, suppose that Assumption 3 does not hold (while
maintaining Assumption 2) such that D(·; θ 0 , θ 00 ) is increasing in both θ 0 and θ 00 . Thisset of assumptions, which we call the “alternative specification” in this discussion, implies
that m(a, t, ·) is quasi-concave for any (a, t). Because intermediate types have the highest
signaling costs under the alternative specification, it is unsurprising that S(·) will not be
quasi-concave, and Proposition 2 does not follow.
To see why Proposition 3 holds under Assumptions 2 and 3, note that the key step of
the proof is equation (9), which relies on Assumption 3. Moreover, Assumption 2 implies
that the equilibrium indifference curve of type θ2 must stay to the left of the dividing lineD(·; θ2 , θ1 ) for any θ1 < θ2 . This fact implies that no allocation can cross the equilibrium
indifference curve of type θ2 from above because a lower type has a higher marginal rateof substitution, which leads to downward incentive compatibility. A similar argument isused to establish upward incentive compatibility.
This argument more generally suggests that the key to ensuring global incentive compatibility under double-crossing preferences is that the allocation and the dividing line
move in opposite directions. Under our current specification (Assumptions 2 and 3), the24allocation increases in type (i.e., S(·) is weakly increasing) for types below θ∗ while thedividing line decreases in type. Under the alternative specification, however, they movein the same direction for types below θ∗ because the dividing line is increases in type. In
this case, even if the allocation is in the SC-domain for all θ ∈ [θ , θ∗ ], we may still have
D(T (θ2 ); θ2 , θ1 ) < S(θ2 ) < D(T (θ2 ); θ2 , θ2 ) for some θ1 < θ2 ≤ θ∗ , in which case equation(9) no longer holds. When this occurs, the marginal rate of substitution is not monotone
in type at some points on the equilibrium indifference curve of type θ2 , such that an indifference curve of a lower type θ1 < θ2 can cross the equilibrium indifference curve oftype θ2 twice to the left of S(θ2 ); therefore, there is no generally tractable way of ensuring
incentive compatibility.
It is easy to see that for each of the four possible specifications, there is always a domainthat is “well-behaved” (in the sense of ensuring incentive compatibility) and another thatis “ill-behaved.”10 If an allocation lies entirely in the well-behaved domain, local incentive
compatibility implies global incentive compatibility as under single-crossing preferences.
This is of course not the end of the story because an allocation may cross the dividing line
and enter the ill-behaved domain. In this case, we take advantage of the pairwise-matchingcondition for types in the ill-behaved domain to ensure global incentive compatibility. Pairwise matching works whenever these types have “more convex” indifference curves than
do the types (in the well-behaved domain) that they are pooling with. Under our set ofassumptions, higher types have strictly more convex indifference curves. Therefore, if
the lower type prefers his own allocation to that of any other type, the higher type alsoprefers his own allocation, meaning that incentive compatibility for the lower type implies
incentive compatibility for the paired higher type.
6.2.

Countersignaling and the relation to the literature

Our characterization shows that we may have two distinct forms of pooling above the gap.
When there is atomless pooling, S(·) must be strictly increasing in one arm and strictly
decreasing in the other, suggesting that equilibrium actions are nonmonotone. This phenomenon, where higher types separate from intermediate types by pooling with lower
10

In the screening literature, Araujo and Moreira (2010) and Schottmüller (2015) both adopt a specification in which the RSC-domain is to the left of the dividing line. If the dividing line decreases with type,
the RSC-domain is ill-behaved, as in Araujo and Moreira (2010). If the dividing line increases with type, theSC-domain is ill-behaved, as in Schottmüller (2015). Therefore, local incentive compatibility is insufficient
to ensure incentive compatibility. We provide a more detailed discussion on these possible specifications inAppendix F.

25

types at low signaling levels, is known as “countersignaling” in the literature. With acontinuum of types, atomless pooling and countersignaling are equivalent: there is countersignaling if and only if there is atomless pooling.
Examples of countersignaling abound.11 We often observe that the most talented individuals deviate from social norms, e.g., CEOs casually wearing jeans in formal occasions
or successful startup entrepreneurs not bothering to finish college. This type of observation can be rationalized by signaling with news, where exceptional talents expect they canreveal their competence through their vision, creativity, and charisma. From the theoretical point of view, it is clear that the single-crossing property must fail in some sense tohave any form of countersignaling. Aside from this, however, little is known about what
restrictions, in terms of primitives, are required to generate countersignaling in generalenvironments. Below, we discuss how our general framework sheds new light on thisperverse yet pervasive phenomenon in relation to the existing literature.
The seminal work on countersignaling is Feltovich et al. (2002). The authors provide a model, whose essence is captured by the signaling with news example, with three
types—high, medium, and low—and construct a countersignaling equilibrium in which
the medium type chooses a higher action while the high type and the low type are pooled.
As enlightening as it is, however, extending this setup to more than three types seems
formidable without the discipline from a more systematic framework. This is problematic
because a model with three types can only give a partial picture of countersignaling. In
particular, our general setup clarifies two points about countersignaling that would not
be obvious if we only had a model with three types. First, the highest-type θ typicallypools with some intermediate type θ0 instead of the lowest-type θ —CEOs may dress casually but they do not dress like a tramp. Second, an equilibrium seldom takes the formof pooling at two actions only: when there is a continuum of types, we have shown that
it is not possible for S(·) to jump between two pooling actions, and when types are discrete but there are many of them, small differences in signaling costs between adjacenttypes impose constraints on how far apart their signaling actions can be. In other words,
fine distinctions through small differences in signaling actions are still the norm, even in
11

Feltovich et al. (2002) provide experimental evidence that subjects do indeed engage in countersignaling. They also raise a number of examples drawn from common observations, such as “The nouveau
riche flaunt their wealth, but the old rich scorn such gauche displays.” Also see Araujo et al. (2007) andthe references therein. Dixit and Nalebuff’s (2008) book, The Art of Strategy, has a section devoted to countersignaling.

26

environments that exhibit countersignaling.12 It is worth emphasizing that both of thesepoints are endogenous features of our model derived from model primitives such as payoffand type distribution functions.
In terms of the form of the equilibrium, our work is also related to Araujo et al. (2007),
but the logic and the mechanism behind it are different. They consider a model in which
workers are characterized by two attributes: intelligence ι and perseverance η.13 Workers
invest in schooling activity a as a signal with marginal signaling cost 1/(ιη). In addition
to observing the schooling signal, firms also observe an interview score, g = αι + η. Thus,
given the interview score g, the two attributes can be reduced to a one-dimensional type,θ = ι, with marginal rate of substitution given by:m(a, t, θ ) =1.
θ (g − αθ )This marginal rate of substitution is quasi-convex in type, as in our model. However, acrucial difference is that m(a, t, θ ) above does not depend on the level of signaling activity
a, so it does not satisfy Assumption 2. For any two types, θ 00 < θ 0 , either (a) type θ 00
has steeper indifference curves for all signaling levels (global single-crossing); (b) type
θ 0 has steeper indifference curves for all signaling levels (global reverse single-crossing);or (c) the two types have identical indifference curves. Under this specification, indifference curves of any pair of types never cross twice, so preferences do not belong to theclass of double-crossing preferences described in our paper. There are no dividing linesthat separate the SC-domain from the RSC-domain. The fact that indifference curves cancross at most once in Araujo et al. (2007) makes the analysis much simpler. Types are
exogenously paired together when they have identical indifference curves, and each pairof types naturally choose the same signaling action, resulting in an equilibrium with only
atomless pooling. This draws a clear contrast to our setting, where the form of p(·), which
determines how agents are paired together, is endogenously determined. Although theanalysis of our model is more involved, the environment we consider is quite general andhas a wide scope of applicability, as we illustrate in the following subsection.
12

Although our characterization is established for the case of continuous types, the logic of this construction applies more generally. Specifically, we can show that S(·) must be quasi-concave, and that a (weakerversion of) pairwise-matching condition must hold in the discrete type case.
13Frankel and Kartik (2019) discuss how two-dimensional types may lead to failure of the single-crossing
property. See also Ball (2020).

27

6.3.

More examples

One of the main contributions of this paper is that it identifies a set of assumptions, mostimportantly Assumptions 2 and 3, which makes the analysis under double-crossing preferences tractable and leads to a clear characterization of equilibrium. Our specification isbut one class of environments that capture how the single-crossing property may fail. In
addition to technical considerations, we argue that the set of assumptions we adopt is economically meaningful and relevant. The signaling with news example is a case in point,
showing a natural economic environment that fits the description of our model. It also
points to the possibility that the single-crossing property is not as robust as it is generally
believed, as it can easily break down with some minor but realistic modifications to theunderlying environment.
Below, we provide three more examples to solidify this point and to make a case for why
double-crossing preferences are pervasive in reality. The common thread of our examples
is that gains from signaling are typically bounded. Moreover, in many economic situations of interest, higher types reach this point of diminishing returns earlier than do lower
types. When these conditions are met, we tend to observe double-crossing preferences as
specified in our framework.
Reputation enhances the chances of success. In many facets of life a person’s chancesof success depend not only on his true ability but also on other people’s perception of
his ability. Take the case of a startup entrepreneur. His reputation in the market affects
the availability of initial funding and the capacity to attract talent to work in his firm.
These factors, together with his true entrepreneurial ability, determine the performanceof his business and its probability of reaching the next milestone (such as developing a
prototype product, or attracting the next round of funding). In this example, the signaling
incentive comes from the fact that reputation matters for improving performance.
Suppose the performance of a startup entrepreneur is θ + β t + ", where β > 0 is aweight that determines the importance of reputation relative to true ability. The term "summarizes the random factors that may affect performance, and its distribution is givenby H(·) with the corresponding log-concave density h(·). The startup business can reachthe next milestone if its performance exceeds some exogenous threshold K, and the valueof reaching the milestone is V . Let a represent the level of signaling activity he chooses toestablish his reputation, such as the level of education or the share of equity held by the28entrepreneur (Leland and Pyle, 1977). The payoff to the entrepreneur is


γa a2
u(a, t, θ ) = V (1 − H(K − θ − β t)) −+,θ2where γ > 0 is a signaling cost parameter. This givesm(a, t, θ ) =γ+θa.
θ β V h(K − θ − β t)

One can verify that m(a, t, θ ) is quasi-convex in θ . Moreover, for θ 0 > θ 00 , if φ(·, u0 , θ 00 ) is
an indifference curve of type θ 00 at some utility level u0 , then the ratio 00  m(a, φ(a, u0 , θ 00 ), θ 0 )h(K − θ 00 − βφ(a, u0 , θ 00 ))γ + θ 0aθ=m(a, φ(a, u0 , θ 00 ), θ 00 )θ0
γ + θ 00 ah(K − θ 0 − βφ(a, u0 , θ 00 ))strictly increases in a by the log-concavity of h(·). Thus, condition (1) holds, and the
double-crossing property is satisfied.
In this example, the payoff from signaling is bounded from above by V . Moreover,the log-concavity of h(·) implies that the density function is unimodal. This means thata higher reputation does not significantly improve the probability of success for very low
types or very high types. The marginal increase in the probability of reaching the target
K is the greatest for intermediate types, and they tend to have the greatest incentives toinvest in signaling.
Risky experimentation. This example is drawn from our previous work (Chen et al.,2021), which deals with two discrete types but can be readily adapted to continuous
types.14 The model is an optimal stopping problem with reputation concerns. If the agent
achieves success at some random time, then he receives a payoff of V . If he abandons the
project at time a, then the outside-option payoff is given by his reputation t at the timeof termination. The key question is whether an agent with superior ability, modeled by ahigher Poisson arrival rate of success, signals his type by staying with a risky project for alonger duration or by quitting early. Bobtcheff and Levy (2017) explore related incentives.
It is easy to show that this model satisfies Assumptions 2 and 3 and thus exhibits the
double-crossing property. The reason for this is intuitive. Higher types are more likely toachieve success if the state is good. This implies that they have more incentive to persist with the risky project compared to lower types at early stages when the difference in
14

In Chen et al. (2021), signaling is beneficial up to some point and hence not always costly. As noted

above, our results can also be applied to this case.

29their beliefs about the state is relatively small. As experimentation continues and yields nosuccess, higher types become pessimistic more quickly than lower types do because theylearn faster that their project is not promising. Past some point, they become more reluctant to persist. This structure suggests that signaling by persisting with the risky projectis relatively more attractive for higher types than for lower types when a is small, but thecomparison flips when a is large.
Productive signaling. Many signaling models assume away any positive benefit of signaling activity to isolate its role in conveying hidden information. While this assumption
may appear innocuous, once we admit the possibility that signals can be directly productive, details of the model specification can have substantial impacts and yield qualitatively
different predictions for signaling outcomes.
Assume that education is directly productive in addition to serving as a signal about
private information. Specifically, let s = aθ represent an agent’s skill, which depends bothon his natural ability θ and on the level of education a. The labor-market benefit fromhaving skill s and reputation t is βs + t, and the cost of acquiring skill through education isC(a, θ ) = γ0 a + γ(aθ )2 . This cost function is unconventional because Caθ > 0, indicatingthat high-ability agents have a higher marginal cost of investing in education—say, due toopportunity costs. However, we may also express the cost of acquiring skill as a functionof the target skill level and write C̃(s, θ ) = γ0 s/θ + γs2 . This formulation shows that the
total cost of reaching skill level s and the marginal cost of increasing skill are lower for
higher types. In this example, the utility function has the form:u(a, t, θ ) = β aθ − γ0 a − γ(aθ )2 + t.
It can be readily verified that this formulation satisfies Assumptions 2 and 3. What appears
to be a minor—and not unreasonable—modification in specification converts the standard
setup into a model that exhibits the double-crossing property.

7.

Conclusion

Despite its widespread use in economic analysis, the single-crossing property imposes
strong restrictions on the structure of preferences, and its validity and robustness are notnecessarily always evident in economic applications. Because many insights about signaling behavior we learn from standard models depend on this property, it is important
to extend the scope of analysis to circumstances that are not constrained by the singlecrossing property. We take a step in this direction by providing a formal framework to30

capture double-crossing preferences in signaling models. Our characterization shows thatequilibrium under double-crossing preferences exhibits a particular form of pooling at thehigher end of types, which we label pairwise-pooling. Pairwise-pooling generalizes andclarifies a phenomenon known as countersignaling in the literature: double-crossing preferences often induce middle types to invest more in signaling whereas higher types arecontent with pooling with lower types. Our model identifies the assumptions on preferences that tend to produce pairwise-pooling, as well as the constraints that affect the form
it takes (i.e., atomless or mass pooling). We provide a simple algorithm to find an LSHPP
equilibrium and show that equilibrium exists under fairly weak conditions. Some of theanalysis in this paper can potentially illuminate broader incentive compatibility issues under double-crossing preferences in a wider class of mechanism design models.
From a practical point of view, it is perhaps not very controversial to say that the singlecrossing property may fail in some situations. The problem is rather that this can occur
in many different ways. Although we argue that our framework covers a broad range of
economically relevant situations and that this framework is relatively tractable, it does notby any means exclude other variations of non-single-crossing preferences. We hope to see
more work along these lines to gain a more comprehensive understanding of signaling
behavior that goes beyond the single-crossing property.

31

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33

Appendix
A. Proof of Theorem 1
A.1.

Preliminaries

Denote the set of types that choose action a in equilibrium as Q(a) = {θ : S(θ ) = a}. If
there is some action a p such that Q(a p ) is neither empty nor a singleton, then we refer toa p as a pooling action and to Q(a p ) as a pooling set. Recall that we define θ p := max Q(a p )and θ p := min Q(a p ).
Consider some pooling action a p . We say that actions below a p are on-path if there
exists a small ε > 0 such that Q(a) 6= ; for all a ∈ (a p − ε, a p ); otherwise, actions below a p
are off-path. Similarly, actions above a p are on-path if there exists a small ε > 0 such that
Q(a) 6= ; for all a ∈ (a p , a p + ε); otherwise, actions above a p are off-path. If there exists
a sequence θ n approaching θ 0 for some θ 0 such that S(θ n ) approaches a p , with eitherS(θ n ) > a p or S(θ n ) < a p for all n, then we call θ 0 a limit type.
The following lemma shows an important property of double-crossing preferences that
we exploit repeatedly. Define q(a, t, θ ) such that m(a, t, q(a, t, θ )) = m(a, t, θ ), withq(a, t, θ ) = θ if θ = θmin (a, t) (where we omit the last argument of θmin (a, t; Q) whenever
it is not confusing). This mapping gives a counterpart type that has the same marginal
rate of substitution at (a, t). If no such counterpart type exists, then let q(a, t, θ ) = θ if
θ < θmin (a, t) and q(a, t, θ ) = θ if θ > θmin (a, t).
Lemma 3. Consider some type θ 0 and two choices (a1 , t 1 ) and (a2 , t 2 ) where a1 > a2 . Suppose
θ 0 > θmin (a1 , t 1 ) and u(a2 , t 2 , θ 0 ) ≥ u(a1 , t 1 , θ 0 ). If q(a1 , t 1 , θ 0 ) > θ and a1 is bounded awayfrom a2 , then u(a2 , t 2 , θ ) > u(a1 , t 1 , θ ) for all θ ∈ [θ , q(a1 , t 1 , θ 0 )].
Proof. If type θ 0 is indifferent between (a1 , t 1 ) and (a2 , t 2 ), then type q(a1 , t 1 , θ 0 ), whose
indifference curve that passes through (a1 , t 1 ) must stay strictly below that of type θ 0 ,strictly prefers (a2 , t 2 ) to (a1 , t 1 ). For all types below q(a1 , t 1 , θ 0 ), the standard argument
suggests that they strictly prefer (a2 , t 2 ) to (a1 , t 1 ).
Lemma 3 states that if two indifference curves are tangent at some (a1 , t 1 ), the indifference curve of the higher type is strictly contained by that of the lower type because ofAssumption 2. This lemma is useful when a1 is bounded away from a2 . When a1 and a2

34

are arbitrarily close to each other, however, the preference ranking between (a1 , t 1 ) and(a2 , t 2 ) depends only on the marginal rate of substitution at that point.
Lemma 4. If actions above a p are on-path with limit type θ 0 ∈ [θ p , θ p ], then no type between
θ 0 and q(a p , t p , θ 0 ) chooses a p . If actions below a p are on-path with limit type θ 0 , then only
types between θ 0 and q(a p , t p , θ 0 ) may choose a p .
Proof. When there is a continuous path S(·) to a p , preferences are determined entirely by
the marginal rate of substitution at (a p , t p ). Since all types between θ 0 and q(a p , t p , θ 0 )have a lower marginal rate of substitution than type θ 0 , they strictly prefer an actionslightly above a p . For the second statement, let θ 0 < θmin (a p , t p ) (the opposite case follows
by the same argument). Then, all types below θ 0 and above q(a p , t p , θ 0 ) have a higher
marginal rate of substitution than type θ 0 . They strictly prefer an action slightly below
ap .
Given this result, we show that any pooling set must span across the dividing line. The
following result holds regardless of whether a pooling set is connected or disconnected.
Lemma 5. If there is pooling at (a p , t p ), then it is in the SC-domain of type θ p and in the
RSC-domain of type θ p . Moreover, m(a p , t p , θ p ) ≥ m(a p , t p , θ p ).
Proof. Suppose actions below and above a p are off-path. In this case, D1 requires thatm(a p , t p , θ p ) ≥ m(a p , t p , θ p ) > m(a p , t p , θmin (a p , t p )). This is possible only if m(a p , t p , ·)decreases at θ p (in the SC-domain) and increases at θ p (in the RSC-domain), i.e., (a p , t p )belongs to SC(θ p ) and to RSC(θ p ).
Now suppose that actions above a p are on-path. Let θ 0 be a limit type. Since Lemma 4
suggests that no type between θ 0 and q(a p , t p , θ 0 ) chooses a p , we must have θ 0 , q(a p , t p , θ 0 ) ∈(θ p , θ p ). This is possible only if (a p , t p ) belongs to SC(θ p ) and to RSC(θ p ). Moreover, if
actions below a p are off-path, then D1 requires m(a p , t p , θ p ) ≥ m(a p , t p , θ p ).
Finally, suppose actions below a p are on-path. Let θ 0 be a limit type. If θ 0 < θmin (a p , t p ),then θ 0 = θ p because no type below θ 0 can choose a p by Lemma 4. Note that since t p > θ p ,there must be another limit type θ 00 = θ p to satisfy incentive compatibility. By Lemma 2,
m(a p , t p , θ 0 ) = m(a p , t p , θ 00 ), which means (a p , t p ) belongs to SC(θ p ) and to RSC(θ p ).
If a pooling set is connected, then the following fact trivially holds: S(·) is continuousand (weakly) quasi-concave on [θ p , θ p ]. Along with Lemma 5, this essentially gives a35

characterization of connected pooling sets. However, the case of disconnected poolingsets is much more complicated, which we will discuss next.
A.2.

Disconnected pooling sets

Consider a disconnected pooling set Q(a p ). Let t p be the corresponding reputation. Additionally, define J(a p ) := {θ : θ ∈/ Q(a p ), θ ∈ (θ p , θ p )}, and let θ j := inf J(a p ) andθ j := sup J(a p ).
Lemma 6. Suppose there is pooling at (a p , t p ) such that the pooling set Q(a p ) is disconnected.
(a) S(θ ) is continuous for all θ ∈ [θ p , θ p ].
(b) S(θ ) ≥ a p for all θ ∈ [θ p , θ p ].
(c) Q(a p ) = Q L (a p ) ∪ Q R (a p ), where Q L (a p ) and Q R (a p ) are two disjoint intervals, with(a p , t p ) ∈ SC(θ ) for θ ∈ Q L (a p ) and (a p , t p ) ∈ RSC(θ ) for θ ∈ Q R (a p ).
Proof. Part (a). Suppose S(·) is discontinuous on [θ p , θ p ]. There are two cases: one inwhich S(·) jumps up and the other in which S(·) jumps down.
Case 1. Suppose S(·) jumps up at some θ1 ∈ (θ p , θ p ). Let (S(θ1+ ), T (θ1+ )) = (a1 , t 1 ). By
continuity, type θ1 must be indifferent between (a p , t p ) and (a1 , t 1 ).
We first argue that there cannot be any pooling at (a1 , t 1 ). Suppose otherwise. By
Lemma 5, we have (a1 , t 1 ) ∈ RSC(max Q(a1 )). This means that a type slightly above
max Q(a1 ) must choose some (a0 , t 0 ) such that a0 < a1 . By continuity, type max Q(a1 ) must
be indifferent between (a1 , t 1 ) and (a0 , t 0 ). If S(·) is continuous at max Q(a1 ) and a0 isarbitrarily close to a1 , then there must be another limit type min Q(a1 ) ≤ θ p . Note thatm(a1 , t 1 , min Q(a1 )) ≥ m(a1 , t 1 , max Q(a1 )) by Lemma 5 and, hence, m(a1 , t 1 , min Q(a1 )) ≥m(a1 , t 1 , θ ) for any θ ∈ (min Q(a1 ), max Q(a1 )). Lemma 3 then implies that type min Q(a1 )must strictly prefer (a p , t p ), which is a contradiction. If a0 is bounded away from a1 ,then we can directly apply Lemma 3 to show that type min Q(a1 ) strictly prefers (a0 , t 0 )to (a1 , t 1 ). As such, there cannot be any pooling at (a1 , t 1 ).
This argument establishes that S(·) must be fully separating in a right neighborhoodof θ1 , meaning that there is some ε > 0 such that both S(·) and T (·) are increasing on(θ1 , θ1 + ε). Additionally, observe that (S(θ ), T (θ )) ∈ SC(θ ) for θ ∈ (θ1 , θ1 + ε), suchthat if there is any jump in S(·), then it must be upward. But then we can apply the36same argument as above, and there cannot be any pooling immediately after the jump.
Furthermore, if S(·) is fully separating again after the jump, then T (·) must be continuous
and incentive compatibility cannot be satisfied. This shows that S(·) must be continuous
and fully separating until it jumps down to a p at some θ2 ∈ (θ p , θ p ). This is a contradiction,
however, because if this is the case, then θ2 ∈ SC(S(θ2− ), T (θ2− )) and S(·) cannot jumpdown to a p at θ2 .
Case 2. Now suppose S(·) jumps down at some θ1 ∈ (θ p , θ p ). Let (S(θ1+ ), T (θ1+ )) =(a1 , t 1 ). Since S(·) jumps down at θ1 , we have (a p , t p ) ∈ RSC(θ1 ). By continuity, type θ1
must be indifferent between (a p , t p ) and (a1 , t 1 ). By Lemma 5, m(a p , t p , θ p ) ≥ m(a p , t p , θ p ),which in turn implies m(a p , t p , θ p ) > m(a p , t p , θ1 ). This is a contradiction because if type
θ1 is indifferent between (a p , t p ) and (a1 , t 1 ), then type θ p must strictly prefer (a1 , t 1 ) to(a p , t p ) by Lemma 3.
Part (b). It directly follows from Proposition 2 and part (a).
Part (c). By parts (a) and (b), if J(a p ) is not empty, we must have S 0 (θ j + ) > 0 and−

S 0 (θ j ) < 0. Therefore, there exists ε > 0 such that for any a ∈ (a p , a p + ε), Q(a) is apooling set. Using part (b) again, all types in [min Q(a), max Q(a)] choose actions higher
than or equal to a and cannot choose a p . This establishes that Q(a p ) = Q L (a p ) ∪ Q R (a p ),with Q L (a p ) = [θ p , θ j ] and Q R (a p ) = [θ j , θ p ]. Furthermore, S 0 (θ +j ) > 0 implies that(a p , t p ) ∈ SC(θ −j ). By Assumption 3, (a p , t p ) is in the SC-domain of all types in Q L (a p ).
−Similarly, S 0 (θ j ) < 0 implies that (a p , t p ) is in the RSC-domain of all types in Q R (a p ).
Lemma 6 also implies that for any disconnected pooling set Q(a p ), there are two limit
types that approach a p from inside the interval (θ p , θ p ), given by θ j and θ j , with θ j <−θmin (a p , t p ) < θ j such that S(θ j + ) = S(θ j ) = a p . Moreover, m(a p , t p , θ j ) = m(a p , t p , θ j ).
Both θ j and θ j belong to Q(a p ), and the types in J(a p ) choose actions higher than a p .
To obtain an LSHPP strategy, we need to ensure that max Q(a p ) = θ for some a p .
Lemma 7. If actions below any pooling action a p are off-path, then Q(a p ) must include θ .
++Proof. Suppose θ p < θ , and let (S(θ p ), T (θ p )) = (a1 , t 1 ) where a1 is bounded awayfrom a p . By Lemma 5, (a p , t p ) ∈ RSC(θ p ) and hence a1 < a p . By continuity, type θ pmust be indifferent between (a p , t p ) and (a1 , t 1 ). Lemma 5 also suggests, however, thatm(a p , t p , θ p ) ≥ m(a p , t p , θ p ), or equivalently θ p ≤ q(a p , t p , θ p ). This is a contradiction37because if type θ p is indifferent between (a p , t p ) and (a1 , t 1 ), then type θ p must strictlyprefer (a1 , t 1 ) to (a p , t p ) by Lemma 3.
A.3.

Below the gap

The argument thus far characterizes the equilibrium above the gap. Below, we deal withthe situation below the gap.
Lemma 8. Suppose S(θ ) = s∗ (θ ) for θ ∈ (θ1 , θ2 ). If S(·) jumps at θ2 , then Q(S(θ2+ )) is apooling set and S(θ2+ ) > s∗ (θ2− ).

Proof. Suppose there is a jump at θ2 to S(θ2+ ). Since (s∗ (θ2− ), θ2 ) ∈ SC(θ2 ), we must haveS(θ2+ ) > s∗ (θ2− ). If Q(S(θ2+ )) is also a singleton, then incentive compatibility must be

violated because T (θ2+ ) = T (θ2− ) = θ2 , while S(θ2+ ) > S(θ2− ).

Suppose there is some pooling at (a0 , t 0 ) where a0 = min{a : Q(a) is a pooling set}.
Lemma 8 suggests that once an equilibrium starts from a fully separating segment s∗ (·),
there are only two possibilities. First, there could be some θ0 such that s∗ (θ0 ) = a0 . Second,
S(·) could jump at θ0 to S(θ0+ ) = a0 . The first possibility is ruled out because T (θ0− ) = θ0while t 0 > θ0 . This means that if there is an equilibrium in which full separation andpooling coexist, then there must be an upward jump at the point of transition between
separation and pooling.
A.4.

Equilibrium characterization

We are now ready to complete the proof that any D1 equilibrium must be LSHPP.
First, if there is no fully separating region, i.e., θ0 = θ , then Q(S(θ )) must be a poolingset. Let a0 = S(θ ) and t 0 = T (θ ). Suppose a0 > 0. By Lemma 6, no type in [θ0 , max Q(a0 )]can choose an action lower than a0 . If there is any type that chooses an action slightly lower
than a0 , then it must be types above max Q(a0 ), but this cannot be incentive compatible
because t 0 < max Q(a0 ). Since actions below a0 are off-path, Lemma 7 suggests θ ∈ Q(a0 ).
Other properties of LSHPP equilibrium follow directly from Lemmas 2 and 6, suggestingthat if Q(a0 ) is a pooling set, then the equilibrium must be LSHPP.
If a0 = 0, then there are no off-equilibrium actions below a0 , and we cannot apply
Lemma 7. We thus need to ensure that Q(a0 ) contains type θ . Suppose to the contrary that θ > max Q(a0 ) and S(θ ) > a0 . If actions above a0 are off-path, then (a0 , t 0 ) ∈38

RSC(max Q(a0 )), and S(·) cannot jump up at max Q(a0 ). Since this argument applies forany max Q(a0 ), it is not possible to have θ > max Q(a0 ). If actions above a0 are onpath, then there must be two limit types θ 0 and θ 00 , such that no type in (θ 00 , θ 0 ) canchoose a0 by Lemma 4. This means that there must be types above θ 0 that choose a0 and(a0 , t 0 ) ∈ RSC(θ ) for θ > θ 0 . Since S(·) cannot increase, this establishes that the poolingset Q(a0 ) must contain type θ .
Now, suppose that Q(S(θ )) is a singleton and T (θ ) = θ , in which case S(·) = s∗ (·)
in a right neighborhood of θ . Lemma 8 implies that S(·) = s∗ (·) for all θ ∈ [θ , θ ] (thecase of a fully separating equilibrium), or S(·) must jump at some point. Note that the
fully separating equilibrium is a special case of LSHPP equilibrium. If S(·) jumps at θ0 to
a0 = S(θ0+ ), then actions below a0 must be off-path. This follows because if there is any
type that chooses an action slightly lower than a0 , then it must be a type above max Q(a0 ),but this cannot be incentive compatible because t 0 < max Q(a0 ). Since actions below a0
are off-path, Lemma 7 suggests θ ∈ Q(a0 ). We can then apply Lemmas 2 and 6 as above
to show that the equilibrium must be LSHPP.

B. Algorithm
We establish the existence of an equilibrium by construction. To this end, we develop an
algorithm to construct pairwise-pooling above the gap. Since there are two different formsof pooling, we first address each case and then combine them together.
B.1.

Atomless pooling

If we begin with the initial condition σ(θB ) = aB , τ(θB ) = t B , p(θB ) = θ̂B , we can summarize the initial state using a 4-tuple, cB = (θB , θ̂B , aB , t B ). For this to be a legitimate initialstate, we requiret B ∈ (θB , θ̂B ) andm(aB , t B , θB ) = m(aB , t B , θ̂B ).

(11)

Suppose there is a well-defined solution to the differential equations (5), (6), and (7)for θ ∈ [θ E , θB ]. We can then obtain the end state summarized by the 4-tuple, c E =(θ E , p(θ E ), σ(θ E ), τ(θ E )). Obviously, the end state will depend on the initial state and on
the value of θ E at which we choose to evaluate the solution functions, so we denote thismapping by c E = ZA(θ E ; cB ). By construction, if the initial state cB satisfies condition (11),then the output c E of this mapping also satisfies (11).
39The main constraint for pairwise matching is that σ(·) must be strictly increasing on(θ E , θB ), reflecting the requirement that S(·) is quasi-concave above the gap. This constraint is given by (8). Once mθ (·) − m̂θ (·)p0 (·) turns from positive to zero, the solution to
the differential equation cannot be extended further back. Let¦©
χA(cB ) := θ E : constraint (8) holds for all θ ∈ (θ E , θB ] and p(θ E ) ≤ θ .
For any cB satisfying (11) and any θ E ∈ χA(cB ), the mapping ZA(θ E ; cB ) is well defined andproduces a valid solution satisfying the monotonicity requirement on the domain (θ E , θB ].
B.2.

Mass pooling

Begin with an initial condition, summarized by cB = (θB , θ̂B , aB , t B ), that satisfies (11). To
construct an equilibrium in which all types in [θ E , θB ] ∪ [θ̂B , θ̂ E ] pool to choose (aB , t B ),the equilibrium conditions require:m(aB , t B , θ̂ E ) − m(aB , t B , θ E ) = 0,E θ | θ ∈ [θ E , θB ] ∪ [θ̂B , θ̂ E ] − t B = 0.


(12)
(13)

Let ψ(·; aB , t B ) represent the implicit function that gives the θ̂ E satisfying (12) for eachθ E . Similarly, let ν(·; θB , θ̂B , t B ) give the θ̂ E satisfying (13) for each θ E . Both functions aredefined on the domain [b, θB ], such that b solves ν(b; θB , θ̂B , t B ) = θ . If no such b exists,
we set b = θ . Whenever ψ(θ E ) is undefined for θ E ∈ [b, θB ], we set ψ(θ E ) = θ . According
to this extended definition, ψ(b; aB , t B ) = θ if and only if m(aB , t B , b) ≥ m(aB , t B , θ ).
A solution to the system of equations (12) and (13) exists if there is a θ E such thatψ(θ E ) = ν(θ E ). By implicit differentiation, one can show that both functions are decreasing for any cB satisfying (11). The condition that mθ (·) − m̂θ (·)p0 (·) is nonnegative
corresponds to ψ0 (·) ≥ ν0 (·). To satisfy the conditions for mass pooling at (aB , t B ), θ E andθ̂ E must satisfy (12) and (13). Furthermore, for any interior crossing point (i.e., θ E > b),
we require that ψ0 (θ E ) ≥ ν0 (θ E ). This would allow the end point of mass pooling θ E toserve as an initial starting point for atomless pooling immediately to the left of θ E .
To summarize, letχ M (cB ) := θ E : ψ(θ E ) = ν(θ E ) and ψ0 (θ E ) ≥ ν0 (θ E ), or θ E = b and ψ(b) ≥ ν(b) .
Given an initial state cB , and for any θ E ∈ χ M (cB ), we can obtain an end state c E =(θ E , ψ(θ E ), aB , t B ). We denote this mapping by c E = Z M (θ E ; cB ). By construction, the40

output of this mapping satisfies (11) except possibly at θ E = b. However, in this case, thepairwise-pooling region is [b, θ ], and m(a E , t E , b) ≥ m(a E , t E , θ ) ensures that there is no
incentive for downward deviation below a E .
B.3.

Algorithm to construct pairwise-pooling above the gap

If S(·) attains a maximum at a unique θ∗ , then there is atomless pooling in a neighborhoodof θ∗ . In this neighborhood, (σ(θ ), τ(θ )) is in the SC-domain of type θ and in the RSCdomain of type p(θ ). This means that (σ(θ∗ ), θ∗ ) must be on the boundary of the SCdomain and RSC-domain of type θ∗ . Therefore, a boundary condition that satisfies (11)in the limit is:σ(θ∗ ) = D(θ∗ ; θ∗ , θ∗ ),τ(θ∗ ) = θ∗ ,p(θ∗ ) = θ∗ .

If there is mass pooling in a neighborhood of θ∗ , then using this boundary condition ensuresthat the off-equilibrium belief for an upward deviation above σ(θ∗ ) is weakly lower thanθ∗ , which does not exceed the equilibrium belief θ∗ .
For any given θ∗ , we go through the following iterative procedure to ensure that theequilibrium conditions for pairwise-pooling are satisfied:
1. Initialize k = 1. Set ck = (θ∗ , θ∗ , D(θ∗ ; θ∗ , θ∗ ), θ∗ ), and set θBk = θ∗ . If inf χA(ck ) < θ∗ ,go to step 2; otherwise go to step 3.
2. Let θ E = inf χA(ck ). Construct the atomless-pooling solution for θ ∈ (θ E , θBk ]. Ifp(θ E ) = θ , stop. Otherwise, let ck+1 = ZA(θ E , ck ) and θBk+1 = θ E , increase k by 1 andgo to step 3.
3. Let θ E = max χ M (ck ). Construct the mass-pooling solution for θ ∈ (θ E , θBk ]. If
θ E = b, stop. Otherwise, let ck+1 = Z M (θ E , ck ) and θBk+1 = θ E , increase k by 1 andgo to step 2.
Once θ∗ is fixed, this algorithm yields a well defined θ E such that p(θ E ) = θ at the endof the procedure, along with σ(θ ), τ(θ ), and p(θ ) for θ ∈ [θ E , θ∗ ]. By construction, theseobjects satisfy Bayes’ rule, incentive compatibility, and pairwise matching. Let ζ : [θ , θ ] →[θ , θ ] denote this mapping, where ζ(θ∗ ) is the θ E obtained at the end of the procedure
starting from θ∗ .

41

C. Proof of Theorem 2
We exploit the algorithm developed in Appendix B to establish equilibrium existence byconstruction. The proof of the existence of an equilibrium consists of three parts. Let ζ(θ∗ )represent the θ E obtained at the end of our algorithm starting from an initial state θ∗ . We
first establish the continuity of ζ(·), which in turn implies that ∆u (·) is also continuous.
The second part establishes the existence of θ∗ such that ∆u (θ∗ ) ≤ 0 (with strict inequality
only if ζ(θ∗ ) = θ ). The candidate solution obtained from such θ∗ satisfies all the local
incentive compatibility constraints. In the final step, we show that the candidate solution
satisfies global incentive compatibility and constitutes an equilibrium.
C.1.

Continuity

Under our algorithm, the solution switches from atomless pooling to mass pooling whenmθ (·)− m̂θ (·)p0 (·) switches from positive to negative, and it switches back from mass pooling to atomless pooling as soon as m(·) − m̂(·) turns from positive to zero. We can rewrite
equation (7) as[m̂a (·) − ma (·)] σ0 + [m̂ t (·) − m t (·)] τ0 = maxmθ (·) − m̂θ (·)p0 I (m (·) = m̂ (·)) , 0 ,which incorporates both atomless pooling and mass pooling. Let x = (p, σ, τ). For theease of notation, we writew̃(θ , x) := mθ (σ, τ, θ ) − m̂θ (σ, τ, p)f (θ ) θ − τ,
f (p) p − τw(θ , x) := max {w̃ (θ , x) I (∆m (θ , x) = 0) , 0} .
where ∆m (θ , x) := m(σ, τ, θ )− m̂(σ, τ, p). Together with (5) and (6), we obtain a systemof differential equations of the form x 0 = W (θ , x), wheref (θ ) θ −τ0
 p = f (p) p−τ ,w(θ ,x)σ0 = m̂a (σ,τ,p)−ma (σ,τ,θ )+m(σ,τ,θ )[m̂t (σ,τ,p)−m t (σ,τ,θ )] ,m(σ,τ,θ )w(θ ,x)τ0 =m̂a (σ,τ,p)−ma (σ,τ,θ )+m(σ,τ,θ )[m̂ t (σ,τ,p)−m t (σ,τ,θ )] .

We solve this system backwards from c1 = (θ∗ , x ∗ (θ∗ )), where x ∗ (θ∗ ) = (θ∗ , D(θ∗ ; θ∗ , θ∗ ), θ∗ ).
The initial value problem we consider is as follows:(
x 0 = W (θ , x),x(θB ) = x B := (pB , τB , σB ),42

where (θB , x B ) is an arbitrary initial state. Let y(·; θB , x B ) denote the solution to this problem. By standard argument, y(·; θB , x B ) is continuous with respect to the initial state in aneighborhood of (θB , x B ) if w(·, ·) is locally Lipschitz at (θB , x B ).15
Suppose first that there is either mass pooling or atomless pooling in a neighborhoodof (θB , x B ). If there is mass pooling, then we have w(·, ·) = 0, and if there is atomlesspooling, then we have w(·, ·) = w̃(·, ·). In either case, w(·, ·) is locally Lipschitz at (θB , x B ).
If there is a transition from atomless pooling to mass pooling at (θB , x B ), then we haveboth w̃(θB , x B ) = 0 and ∆m (θB , x B ) = 0 by construction. In this case, w(·, ·) is still locally
Lipschitz at (θB , x B ).
When there is a transition from mass pooling to atomless pooling, the indicator function
turns from 0 to 1, and w(·, ·) is discontinuous at (θB , x B ) if w̃(θB , x B ) > 0. To address thiscase, consider an initial state (θB , x B ) such that∆m (θB , x B ) = 0,w̃(θB , x B ) > 0,

which represents a transition point from mass pooling to atomless pooling.16 Pick anarbitrary state x from a set X (θB ) such that
X (θB ) := {x : ∆m (θB , x) > 0}.
By this definition, there is mass pooling in a neighborhood of (θB , x) if x ∈ X (θB ). Defineθ T (x) := max {θ : ∆m (θ , y(θ ; θB , x)) = 0} < θB ,for x ∈ X (θB ) if it exists, and let Nδ (x B ) := {x : kx − x B k < δ}.
Lemma 9. For any ε > 0, there is a δ such that θ T (x) exists and θB − θ T (x) < ε forx ∈ Nδ (x B ) ∩ X (θB ).
Proof. We write ψ(·; x) and ν(·; x) to denote their dependence on x. Recall that ∆m (θB , x) >0 is equivalent to ψ(θB ; x) > ν(θB ; x), such that we consider a change in x that makes
ψ(·; x) go above ν(·; x) at θB . Additionally, note that w(θB , x B ) > 0 is equivalent toψ0 (θB ; x B ) > ν0 (θB ; x B ) and therefore that ψ(·; x B ) < ν(·; x B ) in a left neighborhood of
The system is well defined except at θ∗ , where p(θ∗ ) = τ(θ∗ ) = θ∗ is imposed by construction. In thiscase, however, we can show p0 (θ∗ ) = −1 and τ0 (θ∗ ) = σ0 (θ∗ ) = 0 for any θ∗ . See Appendix E.
16
We can have a (nongeneric) case with w̃(θB , x B ) = 0 even when there is a transition from mass poolingto atomless pooling. This occurs if ψ(·) and ν(·) are tangent to each other at θB (and possibly over someinterval that contains θB ). We can disregard this possibility because w(·, ·) is continuous in this case.
15

43θB . Then, since both ψ(·; x) and ν(·; x) are continuous in x, for any ε > 0, we can find a
δ > 0 such that ψ(θB −ε; x) < ν(θB −ε; x) and ψ(θB ; x) > ν(θB ; x) for x ∈ Nδ (x B )∩X (θB ).
According to the continuity of ψ(·) and ν(·), θ T (x) must lie in (θB − ε, θB ).
The lemma shows that θ T (x) converges to θB as x becomes arbitrarily close to x B .
Therefore, the solution induced from (θB , x) also converges to the solution induced from(θB , x B ) as x approaches x B .
This completes the proof that the solution from our algorithm is continuous with respect to the initial state. Suppose that (θB , x B ) represents the first transition point frommass pooling to atomless pooling such that continuity up to that point is ensured. Thismeans that x = y(θB ; θ∗ , x ∗ (θ∗ )) is continuous in θ∗ . Since y(·; θB , x) is also continuous inx, we can ensure that the mapping ζ(·) consistently produces a θ0 that varies continuouslywith θ∗ .
C.2.

Indifference at the gap

Recall that ∆u (·) is defined as∆u (θ∗ ) = u(s∗ (ζ(θ∗ )), ζ(θ∗ ), ζ(θ∗ )) − u(σ(ζ(θ∗ ); θ∗ ), τ(ζ(θ∗ ); θ∗ ), ζ(θ∗ )),where, for clarity, we use (σ(·; θ∗ ), τ(·; θ∗ )) to indicate the action-reputation pair induced
from boundary type θ∗ . Since ζ(·) is continuous, ∆u (·) is also continuous.
Define z as the boundary type such that ζ(z) = θ . Such a type exists due to the continuity of ζ(·). If ∆u (z) ≤ 0, then (σ(·; z), τ(·; z)) with θ0 = θ constitutes a candidate
solution.
Now, suppose that ∆u (z) > 0. Note that ζ(θ ) = θ , and therefore, σ(θ ; θ ) = D(θ ; θ , θ ) <s (θ ) (otherwise we can have a fully separating equilibrium) and τ(θ ; θ ) = θ . This means∗that ((σ(θ ; θ ), τ(θ ; θ )) is strictly preferred to (s∗ (θ ), θ ). We thus have ∆u (θ ) < 0. It
then follows that there exists θ∗ ∈ (z, θ ) such that ∆u (θ∗ ) = 0. For such θ∗ , the solution(σ(·; θ∗ ), τ(·; θ∗ )) with θ0 = ζ(θ∗ ) constitutes a candidate solution.
C.3.

Global incentive compatibility

By construction, the candidate solution satisfies local incentive compatibility and the pairwisematching condition. Proposition 3 implies that no type has an incentive to deviate to any
on-path action. The remaining issue is deviation to some off-equilibrium action.
44

Case 1: Deviation to a > σ(θ∗ ). At (σ(θ∗ ), θ∗ ), all types above θ∗ have a higher marginal
rate of substitution, and moreover their equilibrium indifference curves stay strictly abovethat of type θ∗ for all a > σ(θ∗ ). Under D1, the belief assigned to any deviation to an
action higher than σ(θ∗ ) must be lower than θ∗ . Thus, no type can benefit from deviating
to an action higher than σ(θ∗ ).
Case 2: Deviation to a ∈ [s∗ (θ0 ), σ(θ0 )). Global incentive compatibility for on-path
actions means that the equilibrium indifference curve of any type (other than type θ0 )is strictly above the points (s∗ (θ0 ), θ0 ) and (σ(θ0 ), τ(θ0 )). For a type θ ∈ [θ , θ∗ ], both
points are in SC(θ ); therefore, his equilibrium indifference curve must be entirely abovethat of type θ0 for all a ∈ [s∗ (θ0 ), σ(θ0 )]. For any type θ 0 ∈ (θ∗ , θ ], there is another type
θ 00 ∈ [θ0 , θ∗ ] that chooses the same allocation (σ(θ 0 ), τ(θ 0 )) as type θ 0 , with an equilibrium
indifference curve that is “less convex” at that point. Therefore the equilibrium indifference
curve of type θ 0 is entirely above that of type θ 00 for all a ∈ [s∗ (θ0 ), σ(θ0 )], which in turnis entirely above the equilibrium indifference curve of type θ0 for all such a. This meansthat any deviation to an action between s∗ (θ0 ) and σ(θ0 ) is attributed to type θ0 under D1.
Clearly, type θ0 has no incentive to deviate to such a for no gain in reputation. It follows
that no other type has an incentive to deviate to such a either.

D. Double-Crossing Property and Marginal Rate of Substitution
Lemma 10. If preferences satisfy the double-crossing property, then for θ 0 > θ 00 ,(m(a, t, θ 0 ) − m(a, t, θ 00 )≤ 0 if a ≤ D(t; θ 0 , θ 00 ),≥ 0 if a ≥ D(t; θ 0 , θ 00 ).

Proof. Let u00 and u0 be the utility levels of types θ 00 and θ 0 , respectively, at (a1 , t 1 ). For
a2 < a1 ≤ D(t 1 ; θ 0 , θ 00 ), part (a) of Definition 1 requires that t 2 = φ(a2 , u00 , θ 00 ) implies t 2 <φ(a2 , u0 , θ 0 ). Taking the limit as a2 approaches a1 from below, we obtain φa (a1 , u0 , θ 0 ) ≤φa (a1 , u00 , θ 00 ), which implies that m(a1 , t 1 , θ 0 ) ≤ m(a1 , t 1 , θ 00 ), with equality only if a1 =D(t 1 ; θ 0 , θ 00 ).
If a1 > a2 ≥ D(t 2 , θ 0 , θ 00 ), then we let u00 and u0 represent the utility levels of thecorresponding types at (a2 , t 2 ). Part (b) of the definition requires t 1 = φ(a1 , u00 , θ 00 ) >φ(a1 , u0 , θ 0 ). Taking the limit as a1 approaches a2 from above, we obtain φa (a2 , u00 , θ 00 ) ≤45

φa (a2 , u0 , θ 0 ), which implies that m(a2 , t 2 , θ 00 ) ≤ m(a2 , t 2 , θ 0 ), with equality only if a2 =D(t 2 ; θ 0 , θ 00 ).
Lemma 11. Preferences satisfy the double-crossing property if and only if, for θ 0 > θ 00 , there
exists D(·; θ 0 , θ 00 ) such that(m(a, φ(a, u0 , θ 00 ), θ 0 ) − m(a, φ(a, u0 , θ 00 ), θ 00 )≤0

if a ≤ a0 ≤ D(t 0 ; θ 0 , θ 00 ),≥0

if a ≥ a0 ≥ D(t 0 ; θ 0 , θ 00 );with strict inequality except when a = a0 = D(t 0 ; θ 0 , θ 00 ).
Proof. Suppose that preferences satisfy the double-crossing property. If type θ 00 is indifferent between (a0 , t 0 ) and (a, φ(a, u0 , θ 00 )), then parts (a) and (b) of Definition 1 together
imply that a < a0 ≤ D(t 0 ; θ 0 , θ 00 ) and a ≥ D(φ(a, u0 , θ 00 ); θ 0 , θ 00 ) would lead to a contradiction. Therefore, a < a0 ≤ D(t 0 ; θ 0 , θ 00 ) implies a < D(φ(a, u0 , θ 00 ); θ 0 , θ 00 ). By Lemma
10, we have m(a, φ(a, u0 , θ 00 ), θ 0 ) − m(a, φ(a, u0 , θ 00 ), θ 00 ) ≤ 0, with equality only if a2 =a1 = D(t 1 ; θ 0 , θ 00 ). Similarly, a > a0 ≥ D(t 0 ; θ 0 , θ 00 ) implies a > D(φ(a, u0 , θ 00 ); θ 0 , θ 00 ). By
Lemma 10, we have m(a, φ(a, u0 , θ 00 ), θ 0 ) − m(a, φ(a, u0 , θ 00 ), θ 00 ) ≥ 0, with equality only
if a1 = a2 = D(t 2 ; θ 0 , θ 00 ).
For sufficiency, let u0 represent the utility level of type θ 0 at (a0 , t 0 ). If a < a0 ≤D(t 0 ; θ 0 , θ 00 ), then m(a, φ(a, u0 , θ 00 ), θ 0 ) < m(a, φ(a, u0 , θ 00 ), θ 00 ). Therefore, φ(a0 , u0 , θ 0 ) =φ(a0 , u0 , θ 00 ) implies that φ(a, u0 , θ 0 ) > φ(a, u0 , θ 00 ) for a slightly below a0 . We argue thatφ(a, u0 , θ 0 ) must stay above φ(a, u0 , θ 00 ) for all a < a0 . Suppose otherwise. Then, let a1 be
the largest a < a0 such that the two indifference curves cross. Since φ(a, u0 , θ 0 ) is strictly
above φ(a, u0 , θ 00 ) for a ∈ (a1 , a0 ), we must have φa (a1 , u0 , θ 0 ) ≥ φa (a1 , u0 , θ 00 ). However,this is equivalent to m(a1 , φ(a1 , u0 , θ 00 ), θ 0 ) ≥ m(a1 , φ(a1 , u0 , θ 00 ), θ 00 ), which is a contradiction. Because φ(a, u0 , θ 0 ) stays strictly above φ(a, u0 , θ 00 ) for all a < a0 , and becauseu t (·) > 0, whenever type θ 00 weakly prefers (a0 , t 0 ) to some (a, t) with a < a0 , type θ 0strictly prefers the former.
Similarly, if a > a0 ≥ D(t 0 ; θ 0 , θ 00 ), then m(a, φ(a, u0 , θ 00 ), θ 0 ) > m(a, φ(a, u0 , θ 00 ), θ 00 ).
Therefore, φ(a0 , u0 , θ 0 ) = φ(a0 , u0 , θ 00 ) implies that φ(a, u0 , θ 0 ) > φ(a, u0 , θ 00 ) for an aslightly above a0 . Suppose φ(a, u0 , θ 0 ) does not stay above φ(a, u0 , θ 00 ) for all a > a0 . Then,
let a1 be the smallest a > a0 such that the two indifference curves cross. Since φ(a, u0 , θ 0 )is strictly above φ(a, u0 , θ 00 ) for a ∈ (a0 , a1 ), we must have φa (a1 , u0 , θ 0 ) ≤ φa (a1 , u0 , θ 00 ).
However, this is equivalent to m(a1 , φ(a1 , u0 , θ 00 ), θ 0 ) ≤ m(a1 , φ(a1 , u0 , θ 00 ), θ 00 ), which isa contradiction. Because φ(a, u0 , θ 0 ) stays strictly above φ(a, u0 , θ 00 ) for all a > a0 , and46

because u t (·) > 0, whenever type θ 00 weakly prefers (a0 , t 0 ) to some (a, t) with a > a0 ,type θ 0 strictly prefers the former.

E. The Solution at the Boundary
The solution of our model is characterized by the system of differential equations x 0 =W (θ , x) where x = (p, σ, τ). Observe that the differential equations are not well defined atθ∗ since p(θ∗ ) = τ(θ∗ ) = θ∗ is imposed by construction. Below, we argue that p0 (θ∗ ) = −1and σ0 (θ∗ ) = τ0 (θ∗ ) = 0 hold for any θ∗ so that the system always produces a well-behaved
solution.
When there is mass pooling in a neighborhood of θ∗ , we have σ0 (θ∗ ) = τ0 (θ∗ ) = 0. The
function p(·) is determined by the equal marginal rate of substitution condition, which
gives p0 (θ∗ ) = mθ (·)/m̂θ (·) = −1.
When there is atomless pooling in a neighborhood of θ∗ , the local incentive compatibility constraint for type θ∗ is slightly irregular as this type may mimic either type θ∗ − εor type p(θ∗ − ε). The conditions for this can be written asu(σ(θ∗ ), τ(θ∗ ), θ∗ ) ≥ u(σ(θ∗ − ε), τ(θ∗ − ε), θ∗ ),u(σ(θ∗ ), τ(θ∗ ), θ∗ ) ≥ u(σ(p(θ∗ − ε)), τ(p(θ∗ − ε)), θ∗ ),where (σ(·), τ(·)) = (σ(p(·)), τ(p(·))) by definition. In the limit, we must have
σ0 (θ∗ ) =τ0 (θ∗ )τ0 (θ∗ )p0 (θ∗ )=.
m(σ(θ∗ ), τ(θ∗ ), θ∗ ) m(σ(θ∗ ), τ(θ∗ ), θ∗ )We apply l’Hopital’s rule to equation (5) to obtainp0 (θ∗ ) =1 − τ0 (θ∗ ).
p0 (θ∗ ) − τ0 (θ∗ )Solving this for p0 (θ∗ ) yields
p (θ∗ ) =0

τ0 (θ∗ ) ±p

τ0 (θ∗ )2 + 4(1 − τ0 (θ∗ )) τ0 (θ∗ ) ± (τ0 (θ∗ ) − 2)=.
22

Since p0 (·) must be negative, we must have p0 (θ∗ ) = τ0 (θ∗ ) − 1. Therefore, the only consistent solution is σ0 (θ∗ ) = τ0 (θ∗ ) = 0 and p0 (θ∗ ) = −1.

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F. Other Variants of Double-Crossing Preferences
In the main text we use Assumptions 2 and 3 to specify double-crossing preferences. We
adopt these two assumptions because they are economically relevant for many applications, as shown by our examples. Nevertheless we do not rule out the possibility that
other variants of double-crossing preferences may also be relevant in some contexts. We
provide a brief discussion of these variations below.
Consider the following assumptions.
Assumption 20 . For any θ 0 > θ 00 , there exists a continuous function D(·; θ 0 , θ 00 ) : [θ , θ ] →R+ such that(a) if a < a0 ≤ D(t 0 ; θ 0 , θ 00 ), thenu(a, t, θ 0 ) ≤ u(a0 , t 0 , θ 0 ) =⇒ u(a, t, θ 00 ) < u(a0 , t 0 , θ 00 );(b) if a > a0 ≥ D(t 0 ; θ 0 , θ 00 ), thenu(a, t, θ 0 ) ≤ u(a0 , t 0 , θ 0 ) =⇒ u(a, t, θ 00 ) < u(a0 , t 0 , θ 00 ).
Assumption 30 . For any t, D(t; θ 0 , θ 00 ) is continuous and strictly increases in θ 0 and in θ 00 .
Assumption 20 (A20 ) states that the reverse single-crossing property holds to the left ofthe dividing line (i.e., (a, t) ∈ RSC(θ ) if a < D(t; θ , θ )) while the single-crossing property
holds to the right of it (i.e., (a, t) ∈ SC(θ ) if a > D(t; θ , θ )). This is the opposite ofAssumption 2 (A2). Similarly, Assumption 30 (A30 ) is the opposite of Assumption 3 (A3).
Any combination of (A2) or (A20 ) with (A3) or (A30 ) would lead to a different specificationof double-crossing preferences; therefore, there are four possible specifications.
For each of the four specifications, there is always one domain that is “well-behaved”
and another domain that is “ill-behaved.” It is easy to check which domain is ill-behaved.
Under (A3), the dividing line decreases in type, which makes the RSC-domain ill-behaved
regardless of whether (A2) or (A20 ) holds. Under (A30 ), the dividing line increases in typesuch that the SC-domain is ill-behaved. In the main text, we define θ∗ as the largest typesuch that (S(θ ), T (θ )) is not in the RSC-domain (i.e., S(θ ) ≤ D(T (θ ); θ , θ )) for all θ ≤ θ∗ .
We now extend this definition and letθ∗ := sup{θ 0 : (S(θ ), T (θ )) is not in the ill-behaved domain for all θ ≤ θ 0 }.
48

Similarly, letθ∗∗ := inf{θ 0 : (S(θ ), T (θ )) is not in the ill-behaved domain for all θ ≥ θ 0 }.
The choice of (A2) or (A20 ) determines the direction in which we impose the pairwisematching condition. Under (A2), the SC-domain is to the left of the dividing line, andhigher types have more convex indifference curves; therefore, the pairwise-matching condition is imposed for higher types. In this case, θ∗ is applicable. Under (A20 ), the RSCdomain is to the left of the dividing line, and lower types have more convex indifference
curves; therefore, the pairwise-matching condition is imposed for lower types. In this case,θ∗∗ is applicable.
Our model assumes (A2) and (A3). In this specification, incentive compatibility ispotentially an issue for allocations in the RSC-domain. We use the pairwise-matchingcondition to ensure global incentive compatibility via Proposition 3. Note that an extended
version of the pairwise-matching condition can now be stated as follows.
Definition F.1. An allocation satisfies condition (P) if for any θ 0 > θ∗ , there exists θ 00 < θ∗such that (S(θ 00 ), T (θ 00 )) = (S(θ 0 ), T (θ 0 )) and m(S(θ 00 ), T (θ 00 ), θ 00 ) = m(S(θ 0 ), T (θ 0 ), θ 0 ).
Proposition 3 shows that under (A2) and (A3), an allocation that satisfies local IC forθ < θ∗ and the pairwise-matching condition is incentive compatible. With the extended
definition of (P), this conclusion can also be applied to the case under (A2) and (A30 ).
Proposition F.1. Under Assumption (A2), an allocation that satisfies local IC and condition(P) is incentive compatible.
Proof. The case under (A3) is already discussed in Proposition 3, so we focus on (A30 ).
Because S(θ ) ≥ D(T (θ ); θ , θ ) for all θ ∈ [θ , θ∗ ], local IC implies that S(·) is weakly
decreasing on this interval. Here, we only show that incentive compatibility holds forany pair of types on this interval. Once this is established, the remainder of the proof
immediately follows from the proof of Proposition 3.
Consider types θ1 < θ2 ≤ θ∗ . By (A30 ),S(θ2 ) ≥ D(T (θ2 ); θ2 , θ2 ) > D(T (θ2 ); θ2 , θ1 ).

(14)(A2) requires that φ ∗ (·; θ2 ) cannot cross D(·; θ2 , θ1 ) to the right of S(θ2 ), such that thesingle-crossing property holds along this indifference curve:a > D(φ ∗ (a; θ2 ); θ2 , θ1 ) for a ≥ S(θ2 ).
49

(15)

At any point on φ ∗ (a; θ2 ) for a ≥ S(θ2 ), any lower-type θ1 < θ2 always has a lower
marginal rate of substitution than type θ2 .
We argue that any locally IC allocation must stay below φ ∗ (a; θ2 ) for a ∈ [S(θ2 ), S(θ1 )].
Suppose the opposite is true, and let T (θ 00 ) ≥ φ ∗ (S(θ 00 ); θ2 ) with T (θ ) < φ ∗ (S(θ ); θ2 )for all θ ∈ (θ 00 , θ2 ]. Local IC then implies that φ ∗ (·; θ 00 ) reaches φ ∗ (·; θ2 ) from below
at some a00 ∈ (S(θ2 ), S(θ 00 )]. However, this is a contradiction because by (15), type θ 00
must have a lower marginal rate of substitution at any point on φ ∗ (·; θ2 ). This shows thatT (θ1 ) < φ ∗ (S(θ1 ); θ2 ), and thus, type θ2 has no incentive to mimic type θ1 . Similarly, any
locally IC allocation must stay below φ ∗ (a; θ1 ) for a ∈ [S(θ2 ), S(θ1 )]. Suppose the opposite
is true, and let φ ∗ (S(θ 0 ); θ1 ) ≥ T (θ 0 ) with T (θ 0 ) < φ ∗ (S(θ 0 ); θ1 ) for all θ ∈ [θ1 , θ 0 ).
Local IC implies that φ ∗ (·; θ1 ) reaches φ ∗ (·; θ 0 ) from above at some a0 ∈ [S(θ 0 ), S(θ1 )).
However, this is a contradiction because by (15), type θ1 must have a lower marginal rateof substitution at any point on φ ∗ (·; θ 0 ). This shows that T (θ2 ) ≤ φ ∗ (S(θ2 ); θ1 ), and sotype θ1 has no incentive to mimic type θ2 .
Observe that for condition (P) to hold under (A2) and (A30 ), θ∗ must be strictly greaterthan θ , meaning that (S(θ ), T (θ )) ∈ RSC(θ ). In signaling models, if we assume (0, θ ) ∈SC(θ ), then whenever the lowest type separates, condition (P) would have no bite under(A2) and (A30 ), and global incentive compatibility would become an issue. However, forgeneral mechanism design models, condition (P) may help play a role in ensuring incentive
compatibility.
When we maintain (A20 ) instead of (A2), higher types have “less convex” indifference curves than lower types on the dividing line. Condition (P) needs to be modified
accordingly to ensure incentive compatibility. Recall that we define that θ∗∗ := inf{θ 0 :(S(θ ), T (θ )) is not in the ill-behaved domain for all θ > θ 0 }.
Definition F.2. An allocation satisfies condition (P0 ) if for any θ 0 < θ∗∗ , there exists θ 00 > θ∗∗such that (S(θ 00 ), T (θ 00 )) = (S(θ 0 ), T (θ 0 )) and m(S(θ 00 ), T (θ 00 ), θ 00 ) = m(S(θ 0 ), T (θ 0 ), θ 0 ).
We can now state an analogous result that applies when (A20 ) holds. Because theargument leading to the following result is very similar to that leading to Propositions 3
and F.1, we only provide a brief proof here.
Proposition F.2. Under Assumption (A20 ), an allocation that satisfies local IC and condition(P0 ) is incentive compatible.

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Proof. Suppose (A3) holds. Local IC then implies global incentive compatibility for allocations in the SC-domain (i.e., for any pair of types above θ∗∗ ). The proof of this claim
follows that for Proposition 3. This result, together with the modified pairwise matchingcondition (P0 ), ensures that any type θ 0 ≥ θ∗∗ has no incentive to mimic any other type.
Moreover, under (A20 ), for type θ 00 < θ∗∗ , condition (P0 ) implies that we can find θ 0 > θ∗∗that is “matched to” type θ 00 , with the property that the indifference curve of the lower
type θ 00 is “more convex” than that of type θ 0 . Since incentive compatibility holds for
type θ 0 , the greater convexity of the indifference curve for type θ 00 implies that incentive
compatibility also holds for type θ 00 .
Now suppose (A30 ) holds. Local IC then implies global incentive compatibility for allocations in the RSC-domain (i.e., for any pair of types above θ∗∗ ). The proof of this claim
follows that for Proposition F.1. This result, together with the modified pairwise matchingcondition (P0 ), ensures that any type θ 0 ≥ θ∗∗ has no incentive to mimic any other type.
Moreover, under (A20 ), for type θ 00 < θ∗∗ , condition (P0 ) implies that we can find θ 0 > θ∗∗that is “matched to” type θ 00 , with the property that the indifference curve of the lower
type θ 00 is “more convex” than that of type θ 0 . Since incentive compatibility holds for
type θ 0 , the greater convexity of the indifference curve for type θ 00 implies that incentive
compatibility also holds for type θ 00 .

G. General CaseIn the main text, we suppose that signaling is always costly (i.e., ua (·) < 0) as often assumed in standard signaling models. The assumption is made purely for clarity and notnecessary to obtain our main conclusions, Theorems 1 and 2, although the argument necessarily becomes more involved. In this appendix, we provide an extended version of theproof which allows the possibility that signaling is beneficial up to some point.
G.1.

Characterization

When we do not impose costly signaling, the optimal action for type θ under complete
information may be strictly positive. The initial condition for the fully separating solution
must satisfys∗ (θ ) = a∗ := arg max u(a, θ , θ ).
a

If (a , θ ) ∈ RSC(θ ), s (·) must be strictly decreasing in type. Given this, part (b) of Defi∗∗

nition 4 needs to be modified as follows.
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Definition 40 . A sender’s strategy is LSHPP (Low types Separate and High types Pairwise-Pool)
if there is some θ0 ∈ [θ , θ ] such that:(a) S(θ ) = s∗ (θ ) for θ ∈ [θ , θ0 ).
(b) S(θ ) is discontinuous only at θ = θ0 , with an upward (downward) jump if θ0 < θ ands∗ (·) is increasing (decreasing) on [θ , θ0 ).
(c) There exist θ∗ ∈ (θ0 , θ ) and p : [θ0 , θ∗ ] → [θ∗ , θ ], such that for θ ∈ [θ0 , θ∗ ]: (i) S(·) is
continuous and weakly increasing; (ii) p(·) is continuous and strictly decreasing withp(θ0 ) = θ and p(θ∗ ) = θ∗ ; and (iii) S(·) = S(p(·)).
We proved our characterization result through a series of lemmas, most of which rely
only on Assumptions 2 and 3 and are unaffected by the sign of ua (·). The only part ofthe lemmas which exploits the costly-signaling assumption is part (a) of Lemma 6. In thelemma, we consider two cases, one in which S(·) jumps up and the other in which S(·)
jumps down. When S(·) jumps up at some θ1 to (a1 , t 1 ), we show that it is not possible
to have any pooling at (a1 , t 1 ). We then conclude that S(·) must be increasing in a rightneighborhood of θ1 and derive a contradiction. However, when costly signaling is not
imposed, it is also possible to have S(·) which jumps up at θ1 and then decreases withtype.
Now suppose that S(·) jumps up at θ1 and strictly decreases for θ ∈ (θ1 , θ2 ). Since
S(·) decreases while T (·) increases, the indifference curve of type θ1 must be downward
sloping at (a1 , t 1 ), and (a1 , t 1 ) ∈ RSC(θ1 ). Now consider an action a0 slightly above a1
with corresponding reputation t 0 . Suppose there is some type θ 0 > θ2 who chooses (a0 , t 0 ).
Then, by continuity, type θ 0 must weakly prefer (a1 , t 1 ) to (a p , t p ). However, since (a1 , t 1 ) ∈RSC(θ ) for all θ ≥ θ1 , all types in [θ1 , θ 0 ), including type θ2 , must strictly prefer (a1 , t 1 ) to(a p , t p ), a contradiction. If any type below θ1 chooses a0 , then t 0 < θ1 = t 1 , which cannot
be incentive compatible. We thus conclude that actions slightly above a1 are off-path.
Given this, if there is a deviation to a0 , D1 assigns belief t 0 = θ1 . But then type θ1 would
have an incentive to deviate upward, because t 0 = t 1 and a0 > a1 , and the indifference
curve of θ1 is downward sloping at that point. This argument shows that Lemma 6 holds
as stated even if the costly-signaling assumption is not imposed.
In the final step of the proof, we also need to make sure that Q(a0 ) contains typeθ . If s∗ (·) is decreasing, then S(·) jumps downward at θ0 to a0 = 0; this is a case that
does not occur under costly signaling. Even in this case, however, we can apply the same
argument as when θ0 = θ and a0 = 0 to show that Q(a0 ) contains type θ . This means
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that the equilibrium must be LSHPP and our Theorem 1 continues to hold without the
costly-signaling assumption, under the modified definition of LSHPP equilibrium.
G.2.

Existence

To establish the existence of an equilibrium without the costly-signaling assumption, we
need an additional technical assumption.
Assumption 5. For any θ , u(·, θ , θ ) is quasi-concave, with a unique optimal action a∗ (θ )such that (a∗ (θ ), θ ) ∈ SC(θ ).
Assumption 5 allows the possibility that signaling is always costly as we assume in themain text. It excludes the possibility that u(·, θ , θ ) is strictly increasing, in which case an
optimal action does not exist. When (a∗ (θ ), θ ) ∈ SC(θ ), together with quasi-concavity ofu(·, θ , θ ), the marginal rate of substitution is positive (signaling is locally costly) at any
point (a, t) on the border or in the RSC-domain of type θ .
Even when the costly-signaling assumption is not imposed, we can still employ thesame algorithm and show that ζ(·) is continuous by the same argument. We need a slight
modification in the proof to ensure the existence of the gap type θ0 = ζ(θ∗ ) such that∆u (θ∗ ) = 0, for which Assumption 5 is required. Once this is established, we have acandidate solution which satisfies incentive compatibility for all actions that are chosen in
equilibrium. Since actions below a∗ are off-path when a∗ > 0, we will then make sure thatno type has an incentive to deviate to any action below a∗ under D1.
Indifference at the gap. Recall that ∆u (·) is defined as∆u (θ∗ ) = u(s∗ (ζ(θ∗ )), ζ(θ∗ ), ζ(θ∗ )) − u(σ(ζ(θ∗ ); θ∗ ), τ(ζ(θ∗ ); θ∗ ), ζ(θ∗ )),where for clarity we use (σ(·; θ∗ ), τ(·; θ∗ )) to indicate the action-reputation pair induced
from boundary type θ∗ . Since ζ(·) is continuous, ∆u (·) is also continuous.
Define z to be the boundary type such that ζ(z) = θ ; such type exists dues to continuityof ζ(·). If ∆u (z) ≤ 0, then (σ(·; z), τ(·; z)) with θ0 = θ constitute a candidate solution.
Now suppose ∆u (z) > 0. Note that ζ(θ ) = θ , and therefore σ(θ ; θ ) = D(θ ; θ , θ ) <s (θ ) (otherwise we can have a fully separating equilibrium) and τ(θ ; θ ) = θ . Assumption∗

5 implies that m(D(θ ; θ , θ ), θ , θ ) > 0, and hence ((σ(θ ; θ ), τ(θ ; θ )) is strictly preferred to(s∗ (θ ), θ ). We therefore have ∆u (θ ) < 0. It then follows that there exists θ∗ ∈ (z, θ ) such53

that ∆u (θ∗ ) = 0. For such θ∗ , the solution (σ(·; θ∗ ), τ(·; θ∗ )) with θ0 = ζ(θ∗ ) constitute a
candidate solution.
Deviation to off-equilibrium actions. In the main text, we have dealt with deviation toa > σ(θ∗ ) and to a ∈ [s∗ (θ0 ), σ(θ0 )). When a∗ > 0, we also need to take care of deviation
to an action below a∗ > 0.
Suppose a∗ > 0, which implies m(a∗ , θ , θ ) = 0. We have already shown that all types
above θ strictly prefer their equilibrium choice to (a∗ , θ ). So the equilibrium indifference
curve of any type θ can cross that of type θ at a point to the right of a∗ and from above.
By condition (1) and Assumption 5, this implies that, for all a < a∗ ,m(a, φ(a, u, θ ), θ ) < m(a, φ(a, u, θ ), θ ) < 0,where u is the equilibrium utility of type θ . This means that any deviation to a < a∗ is attributed to the lowest type under D1. Furthermore, since the marginal rate of substitution
is negative, if type θ prefers his equilibrium choice to (a∗ , θ ), he will prefer his equilibrium
choice to (a, θ ) for a < a∗ .

54