Information Transmission with Almost-Cheap Talk∗
Navin Kartik†
First version: November 2003; This version: May 29, 2007
Abstract
Misrepresenting private information is often costly, for technological, legal, or psychological
reasons. I develop a model of strategic information transmission based on Crawford and Sobel
(1982) (CS), but with a convex cost of lying or misreporting. There are three main results.
First, I prove that a sequence of monotone equilibria converges to a CS equilibrium as the cost
of misreporting shrinks to 0 only if the CS equilibrium satisfies the “No Incentive to Separate”
(NITS) condition. In a cheap talk game, NITS requires that the lowest type weakly prefers
the action it elicits in equilibrium to what it would elicit in the complete information game.
In commonly used specifications, only the “most-informative” CS equilibrium satisfies NITS.
Second, I show that under a mild technical condition, the converse is also true: any CS equilibrium satisfying NITS is the limit of a sequence of monotone equilibria as the misreporting
cost shrinks to 0.
This simultaneously proves existence of monotone equilibrium for small
costs, under the technical condition.
The third result provides a complete characterization
of a class of monotone equilibria for arbitrary costs of misreporting in a cheap talk extension
of the model.
These equilibria display language inflation, with a region of low types fully
separating when costs are large.
Keywords:
Cheap Talk, Costly Lying, Misreporting, Signaling, Refinements,
Equilibrium Selection, Babbling, mD1, Exaggeration, Inflated Reporting
J.E.L. Classification: C7, D8
∗
This paper builds on my Ph.D. dissertation; I am indebted to my advisors, Doug Bernheim and Steve Tadelis,
for their generous support and advice. I have benefitted from collaboration with Ying Chen, Marco Ottaviani,
Joel Sobel, and Francesco Squintani on related projects. I am especially grateful to Joel Sobel for advice and
insight over numerous conversations. For helpful comments, I thank David Ahn, Nageeb Ali, Vince Crawford,
Peter Hammond, Cristóbal Huneeus, Jon Levin, Mikko Packalen, Ilya Segal, Jeroen Swinkels, Bob Wilson, and a
number of seminar and conference audiences. It is my pleasure to acknowledge financial support from a John M.
Olin Summer Research Fellowship and a Humane Studies Fellowship. Previous incarnations of this article were
circulated under the title “Information Transmission with Cheap and Almost-Cheap Talk”.
†
Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508.
Email: [email protected]. Web: http://econ.ucsd.edu/∼nkartik.
1
Introduction
The seminal work of Crawford and Sobel (1982) (hereafter, CS) on strategic information transmission though “cheap talk” or costless signaling has been widely applied to problems in bargaining,
monetary policy, political economy, and other areas. The basic setting is a one-shot game where
an informed party, the Sender (he), communicates his one-dimensional private information to an
uninformed decision-maker, the Receiver (she).
Costless misrepresentation of private information is one extreme in a continuum of possibilities.
At the other extreme, one could posit that agents are either unable to manipulate
information or are committed to truth-telling. Plainly, this would eliminate strategic considerations altogether. Evidence suggests that reality typically lies somewhere in between: individuals
can and do misrepresent private information, but bear some cost in doing so. There are various
reasons why such misrepresentation—synonymously, lying or misreporting—can be costly. First,
there may be costs of falsifying a state of the world, such as when a manager has to spend time
“cooking the numbers” to report higher profits, or more generally forego other economic opportunities in order to report non-truthfully (e.g. Lacker and Weinberg, 1989). Second, there may be
probabilistic ex-post state verification resulting in direct penalties or adverse reputation effects
if an agent is caught misreporting, such as a taxpayer running the risk of a random audit (e.g.
Allingham and Sandmo, 1972). Third, a recent body of experimental work documents that people appear to have psychological or moral aversion to lying (Gneezy, 2005; Hurkens and Kartik,
2006).1
This paper studies strategic information transmission with costly misreporting instead of,
or in addition to, cheap talk. Specifically, the model is that of CS with one critical difference:
when a Sender of type t, a one-dimensional variable that represents his private information, sends
a one-dimensional signal or report r to the Receiver, he bears a cost kC(r, t), where C(r, t) is a
misreporting cost function and k is a scalar that parameterizes its magnitude. The case of k = 0
corresponds to pure cheap talk, analyzed by CS; this paper studies “costly talk ”, k > 0. I make
two assumptions on the cost function C(r, t): (i) for any type of the Sender, lying is increasingly
costly the further a report is from the type’s cost-minimizing report (while it is natural to think
of truth-telling, r = t, as minimizing cost, this is not necessary); (ii) the cost-minimizing report is
increasing in the Sender’s type, as is certainly the case when truth is cost-minimizing. Although
the model permits various interpretations of what the costly reports are, it is convenient to think
of these signals as being submitted evidence or literal statements about the Sender’s type in
commonly understood language.
1
Another, indirect, source for misreporting costs is the possibility of naive or boundedly rational behavior by
the Receiver; see Section 5.
1
The model is one of discriminatory signaling in the tradition following Spence (1973),
but it converges to the cheap talk model of CS as the magnitude of cost k → 0.
A notorious
problem with the analysis of cheap talk is that of multiple equilibria. In particular, the CS model
generally possesses a range of equilibria that can be ranked in terms of their “informativeness”,
measured by the Receiver’s ex-ante utility. At one end of the spectrum is a “most-informative”
equilibrium that may convey significant amounts of information; at the other is the completely
uninformative, “babbling” equilibrium. Unfortunately, there is no well-established criterion for
selecting amongst these.2
Though CS is a signaling game, standard theory for equilibrium
selection in signaling games (Cho and Kreps, 1987; Banks and Sobel, 1987) does not apply precisely
because the signals are completely costless to the Sender.3
While there have been alternative
refinements developed specifically for cheap talk games, these have been only partially successful in
general, and particularly unsuccessful when applied to CS. Notably, except in trivial cases, there is
no CS equilibrium that survives criteria like neologism-proofness (Farrell, 1993) or announcementproofness (Matthews, Okuno-Fujiwara, and Postlewaite, 1991); whereas all CS equilibria can
survive credible rationalizability (Rabin, 1990).4
The model proposed here permits a new perspective on this issue.
If communication
does typically possess a dimension with misreporting costs, then CS is best thought of as an
approximation to a setting where these costs are small.
Hence, I refer to the case of k ≈ 0
as “almost-cheap talk ”. The obvious question then is: which cheap talk equilibria are limits of
equilibria with misreporting costs as k → 0? Throughout my analysis, I focus on a natural subset
of (perfect Bayesian) equilibria—monotone equilibria—where the Sender’s report is increasing in
his type and the Receiver’s beliefs about the Sender’s type are increasing, in the sense of first
order stochastic dominance, in the observed report.5
2
The applied literature typically focuses on the most-informative equilibrium. The justification usually given is
that this is the ex-ante Pareto dominant equilibrium under some conditions, hence the one that players should be
expected to coordinate on. Aside from any general concerns, there are at least two reasons why this justification
is not very satisfactory in CS. First, it is necessarily the case that different Sender types have different preferences
over equilibria with different outcomes. In particular, there is always a type whose most-preferred equilibrium
outcome is the uninformative one. Second, since CS is a signaling game, simply invoking the ex-ante Pareto
criterion is troubling because in other widely applied signaling games, e.g. Spence (1973), relatively uncontroversial
refinements such as the Intuitive Criterion of Cho and Kreps (1987) can select equilibria that are Pareto (a fortiori,
ex-ante Pareto) dominated.
3
These equilibrium refinements are based on Kohlberg and Mertens’s (1986) notion of strategic stability, and
operate through dominance arguments applied to out-of-equilibrium signals. Since any cheap talk outcome can be
supported by an equilibrium where all messages are sent on the equilibrium path, they have no power in cheap talk
games.
4
This also applies to general perturbation methods studied by Blume (1994) when the message space is large.
Evolutionary arguments such as those in Blume, Kim, and Sobel (1993) have only limited power as well, although
they do rule out uninformative outcomes in some cases.
5
Signaling games can often support implausible equilibria through the manipulation of off-the-equilibrium-path
beliefs. Monotone equilibria are appealing in the current model because higher types have an intrinsic preference
for higher signals (the cost-minimizing report is increasing in type and costs are convex), and if signals and thus
2
Section 3 establishes a necessary condition for any CS equilibrium to be the limit of
monotone almost-cheap talk equilibria. The condition is easy to check, and is called NITS, for no
incentive to separate: does the lowest type weakly prefer the action it elicits in the CS equilibrium
to the action it would obtain by revealing its type, or equivalently the action it would elicit in the
complete-information game? Under a mild technical assumption, I also prove the converse: any
CS equilibrium satisfying NITS is the limit of a sequence of monotone equilibria of my model as
the misreporting cost shrinks to 0. These results can be interpreted as providing a rationale for
focussing on CS equilibria that satisfy NITS. In a companion paper, Chen, Kartik, and Sobel
(2007) show that NITS is a powerful criterion to select amongst CS equilibria: there is at least one
CS equilibrium that satisfies NITS, and under a regularity condition (Condition M of CS), only
the most-informative CS equilibrium satisfies NITS. To this extent, my analysis provides a novel
argument for selecting the most-informative CS equilibrium in commonly used specifications.
In Section 4, I augment the model by allowing the Sender to send two kinds of signals:
as before, a report which entails a convex cost of lying; and a pure cheap talk message, which is
costless. The costless message may be viewed as either a purely technical device to analyze the
cheap talk extension of the costly lying model (as proposed by Manelli, 1996), or alternatively, as
representing a substantive economic notion that costless signals are also available to the Sender
(analagous to Austen-Smith and Banks, 2000). Adding the costless messages does not affect any
of the convergence analysis discussed above—the results of Section 3 carry over without change—
but their presence allows me to analyze a class of equilibria with appealing properties. Specifically,
I focus on a subset of monotone equilibria that satisfy a strong forward-induction requirement, the
monotone D1 (mD1) criterion proposed by Bernheim and Severinov (2003).6 I prove existence
and provide a complete characterization of mD1 equilibria in the current framework, for arbitrary
costs of misreporting.
In an mD1 equilibrium, there is an interval of fully separating types at
the bottom end of of the type space, each of who uses a distinct costly report, while all types
above some cutoff type pool on the highest available costly report. Nonetheless, above the cutoff
type, there may be further segmentation using the cheap talk messages, similar to CS. This
characterization leads the final result of the paper: under the regularity condition mentioned
before, Condition M of CS, any CS equilibrium satisfying NITS is the limit of a sequence of mD1
equilibria as k → 0.
Aside from the limit results on cheap talk equilibrium selection, the model proposed here
is interesting away from the limit as well, as discussed in Section 5. A pure cheap talk framework
beliefs are monotone on the equilibrium path, it would be perverse to have non-monotone beliefs off the equilibrium
path. In fact, the formal analysis only requires an even weaker notion of belief monotonicity off the equilibrium
path.
6
The mD1 criterion is similar in spirit to but more restrictive than the widely used D1 or divinity criteria of
Cho and Kreps (1987) and Banks and Sobel (1987), which generally do not have much power in the present setting.
3
is not suited to understand issues of language: the equilibrium relationship between messages
and types.
In particular, the CS model is silent on the phenomenon of “inflated language”
or exaggeration, viz. the use of language that deviates from literal truth in the direction of
the Sender’s bias.
Such a phenomenon is prevalent in many real-world situations of strategic
information transmission, such as recommendation letters, advertising, earnings reports from
firms, and stock recommendations.7
Treating communication as costly talk rather than cheap
talk is quite natural in these situations, for the reasons previously mentioned. The current model
naturally generates inflated language in equilibrium. Moreover, the mD1 equilibria I study predict
that as the magnitude of misreporting cost shrinks, the degree of report inflation in equilibrium
weakly rises (strictly so, if the magnitude is not already too close to 0). This is an intuitive result
that has no counterpart in a standard cheap talk framework.
This paper makes contributions to both costly signaling and cheap talk literatures, and
especially their intersection. The remainder of this Introduction discusses in some detail other
recent research at this intersection and their relationship to my work.
1.1
Related Literature
Kartik, Ottaviani, and Squintani (2007) also study the notion of costly talk and language inflation.
However, the focus and analysis is quite different, because we study there an unbounded type and
message space setting. Our results in that paper concern the existence of separating equilibria
for arbitrarily small communication costs, which relies on unboundedness.
Here, I study the
canonical bounded type space version of CS, with a bounded message space, and I show that
separating equilibria do not exist when costs are sufficiently small. Consequently, the attention
here is necessarily on partially-pooling equilibria. Moreover, questions of equilibrium convergence
as costs shrink to 0 and selection of CS equilibria do not arise in Kartik, Ottaviani, and Squintani
(2007).8
In a bounded type and message space model, Ottaviani and Squintani (2006) study the
“uniform-quadratic” case of CS where the prior on types is uniform and utility functions take
quadratic loss form.
They posit that the receiver may be naive or non-strategic with some
probability. This transforms the game from one of pure cheap talk into a costly signaling structure
7
See, for example, Malmendier and Shantikumar (2004), Healy and Wahlen (1999) for empirical evidence, and
Cai and Wang (2006) for laboratory data.
8
Austen-Smith and Banks (2000) analyze a model of cheap talk and costly signaling, except that the costly signal
takes the form of “burned money”. That is, the cost of the signaling instrument does not vary with the Sender’s
type. Their focus is on how the presence of this additional instrument can enlarge the set of CS equilibria, with a
key result that so long as the Sender’s ability to impose costs on himself is unbounded, separating equilibria exist.
Using the same argument as in Lemma 1 of this paper, Kartik (2006) shows that as the ability to burn money
shrinks to 0, the set of equilibria in Austen-Smith and Banks (2000) converges to the entire set of CS equilibria.
See Kartik (2005) for a detailed discussion of the differences between burned money and costly lying.
4
that can be subsumed by my framework here (see Section 5). For some parameterizations, they
construct equilibria that have similar features to the mD1 equilibria here: separation at the low
end of the type space and pooling at the top, with language being inflated. Their construction
does not require the cheap talk extension that I use in Section 4.
However, their parametric
restrictions do not permit the probability of the receiver being naive—isomorphically, the cost of
misreporting—to be close to 0; hence they have no analog to my analysis in Section 3. Instead,
they perform some other comparative statics, and they show that as the type space grows, holding
the probability of naivety constant, their equilibrium converges to the separating equilibrium of
Kartik, Ottaviani, and Squintani (2007).
Chen (2006) furthers the non-strategic approach by allowing not only the Receiver to be
naive, but also the Sender. Specifically, with some probability the Sender just reports his type
truthfully, and with some independent probability the Receiver interprets the report as being
truthful.
As with Ottaviani and Squintani (2006), Chen (2006) studies the uniform-quadratic
case of CS. She shows that there is a unique equilibrium in the cheap talk extension of her
model when required to satisfy a monotonicity property. The equilibrium converges to the mostinformative equilibrium of the uniform-quadratic CS model when the probabilities of naivety of
both Sender and Receiver are taken to 0. This parallels the selection here of CS equilibria that
satisfy NITS when costs of lying shrink to 0, since the only CS equilibrium satisfying NITS in the
uniform-quadratic case is the most-informative one.9
As mentioned earlier, Chen, Kartik, and Sobel (2007) discuss in detail the NITS property,
proving results about which CS equilibria satisfy this property, and providing some justifications
for it beyond the perturbation arguments of Chen (2006) and the current paper.
They also
demonstrate that NITS is useful in one-dimensional cheap talk models aside from CS.
With regards to the broader signaling literature, I use techniques developed by Mailath
(1987) to study separation through signaling with a continuum of types. Whereas his analysis
concerns fully separating equilibria, the main results in this paper concern partially-pooling equilibria, as already noted. Cho and Sobel (1990) were the first to derive incomplete separation in
the manner obtained here: pooling at the top of the type space and separation at the bottom with
respect to the costly signal. Their analysis is for a class of signaling games that have a finite type
space and satisfy a single-crossing property that does not apply in this paper. A more closely
related model is that of Bernheim and Severinov (2003). While theirs is a specific application
9
A strength of Chen’s (2006) approach with Sender naivety is that it ensures that all reports are on the equilibrium path, pinning down Receiver beliefs, and obviating the need to analyze off-the-equilibrium-path beliefs. The
choice of how to specify Sender naivety plays an important role, however, since it imposes particular restrictions
on Receiver beliefs when a report is observed that is not sent by a strategic Sender. In effect, the specification of
Sender naivety substitutes for restrictions on off-the-equilibrium-path beliefs in a fully rational model. Moreover,
the flexibility with off-path beliefs in the current model allows me to prove existence of monotone equilibria when
the perturbation from CS is small without resorting to the cheap talk extension.
5
to public finance, the formal structures share many properties. As previously mentioned, they
introduced the mD1 refinement; my analysis of such equilibria owes much to them, though there
are important differences both in the characterization itself and the proof. Moreover, their focus
is not on convergence as the cost of signaling shrinks to 0; in particular, the analysis here in
Section 3 has no analog in their work.
Finally, I should also mention two other connections: first, there is a small literature
on mechanism design with communication costs, e.g. Deneckere and Severinov (2003).
The
underlying motivation of costly misreporting of private information is similar to that of this
paper; the issues addressed are very distinct.
Second, there is a literature in Accounting that
models signaling explanations for “earnings management”, e.g. Stein (1989). These papers also
motivate costly misreporting, but to my knowledge only consider unbounded type spaces and
special functional forms.
2
Preliminaries
2.1
Model
There are two players, a Sender (S) and a Receiver (R).
The Sender has private information
summarized by his type t ∈ T ≡ [0, 1], which is drawn from a differentiable probability distribution
F (t), with strictly positive density for all t ∈ T . After privately observing his type t, the Sender
sends the Receiver a one-dimensional report, r, about his type.
The report r lies in a non-
degenerate compact interval, which for convenience is assumed to be the same as the type space,
[0, 1].10
After observing the report, R takes an action, a ∈ R.
The payoff for R is given
by V (a, t), and the payoff for S is given by U (a, t) − kC (r, t), where C (r, t) is a misreporting
cost function and k > 0 is a scalar that parameterizes the intensity of misreporting cost. This
formulation entails that r is payoff-relevant only for S. All aspects of the game except the value
of t are common knowledge.
Throughout, the following assumptions on payoffs are maintained. The functions U (a, t)
and V (a, t) are twice continuously differentiable on R × T .11 Using subscripts to denote derivatives, U11 < 0 < U12 and V11 < 0 < V12 , so that payoffs are concave in R’s action and supermodu¡
¢
¡
¢
lar. For any t, there exists aR (t) and aS (t) respectively such that V1 aR (t) , t = U1 aS (t) , t =
0, with aS (t) > aR (t). That is, the most-preferred actions are well-defined for both players, and
the Sender prefers higher actions than the Receiver. These assumptions on U and V imply that
10
It requires only relabeling to extend the analysis to a report space that is an arbitrary compact interval.
That is, there exists an open set T 0 containing T and two twice continuously differentiable functions on R × T 0
which are respectively identical to U and V on R × T .
11
6
for i ∈ {R, S}, ai1 (t) > 0, i.e. both players prefer higher actions when the Sender’s type is higher.
Finally, C (r, t) is twice continuously differentiable on T × T , with C11 > 0 > C12 , and for all t
there exists rS (t) ∈ [0, 1] such that C1 (rS (t), t) = 0. That is, rS (t) is the cost-minimizing report
for type t, higher types have higher cost-minimizing reports, and for any type, the marginal cost
of misreporting is increasing as the report gets further away from its cost-minimizing report. (It
is helpful to keep in mind the special case when C1 (t, t) = 0 for all t ∈ T , in which case the
cheapest report for any type is the truth, i.e. rS (t) = t.)
A few aspects of this simple model are worth emphasizing.
First, for all k > 0, the
report r is a discriminatory signal, in the sense that the cost varies with type, rather than cheap
talk or money-burning (both which are non-discriminatory).
Second, if k = 0, the report is
pure cheap talk and the model is exactly that of CS. Third, the upper bound on the report
space plays a crucial role in the equilibrium convergence results as k → 0 because it ensures that
regardless of how small k is, the function kC(r, t) is bounded and has bounded derivatives (given
the smoothness assumptions). An alternative that would also deliver the ensuing results would
be to allow for an unbounded report space, but place appropriate bounds on the cost function.
Lastly, it may be realistic in some situations to permit the Sender access to not only the costly
report I have modeled so far, but also an additional costless signal.12
While I introduce this
explicitly in Section 4, I have chosen to keep the model simple at the outset, because adding this
does not affect the first part of the analysis at all, but does complicate notation.
2.2
Pure Cheap Talk
It is useful to recall some of the main results of CS at this stage. Abusing notation, for any
R t0
t00 < t0 , let aR (t00 , t0 ) ≡ arg max t00 V (a, t) dF (t) be the optimal action for the Receiver if the only
information she has is that the Sender’s type lies in [t00 , t0 ]. CS studied the case where the report
sent by the Sender is pure cheap talk, i.e. k = 0. In that setting, they proved that each (Bayesian
Nash) equilibrium is characterized by a finite partition of the type space, ht0 ≡ 0, t1 , . . . , tN ≡ 1i,
where tj > tj−1 for all j = 1, . . . , N ,13 and the following “arbitrage” condition holds:
(∀j = 1, . . . , N − 1) U (aR (tj−1 , tj ), tj ) − U (aR (tj , tj+1 ), tj ) = 0.
(A)
In equilibrium, information transmission is coarse: the Sender conveys which element of
12
For example, firms can take costly actions to manipulate earnings, but also make cheap talk statements to the
press with explanations for why earnings may be unexpectedly low.
13
Note that the restriction to a strictly increasing sequence is without any essential loss of generality, because
the only other possibility for a CS equilibrium partition is one where h0 = t0 = t1 < t2 < . . . < tN = 1i. Such an
equilibrium partition is essentially equivalent to one where type 0 does not separate, because the prior distribution
on types puts 0 probability on any particular type.
7
the partition his type lies in, and the Receiver optimally responds to this information, so that
a type t ∈ (tj , tj+1 ) sends a message that leads to action aR (tj , tj+1 ). The arbitrage condition
(A) says that a boundary type, tj , that divides two segments of the partition must be indifferent
between the actions associated with each of these segments. The arbitrage condition (A) combined
with the initial and final conditions t0 ≡ 0 and tN ≡ 1 defines a 2nd order difference equation.
There will in general exist multiple solutions, and thus multiple equilibrium outcomes; this is
particularly severe when the preferences of both players are relatively congruent. CS proved that
there is a finite upper bound N ≥ 1 on the maximum number of steps in a solution, and moreover,
for each N ∈ {1, . . . , N }, there is an N -step equilibrium.
An important regularity condition of CS (p. 1444) will play a role at points of this paper.
Say that a sequence ht0 , t1 , . . . , tJ i is a forward solution to (A) if t1 > t0 and for all j = 1, . . . , J −1,
U (aR (tj−1 , tj ), tj ) − U (aR (tj , tj+1 ), tj ) = 0.
Condition M. If ht0 , t1 , ..., tJ i and ht̃0 , t̃1 , ..., t̃J i are both forward solutions to (A) with t1 > t̃1 >
t0 = t̃0 , then tj > t̃j for all j ∈ {1, . . . , J}.
CS (Theorem 2) provide sufficient conditions on primitives that guarantee that this is
satisfied. Applied papers using the CS model almost always assume Condition M. The reason
is that Condition M ensures two properties: first, there is no more than one partition with a
given number of steps (CS Lemma 3); second, an N -step equilibrium ex-ante Pareto dominates
an (N − 1)-step equilibrium (CS Theorems 4 and 5).
Using the Receiver’s ex-ante expected
utility as a measure of the informativeness of a CS equilibrium, this implies that under Condition
M, cheap talk equilibria can be ranked in terms of informativeness, with the most informative
equilibrium being ex-ante Pareto dominant.
2.3
Monotone Equilibrium
Return now to the current model, where k > 0. As noted earlier, this transforms the game from
one of pure cheap talk to one where signaling is costly and discriminatory across types. The basic
solution concept I employ is (pure strategy) perfect Bayesian Equilibrium.14 The Sender’s (pure)
strategy is a Borel-measurable function ρ : [0, 1] → [0, 1], where ρ(t) is the report sent by type
t. Denote the posterior beliefs of the Receiver given a report r by the cumulative distribution
G (t | r). The Receiver’s (pure) strategy is a Borel-measurable function α : [0, 1] → R, so that
α(r) is the action taken when report r is observed. Equilibrium requires the Sender to be playing
14
The notions of sequential equilibrium (Kreps and Wilson, 1982) and perfect Bayesian equilibrium coincide for
finite signaling games (Fudenberg and Tirole, 1991). The restriction to pure strategy equilibria is standard in the
signaling literature. Indeed, in the current model, it is without loss of generality to only consider pure strategies
for the Receiver, since her utility function V (a, t) is strictly concave in a.
8
optimally given the Receiver’s strategy, and that the Receiver play optimally given his beliefs,
which must satisfy Bayes rule where possible.15
As is well known, signaling games often possess a plethora of equilibria, many of which
can be implausible. I restrict attention to equilibria that display some natural monotonicity.
Definition 1. An equilibrium with strategy profile (ρ, α) is monotone if
1. (Report Monotonicity) ρ (t) is non-decreasing;
2. (Belief Monotonicity) α(r) is non-decreasing.
Report monotonicity says if t0 > t, then the report sent by t0 is no smaller than the report
sent by t. Given report monotonicity, Bayes rule implies that R’s beliefs upon seeing a report,
r, first order stochastically dominates (FOSD) her beliefs upon seeing a report r0 < r, whenever
r and r0 are both on the equilibrium path. That is, report monotonicity implies that R’s beliefs
are weakly monotone in the sense of FOSD on the equilibrium path.
By the assumptions on
V (·, ·), optimality of the Receiver’s strategy requires that weakly higher beliefs in the sense of
FOSD imply weakly higher actions. (The argument is standard: by FOSD and supermodularity,
R
R
V1 (a, t) dG (t|r) is increasing in r. Therefore, V (a, t) dG (t|r) has increasing differences in
R
a, t. By Topkis’ Theorem, the maximizers of V (a, t) dG (t|r) are non-decreasing in r.) It thus
follows that α(r) is non-decreasing on the equilibrium path. The belief monotonicity condition
extends this to off -the-equilibrium-path reports as well.
Let me now discuss why the restriction to monotone equilibria is appealing. First consider
report monotonicity.
In traditional signaling games (as defined in Cho and Sobel, 1990), all
equilibria necessarily feature report monotonicity.
This follows from two usual assumptions:
first, that irrespective of his true type, the Sender’s payoff is increasing in his perceived type;
second, a single crossing condition holds such that for any two types, indifference curves in a − r
space only cross once, if at all. My model generally violates both of these properties: the former
because of the CS preferences over actions and types, i.e. the assumptions on U and V ; the
latter because of these assumptions combined with the assumptions on C.16 This is what opens
the possibility of constructing equilibria that violate report monotonicity.
However, any such
equilibrium—if one exists—seems implausible relative to a report monotone equilibrium, because
higher types intrinsically prefer higher reports in this model (rS (t) is strictly increasing and C(·, t)
is convex around rS (t)).
15
To be precise, G(t|r) must be a regular conditional distribution of t given r, derived from ρ and F .
Specifically, indifference curves in a − r space generally cross either twice or never in the current model. This
is the main property shared with Bernheim and Severinov (2003).
16
9
As already noted, report monotonicity implies that the Receiver’s beliefs must be monotone (in the sense of FOSD) on the equilibrium path. So the requirement of belief monotonicity is
only further restrictive off the equilibrium path. Given monotone beliefs on the equilibrium path,
non-monotonicities in beliefs off the path would seem perverse. Moreover, as will be made precise
in Section 3, my results only require a weaker condition, viz. that beliefs for an off-path report
be “in between” the highest belief held for some lower on-path report and the lowest belief held
for some higher on-path report. An equilibrium where this property is violated uses extremely
elaborate beliefs, making it unattractive.
Finally, it is important to note that the restriction to monotone equilibria is without any
essential loss of generality in the case of pure cheap talk, in the sense that every CS equilibrium
mapping from T to A can be generated by a monotone equilibrium when k = 0.
3
Almost-Cheap Talk
This section deals with the convergence of monotone equilibria as k → 0. While it is not possible
to give a complete characterization of all monotone equilibria, enough can be said about their
structure for small k to obtain a necessary condition that any limit CS equilibrium must satisfy
(Theorem 1).
I then show that under a technical assumption, this condition is also sufficient,
simultaneous proving the existence of monotone equilibria when k is small (Theorem 2).
Some terminology will be useful. With respect to a (generally implicit) strategy profile,
¡ ¡ ¢¢
(ρ, α), say that a type t̃ elicits or induces action a if α ρ t̃ = a; t̃ is separating if there exists
some report r̃ such that {t : ρ (t) = r̃} = {t̃}; and a set of types are pooling if they all send the
same report. Report monotonicity has a basic implication about elicited actions.
Lemma 1. In any monotone equilibrium, the set of types that elicit the same action is convex.
Proof. Suppose to the contrary that for some t1 < t2 < t3 , both t1 and t3 elicit action a1 , whereas
t2 elicits a2 (a2 6= a1 ). By report monotonicity, α (ρ(t1 )) < aR (t2 ), and also α (ρ(t3 )) > aR (t2 ).
This contradicts both t1 and t3 eliciting a1 .
Q.E.D.
Lemma 1 implies that for any monotone equilibrium, there is a unique partition of the
type space, ht0 ≡ 0, t1 , . . . , tJ ≡ 1i (where with some abuse of notation, J ∈ {1, 2, . . . , ∞}) that
satisfies all three of the following properties: (i) for all j < J, tj < tj+1 ; (ii) within the interior
of any element of the partition, (tj , tj+1 ), either all types pool and elicit aR (tj , tj+1 ), or all types
separate and each elicits aR (t); (iii) if (tj , tj+1 ) is a separating interval of types, then (tj−1 , tj )
and (tj+1 , tj+2 ) are pooling intervals (when they exist).
10
Note that CS partitions also satisfy
these properties, in part vacuously. In comparison with CS, however, the monotone equilibrium
partitions for k > 0 may differ in two ways: first, they need not be finite; second, types within an
element of the partition may be separating.
Definition 2. A mapping, β : [0, 1] → R, is a (monotone) outcome if there exists a (monotone)
equilibrium, (ρ, α), such that β = α ◦ ρ. β is said to be supported by (ρ, α).
By Lemma 1, any monotone outcome is associated with a partition of the type space.
While an outcome in this sense disregards the payoff-relevant reports used by each type, it is a
convenient way to frame convergence of equilibria as k → 0, since the payoff impact of reports
vanishes. Analogously, a CS outcome is an outcome supported by a CS equilibrium.
The following Lemma severely restricts the set of monotone outcomes when k is small.
Lemma 2. ∀ε > 0, ∃δ > 0 such that if k < δ, then for any monotone outcome β k ,
(i) no type t > ε is separating;
(ii) {a : ∃t > ε s.t. a = β k (t)} is finite;
(iii) there is a CS outcome, β 0 , such that |β k (t) − β 0 (t)| < ε for all t > ε.
Proof. See Appendix A.
The intuition for the first part stems from the fact that if a monotone equilibrium has a
type t̂ > 0 separating, then as a consequence of Lemma 1, a type t < t̂ cannot be eliciting an
action greater than aR (t, t̂). But then, when k is small enough, t̂ cannot be separating, because
some t < t̂ would deviate and mimic t̂. For the second part, note that for there to be an infinite
number of actions elicited by types above ε, it would have to be that amongst the set of types
above ε, there are arbitrarily small convex pools.
This implies that there must be some type
t > ε who is eliciting an action arbitrarily close to aR (t), i.e. be close to separating. This cannot
obtain when k is small enough for the same reason as the first part of the Lemma.
Finally,
part (iii) is established by showing that given the earlier arguments, the incentive constraints for
types above ε imposes conditions that are arbitrarily close to the CS arbitrage conditions (A)
as k is made arbitrarily small. This is because each of the boundary types between successive
pools in [ε, 1] must be indifferent between joining its two adjacent pools; when k is small, this is
approximately the same condition as the CS arbitrage condition.
Combined, Lemmas 1 and 2 imply that any monotone equilibrium with almost-cheap
talk (i.e. small k) consists of a finite sequence of connected non-degenerate pools for all both
11
a vanishing measure of the lowest types. In particular, it is an immediate corollary that there
cannot be full separation once k is sufficiently small.17
To state the main results, one more definition is needed.
Definition 3. A CS equilibrium with partition, ht00 ≡ 0, t01 , . . . , t0N ≡ 1i, has or satisfies the No
Incentive to Separate (NITS) property if U (aR (0, t01 ), 0) ≥ U (aR (0), 0). It satisfies NITS strictly
if the inequality is strict.
A CS outcome has or satisfies the NITS property (strictly) if it is
supported by a CS equilibrium that satisfies NITS (strictly).
In words, a CS equilibrium satisfies NITS if and only if the lowest type weakly prefers the
action it elicits in the cheap talk equilibrium to the action it would elicit by separating itself (if
it could) at no cost. The ensuing theorem says that only CS outcomes satisfying NITS can be
limits of monotone almost-cheap talk outcomes.
Theorem 1. If a sequence of monotone outcomes, β k → β̂ in measure as k → 0, then β̂ = β 0
a.e., where β 0 is a CS outcome that satisfies NITS.
Because the logic of the proof is central and instructive, a detailed sketch is included in
the text.
Proof Sketch. Fix a sequence of monotone outcomes, β k → β̂ in measure as k → 0, and for each β k ,
let the supporting equilibrium be (ρk , αk ) with (possibly infinite) partition htk0 ≡ 0, tk1 , . . . , t0J(k) ≡
1i.
It is straightforward that there exists a unique CS outcome, β 0 , such that β̂ = β 0 a.e.
(existence follows from Lemma 2 (part iii), and uniqueness follows from the fact that no two CS
outcomes can be a.e. equal). Obviously, β k → β 0 in measure. I argue by contradiction that β 0
satisfies NITS. Suppose β 0 does not satisfy NITS. Let the CS equilibrium that generates β 0 have
supporting partition ht00 = 0, t01 , . . . , t0N = 1i, and denote the action elicited by types in (t0n−1 , t0n )
by a0n . Since NITS is not satisfied, U (a01 , 0) < U (aR (0), 0). In words, type 0 strictly prefers to
elicit aR (0) than elicit a01 at equal cost. The argument proceeds via four Claims outlined below;
details for each are in the Appendix.
Claim 1: For all k sufficiently small, (ρk , αk ) has no separating types. The intuition is
that by Lemma 2 (part iii), for small enough k, ρk must entail types pooling together approximately as in β 0 , except perhaps for types arbitrarily close 0. Hence, if a type is separating, it
must be an eliciting an action arbitrarily close to aR (0). Since NITS is not satisfied, by continuity
and small costs, some type above 0 that is eliciting an action close to a01 will strictly prefer to
deviate and mimic the separating type.
17
That there cannot be full separation once k is sufficiently small holds for the class of all equilibria, including
non-monotone and mixed strategy equilibria. A proof is available upon request.
12
Therefore, for all k sufficiently small, the equilibrium (ρk , αk ) has only pools, and its
equilibrium partition must have exactly N steps. Accordingly, henceforth write the equilibrium
partitions as htk0 ≡ 0, tk1 , . . . , tkN ≡ 1i. For the remainder of the proof, for all j = 1, . . . , N , let rjk
and akj denote the report sent and action elicited by types in interval (tkj−1 , tkj ).
k ≥ r S (1). The intuition is that if not, type 1 can
Claim 2: For all k small enough, rN
profitably deviate by playing r = rS (1) (which is off-path by report monotonicity) and eliciting
an action a ≥ akN (by belief monotonicity) and incurring a strictly lower cost.
Claim 3: For all k sufficiently small, rnk ≥ rS (tkn ) for all n = 1, . . . , N . This is the main
k
step, and the argument is inductive. Take as given that rn+1
> rS (tkn ) (Claim 2 delivers this for
n = N − 1). Then if rnk < rS (tkn ), report monotonicity requires that rS (tkn ) is off the equilibrium
path, and belief monotonicity requires that α(rS (tkn )) ∈ [akn , akn+1 ].
For small k, since akj ≈ a0j
and tkj ≈ t0j for all j = 1, . . . , N , it follows that rS (tkn ) is a profitable deviation for type tkn because
the induced action is weakly preferred to either akn or akn+1 (because U11 < 0) and the report is
k .
strictly less costly than either rnk or rn+1
Claim 4: For all k sufficiently small, r1k ≤ rS (0). The intuition is that if not, then for
k small enough, rS (0) is off the equilibrium path by report monotonicity and α(rS (0)) ≤ ak1 by
belief monotonicity. For small k, since ak1 ≈ a01 , it follows that rS (0) is a profitable deviation for
type 0, because the induced action is weakly preferred (since U11 < 0 and U (a01 , 0) < U (aR (0), 0)
by hypothesis of NITS not being satisfied), and the report is strictly cheaper than r1k .
We have the desired contradiction, since Claims 3 and 4 imply that for all k sufficiently
small, r1k ≤ rS (0) < rS (tk1 ) ≤ r1k .
Q.E.D.
A few remarks about the Theorem are in order.
Remark 1 (Weakening belief monotonicity). An inspection of the proof reveals that the belief
monotonicity restriction in Definition 1 is stronger than necessary.
Instead, all that is needed
is that the action played in response to an out-of-equilibrium report, r, must be weakly higher
than the highest action played in response to any on-path report r0 < r, and weakly lower than
the lowest action played
report r0 > r. To state this formally, given
( in response to any on-path
)
½
¾
R
R
(ρ, α), let l (r) ≡ max a (0), sup α (ρ (t)) and h (r) ≡ min a (1), inf α (ρ (t)) .18 The
t:ρ(t)>r
t:ρ(t)<r
following weak belief monotonicity is sufficient for the Theorem 1: for all (out-of-equilibrium) r,
α(r) ∈ [l(r), h(r)].
It is straightforward to verify that belief monotonicity is strictly stronger
than weak belief monotonicity; the latter allows for α to be non-monotone over an interval of
out-of-equilibrium reports, whereas the former does not.
18
I follow the convention that the supremum (infimum) of an empty set is −∞ (+∞).
13
Remark 2 (Role of (weak) belief monotonicity). The role of [weak] belief monotonicity is illustrated
in Figure 1, where for simplicity I assume aR (t) = rS (t) = t.
In the figure, the dotted curves
represent indifference curves for various types; the thick (red) solid line is the Sender’s strategy;
and the thin (blue) solid line is the Receiver’s strategy.
ht00
≡
0, t01 , t02
≡ 1i, with associated actions
a01
and
a02 .
There is a two-step CS partition,
From the indifference curve of type 0,
I(0), it should be clear that the CS outcome does not satisfy NITS. However, there may be a
sequence of equilibrium outcomes that converge to the CS outcome (of course, they must violate
monotonicity).
For k = ε, the figure illustrates a potential equilibrium profile (ρε , αε ) whose
outcome is close to the CS outcome.
Theorem 1 does not apply because αε does not satisfy
[even weak] belief monotonicity. Notice that [weak] belief monotonicity would require that for all
r ∈ (0, 1), αε (r) ∈ [aε1 , aε2 ], in which case the outcome αε ◦ ρε cannot be supported, because type
tε1 would have a profitable deviation to some r ≈ 1.
Remark 3 (Allowing pure cheap talk). With an eye towards the sequel, let me note at this juncture
that Theorem 1 and its preceding Lemmas are unchanged if the Sender is allowed to send not
only the report, r, but also a pure cheap talk message, m. (Obvious modifications to notation
and definitions need to be made; but the only conceptually important point is that report and
belief monotonicity would continue to be defined with respect to the report, r, alone.) The logic
of the proof reveals this, and the intuition is simple: the additional costless message, m, can only
play a role in further segmenting the set of types that are pooled on some report; moreover, no
types can fully separate using m alone.
Theorem 1 identifies NITS as a necessary condition for a CS equilibrium to be a limit of
monotone outcomes of my model. Theorem 2 below asserts the converse under a mild technical
condition. To state it requires some notation. For any t ∈ [0, 1], define, similar to CS,
K(t) ≡ max{J : ∃ 0 = t0 ≤ t = t1 < . . . < tJ ≤ 1 satisfying (A)}.
Let τ (t) ≡ h0 = τ0 (t), t = τ1 (t), . . . , τK(t) (t) ≤ 1i denote a sequence of length K(t) that
satisfies (A). Plainly, τ (t) is a partition of the type space if and only if τK(t) (t) = 1.
Assumption 1. If τ (t) is a partition for t < 1 , then for any θ > 0, there exists t̃ ∈ (t − θ, t + θ)
such that K(t̃) < K(t).
To understand the assumption, first note that by the continuity of K(·) at any t such that
τK(t) (t) < 1, τ (t) is a partition if K(·) has a discontinuity at t. The import of Assumption 1 is
to strengthen this “if” to an “if and only if”. That is, under the Assumption, CS partitions are
characterized completely by discontinuity points of K(·). Whether Assumption 1 holds or not
is determined by the triple of parameters (U, V, F ), and intuitively, it only fails for very special
14
r
1
ρε(t )
I (0)
ε
1
I (t )
αε(r )
αε(1)
I (1)
I (t1ε )
I (0)
ε
1
I (t )
αε(r )
αε(0)
ρε(t )
ε
0
0= a R(0) a1 a1
I (1)
ε
1
I (t )
t1ε t10
a2ε a20
Figure 1: Role of Belief Monotonicity
15
1
t, a
constellations. In particular, it is easily verified that Assumption 1 is satisfied if there is at most
one CS equilibrium partition with N steps for any integer N , and a fortiori, the Assumption is
implied by Condition M.
Theorem 2. Assume Assumption 1.
A monotone equilibrium exists for all k small enough.
Moreover, if β 0 is CS outcome satisfying NITS, there is a sequence of monotone outcomes, β k →
β 0 in measure as k → 0.
Proof. See Appendix A.
The idea of the proof is as follows: given a CS outcome that satisfies NITS,19 I construct
an arbitrarily small perturbation of the partition and show that this perturbed partition is a
monotone outcome for small enough k, supported by an equilibrium where all the equilibrium
reports are close to highest available report, 1. The crucial role that NITS plays is to ensure that
the unused reports at the bottom (i.e. below the report sent by type 0) can be assigned monotone
beliefs that maintain equilibrium incentives. In particular, one can assign point-mass beliefs of
type 0 to all such reports, so that the Receiver responds with aR (0). As in the proof of Theorem
1, if NITS is not satisfied by the CS outcome, such a construction is not feasible, because the
lowest type would strictly prefer to deviate to a low report for small enough k.
How useful is the NITS criterion in pruning the set of CS equilibria? Chen, Kartik, and
Sobel (2007) prove the following.
Lemma 3. Every CS outcome with the maximal number of induced actions, N , satisfies NITS.
If Condition M holds, only the unique CS outcome with N induced actions satisfies NITS.
Recall that Condition M is a regularity condition that is used to perform comparative
statics in CS with respect to changes in preferences; it is typically invoked in applied papers that
use this framework, and in particular is satisfied by the widely used uniform-quadratic specification
of CS. The following corollary of the earlier results is therefore worth stating explicitly.
Corollary 1. Assume Condition M. There is a sequence of monotone outcomes β k → β̂ in
measure as k → 0 if and only if β̂ is (a.e.) the CS outcome with maximum induced actions, which
is the most-informative outcome.
I end this section by highlighting a limitation of the analysis so far: it has focussed
entirely on the case of almost-cheap talk, i.e. small k. On the other hand, the idea of costly talk
is interesting away from the limit as well. Unfortunately, it appears infeasible to say much about
the class of all monotone equilibria when k is large.
The following section therefore studies a
particular class of monotone equilibria, facilitated by augmenting the model.
19
one exists, by Lemma 3 below.
16
4
Cheap and Costly Talk
For the remainder of the paper, the Sender is allowed to to send a signal pair, (r, m), where
r continues to be the costly report as before, but m is now a payoff-irrelevant pure cheap talk
message.
Note that for terminological clarity, I always refer to r as the report and m as the
message.
It is assumed that m ∈ M , where M is an arbitrary infinite space.
The Sender’s
strategy henceforth consists of two components: a reporting strategy, ρ(t), and a message strategy,
µ(t). The Receiver’s strategy continues to be denoted α, but is now a function of two arguments,
hence written as α(r, m).
The message m can be thought of as either a technical device to study the cheap talk
extension of the costly misreporting model (as suggested by Manelli, 1996), or as representing a
substantive economic notion that costless signals are also available to the Sender (as in AustenSmith and Banks, 2000). Each of these interpretations is appealing in different applications.20
In this setting, a single report may map into multiple actions from the Receiver, depending
on the accompanying message. Hence, the definition of belief monotonicity needs a modification:
if r > r0 , then inf m α(r, m) ≥ supm α(r, m). With this change, the monotonicity definition for
equilibrium (Definition 1) continues to apply.
Note also that an outcome is now a function
β : [0, 1] → [0, 1] such that there is an equilibrium ((ρ, µ), α) such that β = α ◦ (ρ, µ).
It is
important to emphasize that as noted in Remark 3, Theorem 1 goes through unchanged, and so
does Theorem 2 because cheap talk can always be made irrelevant.
The first part of this section introduces a forward-induction refinement of monotone equilibria. The second part characterizes the class of such equilibria, and proves existence within this
class. I use these results in the third part of the Section to prove prove an analog of Theorem 2
within the narrower class of forward-induction equilibria.
4.1
The Monotonic D1 Criterion
The refinement I adopt is the monotonic D1 (mD1) criterion, due to Bernheim and Severinov
(2003).
The underlying idea is the same as Cho and Kreps’s (1987) D1 criterion, which says
that the Receiver should not attribute a deviation to a particular type if there is some other type
that is willing to make the deviation for a strictly larger set of inferences/responses. The mD1
criterion strengthens this by applying the test to only those responses from the Receiver that
satisfy belief monotonicity. To state this formally, some notation is needed. With respect to a
20
See fns. 12 and 25.
17
given profile ((ρ, µ), α), define
ξl (r) ≡ max{aR (0), sup α (ρ (t) , µ(t))},
t:ρ(t)<r
R
ξh (r) ≡ min{a (1), inf
t:ρ(t)>r
α (ρ (t) , µ(t))}.
Suppose ((ρ, µ), α) is an equilibrium. For an out-of-equilibrium report r0 such that some report
r < r0 (resp. r > r0 ) is sent in equilibrium, ξl (r0 ) (resp. ξh (r0 )) is the “highest” (resp. “lowest”)
action taken by the Receiver in response to an equilibrium report lower (resp. higher) than r0 .
If r0 is such that there is no report r < r0 (resp. r > r0 ) sent in equilibrium, then ξl (r0 ) (resp.
ξh (r0 )) just specifies the lowest (resp. highest) rationalizable action for the Receiver.
With implicit respect to some profile ((ρ, µ), α), let
A(r, t) ≡ [ξl (r), ξh (r)] ∩ {a : U (a, t) − kC (r, t) ≥ U (α (ρ (t) , µ(t)) , t) − kC (ρ (t) , t)},
A(r, t) ≡ [ξl (r), ξh (r)] ∩ {a : U (a, t) − kC (r, t) > U (α (ρ (t) , µ(t)) , t) − kC (ρ (t) , t)}.
To interpret, consider an unused report r in some equilibrium. A(r, t) (resp. A(r, t)) is
the set of responses within the set [ξl (r), ξh (r)] that give type t a weak (resp. strict) incentive to
deviate to r.
Definition 4. A (monotone) equilibrium, ((ρ, µ), α), satisfies the mD1 criterion if α(r, m) =
aR (t0 ) for any m and any out-of-equilibrium report r for which A(r, t0 ) 6= ∅ and A(r, t) ⊆ A(r, t0 )
for all t 6= t0 .
Note that in the Definition, the requirement that α(r, m) = aR (t0 ) could alternatively
be posed as support[G(·|r, m)] = {t0 }. If we replace [ξl (r) , ξh (r)] in the definitions of A and
£
¤
A with aR (0) , aR (1) , then the above test is like the D1 criterion (cf. Cho and Sobel, 1990,
p. 385).21
However, given belief monotonicity, for any out-of-equilibrium report r and any
message m, α (r, m) ∈ [ξl (r) ξh (r)].
Accordingly, the definition above applies the idea behind
the D1 criterion on the restricted action space [ξl (r) , ξh (r)]. That is, it requires that for some
out-of-equilibrium report r, if there some type t0 who would strictly prefer to deviate to r for
any response a ∈ [ξl (r) , ξh (r)] that a type t 6= t0 would weakly prefer to deviate for, then upon
observing the deviation r (coupled with any message), the Receiver should believe it is type t0 .
21
Actually, the test would be weaker than the D1 criterion. This is because D1 requires distinct types t1
and t2 to both be pruned from the support of R’s beliefs if there exist distinct t01 and t02 such that for i = 1, 2,
A(r, ti ) ⊆ A(r, t0i ). The current formulation does not require this; it only requires pruning if there is a single t0
that “covers” all t 6= t0 . For exactly the same reason, the mD1 test formulated here is weaker than Bernheim and
Severinov’s (2003). However, it can be proved that the sets of refined equilibria in the current model are identical
under either formulation. For the same reason, even though the formulation of the mD1 criterion I use is strictly
speaking weaker than Bernheim and Severinov’s (2003), they are equivalent here.
18
The mD1 criterion obviously imposes a strong restriction on off-the-equilibrium path beliefs of the Receiver.
However, this makes them particularly appealing, and strengthens the
ensuing results about existence of such equilibria and their convergence. Moreover, it permits a
tight characterization of a class of equilibria for any k > 0.
Remark 4. Not surprisingly, the mD1 criterion does not directly help restrict the set of equilibrium
outcomes in the pure cheap talk game (i.e. when k = 0). To see this, consider an equilibrium when
k = 0. Given that M is uncountable and k = 0, there is an essentially equivalent equilibrium (i.e.
one that induces the same outcome mapping from types to actions) where all types send the same
report, call it r∗ , and use possibly different cheap talk messages. I claim that this equilibrium
can be supported by strategies that satisfy mD1. As in CS, there can only be a finite number of
actions induced in equilibrium when k = 0; hence the equilibrium can be supported with finitely
many distinct cheap talk messages. Denote the highest and lowest actions induced in equilibrium
by ah and al respectively with corresponding messages mh and ml . For any m and t, define for
all r < r∗ , G(t | r, m) ≡ G(t | r∗ , ml ), and for all r > r∗ , G(t | r, m) ≡ G(t | r∗ , mh ). Then for
all m, if r < r∗ , α(r, m) = al and if r > r∗ , α(r, m) = ah . Clearly these strategies and beliefs
form an equilibrium that supports the same outcome as the original equilibrium, and moreover
the strategies satisfy report and belief monotonicity. To see that mD1 is satisfied, consider any
r > r∗ (the argument is analogous for r < r∗ ).
We have ξl (r) = ah and ξh (r) = aR (1), hence
ah ∈ [ξl (r), ξh (r)]. Since no type strictly prefers ah over what it elicits in equilibrium, mD1 is
satisfied.
4.2
Characterization and Existence
For any report r, let tl (r) ≡ inf {t : ρ (t) = r} and th (r) ≡ sup {t : ρ (t) = r}.
(As usual, the
dependence on the reporting strategy ρ is left implicit.) If there is pooling on r, then tl (r) < th (r)
and every type t ∈ (tl (r) , th (r)) sends report r. It is important to note that within the set of
types pooling on r, there may be some information conveyed via cheap talk messages, but just as
in CS, cheap talk cannot induce any full separation within (tl (r), th (r)).
The key step towards characterizing mD1 equilibria is the following Lemma.
Lemma 4. In any mD1 equilibrium, ((ρ, µ), α),
(i) If there is pooling on report rp < 1, there exists some θ (rp ) > 0 such that reports r ∈
(rp , rp + θ (rp )) are unused;
(ii) For all such r ∈ (rp , rp + θ (rp )), and any message m, α (r, m) = aR (th (rp ));
(iii) If type 0 is separating, then ρ(0) = rS (0).
19
Proof. See Appendix C.
Here is the intuition behind the result. Fix a report, rp < 1, that a set of types are pooling
on. For this discussion, let th be shorthand for th (rp ), and assume for simplicity that ρ(th ) = rp .
The first part of the Lemma follows from the fact that because it is pooling, type th induces an
action strictly smaller than aR (th ). If ρ were continuous at th , then types immediately above th
must be separating. Since the utility gain for th from mimicking th + ε is positive and bounded
away from 0 as ε → 0, whereas any additional cost of reporting goes to 0, it follows that th has a
profitable deviation for small enough ε > 0; a contradiction.
Now turn to the second part of the Lemma.
The proof shows that the mD1 criterion
requires the Receiver to play aR (th ) in response to any report r ∈ (rp , rp + θ (rp )), regardless
of the accompanying cheap talk message.
elicited by any type in the pool on rp , and
To see how this works, let a− be the highest action
a+
be the lowest action elicited in equilibrium by types
above th . If type th would want to strictly deviate from its equilibrium play for every response
a ∈ [a− , a+ ] to report r that any other type t would want to weakly deviate for, then mD1 requires
the Receiver to place probability 1 on th upon seeing r. Consider some type t < th (the logic is
symmetric for t > th ).
In equilibrium, t induces an action weakly lower than that induced by
th , and weakly lower than a− . Moreover, t’s preferred action and t’s preferred report are both
strictly lower than th ’s. The supermodularity of U (·, ·) and the submodularity of C(·, ·) imply
that if t would weakly prefer to deviate to r for some response of the Receiver in [a− , a+ ], then
th would strictly prefer to deviate.
The intuition is illustrated in Figure 2, which is drawn with the specification k = 1,
C(r, t) = −(r − t)2 , U (a, t) = −(a − t − x)2 (x > 0), and V (a, t) = −(a − t)2 . In the figure, the
thick solid (red) line is the Sender’s reporting strategy; the thick dotted (blue) line is the Receiver’s
strategy; and the thin solid (blue) curves are indifference curves for various types.
Note that
due to the quadratic specification, indifference curves are circles. The set of report-action pairs
that type t0 < t1 would weakly prefer to deviate to (satisfying r ∈ (rp , θ(rp )) and a ∈ [a− , a+ ])
is shaded with (green) stripes in one direction. The set of report-action pairs that type t1 > th
would weakly prefer to deviate to (satisfying r ∈ (rp , θ(rp )) and a ∈ [a− , a+ ]) is shaded with
(orange) stripes in the other direction.
Clearly, both these sets are strict subsets of the set of
report-action pairs that type th would weakly prefer to deviate to (satisfying r ∈ (rp , θ(rp )) and
a ∈ [a− , a+ ]).
Lemma 4 implies that in an mD1 equilibrium, there can only be pooling on the highest
report.
If there were pooling on some other report, r, then for small enough ε > 0, type
th (r) − ε would have a profitable deviation to a report r + ε, which by the Lemma must induce
the response aR (th ). This reveals the basic structure of any mD1 equilibrium: there could be an
20
r
1
ρ(t )
I (t 1 )
α(r,m)
θ (rp)
α(rp ,m2)
α(rp ,m1)
rp
I (t h )
ρ(t )
I (t 0 )
m2
α(r,m)
0
m1
tl
a- t0 th
t1 a+
Figure 2: mD1 rules out pooling on rp < 1 (Lemma 4)
21
1
t, a
interval of separating types at the bottom end of the type space (separation must of course occur
through distinct reports); whereafter all remaining types pool on r = 1, but may further segment
themselves using cheap talk messages.
To characterize the separating region, consider the following differential equation:
¡
¢
U1 aR (t) , t aR
1 (t)
ρ (t) =
.
kC1 (ρ (t) , t)
0
(DE)
Lemma 5. There is a unique solution to the problem of finding a t ∈ (0, 1] and ρ : [0, t] → [0, 1]
such that (i) ρ is strictly increasing and continuous on [0, t], (ii) ρ(0) = rS (0), (iii) ρ solves (DE)
on (0, t), and (iv) ρ(t) = 1 if t < 1.
Proof. See Appendix C.22
For any open interval of types that are separating, ρ must solve (DE). This is straightforward assuming differentiability of ρ, since in that case, ρ0 in (DE) is the solution to the first
order condition for type t. Arguments extending those of Mailath (1987) establish that in fact
conditions (i)-(iv) of the Lemma are necessary and sufficient for a “separating function”. That
the initial value must be ρ(0) = rS (0) is a consequence of Lemma 4 (part iii).
Henceforth, let ρ∗ and t denote the objects identified in Lemma 5.
£ ¤
Theorem 3. In any mD1 equilibrium, ((ρ, µ), α), there exists some t ∈ 0, t and a finite partition
of [t, 1], ht0 ≡ t, t1 , ..., tJ ≡ 1i, such that
(BIN) ht0 , . . . , tJ i is a forward solution to (A)
(CIN) t ∈ (0, 1) =⇒ U (aR (t), t) − kC(ρ∗ (t), t) = U (aR (t, t1 ), t) − kC(1, t)
(ZWP) t = 0 =⇒ U (aR (0), 0) − kC(rS (0), 0) ≤ U (aR (0, t1 ), 0) − kC(1, 0)
(a.1) ∀t < t, ρ(t) = ρ∗ (t)
(a.2) ρ(t) ∈ {ρ∗ (t), 1}
(a.3) ∀t ∈ (t, 1], ρ(t) = 1
(b) ∀j = 1, ..., J, ∀t ∈ (tj−1 , tj ), µ(t) = mj
(mj 6= mn ∀n 6= j)
£
¢
(c.1) ∀m and ∀r ∈ 0, rS (0) , α(r, m) = aR (0)
22
Since C1 (rS (t), t) = 0 for all t ∈ [0, 1], there is no Lipschitz condition on ρ0 in (DE). Thus, standard results on
the existence of solutions to differential equations do not apply.
22
£
¢
(c.2) ∀m and ∀r ∈ rS (0), ρ∗ (t) , α(r, m) = aR ((ρ∗ )−1 (t))
(c.3) ∀m and ∀r ∈ [ρ∗ (t), 1), α(r, m) = aR (t)
(c.4) ∀j = 1, ..., J, α(1, mj ) = aR (tj−1 , tj )
£ ¤
Conversely, for any t ∈ 0, t and a finite partition of [t, 1], ht0 ≡ t, t1 , ..., tJ ≡ 1i, that
satisfy (BIN),(CIN), and (ZWP), there is an mD1 equilibrium, ((ρ, µ), α), that satisfies (a)-(c),
with ρ(0) = 1 if t = 0.
Proof. See Appendix C.
The Theorem says that any mD1 equilibrium can be fully characterized by a cutoff type,
t ∈ [0, t], and a partition of [t, 1], ht0 ≡ t, t1 , ..., tJ ≡ 1i, such that three conditions are satisfied:
(i) for any j ∈ {1, ..., J − 1}, holding the report cost fixed, the boundary type tj is indifferent
between being perceived as a member of [tj−1 , tj ] or a member of [tj , tj+1 ] (this is condition BIN,
for boundary indifference); (ii) if the cutoff type t is strictly interior, then t is indifferent between
being perceived as a member of [t, t1 ] and incurring the cost of report 1, or separating itself and
incurring the cost of report ρ∗ (t) (this is condition CIN, for cutoff indifference); (iii) if the cutoff
type is 0 then type 0 weakly prefers being perceived as a member of [0, t1 ] and incurring the cost
of report 1 to separating itself and incurring the cost of report 0 (this is condition ZWP, for zero
weak preference). All types below t separate themselves using the report strategy ρ∗ (t), whereas
all types above t send report 1, but may further segment themselves into a partition of [t, 1] by
using cheap talk messages as in CS.
Figure 3 illustrates the structure of an mD1 equilibrium with a strictly positive cutoff
type and two distinct cheap talk messages that can accompany report r = 1 in equilibrium. It
is drawn for a case where aR (t) = rS (t) = t. The function ρ∗ is plotted as the solid thin (black)
line, the equilibrium reporting strategy, ρ, is the solid thick (red) line, and the Receiver’s strategy,
α, is the dotted (blue) line.
As stated in part (c.3) of the Theorem, all reports r ∈ (ρ∗ (t), 1)
induce the action aR (t). There is segmentation within [t, 1] through cheap talk: types t ∈ (t, t1 )
send cheap talk message m1 , whereas types t ∈ (t1 , 1) send m2 (m2 6= m1 ). (If Sender types are
distributed ex-ante uniformly over [0, 1], then α(1, m1 ) =
t+t1
2
and α(1, m2 ) =
1+t1
2 .)
Theorem 3 characterizes necessary and sufficient conditions for an mD1 equilibrium. The
following result assures that these conditions can always be met.
Theorem 4. An mD1 equilibrium exists for any k. Moreover, for all k sufficiently large, there
is an mD1 equilibrium with a positive measure of separating types.
Proof. See Appendix C.
23
r
α(1,m1)
α(1,m 2)
ρ(t )
1
α(r,m)
ρ∗(t )
ρ (t )
∗
ρ(t )
α(r,m)
m2
m1
0
t
t t1
Figure 3: An mD1 equilibrium
24
1
t, a
4.3
Convergence
Say that an outcome is an mD1 outcome if it can be supported by an mD1 equilibrium.
The
following result basically strengthens the conclusion of Theorem 2 in the current environment: it
says that a CS equilibrium satisfying NITS is the limit of not just some sequence of monotone
equilibria, but rather a sequence of mD1 equilibria.
Theorem 5. Let β 0 be a CS outcome. If β 0 satisfies NITS strictly, it is an mD1 outcome for
all k sufficiently small. If β 0 satisfies NITS (but not strictly) and Condition M holds, there is a
sequence of mD1 outcomes, β k → β 0 pointwise as k → 0.
Proof. The second part is deferred to Appendix C. For the first part, fix a CS outcome, β 0 ,
that satisfies NITS strictly, and let the supporting partition of any generating equilibrium be
h0 ≡ t00 , t01 , . . . , t0N ≡ 1i. Then U (aR (0, t01 ), 0) > U (aR (0), 0), and there exists k̂ > 0 such that if
k < k̂, U (aR (0, t01 ), 0) − kC(1, 0) > U (aR (0), 0) − kC(rS (0), 0). The sufficiency part of Theorem
3 implies that for all k < k̂, there is an mD1 equilibrium with cutoff type t = 0 whose outcome is
β0.
Q.E.D.
While there is no simple way to characterize exactly when there is a CS outcome that
satisfies NITS but not NITS strictly, it is intuitive that if a triple of preferences and prior, (U, V, F ),
is such that there is a CS outcome that satisfies NITS but not strictly, any small perturbation to
either preferences or the prior will ensure that all CS outcomes satisfy NITS either strictly or not
at all.23 The following Corollary is an immediate consequence of Theorem 5 and Lemma 3.
Corollary 2. Assume Condition M. There is a sequence of mD1 outcomes β k → β 0 pointwise
if β 0 is the most-informative CS outcome.
5
Discussion
I have studied a model of communication between an informed Sender and an uninformed Receiver,
where the Sender has a convex cost of misreporting his private information. Using a scalar to
parameterize the importance of the lying cost, the pure cheap talk model of CS can be viewed as
the limit of the costly lying games studied here. The main insight is that a CS equilibrium is the
limit of a sequence of monotone equilibria as talk gets almost-cheap only if the CS equilibrium
satisfies NITS, i.e. the lowest type weakly prefers the action elicited in the CS equilibrium to
23
For example, in the commonly applied uniform-quadratic setting with a single parameter b > 0 such that
U (a, t) = −(a − t − b)2 , V (a, t) = −(a − t)2 and F (t) = t, there is a CS outcome that satisfies NITS but not NITS
1
strictly if and only if b = 2J(J+1)
for some integer J > 0. The set of such b is of Lebesgue measure 0.
25
what it would get in the complete-information game. Under a regularity condition (Condition
M), only the most-informative cheap talk equilibrium satisfies NITS. A converse result that there
exists a sequence of monotone equilibria that converge to any CS equilibrium satisfying NITS was
also derived under a mild technical assumption.
The results obtained here provide a novel justification for focussing on the most-informative
equilibrium of CS (when regularity Condition M holds). They also indicate that little is lost by
analyzing this equilibrium of the pure cheap talk game rather than studying the more complicated
costly misreporting game, provided that the costs are small. On the other hand, when costs are
large, there exist equilibria where significantly more information can be conveyed than in the case
of pure cheap talk. Indeed, there may be a large regions of full separation when these costs are
sufficiently large. I conclude by briefly discuss two broader aspects of the theory.
Inflated Reporting. In any mD1 equilibrium, all types—except possibly the endpoints, 0 and
1—use reports that are higher than what they would use in the absence of private information,
rS (t). (See Lemma B.6 in the Appendix for a formal statement.) This is also the case for the
monotone almost-cheap talk equilibria constructed in the proof of Theorem 2. In this sense, the
equilibria I have studied feature inflated language. The underlying intuition is that every type
would like to perceived as being larger than it truly is, and since higher types prefer higher reports,
this leads to reporting strategies that are shifted up from the complete information benchmark.
This is broadly consistent with documented evidence about strategic information transmission,
e.g. analysts’ stock recommendations (Malmendier and Shantikumar, 2004), managers’ reports
about firm earnings (Healy and Wahlen, 1999), and laboratory experiments (Cai and Wang, 2006).
A testable implication of the mD1 equilibria analysis is that over a large range of k (the scalar
parameter of cost of misreporting), a decrease in k should lead to an increase in the degree of
report inflation.
Despite the presence of language inflation in equilibrium, the Receiver, being fully strategic, is not systematically deceived in any way: in equilibrium, she correctly deflates the Sender’s
report so that her conditional expectation of the Sender’s type given the report is in fact correct.
Indeed, the report inflation in an mD1 equilibrium is self-defeating in this sense, because almost
all types of the Sender incur a cost from lying. As in CS, for many parameterizations such as
the uniform-quadratic example, the Sender would prefer to commit himself ex-ante to telling the
truth.
Behavioral Types and Naive Receivers. In addition to the motivations already given for
the presence of misreporting costs, there is another that is less direct but arguably important: the
Receiver may be naive or boundedly rational with some probability, as in Chen (2006), Ottaviani
26
and Squintani (2006), and Kartik, Ottaviani, and Squintani (2007).24 To be concrete, suppose
that the report is in fact pure cheap talk, but with probability q ∈ (0, 1), the Receiver is especially
naive and treats the report r as a recommendation of action, and just follows it by playing
a = r; with probability 1 − q, the Receiver is strategic and plays a best response to the Sender’s
strategy.
Then, for a type t Sender, the expected utility from sending report r and receiving
the action a from a strategic Receiver (and mechanically, the action r if the Receiver is naive)
is: Π(a, r, t) ≡ (1 − q)U (a, t) + qU (r, t). By setting k ≡
q
1−q
and C(r, t) ≡ −U (r, t), this can be
normalized and rewritten as Π(a, r, t) = U (a, t) − kC(r, t), where C satisfies the assumptions of
this paper, so long as the report space contains [aS (0), aS (1)]. As noted in Section 2, the analysis
carries over with straightforward relabeling when the report space is any compact interval; hence,
this naive Receiver specification is a special case of costly misreporting.25
In fact, it requires only minor modifications to the arguments to show that that all the
main results of this paper extend to cost functions of the form C(ν(r), t), where C is as before
but ν is now any twice differentiable and strictly increasing function. The reason is that when
ν is strictly increasing, the most-preferred report function, rS (t), is strictly increasing, which
is the key to the analysis.
This extension is useful because it allows embedding much richer
behavioral types of Receivers.
So long as we can represent the behavioral type’s mechanical
response to a report by a function ν(r) satisfying the above conditions, a pure cheap talk model
with the behavioral type of Receiver translates into the [extended] costly misreporting model
by setting C(ν(r), t) ≡ −U (ν(r), t). For example, the behavioral type of Receiver may believe
that the Sender is telling the truth, and hence play aR (r) upon hearing report. This is simply
the case of ν(r) = aR (r).
type t sends report
Perhaps the behavioral type of Receiver believes that the Sender of
aS (t)—as
would be optimal for the Sender if he thought the Receiver would
blindly follow his recommendation and take action r upon hearing report r—in which case the
¡
¢
behavior type optimally plays aR ◦ (aS )−1 (r) in response to a report r. This translates via
¡
¢
ν(r) ≡ aR ◦ (aS )−1 (r).
The possibility of the Receiver being naive also circumvents the self-defeating nature of
equilibrium lying.
It is easy to verify that even in the uniform-quadratic example, the Sender
may ex-ante prefer the equilibrium play over committing himself to truth-telling if the Receiver
may be appropriate naive. In effect, the Sender and fully-rational Receiver exploit the presence
of the naive Receiver.
24
Crawford (2003) explores a related idea in a different setting.
In this interpretation, I prefer to think of the augmented cheap talk dimension in Section 4 as a technical device;
while it is possible that a Receiver may be naive on one dimension but sophisticated on another, this is strained.
The results of Section 3 apply when the probability of naivety, q, is small.
25
27
Appendix A: Proofs for Section 3
Proof of Lemma 2 on page 11. In proving the first two parts of the Lemma, the following notation will
be useful. For any t, denote by γ(t) the type such that aS (γ(t)) = aR (t) if it exists, or else let γ(t) = 0.
Clearly, if t > 0, γ(t) < t.
(i) It suffices to show that no t > 0 can be separating when k is sufficiently small. Pick any t̂ > 0,
and suppose there is an equilibrium, (ρk , αk ), supporting outcome β k where t̂ is separating. I will argue
to a contradiction
when
small. Since t̂ is separating, report monotonicity implies that
¡
¢ k is¡ sufficiently
¢
β k (γ(t̂)) ≤ aR γ(t̂), t̂ < aR t̂ . For type γ(t̂) not to imitate t̂ requires
¢
¡
¡¢
¡ ¡
¢
¢¤
¡ ¡¢
¢
£
U aR t̂ , γ(t̂) − U β k (γ(t̂)), γ(t̂) ≤ k C(ρk t̂ , γ(t̂)) − C ρk γ(t̂) , γ(t̂) .
But the right hand side of this inequality is converging to 0 as k → 0 ¡because
¡ ¢ C(·,¢·) is ¡bounded, whereas
¢
the left hand side is bounded below by the strictly positive constant U aR t̂ , γ(t̂) − U aR (γ(t̂), t̂), γ(t̂) ;
thus the inequality fails for small enough k.
(ii) Fix ε > 0. Suppose there is an equilibrium, (ρk , αk ), supporting outcome β k , where
Ak ≡ {a : ∃t > ε s.t. a = β k (t)}
is infinite. I will argue to a contradiction when k is sufficiently small. By part (i), when k is sufficiently
small, ever type above ε is part of a non-singleton convex pool. Without loss of generality, assume that
the infimum of types in a pool is part of the pool. Ak can be infinite only if there are arbitrarily small
pools within the truncated type space [ε, 1]. It follows that for any θ > 0, there exists a t̂k (θ) > ε such
k
t̂¯k (θ) is the
¯ lowest type amongst the set of types it pools with, and if some type t pools with t̂ (θ), then
¯t − t̂k (θ)¯ < θ. The following ordering obtains:
¡
¢
¡
¢
¡
¢
β k (γ(t̂k (θ))) ≤ aR γ(t̂k (θ)), t̂k (θ) < aR t̂k (θ) < β k t̂k (θ) .
For type γ(t̂k (θ)) not to imitate t̂k (θ) requires
¡ ¡
¢
¢
¡
¢
U β k t̂k (θ) , γ(t̂k (θ)) − U β k (γ(t̂k (θ))), γ(t̂k (θ))
£
¡
¢
¡ ¡
¢
¢¤
≤ k C(ρk t̂k (θ) , γ(t̂k (θ))) − C ρk γ(t̂k (θ)) , γ(t̂k (θ)) .
(A-1)
The right hand side of (A-1) is converging to 0 as k → 0 (because C(·, ·) is bounded). On the
other hand, by picking θ small enough,
the left¢ hand¡ side of (A-1) can
¡
¢ be bounded below by a strictly
positive constant because mint≥ε U aR (t) , γ(t) − U aR (γ(t), t), γ(t) > 0, and by picking θ sufficiently
¡
¢
¡
¢
small, β k t̂k (θ) − aR t̂k (θ) can be made as small as desired (uniformly over k). Therefore, (A-1) fails
for small enough k.
(iii) With respect to an equilibrium (ρk , αk ), let τ0k be the smallest type such that the types in
form a finite number of connected pools in the equilibrium. Lemma 1 and parts (i) and (ii) of
the current Lemma ensure that τ0k is well-defined once k is sufficiently small, and moreover, τ0k → 0 as
k
k
k → 0. Denote by hτ0k , τ1k , . . . , τN
(k) ≡ 1i (N (k) ≥ 1 and finite) the partition of [τ0 , 1] such that for all
k
k
k
j = 0, . . . , N (k)−1, types in (τjk , τj+1
) elicit aR (τjk , τj+1
). Let rjk be the report used by the pool (τjk , τj+1
).
(τ0k , 1]
It suffices to prove that for any ε > 0, for all k small enough, any equilibrium, (ρk , αk ) with
k
partial partition hτ0k , τ1k , . . . , τN
CS equilibrium with partition
(k) ≡ 1i as just defined has ¯a counterpart
¯
k
¯
¯
ht0 ≡ 0, t1 , . . . , tN (k) ≡ 1i such that for all j = 0, . . . , N − 1, τj − tj < ε.
28
Fix an ε > 0. Incentive compatibility of equilibrium (ρk , αk ) requires
k
k
k
(∀j = 1, . . . , N (k) − 1) U (aR (τj−1
, τjk ), τjk ) − U (aR (τjk , τj+1
), τjk ) = k[C(rjk , τjk ) − C(rj+1
, τjk )].
This is a system of N (k) − 1 equations. As k → 0, the right hand side converges to 0, since C(·, ·)
k
k
is bounded. Moreover, τN
(k) ≡ 1, and as already noted, τ0 → 0 as k → 0. By inspection against the
CS equilibrium conditions (t0 ≡ 0, tN ≡ 1, and equations (A)), for any sufficiently small k, there must
be
¯ k an N¯ (k)-step CS equilibrium partition, ht0 ≡ 0, t1 , . . . , tN (k) ≡ 1i, such that for all j = 0, . . . , N − 1,
¯τ − tj ¯ < ε.
Q.E.D.
j
Proof of Theorem 2 on page 16. I fill in the details here for the Claims in the proof sketch provided
in the text.
Proof of Claim 1: Suppose to the contrary that for arbitrarily small k, there are separating
types in (ρk , αk ). Then there is a sequence q → 0 and a sequence of types {τ q } such that β q (τ q ) = aR (τ q ).
By Lemma 2 (part i), τ q → 0. Lemma 2 and β k → β 0 in measure imply that for small enough k,
there exists j ∗ (k) < J(k) such that tkj∗ (k) and tkj∗ (k)+1 are arbitrarily close to 0 and t01 respectively, and
the types in (tkj∗ (k) , tkj∗ (k)+1 ) are pooling in the equilibrium (ρk , αk ). Since τ q is separating, for small
enough q, τ q < tqj∗ (q) . But then, when q is sufficiently small, continuity of U (·, ·) and the hypothesis
that U (a01 , 0) < U (aR (0), 0) imply that there exists a type t ∈ (tqj∗ (q) , tkj∗ (q)+1 ) such that for any r and r0 ,
U (aR (tqj∗ (q) , tqj∗ (q)+1 ), t) − mC(r, t) < U (aR (τ q ), t) − mC(r0 , t). Thus, type t strictly prefers to elicit aR (τ q )
rather than elicit aR (tqj∗ (q) , tqj∗ (q)+1 ), a contradiction with (ρq , αq ) being an equilibrium. ¦
Proof of Claim 2: Suppose this were not true, i.e. there exists k arbitrarily small such that
k
S
rN
< rS (1). For all such k, report monotonicity
£ k R ¤ implies that r (1) is unused in equilibrium, andS belief
k S
monotonicity implies that α (r (1)) ∈ aN , a (1) . But then, when k is sufficiently small, since a (1) >
k
aR (1) > akN and C(rN
, 1) > C(rS (1), 1), type 1 strictly prefers to send rS (1) and elicit α(rS (1)) rather
k
than send rN and elicit akN , a contradiction with equilibrium. ¦
k
Proof of Claim 3: Claim 2 established rN
≥ rS (tkN ) for all k sufficiently small . I argue it
k
≥ rS (tkn+1 ) (n = 1, . . . , N − 1). This
now by induction for all n < N . Assume inductively that rn+1
S k
k
implies that rn+1 > r (tn ). Suppose towards contradiction that rnk < rS (tkn ). By report monotonicity,
rS (tkn ) is unused in equilibrium, and thus belief monotonicity requires αk (rS (tkn )) ∈ [akn , akn+1 ]. It is
straightforward to verify that equilibrium requires type tkn to be indifferent between “pooling up or down”,
k
, tkn ). (Otherwise, a type close to tkn would have
i.e. U (akn , tkn ) − kC(rnk , tkn ) = U (akn+1 , tkn ) − kC(rn+1
a profitable deviation.) When considering k sufficiently small, akn < aS (tkn ) < akn+1 , because tkn−1 and
tkn are arbitrarily close to t0n−1 and t0n respectively (and a0n < aS (t0n ) < a0n+1 by CS arbitrage condition
(A)). By U11 < 0, this implies that for any action a ∈ [akn , akn+1 ], U (a, tkn ) ≥ U (akn , tkn ). Therefore, since
C(rS (tkn ), tkn ) < C(rnk , tkn ), it follows that U (α(rS (tkn )), tkn ) − C(rS (tkn ), tkn ) > U (akn , tkn ) − C(rnk , tkn ), and
hence tkn has a profitable deviation, a contradiction. ¦
Proof of Claim 4: Suppose this were not true, i.e. there exists k arbitrarily small such that
r1k > rS (0). For all such k, report monotonicity
implies
that rS (0) is unused in equilibrium, and be£ R
¤
S
k
lief monotonicity implies that α(r (0)) ∈ a (0), a1 . Since for k sufficiently small, tk1 is arbitrarily
R
R
0
close to t01 , the hypothesis that U
(0), 0)
£ (a
¤ > U (a (0, t1 ), 0) implies that when k is sufficiently small,
R
k
R
k
U (a, 0) > U (a (0, t1 ) for all a ∈ a (0), a1 (applying continuity of U , U11 < 0, and the hypothesis that
U (aR (0, t01 ), 0) < U (aR (0), 0)). Therefore, since C(rS (0), 0) < C(r1k , 0), U (α(rS (0)), 0) − C(rS (0), 0) >
U (ak1 , 0) − C(r1k , 0), and hence type 0 has a profitable deviation, a contradiction. ¦
Q.E.D.
The following definitions and Lemma are needed to prove Theorem 2. Define for any ε ≥ 0, the
29
following modified arbitrage condition for any sequence ht0 , t1 , . . . , tJ i:
∀j = 1, . . . , J − 1
U (aR (tj , tj+1 ), tj ) − U (aR (tj−1 , tj ), tj ) = ε.
(A-2)
For any ε ≥ 0 and any t ∈ [0, 1], define
K(t; ε) ≡ max{J : ∃ 0 ≡ t0 ≤ t ≡ t1 < . . . < tJ ≤ 1 satisfying (A-2)}.
Let τ (t; ε) ≡ h0 ≡ τ0 (t; ε), t ≡ τ1 (t; ε), . . . , τK(t;ε) (t; ε) ≤ 1i denote a sequence of length K(t; ε) that
satisfies (A-2). Note that for any K (t; ε) ≥ j > 1, τj (t; ε) > τj−1 (t; ε); also, τ (t; ε) is a partition if and
only if τK(t;ε) (t; ε) = 1.
Lemma A.1. Assume Assumption 1. For any δ > 0, ε > 0, and a non-decreasing sequence ht00 ≡
0, t01 , . . . , t0N ≡ 1i satisfying (A), there exists a strictly increasing sequence ht0 ≡ 0, t1 , . . . , tN ≡ 1i such
that for all j = 1, . . . , N − 1,
¡ ¢
1. |tj − t0j | ∈ 0, δ ;
2. U (aR (tj , tj+1 ), tj ) − U (aR (tj−1 , tj ), tj ) ∈ (0, ε);
3. If t01 = 0, then U (aR (0, t1 ), 0) > U (aR (0), 0).
(A). The
Proof. Fix δ > 0, ε > 0, and the non-decreasing sequence ht00 ≡ 0, t01 , . . . , t0N ≡¡ 1i satisfying
¢
0
0
;
δ,
ε
denote
the ball
<
1.
Let
B
t
lemma
is
trivially
true
if
N
=
1,
so
assume
N
>
1,
i.e.
t
1
1
¢ © ª
¡0
t1 − δ, t01 + δ \ t01 × (0, ε). By the same arguments as in CS Theorem 1, K(·; ε) changes by one
at any point of discontinuity, and by the continuity of solutions to (A-2) in initial conditions, K(·; ε) is
discontinuous at t only if τ (t; ε) is a partition. Similarly, K(t; ·) changes by one at any point of discontinuity, and is discontinuous at ε only if τ (t; ε) is a partition. The continuity of solutions³to (A-2)
´ in ε
¡ ¢
0
and initial conditions implies that there exist δ̃ ∈ 0, δ and ε̃ ∈ (0, ε) such that if (t, ε) ∈ B t1 ; δ̃, ε̃ then
¯
¯
K (t; ε) ∈ {N − 1, N } and ¯τj (t; ε) − t0j ¯ < δ for all j ∈ {0, 1, . . . , K (t; ε)}. If t01 = 0, then also choose
´
³
¡ ¢
¡
¢
¡
¢
δ̃ ∈ 0, δ small enough to ensure that U aR (0, t) , 0 > U aR (0) , 0 for all (t, ε) ∈ B t01 ; δ̃, ε̃ ; this can
be done because aS (0) > aR (0, t) > aR (0) for all t small enough. I now consider two exhaustive cases.
´
³
¢
¡
Case 1: K (t; ε) = N − 1 for all t, ε ∈ B t01 ; δ̃, ε̃ . Then, for all such (t, ε) close enough to t01 , 0 ,
¡
¢
¡
¢
U aR (τN −1 (t; ε) , 1) , τN −1 (t; ε) − U aR (τN −2 (t; ε) , τN −1 (t; ε)) , τN −1 (t; ε) > ε.
¢
¡
By¡ continuity,
the left hand side above converges to 0 as (t, ε) → t01 , 0 , hence for (t, ε) close
¢
enough to t01 , 0 ,
¡
¢
¡
¢
U aR (τN −1 (t; ε) , 1) , τN −1 (t; ε) − U aR (τN −2 (t; ε) , τN −1 (t; ε)) , τN −1 (t; ε) ∈ (0, ε) .
For any such (t, ε), the partition τ (t; ε) satisfies the requisite properties.
³
´
Case 2: K (t; εt ) = N for some t, εt ∈ B t01 ; δ̃, ε̃ . If K (t; ·) has a discontinuity at any ε < εt ,
then τ (t, ε)
³ satisfies
´ the requisite properties; so suppose that K (t; ε) = N for all ε ∈ (0, ¡εt ]. ¢ Pick any
¡ ¢
0
t̂, ε̂ ∈ B t1 ; δ̃, ε̃ such that K(t̂; 0) = N − 1 and τN −1 (t̂; ε) < 1 for all ε ∈ (0, ε̂); such a t̂, ε̂ exists by
Assumption 1 and the continuity of
£ solutions
© ª to (A-2)
© in
ª¤ initial conditions. For any ε < min {εt , ε̂}, K (·; ε)
has a discontinuity at some tε ∈ min t, t̂ , max t, t̂ . For any ε < min {εt , ε̂} , τ (tε ; ε) is a partition
satisfying the requisite properties.
Q.E.D.
30
Proof of Theorem 2 on page 16. For any CS outcome satisfying NITS, I will construct a sequence of
monotone outcomes that converge to it in measure. Specifically, let the CS outcome β 0 satisfy NITS with
partition ht00 ≡ 0, t01 , . . . , t0N ≡ 1i. Say that a monotone equilibrium (ρk , αk ) is a monotone partitionalpooling equilibrium (MPPE) if it has a partition htk0 ≡ 0, tk1 , . . . , tJ ≡ 1i and each interval of types (tj , tj+1 )
forms a distinct pool.
R
0
R
Case 1: Assume that NITS is satisfied strictly, so that U (a
¡ (0,
¢ t1 ), 0) > U (a (0), 0). k It ksuffices
to show that for any δ > 0, there exists kδ > 0 such that if k ∈ 0, k , there is a MPPE (ρ , α ) with
partition htk0 ≡ 0, tk1 , . . . , tN ≡ 1i such that ∀j = 1, . . . , J − 1, |tkj − t0j | < δ.
¡ R¡ 0
¢ ¢
¡ R
¢
Without loss of generality,
assume
that
δ
is
small
enough
such
that
U
a
0,
t
+
δ
,
0
>
U
a
(0)
,
0
.
1
¡ R¡ 0
¢ ¢
¡ R
¢
S
S
Let kδ be the solution to U a 0, t1 + δ , 0 − kC(r (1) , 0) = U a (0), 0 − kC(r (0) , 0). Plainly,
kδ > 0. For any k < kδ , I construct a MPPE with partition ht0 ≡ 0, t1 , . . . , tN ≡ 1i that satisfies the
desired properties. For such an N -step partition, let εj ≡ U (aR (tj , tj+1 ), tj ) − U (aR (tj−1 , tj ), tj ) for any
j = 1, . . . , N − 1. First I will define the reports that will be used on the equilibrium path, starting from the
upper end of the type space and proceeding inductively. Initialize rN = 1. To define rj for j = 1, . . . , N −1,
assume inductively that rj+1 > rS (tj ) has been defined, and then define rj ∈ (rS (tj ), rj+1 ) as the solution
to
U (aR (tj−1 , tj ), tj ) − kC(rj , tj ) = U (aR (tj , tj+1 ), tj ) − kC(rj+1 , tj ).
(A-3)
ε
Since (A-3) is equivalent to C(rj+1 , tj ) − C(rj , tj ) = kj , (A-3) has a unique solution rj ∈
(r (tj ), rj+1 ) as long as εj is strictly positive and small enough. Therefore, Lemma A.1 implies that
for any k > 0, there is a partition ht0 ≡ 0, t1 , . . . , tN ≡ 1i and a strictly increasing sequence of reports
hr1 , . . . , rN = rS (1)i such that for each j = 1, . . . , N − 1, |tj − t0j | ∈ (0, δ), rj ∈ (rS (tj ), rj+1 ), and (A-3)
holds. To complete the description of equilibrium, set ρ(0) = r1 and for each j = 1, . . . , N , set every
t ∈ (tj−1 , tj ] to play ρ(t) = rj . For all on-the-path reports rj , let α(rj ) = aR (tj−1 , tj ). For off-the-path
reports r < r1 , set α(r) = aR (0), for off-path r > rN set α(r) = aR (tN −1 , tN ), and for any other off-path
report r ∈ (rj , rj+1 ) with j = 1, . . . , N − 1, set α(r) = aR (tj−1 , tj ). The appropriate belief specification,
consistent with Bayes rule, are easy to construct, hence suppressed.
S
It remains to prove that this construction is a monotone equilibrium. It is obvious that message
and belief monotonicity hold, and that the Receiver is playing optimally. To check Sender optimality,
make the following three observations:
1. there are no off-the-equilibrium path reports at the top, since rN = 1.
2. for any j = 1, . . . , N − 1, type tj is indifferent between sending reports rj and rj+1 , and moreover,
strictly prefers sending rj to any r ∈ (rj , rj+1 ) because any such r costs more (rj > rS (tj )) and
elicits the same action as rj .
3. since k < kδ and t1 < t01 + δ, the lowest type strictly prefers sending r1 and eliciting aR (0, t1 ) to
sending any report r < r1 and eliciting aR (0).
These three facts are sufficient to prove optimality for the Sender, following the arguments for
which types have the “biggest incentive to deviate to any unused report” used in the characterization of
mD1 equilibria, Lemmas B.2 and B.3 in Appendix B. Specifically, it is clear from the construction that
there are no profitable deviations to on-path reports. Lemma B.2 implies that if any type has an incentive
to deviate to an unused report r ∈ (rj , rj+1 ) for some j = 1, . . . , N − 1, then type tj also has the incentive;
but as noted in point 2 above, type tj does not have such an incentive. Lemma B.3 implies that if any
type has an incentive to deviate to an unused report r < r1 , then type 0 also has the incentive; but as
noted in point 3 above, type 0 does not have such an incentive.
R
0
Case 2: Now suppose that the CS outcome satisfies NITS with equality, so that U (a
¡ (0,
¢ t1 ), 0) =
U (a (0), 0). It suffices to show that for any δ > 0, there exists kδ > 0 such that if k ∈ 0, k , there is
R
31
a monotone partitional-pooling equilibrium (ρk , αk ) with partition htk0 ≡ 0, tk1 , . . . , tN +1 ≡¡ 1i such ∀j¢ =
1, . . . , N , |tkj − t0j−1 | < δ. For any k > 0, let δk be the smaller of δ and the solution to U aR (0, ·) , 0 −
¡
¢
kC(rS (1) , 0) = U aR (0), 0 − kC(rS (0) , 0); that NITS is satisfied with equality ensures that δk > 0 is
well-defined. For any k > 0, I construct a MPPE with partition ht̃0 ≡ 0, t̃1 , . . . , t̃N +1 ≡ 1i that satisfies
the desired properties.
Consider the partition ht̃00 ≡ 0, t̃01 = 0, . . . , t̃0N +1 ≡ 1i defined by t̃0j+1 = t0j for all j = 1, . . . , N − 1;
this partition is essentially equivalent to ht0 ≡ 0, t1 , . . . , tN ≡ 1i in the sense that all types except 0 elicit
the same action from the Receiver. Following the same logic as the case where NITS is satisfied strictly,
for any k > 0, Lemma A.1 delivers a strictly increasing partition ht̃0 ≡ 0, t̃1 , . . . , t̃N +1 ≡ 1i and a strictly
increasing sequence of reports hr̃1 , . . . , r̃N +1 = rS (1)i such that for each j = 1, . . . , N , |t̃j − t̃0j | ∈ (0, δk ),
r̃j ∈ (rS (tj ), r̃j+1 ), and (A-3) holds. Given these objects, monotone strategies are described analogous to
earlier. That this construction is an equilibrium is verified as before, with the only change that step 3
(the lowest type strictly prefers sending r1 and eliciting aR (0, t1 ) to sending any report r < r1 and eliciting
aR (0)) follows from the fact that t̃1 < δk .
Q.E.D.
Appendix B: Steps to Characterizing mD1 Equilibria
This Appendix lays out a series of lemmata that are needed to prove the results in Section 4.
Let σ be shorthand for the tuple (ρ, µ); i.e. σ(t) ≡ (ρ(t), µ(t)). SLet T (r, m) ≡ {t : σ(t) = (r, m)}.
(As usual, the dependence on σ is left implicit.) Moreover, let T (r) ≡ m T (r, m). Recall from the text
that tl (r) ≡ inf{t : ρ(t) = r} and th (r) ≡ sup{t : ρ(t) = r}.
I start with two simple observations about any mD1 equilibrium, (σ, α). Fix any r. First, the
set of types using r (i.e. T (r)) must be convex (this follows from report monotonicity); second, there must
be a finite partition of the set of types using r into connected non-degenerate intervals,26 such that all the
types in an element of the partition must use the same cheap talk message. This latter point is simply
the logic of CS holding the report, r, fixed. An implication that will be extensively used is that for any
(r, m), T (r, m) must be convex, and moreover |T (r, m)| ∈ {0, 1, ∞}. Note that there is pooling on some
r if and only if |T (r)| > 1; r is unused if and only if |T (r)| = 0; and a type t is separating if and only if
|T (σ(t))| = 1.
Lemma B.1. In any equilibrium, (σ, α), if there pooling on some rp < 1, then there exists θ(rp ) > 0 such
any r ∈ (rp , bp + θ(rp )) is unused.
Proof. Suppose there is pooling on rp < 1. As shorthand, write th instead of th (rp ). If ρ (th ) > rp , then
by report monotonicity, we are done, since reports in (rp , ρ (th )) are unused. So assume that ρ (th ) = rp .
If th = 1, then we are done, since reports r ∈ (rp , 1) are unused. So assume th < 1. Let mh ≡ µ (th ).
Claim: |T (rp , mh ) | > 1.
Proof: If not, type th is separating. But then, for small enough ε > 0, some type th − ε would
prefer to mimic type th , contradicting equilibrium. ¦
It follows that α (rp , mh ) < aR (th ). Let ρ0 ≡ limt↓th ρ (t).
tonicity, though it may not be played in equilibrium.)
(ρ0 is well-defined by report mono-
26
It should be noted that the lowest type using r may be just indifferent between separating itself and pooling
with the first non-degenerate interval of the partition. I assume without any essential loss of generality that in
such a case, this type does not reveal itself.
32
Claim: ρ0 > rp .
Proof: Suppose not. By report monotonicity, it must be that ρ0 = rp = ρ (th ). Note that ρ is
then continuous at th . Since ρ (t) > rp for all t > th , it follows that ρ is strictly increasing on (th , th + δ)
for some δ > 0. Hence, defining aε ≡ α (σ (th + ε)), we have aε = aR (th + ε) for small enough ε > 0.
By picking ε > 0 small enough, we can make C (ρ (th + ε) , th ) − C (rp , th ) arbitrarily close to 0, whereas
U (aε , th ) − U (α (rp , mh ) , th ) is positive and bounded away from 0, because α (rp , mh ) < aR (th ) < aε <
aS (th ). Therefore, for small enough ε > 0, th prefers to imitate th + ε, contradicting equilibrium. ¦
This completes the argument because reports in (rp , ρ0 ) are unused.
Q.E.D.
Lemma B.2. In any mD1 equilibrium, (σ, α), if |T (r)| = 0 for some r > ρ(0), then for any m, α(r, m) =
aR (sup{t : ρ(t) ≤ r}).
Proof. Fix a r̂ > ρ(0) such that |T (r̂)| = 0. Let t̂ ≡ sup{t : ρ(t) ≤ r̂}. Also, define at ≡ α(σ(t)). There
are two distinct cases: either t̂ < 1, or t̂ = 1.
Case 1: t̂ < 1
Let r+ ≡ inf t>t̂ ρ (t), r− ≡ supt<t̂ ρ (t), a+ ≡ ξh (r̂), and a− ≡ ξl (r̂).
Claim: The inequalities below, (B-1) and (B-2), hold for all t, with equality for t = t̂:
¡
¢
¡
¢
U (at , t) − kC (ρ (t) , t) ≥ U a− , t − kC r− , t ;
¡
¢
¡
¢
U (at , t) − kC (ρ (t) , t) ≥ U a+ , t − kC ρ+ , t .
(B-1)
(B-2)
Proof: I prove it for (B-1); it is analogous for (B-2). Suppose first that ρ(t̂) > r̂. Then
a− = limt↑t̂ at and (at , ρ(t)) ↑ (a− , r− ) as t ↑ t̂. (In fact, if |T (r− )| > 0, then for small enough ε >
0, at̂−ε = a− and ρ(t̂ − ε) = r− .) Continuity of U and C imply that any t for which (B-1) does
not hold has a profitable deviation to σ(t̂ − ε) for small enough ε > 0. For type t̂, suppose towards
contradiction
that¢ (B-1) ¡holds
Continuity
of¢ U and
¡
¡ ¢ strictly.
¢
¡
¡ C then imply
¢ that for all sufficiently small
ε > 0, U at̂ , t̂ − ε − kC ρ t̂ , t̂ − ε > U at̂−ε , t̂ − ε − kC ρ(t̂ − ε), t̂ − ε , which contradicts optimality
of σ(t̂ − ε).
Now suppose that ρ(t̂) < r̂. Then a− = at̂ and r− = ρ(t̂), and it is trivial that (B-1) holds with
equality for t̂. Moreover, any t for which (B-1) does not hold has a profitable deviation to σ(t̂). ¦
¡¢
Claim: For all m, α (r̂, m) = aR t̂ .
Proof: It suffices to show that A(r̂, t̂) 6= ∅ and for all t 6= t̂, A(r̂, t) ⊆ A(r̂, 1). Consider the first
item. By the previous Claim,
¡
¢
¡ ¡¢ ¢
¡
¢
¡
¢
¡
¢
¡
¢
U at̂ , t̂ − kC ρ t̂ , t̂ = U a− , t̂ − kC r− , t̂ = U a+ , t̂ − kC ρ+ , t̂ .
Hence, if r̂ ≥ rS (t̂) then a+ ∈ A(r̂, t̂); if r̂ < rS (t̂) then a− ∈ A(r̂, t̂). In either case, A(r̂, t̂) 6= ∅.
Now turn to the second item. I must show that ∀a ∈ [a− , a+ ] and ∀t 6= t̂,
U (a, t) − kC (r̂, t) ≥
⇓
¡ ¢
¡ ¢
U a, t̂ − kC r̂, t̂ >
U (at , t) − kC (ρ (t) , t)
¢
¡ ¡¢ ¢
¡
U at̂ , t̂ − kC ρ t̂ , t̂ .
33
I provide the argument for t < t̂; it is analogous for t > th (using (B-2)) instead of (B-1)). Since
(B-1) holds for all t, and with equality for t̂, it suffices to show that
¡
¢
¡
¢
U (a, t) − kC (r̂, t) ≥ U a− , t − kC r− , t
⇓
¡ ¢
¡ ¢
¡
¢
¡
¢
U a, t̂ − kC r̂, t̂ > U a− , t̂ − kC r− , t̂ .
This is true if
¡ ¢
¡ ¢ £ ¡
¢
¡
¢¤
£ ¡
¢
¡
¢¤
U a, t̂ − kC r̂, t̂ − U a− , t̂ − kC r− , t̂ > U (a, t) − kC (r̂, t) − U a− , t − kC r− , t ,
which can be be rewritten as
Z a Z t̂
Z
U12 (y, z) dzdy > k
a−
r̂
Z
C12 (y, z) dzdy,
r−
t
t̂
t
which is true because U12 > 0 > C12 . ¦
Case 2: t̂ = 1
It needs to be shown that for all m, α(r̂, m) = aR (1). If a1 = aR (1), this is a trivial consequence
of belief monotonicity, so assume henceforth that a1 < aR (1). If ρ (1) > r̂, then the same proof as in Case
1 works, except that one now defines r+ ≡ ρ (1). So consider ρ (1) < r̂. Then a− = a1 and a+ = aR (1).
Claim: For all t < 1, A(r̂, t) ⊆ A(r̂, 1).
£
¤
Proof: I must show that ∀a ∈ a1 , aR (1) , ∀t < 1,
U (a, t) − kC (r̂, t) ≥ U (at , t) − kC (ρ (t) , t)
⇓
U (a, 1) − kC (r̂, 1) > U (a1 , 1) − kC (ρ(1), 1) .
Since optimality of σ(t) implies U (at , t) − kC (ρ (t) , t) ≥ U (a1 , t) − kC (ρ(1), t), it suffices to show
that for all t < 1,
U (a, 1) − kC (r̂, 1) − [U (a1 , 1) − kC (ρ(1), 1)] > U (a, t) − kC (r̂, t) − [U (a1 , t) − kC (ρ(1), t)] ,
which can be rewritten as
Z aZ 1
Z
r̂
Z
U12 (y, z) dzdy > k
a1
t
1
C12 (y, z) dzdy.
ρ(1)
t
This inequality holds because C12 < 0 < U12 . ¦
Now observe that if r̂ is sufficiently close to ρ(1), then aR (1) ∈ A(r̂, 1), and we are done because
A(r̂, 1) 6= ∅. But then, belief monotonicity implies the result even if r̂ is larger.
Q.E.D.
£ S
¢
Lemma B.3. In any mD1 equilibrium, (σ, α), for all (r, m) such that r ∈ r (0), ρ(0) , α (r, m) = aR (0).
Proof. To avoid trivialities, assume rS (0) < ρ(0). Pick any r̂ ∈ [rS (0), ρ (0)) and any m. For any t, let
at ≡ α(σ(t)). Then ξl (r̂) = aR (0) and ξh (r̂) = a0 .
34
Claim: A(r̂, 0) 6= ∅.
Proof: Since r̂ > rS (0), a0 ∈ A(r̂, 0). ¦
Claim: For all t > 0, A(r̂, t) ⊆ A(r̂, 0).
£
¤
Proof: It suffices to show that ∀a ∈ aR (0) , a0 , ∀t > 0,
U (a, t) − kC (r̂, t) ≥ U (at , t) − kC (ρ (t) , t)
⇓
U (a, 0) − kC (r̂, 0) > U (a0 , 0) − kC (ρ (0) , 0) .
That (σ, α) is an equilibrium implies
U (at , t) − kC (ρ (t) , t) ≥ U (a0 , t) − kC (ρ (0) , t) ,
and hence it suffices to show that
U (a, 0) − kC (r̂, 0) − [U (a0 , 0) − kC (ρ (0) , 0)] > U (a, t) − kC (r̂, t) − [U (a0 , t) − kC (ρ (0) , t)] .
This inequality can be rewritten as
Z
a0
Z
Z
t
ρ(0)
Z
U12 (y, z) dzdy > k
a
0
t
C12 (y, z) dzdy,
r̂
0
which holds because C12 < 0 < U12 . ¦
Q.E.D.
Lemma B.4. In any mD1 equilibrium, (σ, α),
(i) there is a cutoff type t ∈ [0, 1] such that all types t < t are separating, and all types t > t are pooling
with ρ(t) = 1.
(ii) there is a finite partition of [t, 1], ht0 ≡ t, t1 , ..., tJ ≡ 1i, that is a forward solution to (A);
(iii) ∀j = 1, . . . , J, ∀t ∈ (tj−1 , tj ), µ(t) = mj (mj 6= mn ∀n 6= j) and α(1, mj ) = aR (tj−1 , tj ).
Proof. Given part (i), parts (ii) and (iii) are straightforward applications of the CS arguments, hence
omitted. For part (i), suppose to the contrary that is some rp < 1 with |T (rp )| > 1. Let â ≡
limε→0 α(σ(th (rp ) − ε)).
Note that â < aR (th (rp )).
By Lemma B.1, ∃θ > 0 such that for any
r ∈ (rp , rp + θ), |T (r)| = 0. By Lemma B.2, for any r ∈ (rp , rp + θ) and any m, α(r, m) = aR (th (rp )). It
follows that for small enough ε > 0 and δ > 0, a type th (rp ) − ε strictly prefers to report rp + δ and induce
aR (th (rp )) rather than play σ(th (rp ) − ε) and induce â, a contradiction.
Q.E.D.
Lemma B.5. In any mD1 equilibrium, (σ, α) with cutoff t = 0, ρ (0) ∈ {rS (0), 1}.
Proof. Assume t = 0 and ρ(0) 6= 1. Note that type 0 is thus separating. I will argue that ρ(0) = rS (0)
via two Claims.
Claim: ρ(0) ≤ rS (0).
Proof: If ρ(0) > rS (0), then by Lemma B.3, for any m, α(rS (0), m) = aR (0) = α(σ(0)). Since
C(r (0), 0) < C(ρ(0), 0), type 0 has a profitable deviation, a contradiction. ¦
S
35
Claim: ρ(0) ≥ rS (0).
Proof: Suppose ρ(0) < rS (0). Let â ≡ limε↓0 α(σ(ε)) (this is the lowest action elicited by
the types just above 0). It is straightforward that U (â, 0) − kC(1, 0) = U (aR (0), 0) − kC(ρ(0), 0), because otherwise, a small enough type ε > 0 would have a profitable deviation to ρ(0). Pick an arbitrary m. Observe that by belief monotonicity, α(rS (0), m) ∈ [aR (0), â]. Thus, U (α(rS (0), m), 0) ≥
min{U (aR (0), 0), U (â, 0)}. Since rS (0) ∈ (ρ(0), 1) implies C(rS (0), 0) < min{C(ρ(0), 0), C(1, 0)}, it follows that U (α(rS (0), m))−kC(rS (0), 0)) > U (â, 0)−kC(1, 0). Therefore, type 0 has a profitable deviation,
a contradiction. ¦
Q.E.D.
Lemma B.6. In any mD1 equilibrium, (σ, α) with cutoff t, ρ (t) > rS (t) for all t ∈ (0, 1).
Proof. The result is trivial for all t ∈ (t, 1), since ρ(t) = 1 > rS (t) for all types. So it suffices to assume
that t > 0 and prove the Lemma for t ∈ (0, t]. The proof is via three Claims.
Claim: If t̂ > 0 is such that all types in [0, t̂] are separating, then ρ(t̂) 6= rS (t̂).
Proof: Suppose not, i.e. suppose that there exists a t̂ > 0 such that all types in [0, t̂] are separating
S
and yet¡ ρ(t̂) =
¢ r (t̂).
¡ ¢ For small ε ≥ 0, define g (ε) as the expected utility gain for a type t̂ − ε by deviating
from ρ t̂ − ε to ρ t̂ . Since we are on a separating portion of the type space,
£ ¡ ¡¢
¢
¡
¢
¢
¡ ¡
¢
¢¤
¢¤ £ ¡ ¡
g (ε) ≡ U aR t̂ , t̂ − ε − kC rS (t̂), t̂ − ε − U aR t̂ − ε , t̂ − ε − kC ρ t̂ − ε , t̂ − ε .
¡ ¡
¢
¢
¡
¢
Since C ρ t̂ − ε , t̂ − ε ≥ C rS (t̂ − ε), t̂ − ε ,
£ ¡ ¡¢
¢
¡
¢¤ £ ¡ ¡
¢
¢
¡
¢ ¤
g (ε) ≥ φ (ε) ≡ U aR t̂ , t̂ − ε − kC rS (t̂), t̂ − ε − U aR t̂ − ε , t̂ − ε − kC rS (t̂ − ε), t̂ − ε .
Differentiating yields
¡ ¡¢
¢
¡
¢
¡
¢
φ0 (ε) = −U2 aR t̂ , t̂ − ε + kC2 rS (t̂), t̂ − ε − kC1 rS (t̂ − ε), t̂ − ε r1S (t̂ − ε)
¡
¢
¡ ¡
¢
¢
¡
¢
¡ R¡
¢
¢
−kC2 rS (t̂ − ε), t̂ − ε + U1 aR t̂ − ε , t̂ − ε aR
t̂ − ε , t̂ − ε .
1 t̂ − ε + U2 a
¡¢
¡
¢
¡ ¡¢ ¢
Since
Clearly, φ0 is continuous, and since C1 rS (t̂), t̂ = 0, φ0 (0) = U1 aR t̂ , t̂ aR
1 t̂ > 0.
φ (0) = 0, for sufficiently small ε > 0, g (ε) ≥ φ (ε) > 0, implying that a type t̂ − ε strictly prefers to imitate
t̂, contradicting equilibrium separation of t̂. ¦
Claim: ρ(t) > rS (t) for all t ∈ (0, t).
Proof: Suppose the result is not true. By the first Claim, it must be that there exists a t0 ∈ (0, t)
such that ρ (t0 ) < rS (t0 ). By report monotonicity, ρ (t0 − ε) < ρ (t0 ) for all ε > 0. It follows that for small
enough ε > 0, C (ρ (t0 − ε) , t0 − ε) > C (ρ (t0 ) , t0 −¡ε). On the other
hand,
since we are on
¢
¡
¢ the separating
part of the type space, for small enough ε > 0, U aR (t0 ) , t0 − ε > U aR (t0 − ε) , t0 − ε . Therefore, for
small enough ε > 0, a type t0 − ε strictly prefers to imitate t0 , contradicting equilibrium separation of t0 . ¦
Claim: ρ(t) > rS (t).
Proof: If ρ(t) = 1, the result is true because rS (t) < 1 (since t < 1). So suppose ρ(t) < 1. In
this case, t is separating. By the first Claim, ρ(t) 6= rS (t). Report monotonicity, continuity of rS , and
ρ (t) > rS (t) for all t ∈ (0, t) then imply that ρ (t) > rS (t). ¦
Q.E.D.
Lemma B.7. In any mD1 equilibrium, (σ, α), with cutoff t,
(i) ρ is continuous at all t 6= t;
36
(ii) if t > 0, then ρ is either right- or left-continuous at t;
Proof. (i) Trivially, ρ is continuous above t, since ρ(t) = 1 for all t > t. Suppose towards a contradiction
that there is a discontinuity at some t0 < t. First assume ρ (t0 ) < limt↓t0 ρ (t) ≡ ρ. By the continuity of C
and the monotonicity of ρ, as ε ↓ 0,
C (ρ (t0 + ε) , t0 + ε) − C (ρ (t0 ) , t0 + ε)
→
C (ρ, t0 ) − C (ρ (t0 ) , t0 ) > 0,
0
S 0
where the inequality above follows
) (the weak
¡ R 0from ρ 0> ρ (t
¢ ) ≥ r¡ (t
¢ inequality here coming from
Lemma B.6). On the other hand, U a (t + ε) , t + ε − U aR (t0 ) , t0 + ε → 0; hence, for small enough
ε > 0, t0 + ε prefers to imitate t0 , contradicting equilibrium separation.
The argument for the other case where ρ (t0 ) > limt↑t0 ρ (t) is similar, establishing that t0 prefers
to imitate t0 − ε for small enough ε > 0.
(ii) Suppose not. If t = 1, t is separating because all types below t are separating. If t < 1, t is
separating because ρ is not right-continuous at t. Since ρ is not left-continuous at t, report monotonicity
implies ρ (t) > limt↑t ρ (t) ≡ ρ. Since for all t ∈ (0, t), ρ(t) is continuous (by part (i) of this Lemma) and
ρ(t) > rS (t) (by Lemma B.6), we have ρ ≥ rS (t). Since t is separating, it must be that for all ε > 0,
¡
¢
¡
¢
U aR (t − ε) , t − kC (ρ (t − ε) , t) ≤ U aR (t) , t − kC (ρ (t) , t) .
(B-3)
¢
¡ ¢
¡
Since limε↓0 ρ (t − ε) = ρ, the left hand side of (B-3) is converging to U aR (t) , t − kC ρ, t . So (B-3)
¡ ¢
¡ ¢
can hold for all ε > 0 only if C ρ, t ≥ C (ρ (t) , t). But ρ (t) > ρ ≥ t implies C (ρ (t) , t) > C ρ, t , a
contradiction.
Q.E.D.
Lemma B.8. In any mD1 equilibrium, (σ, α) with cutoff t, let r1 ≡ limt↑t ρ (t) and m1 ≡ limt↓t µ (t).27
Then,
(i) if t ∈ (0, 1), U (aR (t), t) − kC(r1 , t) = U (α(1, m1 ), t) − kC(1, t);
(ii) if t = 0, U (aR (0), 0) − kC(rS (0), 0) ≤ U (α(1, m1 ), 0) − kC(1, 0).
Proof. (i) Assume t ∈ (0, 1). Fist consider r1 = 1. Then type t plays (1, m1 ) and elicits α(1, m1 ). Since
all types below t are separating, equilibrium requires that for all ε > 0,
U (aR (t − ε)), t) − kC(ρ(t − ε), t) ≤ U (α(1, m1 ), t) − kC(1, t).
The Lemma follows from the above inequality, the continuity of U , C, aR , and r1 = 1 = limt↑t ρ (t).
So assume r1 < 1. By Lemma B.7, either ρ (t) = r1 or ρ (t) = 1. Suppose first ρ (t) = r1 , in
which case t is separating. Define for ε ≥ 0,
¡
¢
W (ε) ≡ U aR (t) , t + ε − kC (r1 , t + ε) − [U (α (1, m1 ) , t + ε) − kC (1, t + ε)] .
If the Lemma does not hold, W (0) > 0 (the reverse inequality is inconsistent with optimality of
σ(t)). But then, by continuity of W , for small enough ε > 0, a type t + ε would prefer to imitate t by
playing σ(t) rather than pool on (1, m1 ), contradicting equilibrium.
It remains to consider ρ (t) = 1, in which case t is pooling. Note that in this case µ (t) = m1 .
Lemma B.2 implies that for all m, α(r1 , m) = aR (t). Thus, if the Lemma does not hold, optimality of
27
These are well-defined respectively when t > 0 and t < 1.
37
¡
¢
σ(t) = (1, m1 ) implies that U (α (1, m1 ) , t) − kC (1, t) > U aR (t) , t − kC (r1 , t). But then, by continuity
of U , C, and aR , and the fact that r1 = limt↑t ρ (t), we have that that for small enough ε > 0,
¡
¢
U (α (1, m1 ) , t − ε) − kC (1, t − ε) > U aR (t − ε) , t − ε − kC (ρ (t − ε) , t − ε) ,
implying that a type t−ε prefers pooling on (1, m1 ) rather than separating, contradicting optimality
of its equilibrium play.
(ii) Assume t = 0. Suppose first ρ(0) = 1. Then α(σ(0)) = α(1, m1 ), and by Lemma B.3, for
any m, α(rS (0), m) = aR (0). The desired result now follows from from optimality of σ(0). So consider
ρ(0) < 1. By Lemma B.5, type 0 is separating with ρ(0) = rS (0). I claim that type 0 must be indifferent
between (1, m1 ) and σ(0), which is sufficient to prove the Lemma. If this were not true, then optimality
of σ(0) implies U (aR (0), 0) − kC(ρ(0), 0) > U (α(1, m1 ), 0) − kC(1, 0). By continuity of U and C, for small
enough ε > 0, a type ε has a profitable deviation to σ(0), contradicting equilibrium.
Q.E.D.
Lemma B.9. In any mD1 equilibrium, (σ, α), with cutoff t, ρ(t) = ρ∗ (t) for all t < t.
Proof. The Lemma is vacuously true if t = 0, so assume that t > 0. Note that since there is separation
below t, ρ must be strictly increasing on [0, t).
Claim: ρ(0) = rS (0).
Proof: If ρ(0) > rS (0), then by Lemma B.3, type 0 has a profitable deviation to rS (0). If
ρ(0) < rS (0), then by Lemma B.7, there exists an ε ∈ (0, t) such that ρ(ε) ∈ (ρ(0), rS (0)) and U (aR (ε), 0) >
U (aR (0), 0); hence type 0 has a profitable deviation to ρ(ε). ¦
Claim: ρ is differentiable on (0, t).
Proof: By Lemma B.7 and Lemma B.6, for all t ∈ (0, t), ρ(t) > rS (t) and ρ is continuous at
t. Given these two facts, differentiability follows from the argument of Mailath (1987, Proposition 2 in
Appendix). ¦
Since ρ is differentiable on (0, t), it must satisfy the first order condition for optimality and thus
solve (DE) there. Lemma B.7 implies that ρ must be also be continuous on [0, t). The proof is completed
by noting that as proved in Lemma 5, ρ∗ is the unique strictly increasing function that satisfies these
properties.
Q.E.D.
£ S ¢
Lemma B.10. In any mD1 equilibrium, (σ, α), with cutoff t, ∀m and ∀r ∈ 0, r (0) , α(r, m) = aR (0).
Proof. It suffices to prove that there exists an m such that α(rS (0), m) = aR (0), since the result then
follows from belief monotonicity. Suppose first that t > 0. Then type 0 is separating with ρ(0) = rS (0)
(by Lemma B.9), hence α(rS (0), µ(0)) = aR (0). So now suppose t = 0. By Lemma B.5, ρ(0) ∈ {rS (0), 1}.
If ρ(0) = rS (0), α(rS (0), µ(0)) = aR (0). If ρ(0) = 1, then by Lemma B.3, for all m, α(rS (0), m) =
aR (0).
Q.E.D.
Appendix C: Proofs for Section 4
Proof of Lemma 4 on page 19. Part (i) is proved as Lemma B.1; part (ii) is a special case of Lemma
B.2; and part (iii) follows from Lemma B.5 and that separation requires ρ(0) < 1.
Q.E.D.
38
Proof of Lemma 5 on page 22. As noted in the text, standard results do not apply because of the
lack of Lipschitz condition on the relevant domain. Thus, I proceed as follows, similar to Mailath (1987).
Step 1: Local existence. Consider the inverse initial value problem to find τ (r) such that:
τ 0 = g(r, τ ) ≡
kC1 (r, τ )
,
U1 (aR (τ ), τ )aR
1 (τ )
τ (rS (0)) = 0.
(C-1)
By the assumptions on C and U , g is continuous and Lipschitz on [0, 1] × [0, 1]. Hence, standard existence
theorems (e.g. Coddington and Levinson, 1955, Theorem 2.3, p. 10) imply that there is a unique location
solution, τ̃ , to (C-1) on [rS (0), rS (0) + δ), for some δ > 0; τ̃ ∈ C 1 ([rS (0), rS (0) + δ)).28 Note that τ 0 (r) > 0
S −1
if and only if C1 (r, τ (r)) > 0, or equivalently, r > rS (τ (r)). Since g(rS (0), 0) = 0 < dr dr (0) , δ can be
chosen small enough such that for all r ∈ (rS (0), rS (0) + δ), r > rS (τ̃ (r)) and thus τ̃ 0 (r) > 0. Defining
ρ̃ ≡ τ̃ −1 gives a solution to (DE) on [0, t̃), for some t̃ ∈ (0, t̃); ρ̃ ∈ C 1 ([0, t̃)) and ρ̃0 > 0. Since the inverse
of any increasing (local) solution to (DE) is a (local) solution to (C-1) on [rS (0), rS (0) + θ) for some θ > 0,
(local) uniqueness of an increasing solution to (DE) follows from the fact that τ̃ is unique (local) solution
to (C-1) above rS (0).
Step£2: ¤The unique extension. To prove that there is a unique t and a unique extension of ρ̃
from [0, t̃) to 0, t such that either t = 1 or ρ̃(t) = 1, it is sufficient to prove the following inductive step:
given the solution ρ̃ ∈ C 1 ([0, δ)) with ρ̃0 > 0, if limt↑δ ρ̃ (t) < 1, then there is a unique extension of ρ̃ to
[0, δ + θ) for some θ > 0, while maintaining ρ̃0 > 0 and ρ̃ ∈ C 1 ([0, δ + θ)). To see that this is sufficient,
observe that there must be some t ≤ 1 that is the supremum over all t such that ρ̃ can be extended to
[0,
£ t) ⊆ [0,
¢ 1]. If t = 1, we are done by setting ρ̃(t) = limt↑1 ρ̃(t) ≤ 1. If t < 1 then ρ̃ cannot be extended to
0, t + ν for any ν > 0, which by the inductive step implies that limt↑t ρ (t) = 1. We are done by setting
ρ̃(t) = limt↑t ρ̃(t) = 1. In either case, ρ̃ ∈ C 1 ([0, t]).
It remains to prove the inductive step. Suppose ρ̃ is a solution to (DE) on [0, δ), with ρ̃ ∈ C 1 ([0, δ))
and ρ̃ > 0. Let rδ ≡ limt↑δ ρ̃ (t).
0
Claim: rδ > rS (δ).
Proof: Suppose not, towards contradiction. Then rδ = rS (δ) (because ρ̃(t) > rS (t) for all
t ∈ (0, δ)) and limt↑δ ρ̃0 (t) = ∞. Let b ≡ max r1S (t). By the assumptions on C(·, ·), b < ∞. Since
ρ̃ ∈ C 1 ([0, δ)), there exists t̂ < δ such that ρ̃0 (t) > b for all t ∈ [t̂, δ). Pick ε > 0 such that ρ̃(t̂) > rS (t̂) + ε.
We have
Z t
rδ = ρ̃(t̂) + lim
ρ̃0 (y)dy
t↑δ
t̂
Z
S
δ
> r (t̂) + ε +
ρ̃0 (y)dy
t̂
Z
> rS (t̂) + ε +
t̂
δ
r1S (y)dy
= rS (δ) + ε,
which contradicts rδ = rS (δ). ¦
U (aR (t),t)aR (t)
Given the Claim, if rδ < 1, 1 kC1 (r,t)1
is continuous, Lipschitz, and bounded in a neighborhood
of (δ, rδ ). Standard extension theorems (e.g. Coddington and Levinson, 1955, Theorem 4.1 and preceeding
dicussion, p. 15) imply that if rδ < 1, there is a unique extension of ρ̃ to [0, δ + θ) for some θ > 0; this
28 1
C ([a, b)) is the set of all functions on [a, b) that have continuous derivatives at all t ∈ (a, b) and in addition
have a right-hand derivative at a that is continuous from the right at a. The obvious analogue applies to C 1 ([a, b]).
39
extension is in C 1 ([0, δ + θ)). Since ρ̃0 can never hit 0 and solve (DE), ρ̃0 > 0 on [0, δ + θ).
Q.E.D.
Proof of Theorem 3 on page 22.
Necessity. I indicate below which of the intermediate Lemmas in Appendix B prove necessity of
each part of the Theorem. (Note that t ≤ t follows from (a.1) and that ρ∗ is only defined on [0, t].)
(BIN )
(CIN )
(ZW P )
(a.1)
(a.2)
(a.3)
(b)
(c.1)
(c.2)
(c.3)
(c.4)
Lemma B.4
Lemma B.8
Lemma B.8
Lemma B.9
Lemmas B.5 and B.7
Lemma B.4
Lemma B.4
Lemma B.10
Consequence of equilibrium and (a.1),(a.2),(a.3)
Lemmas B.2 and B.3
Consequence of equilibrium and (a.1),(a.2),(a.3),(b)
Sufficiency. Given the Lemmata for necessity, this is straightforward once it is proved that
types t < t are playing optimally. Assume that t > 0. In what follows, let ψ(r) ≡ aR ((ρ∗ )−1 (r)), and to
reduce notation, write ρ∗ as just ρ. It only needs to be shown that for all t < t,
ρ(t) ∈ arg
max
r∈[ρ(0),ρ(t)]
U (ψ(r), t) − kC(r, t).
Suppose to the contrary that for some t̃ < t, r̂ ∈ [ρ(0), ρ(t)] \ {ρ(t̃)} is a maximizer of the above
expression, and let t̂ ≡ ρ−1 (r̂). The first order condition for t̃ and evaluating (DE) at t̂ imply
U1 (ψ(r̂), t̃)ψ 0 (r̂) − kC1 (r̂, t̃) = 0 = U1 (ψ(r̂), t̂)ψ 0 (r̂) − kC1 (r̂, t̂).
But this is a contradiction because U12 > 0 > C12 implies that U1 (ψ(r̂), t)ψ 0 (r̂) − kC1 (r̂, t) is
strictly increasing in t.
Q.E.D.
Proof of Theorem 4 on page 23. If t = 1, there is an mD1 equilibrium where all types separate by
playing ρ∗ (t), so assume that t < 1. The proof is constructive.
Step 0: Preliminaries
Start by defining the function
¡
¢
£ ¡
¢
¤
φ (t) ≡ U aS (t) , t − kC (1, t) − U aR (t) , t − kC (ρ∗ (t) , t) .
φ (t) is the gain for type t from sending the highest report and receiving its ideal action over
separating itself (thus inducing aR (t)) with report ρ∗ (t). Note that in equilibrium, the gain from pooling
over separating can be no more than φ (t), and will generally be strictly less. There are two conceptually
distinct cases: one where φ (t) = 0 for some t ≤ t, and the other where φ (t) > 0 for all t ≤ t. Define
½
0
if φ (t) > 0 for all t ≤ t
t̂ ≡
supt∈[0,t] {t : φ (t) = 0} otherwise.
40
¡¢
¡ ¤
φ is continuous and φ t > 0; hence t̂ < t and for all t ∈ t̂, t , φ(t) > 0. In everything that
£ ¤
follows, we are only concerned with t ∈ t̂, t . So statements such as “for all t” are to be read as “for all
t ∈ [t̂, t]” and so forth unless explicitly specified otherwise.
Step 1: Constructing the necessary sequences.
Initialize pl0 (t) = pr0 (t) = t, and al0 (t) = ar0 (t) = aR (t). Define
£ ¡
¢
¤
∆ (a, t) ≡ U (a, t) − kC (1, t) − U aR (t) , t − kC (ρ∗ (t) , t)
S
Clearly,
∆¢ is continuous
¡ R
¡ Sin both
¢ arguments, £and¤ strictly concave in a with a maximum at a (t).
Since
∆ a (t) , t¤ ≤ 0 ≤ ∆ a (t) , t for all t ∈ t̂, t , it follows that for any such t, in the domain
£ R
a ∈ £a (t) , aS ¢(t) there exists a unique solution to ∆ (a, t) = 0. Call this al1 (t). Similarly, on the domain
a ∈ aS (t) , ∞ , there exists a unique solution to ∆ (a, t) = 0. Call this ar1 (t). By continuity of ∆,
¡¢
¡¢
¡¢
¡¢
¡¢
al1 and ar1 are continuous, al1 t = ar0 t , and ar1 t̂ = al1 t̂ = aS t̂ if t̂ > 0. By the monotonocity of
aR (·, ·), for q ∈ {l, r} and t, there is either no solution or a unique t0 that solves aR (t, t0 ) = aq1 (t). If there
is a solution, call it pq1 (t), otherwise set pq1 (t) = 1. It is straightforward that pl1 and pr1 are continuous
l
l
r
r
r
l
functions,
¡ ¢ p1 (t)
¡≥¢ p0 (t) with equality if and only if t = t, and p1 (t) > p0 (t). Note that p1 (t) ≥ p1 (t),
l
r
and p1 t̂ = p1 t̂ if t̂ > 0.
For j ≥ 2 and q ∈ {l, r} , recursively define pqj (t) as the solution to
¡ ¡
¢
¢
¡ ¡
¢
¢
U aR pqj−1 (t) , pqj (t) , pqj−1 (t) − U aR pqj−2 (t) , pqj−1 (t) , pqj−1 (t) = 0
if a solution exists that is strictly greater than pqj−1 (t), and otherwise set pqj (t) = 1. By the monotonicity
¡
¢
of aR and U11 < 0, pqj (t) is well-defined and unique. Define aqj (t) ≡ aR pqj−1 (t) , pqj (t) . Note that for
all j ≥ 2, pqj (t) > pqj−1 (t) and aqj (t) > aqj−1 (t) if and only if pqj−1 (t) < 1. For all j and q ∈ {l, r}, pqj (t)
¡¢
¡¢
¡¢
¡¢
is continuous, prj (t) ≥ plj (t) for all t, plj t̂ = prj t̂ if t̂ > 0, and plj+1 t = prj t (these follow easily by
induction, given that we noted all these properties for j = 1).
Step 2: The critical segment B.
¡¢
¡¢
I claim there exists B ≥ 1 such that prB−1 t < 1 = prB t . (Obviously, if it exists, it is unique.)
¡
¢
To see this, first note that by definition, pr0 t = t < 1. ¯ Let K = inf{K : prK (t) = 1}.29 It is
¡¢
¡ ¢¯
sufficient to show that ∃ε > 0 such that for any j < K, ¯arj+1 t − arj t ¯ ≥ ε.¯ By construction,
¯
¡¢
¡¢
¡¢
¡¢
¡ ¡ ¢¢
¡ ¡ ¢¢
arj t < aS prj t < arj+1 t and arj t ≤ aR prj t ≤ arj+1 t . Since mint∈[0,1] ¯aS (t) − aR (t)¯ > 0,
we are done.
Step 3: Existence when t̂ > 0.
¡¢
¡¢
¡¢
Consider the functions plB and prB . These are continuous, and plB t = prB−1 t < 1 = prB t .
¡
¢
¡
¢
Moreover, plB t̂ = prB t̂ ; hence either plB (t̂) = 1 or prB (t̂) < 1 . It follows that there is some type t̃ ∈ [t̂, t]
such that either (i) plB (t̃) = 1 and plB (t) < 1 for all t > t̃; or (ii) prB (t̃) = 1 and prB (t) < 1 for all t < t̃. Let
q = l if (i)£ is ¢the case; q = r if (ii) is the case.
£ ¤ By construction, there is an mD1 equilibrium where all
types t ∈ 0, t̃ play ρ∗ (t), and ¡all
types
t
∈
¢
¡ ¢t̃, 1 play ρ (t) = 1, and further segment themselves using the
cheap talk messages into ht̃, pq1 t̃ , . . . , pqB t̃ = 1i.
Step 4: Existence when t̂ = 0.
By the continuity of plB and prB , the logic in Step 3 can fail when t̂ = 0 only if£ plB (0) < 1 =
¤
l
r
l
r
So suppose
this
is
the
case.
Note
that
this
requires
p
(0)
<
p
(0).
For
any
t
∈
p
(0)
,
p
(0)
,
1
1
1
1
¢
U a (0, t) , 0 − kC (1, 0) − [U (ar (0) , 0) − kC (0, 0)] ≥ 0, with strict inequality for interior t. In words,
prB¡(0).
R
29
Recall that the infimum of an empty set is +∞.
41
£
¤
when t ∈ pl1 (0) , pr1 (0) , type 0 weakly prefers (indifference at the endpoints and strict preference for
interior t) inducing aR (0, t) with report 1 over inducing aR (0) with report rS (0). This follows from the
construction of pl1 and pr1 , and U11 < 0. Given any t ∈ [0, 1], define τ0 (t) = 0, τ1 (t) = t, and recursively,
for j ≥ 2, τj (t) as the solution to
¡
¢
¡
¢
U aR (τj−1 (t) , τj (t)) , τj−1 (t) − U aR (τj−2 (t) , τj−1 (t)) , τj−1 (t) = 0
if a solution exists that is strictly greater than τj−1 (t),
= 1. It is straightforward
¡ and¢ otherwise set τj (t)
r
r
that for all j ≥ 0, τj (t) is continuous in t. Since τB pl1 (0) = plB (0)
<
1
=
p
(0)
B ¤ = τB (p1 (0)), it follows
¡ l
that t̃ = mint∈[pl (0),pr (0)] {t : τB (t) = 1} is well-defined and lies in p1 (0) , pr1 (0) . By construction, there
1
1
is an mD1 equilibrium where all types ¡send
¢ the
¡ ¢costly report
¡ ¢ of 1, and segment themselves using cheap
talk messages into the partition h0 = τ0 t̃ , τ1 t̃ , . . . , τB t̃ = 1i.
Finally, the second statement of the Theorem follows from the above by noting that t̂ > 0 for all
k large enough, because φ(0) < 0 for all k large enough.
Q.E.D.
Proof of Theorem 5 on page Page 25. The first part of the Theorem was proved in the text. It remains
to show the second part. Chen, Kartik, and Sobel (2007) show that under Condition M, there is exactly
one CS outcome that satisfies NITS. Denote this CS outcome β 0 , with partition ht00 ≡ 0, t01 , . . . , t0N ≡ 1i,
and assume it does not satisfy NITS strictly. Therefore, there is no CS equilibrium that satisfies NITS
strictly. In what follows, I use superscripts to denote dependence of various objects on k in an obvious
fashion.
In any mD1 equilibrium, the cutoff type must be tk > 0, because if not, types are segmenting just
as in some CS equilibrium, and type 0 would have a strict incentive to deviate to r = rS (0), since no CS
k
equilibrium satisfies NITS strictly. It is easy to verify from (DE) that t → 0 as k → 0. Consequently, in
k
k
any sequence of mD1 equilibria as k → 0, t → 0. Moreover, letting tj denote the j th boundary type of
the mD1 equilibrium,
U (aR (tk ), tk ) − kC(ρ∗k (tk ), tk ) = U (aR (tk , tk1 ), tk ) − kC(1, tk ).
Hence, by continuity of U and C, tk → 0, and U (aR (0), 0) = U (aR (0, t01 ), 0), it follows that tk1 → t01 . By
the arbitrage condition (A), each tkj → t0j for j ∈ {2, . . . , N − 1}. Therefore, the sequence of mD1 outcomes
converges pointwise to β 0 .
Q.E.D.
42
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