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Theory of signaling games
Charles Roddie∗
August 28, 2011
Abstract
This paper studies signaling games under weak conditions, generalizing ex-
isting results, and finding new continuity and comparative statics properties.
Spence-Mirrlees single crossing is replaced with a more applicable order the-
oretic condition that is implied by supermodularity. Differentiability of payoffs
is also relaxed. Separating equilibria still exist, there is uniqueness with a con-
tinuum of types, and the dominant separating equilibrium is selected by the D1
refinement.
Separating and dominant separating equilibria are continuous in the type
space as it becomes a continuum, and in the payoff function. Signaling games
have strong comparative statics properties, and conditions are found for strate-
gies to be increasing in a parameter, and for value functions to be supermodular.
These results allow for the analysis of signaling within larger dynamic games.
Keywords: Signaling, single crossing, supermodular games
Contents
1 Introduction 2
2 Signaling payoff spaces 10
3 Topologies 13
∗Nuffield College, Oxford. I am grateful to Meg Meyer for helpful discussions and Gabriel Kriendlerfor research assistance.
1
4 Separating equilibria 14
5 The dominant separating equilibrium 17
6 Incentive compatibility and uniqueness whenΘ is a continuum 19
7 Comparative statics 21
8 Equilibrium refinement 23
A Key: global definitions 26
B Proofs 27
1 Introduction
Signaling models have been used to understand a variety of phenomena in eco-
nomics, biology, and political science. Tractable signaling models satisfy some form
of single crossing condition, which allows higher/better types to separate from lower
types by choosing a higher signal.
Most theoretical results have assumed strong forms of single crossing, normally
Spence-Mirrlees single crossing (SMSC). But this often fails in applications when
there are intrinsic reasons for taking positive signals. This paper shows that the cen-
tral results hold under a weaker and more applicable form of single crossing: exis-
tence of separating equilibria and a dominant separating equilibrium, and selection
of this equilibrium by the D1 refinement. The weaker condition also relaxes differ-
entiability, and even without this we find a unique separating equilibrium under a
continuum of types.
Secondly, we find new comparative statics properties and continuity properties
of separating equilibria, useful for analysis of signaling that takes place within larger
games and in particular supermodular games.
These results are developed in a unified mathematical framework, taking a topo-
logical approach to deal with a continuum of types.
2
1.1 Weakening single crossing, with applications
1.1.1 Strong single crossing and (weak) single crossing
Single crossing is key to the structure of signaling games because it gives higher types
a greater willingness to choose higher signals than lower types in return for a better
response. This is precisely the weak single crossing condition assumed here.
Let θ ∈ Θ ⊆ R denote the signaler’s type, a ∈ A ⊆ R the signal chosen, and b ∈ B
either a response or an inferred type, in either case partially ordered.
Definition. A function u (θ, a,b) satisfies single crossing if when θ1 < θ2 and b1 ≤ b2:
u (θ1, a1,b1) ≤ u (θ1, a2,b2) and a1 ≤ a2 imply u (θ2, a1,b1) ≤ u (θ2, a2,b2), and strict-
ness in either inequality implies u (θ2, a1,b1) < u (θ2, a2,b2).1
Most existing theory has assumed a strong single crossing condition, where a
higher type is always more willing to choose a higher signal even if it results in a worse
response or inference.
Definition. A function u (θ, a,b) satisfies strong single crossing if when θ1 < θ2:
u (θ1, a1,b1) ≤ u (θ1, a2,b2) and a1 ≤ a2 imply u (θ2, a1,b1) ≤ u (θ2, a2,b2), and strict-
ness in either inequality implies u (θ2, a1,b1) < u (θ2, a2,b2).2
The difference from the stronger form is requirement in the comparison that
b1 ≤ b2. Figure 1 illustrates this.3 Strong single crossing implies that in any perfect
Bayesian equilibrium, choice of signal is weakly increasing in type.
There are two cases in which strong and weak forms are equivalent:
Case 1. u (θ, a,b) is strictly increasing in b and strictly decreasing in a. Then without
signaling incentives (under complete information) the minimum signal is always
taken. Then u (θ1, a1,b1) ≤ u (θ1, a2,b2) and a1 ≤ a2 implies b1 ≤ b2, so single
crossing implies strong single crossing.
1This condition is given in Bagwell and Wolinsky (2002), in the context of a limit pricing model witha binary type and response, and called the “single crossing property”. It is close to the notion of singlecrossing defined in Milgrom and Shannon (1994), in between their single crossing and strict singlecrossing, when the partial order in their definition is taken to be the usual (a1,b1) ≤ (a2,b2) if a1 ≤ a2
and b1 ≤ b2.2This is assumption A4 of Cho and Sobel (1990).3Note that the (probably unavoidable) term “single crossing” has become misleading: there can be
more than one crossing of indifference curves, as long as these points are not ordered in R2.
3
𝜃1
𝜃2>𝜃1
𝑏
𝑎
Figure 1: Indifference curves allowed by single crossing but not by strong single cross-ing.
Case 2. The signaling benefit enters payoffs additively: u (θ, a,b) =α (θ, a)+β (b).
There are examples within the first case, but it is more common for signals to have
some intrinsic value. Outside of these special cases, strong single crossing is typically
hard to guarantee in applications.4
The common and stronger SMSC condition invokes differentiability and asserts
that ua/ub is strictly increasing in θ (where A and B are intervals). It implies strong
single crossing (Cho and Sobel (1990)).
1.1.2 Single crossing and supermodularity
Recall the standard definitions of increasing differences and supermodularity.
Definition. f(x, y
)has weakly increasing differences if for x ′ ≥ x and y ′ ≥ y ,
f(x ′, y ′)+ f
(x, y
) ≥ f(x ′, y
)+ f(x, y ′). If additionally strictness in the assumptions
implies a strict conclusion, f has strictly increasing differences. f(x, y, z, ...
)has
weakly/strictly increasing differences in(x, y
)if for any (z, . . . ),
(x, y
) 7→ f(x, y, z, . . .
)has weakly/strictly increasing differences. It is weakly/strictly supermodular if it has
weakly/strictly increasing differences in all pairs of variables.
An important advantage of single crossing over strong single crossing is that it is
implied by supermodularity, as noted in Bagwell and Wolinsky (2002):
4I am not aware of any such applications satisfying strong single crossing, despite the benefit ofbeing able to use existing theoretical results.
4
Fact. If u (θ, a,b) has strictly increasing differences in (θ, a) and weakly increasing dif-
ferences in (θ,b) then it satisfies single crossing in (θ; a,b).
Supermodularity is a common and effective method for construction of signaling
games, and more generally multi-stage games with strategic complements or substi-
tutes, since it can describe these and is closed under addition. A supermodular value
function V(θ,θ′
)from a second stage, combined with a supermodular payoff w (θ, a)
in the first stage results in a supermodular signaling payoff w (θ, a)+δ ·V (θ,θ′
).
1.1.3 Examples and applications
Example 1. Consider a signaling model of education. Payoffs for an agent with
ability θ from taking education a and being believed to be type θ′ are U(θ, a,θ′
) =− (a − (θ+5))2 +9θ ·θ′. Higher types have a higher reward or lower cost of education;
the complete information choice of education is positive, reflecting an intrinsic or job
market value5; the benefit of signaling higher θ′, which opens up higher-level jobs, is
higher for a higher type. U is supermodular, so satisfies weak single crossing.
With two types, 1 and 2 the separating equilibria have type 1 taking the com-
plete information choice 6, and type 2 separating by taking education in[7−3
p2,3
]∪[9,7+3
p2] ≈ [2.8,3]∪ [9,11.2]. The existence of decreasing equilibria shows that U
does not satisfy strong single crossing. Existing theoretical results therefore do not
apply to this model.
Example 2. In a two-stage Cournot limit pricing model, in the second stage there
are profits(P
(q2
I +q2E
)− ci) · q2
i , which are supermodular for the incumbent in(−cI , q2I ,−q2
E
)if the inverse demand function P is weakly concave, and similarly
for the entrant. The payoff of the maximizing incumbent is then supermodular in(−cI ,−q2E
), giving a value function supermodular in
(−cI ,−c ′I), where c ′I is signaled
type, since q2E is increasing in c ′I . This gives a signaling payoff supermodular in(−cI , q1
I ,−c ′I)
in the first stage, implying single crossing. But strong single crossing
has been verified to fail here even with a linear demand function.6
Maintaining the supermodular structure, we can add additional strategic phe-
nomena between stages, for example a learning by doing or brand loyalty effect of
5Spence (1973), the earliest analysis of signaling in economics, assumed that education did nothave intrinsic or productive value.
6Example on request; I am grateful to Henri Savolainen for calculating this.
5
market share. Arbitrary type spaces can also be used without difficulty, and uncer-
tainty in the entrant’s cost doesn’t affect the analysis.
Currently applications often have to reinvent the wheel by deriving existence of
separating equilibrium and applying equilibrium refinements from first principles,
in simplified settings, since they are unable to use theoretical results that depend on
SMSC.
The result is that applications are typically unable to rely on theoretical results,
instead duplicating a subset of results in their specific contexts. In order to do this
easily, the models are simplified, with two-type models predominant and strategic
interactions limited.
1.2 Relaxing technical conditions
1.2.1 Relaxing monotonicity, differentiability, and continuity
Existing theory has assumed differentiability of payoffs (even when SMSC is not
used), and often stronger assumptions are used: smooth or C2 payoffs. If the pay-
off function is U(θ, a,θ′
), a usual further assumption is ∂U
∂θ′ > 0. Here differentiability
is removed, and U is allowed to have discontinuities in θ′. Also, U is only required
to be weakly increasing in θ′, if we correspondingly weaken the notion of separating
equilibrium.
There are two types of applications of these these generalizations. Direct applica-
tions, where simply specified games violate the stronger conditions. And topological
applications, where the improved closure properties of the signaling space allow find-
ing equilibrium of a game as a fixed point.
1.2.2 Direct applications
Discontinuities and non-strictness of monotonicity in θ′ can both arise in applica-
tions where the response is binary (entry of competitor, agreement to a contract),
which leads to a cutoff in θ′, with U being discontinuous there and otherwise con-
stant in θ′. This also implies non-differentiability.
Non-differentiability can also result from some effect that starts or stops at a cer-
tain point. If the response meets a lower or upper bound: for example in Example 2,
6
the entrant’s production will typically be 0 up to a certain signaled cost c∗. This also
causes non-monotonicity when the bound is reached. If the response has multiple
“causes”, there can be non-differentiable points: a firm advertising and selling to two
groups of customers, or a government signaling competence to a market that has reg-
ular reasons for responding favorably, but an extra effect of illiquidity takes place at
low signals.
Multiple equilibria subsequent to the signaling stage can also generate disconti-
nuity or non-differentiability at the point where one type of equilibrium switches to
another.
1.2.3 Topological advantages
Taking limits of signaling games is useful for (at least) two reasons. Firstly in the the-
oretical development here, taking limits is used to extend results, most commonly
from the finite type case to the continuum. Secondly in applications fixed point argu-
ments can be used to show existence of equilibrium: for example to show existence
of Markov perfect equilibrium in an infinite horizon, we can use a continuous opera-
tor on a compact space of value functions. In taking these limits the L1 norm is used
here7, and the theoretical development is proof of its appropriateness.
Differentiability, continuity, and strict monotonicity are not closed in L1: a limit of
C∞ payoff functions Ui(θ, a,θ′
), strictly monotonic in θ′, may be discontinuous and
only weakly monotonic. This is a problem when using fixed point theorems that re-
quire compact spaces. Also when taking a continuum limit, it is important the limit
point should be a valid signaling payoff, which closure guarantees. The relaxed con-
ditions here avoid these issues.8
Relaxing unnecessary conditions displays the essential structure of signaling
games more clearly.
7In our context all the Lp norms are identical, for p <∞.8Precisely, useful subspaces are closed. If we restrict to payoffs that are uniformly Lipschitz in θ and
a, with a lower bound on the concavity of a (or quasi-concavity, in the manner of Mailath (1987)) andincreasing differences in (θ, a), we get a closed space of signaling payoffs.
7
1.3 Generalized results
Signaling games admit both separating and pooling equilibria, and existing theory
and applications have been particularly interested in the separating equilibria. Riley
(1979) showed existence of separating equilibrium in the context of a closely related
sorting model, assuming SMSC. One equilibrium is dominant in payoffs for the “sig-
naler”.
These equilibria exist without differentiability, under just weak single crossing.
Some of the separating equilibria not be increasing, as Example 1 shows, but the dom-
inant equilibrium is increasing.
To cut down on equilibria, theoretical and applied work invokes additional re-
strictions on beliefs. Cho and Sobel (1990) showed that the D1 refinement selects
the dominant separating equilibrium. This assumed finite types, strong single cross-
ing and differentiability. Ramey (1996) showed this for a continuum of types under
SMSC.9 However applied work is often unable to use these results; and often has to
apply a refinement manually in simplified two-type settings (usually the “intuitive
criterion” of Cho and Kreps (1987)).
Here we show that the D1 refinement works under the weaker form single cross-
ing.
The natural specification for most signaling applications is a continuum of types,
rather than a finite type space. With a continuum of types, Mailath (1987) showed
there is a unique separating equilibrium. The argument was that an incentive com-
patible strategy must be differentiable and satisfy a differential equation. Here we
show the result applies in the absence of differentiability.
1.4 New results and applications
1.4.1 Continuity properties
In most settings a continuum is the natural specification of the type space, and here a
topological approach is used to deal with this. Central to the analysis of a continuum
of types are continuity properties about separating and dominant separating equilib-
9Cho and Sobel (1990) and Ramey (1996) extend to multidimensional signals, the latter with aweaker condition from Engers (1987), but it is hard to have multidimensional conditions that are bothtractable and applicable.
8
ria. (See 3.)
The set of separating equilibria and the dominant separating equilibrium are
(upper-semi)continuous in the type space as it becomes dense in a continuum. This
extends Mailath (1988), which found the result for separating equilibria, generalizing
it from SMSC to single crossing.10 The result can be used to extend equilibria for finite
type spaces to a continuum: both existence of equilibria and results about equilibria.
Both separating and dominant separating equilibria are also continuous with re-
spect to the payoff function. This result can be used for existence of equilibrium, if
this is found with a fixed point argument: for example, it is useful if other players
move at the same time as the signaler. (These other players may also be signaling.) It
is also used to show uniqueness of separating equilibrium for a continuum of types.
The continuity holds when type space and payoffs are varied simultaneously,
which is also useful in existence of equilibrium arguments (see Section 5.3).
1.4.2 Comparative statics
What if the signaler is not moving alone in the first stage? What if more than one agent
is signaling simultaneously? What if there are stages before or after the signaling and
response stages? Regarding the action taken before of simultaneously with the signal
as a parameter z, we ask we ask how (dominant separating) strategies and payoffs
depend on z.
A condition related to single-crossing is found for the strategy to be increasing in
z. This property can give the extended games described above a tractable directional
structure.
The condition is satisfied if the payoff function is supermodular, and then we get
a value function which is a supermodular function of z and type. This allows us to
embed signaling games within supermodular dynamic games: future supermodular
stages generate single crossing in the signaling stage, and this preserves a supermod-
ular value function for previous stages. Roddie (2010a) uses this property to study
repeated signaling games. Regularity conditions are also given which bound the rate
of increase of strategies and the degree of supermodularity.
10Manelli (1996) finds that arbitrary equilibria also converge to an equilibrium of the continuumcase.
9
1.5 Contents
Section 2 defines the space of signaling games. The necessary topologies are specified
in Section 3.
Section 4 defines separating equilibria and characterizes the (closure of the) set
of increasing separating equilibria. This set is (upper semi-)continuous in the type
space as it becomes dense in an interval and in the payoff function (Proposition 1).
Existence of separating equilibrium is shown (Proposition 3) without requiring differ-
entiability or strong single crossing.
Section 5 studies the payoff-dominant (“Riley”) separating equilibrium. This ex-
ists uniquely (Proposition 2) and is increasing. Lemma 4 gives further characteriza-
tion. Section 5.3 extends the continuity results for separating equilibria to the Riley
equilibrium (Proposition 3).
Section 6 (Proposition 4) shows uniqueness of separating equilibrium for a con-
tinuum of types. This extends Mailath (1987) to non-differentiable settings.
Section 7 studies the comparative statics of the Riley equilibrium. Suppose a sig-
naling payoff is parametrized by z. A condition related to single crossing implies that
the Riley equilibrium is increasing in z (Proposition 5). In particular this happens if
the signaling payoff is supermodular, and then equilibrium utility is a supermodular
function of z and θ (6).
Section 8 shows that the refinement D1 selects the Riley equilibrium, weakening
the assumption of strong single crossing in Cho and Sobel (1990).
2 Signaling payoff spaces
Types, signals, beliefs, and payoffs
The type space Θ is either the interval Θ= [θmin,θmax] or a finite subset with minimal
and maximal elements θmin,θmax. The set of signals A is an interval [amin, amax]. A
belief about types is an element θ of 4Θ, which is ordered by first-order stochastic
dominance.
A signaling game is specified in reduced form by a utility function u :Θ×A×4Θ→R, for the signaler, giving payoff u
(θ, a, θ
)to type θ of the signaler when he chooses
signal a and is consequently believed to be type θ.
10
To study separating equilibria, we only need to know utility for degenerate pos-
terior beliefs. So we can restrict attention to U : Θ× A ×Θ→ R, where U(θ, a,θ′
):=
u(θ, a,
[θ′
]), where[θ] denotes the probability measure placing probability 1 on θ. We
will be using these spaces for most of this paper, but will return to define and analyze
the fully specified spaces of u : Θ× A ×4Θ→ R in Section 8, where non-separating
equilibria are considered.
Notation. When U is unambiguous,(a1,θ′1
) ¹θ(a2,θ′2
)will mean U
(θ, a1,θ′1
) ≤U
(θ, a2,θ′2
), and similarly for º, ∼, ≺ and Â.
The main signaling payoff space
We define a setΦ of signaling game payoffs satisfying the basic conditions; all payoffs
in this paper will be in Φ.
Definition. Let Φ be the set of U :Θ× A×Θ→Rwith U(θ, a,θ′
):
1. uniformly continuous in (θ, a)1112
2. weakly increasing in θ′, with ((a,θ1) ∼θ (a,θ2)) independent of a
3. strictly quasi-concave in a
4. satisfying single crossing
The first condition is a weakening of the usual assumption of differentiability (or
more commonly U ∈C2 or C∞). It is weaker than continuity of U , allowing for discon-
tinuities with respect to θ′.The second condition weakens the usual assumption ∂U
∂θ′ > 0. If U is only weakly
increasing, it is possible that some type θ, taking some signal a, does not strictly ben-
efit from being believed to be θ2 < θ1 over θ1. The condition requires that when this
happens does not depend on a. This automatically holds if U is strictly increasing in
θ′, or if U =α (θ, a)+β(θ,θ′
), a common property of signaling payoffs in a multi-stage
game.
11This means that for any ε > 0, there exists δ > 0 such that for ‖(dθ,d a)‖ < δ,∣∣U (θ+dθ, a +d a,θ′)−U (θ, a,θ′)∣∣< ε for any (θ, a,θ′).
12Of course, adding any function of θ to U doesn’t affect it’s real properties.
11
The third condition is relatively standard. It implies existence of a unique optimal
strategy under complete information.
The fourth condition asserts the weak form of single crossing defined in the intro-
duction.
Bounded payoffs
An extra condition makes the highest signal sufficiently costly that taking it would
contradict IR even for the highest type. This will ensure there are costly enough sig-
nals for a separating equilibrium to exist.
Definition. Let ΦB be the set of U ∈ Φ with supa U (θmax, a,θmin) ≥U (θmax, amax,θmax).
Responses
Usually the signaling payoff is derived as u(θ, a, θ
) = v(θ, a,ρ
(θ, a
)), where ρ
(θ, a
)is
a best-response of a respondent. Single crossing in v , combined with monotonicity
properties of v and ρ guarantees single crossing for U .
Fact. Suppose v (θ, a,r ) is continuous, weakly increasing in r and satisfies single cross-
ing. Suppose ρ(θ, a
)is weakly increasing in θ and a and uniformly continuous in a.
Then U(θ, a,θ′
):= v
(θ, a,ρ
([θ′
], a
))is weakly increasing in θ′, satisfies single crossing,
and is uniformly continuous in (θ, a). If v is strictly increasing in r , then condition 2 is
automatically satisfied. If ρ is a function of θ′ only, then quasi-concavity in a follows
from quasi-concavity of v in a.
Separating payoffs and the complete information strategy
If the signaler of type θ takes a strategy f : Θ→ A, and is subsequently known to be
type θ, its payoff is:
Definition. Given U ∈Φ and f :Θ→ A, let U f (θ) :=U(θ, f (θ) ,θ
).
This will be its payoff in a separating equilibrium described by f . The complete
information strategy a∗U maximizes this:
Definition. For U ∈Φ, let a∗U (θ) = argmaxa U (θ, a,θ). Where a∗ is used U is implicit.
12
3 Topologies
We shall be using L1 norms on strategies and signaling payoffs, which requires mea-
sures on the initial spaces. On A we use standard Lebesgue measure, normalized to 1.
IfΘ is finite, we use counting measure. If it is a continuum, we use Lebesgue measure,
but altered to give θmin and θmax positive measure. This ensures that L1 convergence
of strategies implies pointwise convergence on {θmin,θmax}. Let µΘ be this measure
on Θ, again normalized to 1.
We also want some notion of convergence in Θ. Let the metric on Θ be as usual,
but with θmin and θmax separated from the other points (d (θmin,θ) = θ−θmin+1 when
θ 6= θmin for example).
Signaling payoffs in Φ are measurable functions U :Θ× A×Θ→R, and we endow
the space Φ with the L1 norm, using the above measures on Θ and A. Strategies are
functions Θ→R, and we will be particularly interested in the weakly increasing func-
tions, Inc(Θ, A). These are measurable. Endow Inc(Θ, A) with the L1 norm; it is then
a compact space.
VaryingΘ
In order to move between finite types and the continuum, both in showing continuity
results and extending results to the continuum case, we need a notion ofΘ converging
to Θ.
Definition. Θi ↗ Θ∞ if {θmin,θmax} ⊆ Θi ∈ Θ is an increasing sequence of sets, with⋃Θi dense in Θ∞.
Suppose fi ∈ Inc(Θi , A) are strategies on spaces Θi ↗ Θ∞. We want some notion
of convergence “ fi ↗ f∞”, despite the fact that they lie in different spaces:
Definition. Suppose Θi ↗ Θ∞ and fi ∈ Inc(Θi , A). Write fi ↗ f∞ if for any fi ∈Inc(Θ∞, A) with fi |Θi = fi , fi → f∞.13
13Lemma B.4 implies that it is sufficient for this to hold for just one sequence(gi
). It is equivalent to
convergence under the metric d(
fi , f j)
:= sup{d
(gi , f j
): gi ∈ Inc
(Θ j , A
), gi = fi on Θi
}.
13
4 Separating equilibria
Section 4.1 defines the sets of separating and increasing separating equilibria. For
U ∈ ΦB , such equilibria exist. The approach to existence is as follows: Section 4.2
characterizes the closure of the set of separating equilibria, allowing Section 4.3 to
find continuity properties, including continuity in the type space. Continuity in the
type space allows us to extend a construction (4.4) of separating equilibrium for finite
types to a continuum.
4.1 Weakly separating strategies
Incentive compatibility is the condition that no type has an incentive to mimic an-
other type. Individual rationality is the condition that no type receives such low utility
that he would prefer to take some other action even if the worst possible beliefs result.
Definition. A strategy f : Θ→ A satisfies individual rationality (over S ⊆Θ) if for all
θ (∈ S) and a,(
f (θ) ,θ) ºθ (a,θmin). It satisfies incentive compatibility (over S) if for
all θ,θ′ (∈ S),(
f (θ) ,θ)ºθ (
f(θ′
),θ′
). If both are satisfied, f is weakly separating (over
S).14
Definition. Let Sep± (U ) denote the set weakly separating strategies.
We shall be particularly interested in strategies that are weakly increasing:
Definition. Let Sep(U ) = Sep± (U )∩ Inc(Θ, A).
If U satisfies strong single crossing, Sep(U ) = Sep± (U ).
Relationship to Perfect Bayesian equilibria
Call f separating if it is weakly separating and injective. Provided f is measurable, it
is then a perfect Bayesian equilibrium, when combined (for example) with beliefs:
14It is possible to remove the IR requirement. This would streamline the theory in some minor ways.But IC+IR corresponds to (separating) perfect Bayesian equilibrium as noted below, the most stan-dard notion of equilibrium, whereas IC alone corresponds to something weaker than Nash equilib-rium. IC alone corresponds to Nash equilibrium if for any θ, a′, v
(θ, a′,rmin
)≤ infa,θ′ U(θ, a,θ′
), where
U(θ, a,θ′
)= v(θ, a,ρ
([θ′
], a
)).
14
β (a) = f −1 (a) a ∈ Im
(f)
[θmin] a 6∈ Im(
f)
Fact 1. If U(θ, a,θ′
)is strictly increasing in θ′, all weakly separating strategies are sep-
arating.
Proof. IC implies a weakly separating function must be injective.
If this does not hold, it is possible for a weakly separating strategy to be non-
injective. Pooling is allowed to occur if within a pool no type has a strict incentive
to appear to be a higher type within the pool. If all types agree on when it is strictly
better to signal θ′2 than θ′1, then a weakly separating equilibrium with pools is a Perfect
Bayesian equilibrium:
Fact 2. Suppose u(θ, a, θ
)is weakly increasing in θ, and
((a,θ′1
)∼θ (a,θ′2
))is a func-
tion of(θ′1,θ′2
); i.e. the ranges of θ′ that do not affect utility are constant in θ. If
f ∈ Sep± (U ) is measurable, then f describes a Perfect Bayesian equilibrium with U f
describing equilibrium utility.
One such equilibrium sets beliefs to [θmin] for actions that are not taken, and a
conditional probability of θ given a for actions that are; such a conditional probability
is easy to construct if f ∈ Sep(U ).
4.2 The closure of separating strategies
A weakly separating f ∈ Sep(U ) may have discontinuities, if U is discontinuous or
has regions of constancy in θ′. Then there are weakly increasing functions equal to f
except at these discontinuities that are not separating. So we cannot say that Sep(U )
is closed in Inc(Θ, A). But when Sep(U ) is adjusted to include these functions, we get
a closed set.
Definition. Let Sep(U ) := {g ∈ Inc(Θ, A) : g = f a.e. for some f ∈ Sep(U )
}.
Lemma 1. Sep(U ) is the closure of Sep(U ) in Inc(Θ, A).
15
4.3 Continuity with respect toΘ and U
In the above Θ has been taken as fixed, but now allow it to vary, with the above func-
tions depending implicitly onΘ. The correspondence Sep is (upper semi-)continuous
with respect to U and the type space Θ. We will use a general argument to show this
simultaneously. All utility functions will be given on the largest type space Θ∞, and
restricted as necessary to smaller ones. Parts 2 and 3 are special cases of part 1.
Proposition 1. 1. (Continuity in U and Θ) Suppose Θi ↗ Θ∞. Suppose Ui ∈Φ (Θ∞), with Ui → U∞ (in L1) and Ui
(θ, a,θ′
)uniformly continuous in (a,θ)
over(i ,θ′
). Suppose fi ∈ Sep(Ui ) and fi ↗ f . Then f ∈ Sep(U∞).
2. (Continuity inΘ; extends Mailath (1988)) Take U ∈Φ(Θ
), and let θmin ∈Θi ↗ Θ.
Suppose fi ∈ Inc(Θi , A) with fi ↗ f . Then f ∈ Sep(U ).
3. (Continuity in U :) Suppose Ui ∈ Φ, with Ui → U∞ in L1 and Ui(θ, a,θ′
)uni-
formly continuous in (a,θ) over(i ,θ′
). Suppose fi ∈ Sep(Ui ) and fi → f . Then
f ∈ Sep(U∞).
4.4 Existence of separating equilibria
In any separating equilibrium, the lowest type must take the complete information
signal a∗U (θmin). For finite Θ, we can define a separating equilibrium χ inductively:
subsequent types maximize utility Uχ (θ) subject to separating from lower types.
Lemma 2. For finite Θ = {θ0,θ1, . . .}, if U ∈ Φ, the following definitions are equivalent
and uniquely define χ ∈ Sep(U ) if a solution exists. If U ∈ΦB a solution must exist.
1. χ (θ0) = a∗ (θ0). For i ≥ 1, χ (θi ) maximizes U (θi , a,θi ) over Bi :={a ≥χ (θi−1) :
(χ (θi−1) ,θi−1
)ºθi−1 (a,θi )}.
2. For each θ′, χ(θ′
)maximizes U
(θ′, a,θ′
)over{
a : a ≥χ (θ) ,(χ (θ) ,θ
)ºθ (a,θ′
)for θ < θ′}.
The construction is standard in the setting of strong single crossing (see Cho and
Sobel (1990) or Sobel (2009)) and here we show it works under just single crossing.
Let ΦSep be the set of signaling payoffs in Φ for which a separating equilibrium
exists:
16
Definition. ΦSep := {U ∈Φ : Sep(U ) 6=∅
}.
This set contains ΦB , so that ΦB ⊆ ΦSep ⊆ Φ. So we have existence of separating
equilibrium without differentiability or strong single crossing:
Lemma 3. ΦB ⊆ΦSep.
Proof. Lemma 2 shows this for finite Θ. For Θ = Θ, take finite Θi ↗ Θ and take a
convergent sub-sequence of separating equilibria. Proposition 1.2 shows this is an
element of Sep(U ), soSep(U ) must be non-empty.
5 The dominant separating equilibrium
When a separating equilibrium exists, there exists a dominant separating equilibrium,
often known as the Riley equilibrium, which maximizes utility for all types of the sig-
naler.
5.1 Existence and uniqueness
Given a finite collection S of weakly separating strategies f , we can take g (θ) for each
θ to be the f (θ) that maximizes U f (θ) over f ∈ S. This results in a weakly separating
g that dominates all the f . With a little care the argument extends to when S is all the
separating equilibria. The dominant equilibrium is weakly increasing, and unique in
Sep(U ). It is essentially unique in Sep± (U ).
Proposition 2. 1. If U ∈ΦSep, ∃g ∈ Sep(U ) such that Ug ≥U f for any f ∈ Sep± (U ).
2. (a) g is unique in Sep(U ).
(b) g is (i.) unique in Sep± (U ) if Θ is finite, and (ii.) unique up to differences
on a countable set if Θ= Θ.
Let R (U ) be this equilibrium:
Definition. R :ΦSep → Sep(Θ, A) gives for each U ∈ΦSep the above element g .
17
5.2 Further characterization
We can give some additional characterizations and properties of R (U ).
Lemma 4. For U ∈ΦSep:
1. R (U ) is the unique smallest function f ≥ a∗
(a) which satisfies IC
(b) which is in Sep(U )
2. If Θ is finite, fixing U ∈ΦSep, R (U ) =χ as defined in Lemma 2.
5.3 Continuity with respect toΘ and U
In Section 5.3 we obtained continuity results for Sep. The Riley equilibrium is (jointly)
continuous in the type space and payoff function:
Proposition 3. 1. Suppose Ui ∈ΦSep (Θ∞), (i ∈N∪ {∞}) with Ui →U∞ (in L1) and
Ui(θ, a,θ′
)uniformly continuous in (a,θ) over
(i ,θ′
). Suppose Θi ↗ Θ∞. Then
R(Ui |Θi
)↗R (U∞).
2. Suppose U ∈ΦSep (Θ∞), and Θi ↗Θ∞. Then R(U |Θi
)↗R (U ).
3. Take Ui ∈ΦSep (Θ), with Ui →U∞ and Ui(θ, a,θ′
)uniformly continuous in (a,θ)
over(i ,θ′
). Then R (Ui ) →R (U∞).
These results are useful for dealing with a continuum of types. Continuity in Θ is
useful for extending results from the finite type case to the continuum. It can also be
used to describe equilibria when types are finite but close to a continuum.
Continuity in U will be used to show uniqueness of separating equilibrium when
Θ = Θ (Section 6). It can also be used to show existence of equilibria where more
than one player is moving in the signaling stage, giving a continuous map from other
players’ actions to the signaler’s strategy.
Simultaneous continuity in Θ and U is stronger than separate continuity, since
neither has been shown to be uniform. It can be used in settings with continuous
types when more than one player is moving in the signaling stage. It can be used
18
to show that a limit of finite equilibria is an equilibrium for continuous types, giving
existence of equilibrium for a continuum of types.15
6 Incentive compatibility and uniqueness when Θ is a
continuum
Mailath (1987) argued that if the type space is a continuum, there is uniqueness of
separating equilibrium, under differentiability and regularity conditions. The main
conditions are that U ∈ C2, with ∂U∂θ′ > 0 and ∂2U
∂θ∂a > 0. This uniqueness comes almost
entirely from the IC conditions. To satisfy IC, a strategy f must satisfy the differential
equation d fdθ
∂U∂a + ∂U
∂θ′ = 0 on (θmin,θmax]. If it also satisfies IR at θmin, i.e. f (θmin) =a∗ (θmin), the solution to the initial value problem is unique. In other words f =R (U )
and is the unique separating equilibrium.
With an arbitrary initial value, f may be discontinuous at θmin, jumping up to
some a that is determined by indifference for θmin. It then is uniquely determined,
satisfying the differential equation with initial value a. In other words it is the Riley
equilibrium when the signal space is restricted to [a, amax].
These conclusions do not require differentiability.
Proposition 4. Suppose Θ= Θ and U ∈Φ is continuous16. Suppose f :Θ→ A satisfies
IC, and f ≥ a∗ on (θmin,θmax].
1. If f (θmin) = a∗ (θmin), then U ∈ΦSep and f =R (U ).
2. (extends Mailath (1987)) For arbitrary f (θmin) there exists a unique a ≥ a∗ (θmin)
with(
f (θmin) ,θmin) ∼θmin (a,θmin). Take A := [a, amax]. Then U |A ∈ ΦSep
(A
),
and f =R(U |A
)on (θmin,θmax].
15Suppose the actions of the other players are z, and equilibrium of a game with finite types satisfiesan equation of the form Fi (zi ) = zi . Suppose zi → z∗. Then we would like Fi (zi ) → F∞ (z∗), whichgives z∗ = F∞ (z∗). Since in Fi (z), i affects the type space, and z affects the signaling payoff function, ifthe Riley equilibrium is involved in Fi , simultaneous continuity in Θ and U is needed.
16This is also true for discontinuous U , as the following informal argument shows: take a separatingequilibrium f of U ; at each discontinuity point θ′, expand the type space into an interval, so that thesum of the interval lengths is finite, giving a continuous payoff function U ′. We can extend f to aseparating f ′ for U ′, argue for uniqueness of f ′ as U ′ is continuous, and then get unique f . This wouldrequire work to formalize as U ′ does not satisfy single crossing with the required strict inequality, soU ′ ∉Φ.
19
B
F
G
C
𝜃1 𝜃3 𝜃2
Indifference curves of
𝑈(𝜃1,∙,∙)
𝜃2
𝜃3 D
E
A
Figure 2: Adding types reduces separating equilibria. Suppose A is the complete in-formation optimum for all types, so f (θ1) = A. With types {θ1,θ3}, the set of incentivecompatible values of f (θ3) is the interval BC, with AB the Riley equilibrium. With theextra type θ2 this reduces to FG.
Note that the assumption f ≥ a∗ must hold if U is strictly increasing in θ′:17
Fact 3. If U(θ, a,θ′
)is continuous and strictly increasing in θ′, and f is weakly increas-
ing and satisfies IC, then f ≥ a∗on (θmin,θmax].
The main use of part 1 is the implication of uniqueness of separating equilibrium.
In a standard model with an optimizing respondent who makes a cognitive inference,
IR is a reasonable assumption so the initial value condition will hold. Part 2 eliminates
IR. It can be used in settings where the respondent is known to play a Nash but not
Bayesian perfect equilibrium strategy.
6.1 Argument
Adding types imposes additional constraints on separating equilibria, so the set of
strategies that are separating over a set S of types is decreasing in S. This is shown
graphically in Figure 2 on page 20.
Consider Figure 2 on page 20 with all 3 types active. The Riley equilibrium is ADF.
The incentive compatible functions f starting at A all lie between ADF and AEG. AEG
is the Riley equilibrium of the altered game U ′, where U ′ (θi , a,θ′)
:=U(θi+1, a,θ′
). So
any incentive compatible f lies between R (U ) and R(U ′).
17If for example U is independent of θ′, then taking any closed set S ⊆ A, the strategy f (θ) =argmaxa∈S U (θ, a) satisfies IC. Then f ≥ a∗ typically does not hold and there is non-uniqueness.
20
For a continuum of types, suppose f satisfies IC and the initial condition. For
any δ> 0, let Uδ be the perturbed payoff Uδ :(θ, a,θ′
) 7→U(θ+δ, a,θ′
). Then we can
bound f between R (U ) and R (Uδ). Now Uδ → U as δ→ 0, so R (Uδ) → R (U ) by
Proposition 3, so we must have f =R (U ). This shows part 2.
To relax the initial condition (part 2), we find that f must jump up to a, and note
that the conditions for part 1 are satisfied when we restrict A to [a, amax].
7 Comparative statics
Here we make statements about how Riley equilibrium strategies and payoffs vary
with the payoff function.
7.1 When is R (U1) ≤R (U2)?
Since incentives in typical signaling games are directional, we often want to know the
direction of movement of one variable with respect to another. So here we ask when
one signaling game leads to a higher equilibrium than another. Just as single crossing
causes higher types to take higher actions than lower types in the Riley equilibrium, a
single crossing relationship between U1 and U2 can imply this.
We can decompose single crossing in terms of the property of “having greater pref-
erence for higher a and θ′”:
Definition. Suppose A, B are weakly ordered sets. If vi : A×B →R for i ∈ {1,2}, v1 ≤SC
v2 if when b1 ≤ b2, (1) v1 (a1,b1) ≤ v2 (a2,b2) and (2) a1 ≤ a2 imply (3) v2 (a1,b1) ≤v2 (a2,b2).
Then u (θ, a,b) satisfies single crossing if for θ1 < θ2, u (θ1, ·, ·) ≤SC u (θ2, ·, ·), and
inequality in (1) or (2) implies inequality in (3).
If the preferences of all types are increased in the sense of ≤SC , then the Riley
equilibrium is weakly increased:
Definition. For U1,U2 ∈Φ, U1 ≤′SC U2 iff for any θ, U1 (θ, ·, ·) ≤SC U2 (θ, ·, ·).
Proposition 5. If U1 ∈ Φ and U2 ∈ ΦSep and U1 ≤′SC U2 then U1 ∈ ΦSep and R (U1) ≤
R (U2).
21
7.2 Supermodular comparative statics
As explained in the introduction, supermodular payoffs are a valid input of signal-
ing games and a productive way to model signaling applications. Supermodularity
can also be an output of signaling games, and this allows for analysis of signaling
within larger supermodular dynamic games. This property is used in Roddie (2010a)
to study repeated signaling games.
Suppose U (z)(θ, a,θ′
)is parametrized signaling payoff. The parameter z could be
an action played before or simultaneously with the signaler’s action. It could also be
a state of the world, or capital stock. If U is supermodular, then the resulting payoff
v (z,θ) in the Riley equilibrium is supermodular:
Proposition 6. Let Z ⊆R be an interval and U : Z →ΦSep. Suppose that U (z)(θ, a,θ′
)is weakly supermodular. Let f (z) := R
(U (z)
)and let v (z,θ) := U (z) f (z) (θ) be utility
of type θ in the Riley equilibrium.
1. f is weakly increasing
2. v is weakly supermodular.
3. If U (z)(θ, a,θ′
)is weakly (strictly) increasing in z, then v (z,θ) is weakly (strictly)
increasing in z.
7.2.1 Bounds
Here we give conditions bounding the increase in the Riley equilibrium with the pa-
rameter z, and bounding the supermodularity of the resulting value function.
The first result uses natural condition for a best response under complete infor-
mation to be Lipschitz in z, showing that this still applies with signaling incentives.
The second result requires a notion of bounds on supermodularity.
Definition. Suppose f(x, y
) ∈ R, where x and y lie in R. f has an increasing differ-
ences bound of β if for x1 ≤ x2 and y1 ≤ y2:
f(x2, y2
)+ f(x1, y1
)− f(x1, y2
)− f(x2, y1
)≤β (x2 −x1)(y2 − y1
)18
If the increasing differences of U are bounded, then the resulting value function
has an increasing difference bound.
18If f is twice differentiable, this is equivalent to ∂ f∂x∂y ≤β.
22
Lemma 5. Let Z ⊆ R be an interval and U : Z → ΦSep. Again let f (z) := R(U (z)
)and v (z,θ) := U (z) f (z) (θ). Suppose that for d > 0, the difference U (z)
(θ, a +d ,θ′
)−U (z)
(θ, a,θ′
)is decreasing on any line {(a, z) = (a0 +α ·λ, z0 +λ)}.19
1. R(U (z)
)(θ) has Lipschitz constant α in z.
2. Suppose there exists some U ∈ Φ(Θ
)extending each U (z). Suppose that
U (z)(θ, a,θ′
)has increasing difference bounds of βzθ,βzθ′ ,βza ,βθa ,βaθ′ in (z,θ),(
z,θ′)
etc. respectively. Suppose f (z) (θ) has Lipschitz constantα in z. Then v has
an increasing difference bound of βθz +βθ′z +α ·βza .
These are regularity properties. The first an be used for uniqueness of equilibrium
if equilibrium z is a best response to the signaler’s strategy. The second result can
be used to give Lipschitz conditions in value function iteration in dynamic signaling
games, helping to preserve a compact space.
8 Equilibrium refinement
Equilibrium refinements cut down on the large number of perfect Bayesian equilib-
ria that exist in signaling games. These games admit a multiplicity of pooling equi-
libria, and for finite types a multiplicity of separating equilibria. Equilibrium refine-
ments work by restricting beliefs off the equilibrium path. They do not fully justify
the equilibria they select, but they give a rationale for them by partly modeling out-
of-equilibrium behavior.20 They are an extension of the modeling process, and have
become a common tool of applied signaling papers.
Payoffs and equilibrium
So far we have been considering reduced utility functions that depend on a revealed
type θ′ ∈Θ; this is all that is needed to analyze separating equilibria. In order to con-
sider refinements, we need to allow for pooling equilibria - if only to rule them out -
19This is implied by the differential condition α · ∂2U(∂a)2 + ∂U
∂a∂z ≤ 0.20In the case of D1, this modeling can be seen as specifying infinitesimal probabilities of mistakes by
the signaler as a function of errors in beliefs about the subsequent response required to justify subop-timal signals.
23
and so need specify payoffs on non-degenerate beliefs θ ∈4Θ. We also assume that
payoffs result, as they usually do in signaling models, from a response ρ(θ, a
) ∈R.
Definition. Let Φ∗ be the set of u : Θ× A ×4Θ→ R with u(θ, a, θ
) = v(θ, a,ρ
(θ, a
)),
where:
1. ρ(θ, a
) ∈R is continuous, weakly increasing in a and strictly increasing in θ
2. v (θ, a,r ) is continuous, strictly increasing in r and satisfies strict single crossing
3. u(θ, a, θ
)is strictly quasi-concave in a
4. u (θmax, amax, [θmax]) ≤ u (θmax, a, [θmin]) for some a
A fully specified payoff u ∈ Φ∗ gives rise to a signaling payoff U ∈ ΦB with
U(θ, a,θ′
)continuous21 and strictly increasing in θ′. The reason for requiring a re-
sponse r , rather than working directly with beliefs θ is that any two responses are
comparable, unlike any two beliefs: so different types have identical preferences over
responses (they prefer higher ones), but not over beliefs.
Letψ∗ ∈4Θ be the initial distribution of types. We assume perfect Bayesian equi-
librium:
Definition. A perfect Bayesian equilibrium is given by a strategy for the signaler f :
Θ→4A and a belief function β : A →4Θ with:
1. θ→ f (θ) (S) measurable for any measurable S ⊆ A
2. a →β (a) (T ) measurable for any measurable T ⊆Θ
3. β (a) giving a conditional distribution of θ given a = f (θ)
4. f (θ) assigns probability 1 to the set of signals a maximizing u(θ, a,β (a)
)Almost all actions a in the support of f (θ) give type θ the same equilibrium payoff
u f ,β (θ) := u(θ, a,β (a)
).
21Applying D1 to discontinuous U (forΘ= Θ) may be problematic. In the Riley equilibrium we couldhave types < θ∗ taking signals below a1, while θ∗ separates from them by taking a2 > a1; ascribing abelief compatible with D1 to actions in (a1, a2) can then be impossible.
24
The equilibrium refinement D1
The refinement D1 was originally given in Cho and Kreps (1987). The following ver-
sion can apply to finitely many or a continuum of types:
Definition. An equilibrium f ,β satisfies D1 if for any a ∈ A and any S ⊂Θmeasurable
withψ∗ (S) > 0,⋃θ∈S
{θ : u
(θ, a, θ
)> u f ,β (θ)}⊆⋂
θ′∉S{θ : u
(θ′, a, θ
)≥ u f ,β(θ′
)}implies
β (a) (S) = 1.
Given an action a, if any belief θ associated with a that would give some type out-
side S a weak incentive to deviate and play a would give all types in S strict incentives
to deviate, then beliefs after observing a must lie in S.
Cho and Sobel (1990) showed that under strong single crossing (and differen-
tiability), for a finite set of types, D1 selects the Riley equilibrium uniquely.22 Ramey
(1996) weakened the condition for multidimensional signals from SMSC to the condi-
tion given in Engers (1987); in 1 dimension this reduces to SMSC. He found the same
result for a continuum of types. Here we restrict to a 1-dimensional signal, but weaken
the order theoretic notion of single crossing and allow for non-differentiability.
Proposition 7. 1. Suppose that u ∈ Φ∗ and that ψ∗ has full support. Suppose at
least one of the following holds: Θ is finite; ρ ([θ] , a) is strictly increasing in a; or
U satisfies strong single crossing. If f ,β satisfy D1 then f =R (U ) (a.e.).
2. Suppose that u ∈Φ∗. f =R (U ) satisfies D1 for some β.
The arguments extend those of Cho and Sobel (1990). Differentiability can be set
aside easily as it is not important to the argument of Cho and Sobel (1990). Relaxing
the order theoretic notion of single crossing requires some alterations. If there is a
pool of types P taking signal a, under strong single crossing it is possible to separate
from it by taking a slightly higher signal, resulting in beliefs at least supP . This is not
true under our single crossing, when the pool is not above the complete information
signals a∗. A larger move, above a∗, must be taken to separate. And downwards sep-
arating moves need to be considered in addition to upwards moves, and ruled out.
22In fact they allowed for payoffs that do not satisfy the bound condition used here, where separat-ing equilibria may not exist: then D1 uniquely selects an equilibrium which has some pooling by thehighest types at the highest action.
25
The notation for a continuum of types is similar to Ramey (1996), although for a
continuum of types the differential methods there are replaced with arguments from
uniqueness of separating equilibrium.
A Key: global definitions
Θ ⊆ Θ= [θmin,θmax] Space of types (finite or interval)
A = [amin, amax] Space of signals
Inc(Θ, A) ⊆ {f :Θ→R
}Weakly increasing functions
Φ ⊆ {U :Θ× A×Θ→R} Signaling payoffs
ΦSep ⊆ Φ Payoffs for which separating equilibria exist
ΦB ⊆ ΦSep Payoffs with sufficiently costly signals
Φ∗ ⊆ {u :Θ× A×4Θ→R} Fully-specified signaling payoffs
a∗ ∈ Inc(Θ, A) Complete information optimal strategy
Sep± : Φ ,→ AΘ Weakly separating equilibria
Sep : Φ ,→ Inc(Θ, A) Weakly increasing weakly separating eq.
Sep The closure
R : ΦSep → Inc(Θ, A) Riley equilibrium
26
B Proofs
B.1 Map of results
L B1-B4: Continuity properties of
strategies and signaling payoffs
L B5: Above , IC ⇒ increasing
Separating equilibria
L B6: characterization
L 1: closed
P 1: continuous in and
L 2: A sep. eq. for finite
L 3: Existence of sep. eq.
Dominant sep. eq.
P 2: Existence & uniqueness
L 4: Characterization of
L B7: average payoffs
P 3: continuous in and
Uniqueness of sep. eq. on
P 4: IC and uniqueness
Comparative statics
P 5: Condition for ( ) ( )
P 6: Supermodular value functions
L 5: Bounds on increase and
supermodularity
Equilibrium refinement
L B8: satisfies “strong IR”
P 7: D1 uniquely selects
Figure 3: (Proposition 2 is assumed in all subsequent results.)
B.2 Proofs of Facts
These are independent of the main development.
Proof of Fact 2 There is some initial distribution θ0 overΘ. A measurable f results in a distribution
over Θ× A. Let β′ : A →4Θ be a conditional probability.
Let A∗ be the set of signals a ∈ Im(
f)
such that β′ (a) gives weight to types not taking a. The
probability of A∗ is 0. On this set, using the axiom of choice, let g (a) be some θ with f (θ) = a.
β (a) =
[θmin] a 6∈ Im
(f)
β′ (a) a ∈ Im(
f)
\A∗
g (a) a ∈ A∗
Then β is measurable, and is a conditional probability. It gives a perfect Bayesian equilibrium
when combined with f .
27
Proof of Fact 3 It is immediate that f must be strictly increasing. a∗ is continuous since U is. So
if f (θ∗) < a∗ (θ∗) for some θ∗ > θmin, then f (θ∗) < a∗ (θ) for some θ < θ∗. Then f (θ) < f (θ∗) < a∗ (θ).
Then(
f (θ) ,θ)≺θ∗ (
f (θ∗) ,θ)≺θ (
f (θ∗) ,θ∗), contradicting IC.
B.3 Topological lemmas
Both f (θ) where f ∈ Inc(Θ, A), and U(θ, a,θ′
)where U ∈Φ, have common property of being weakly in-
creasing in the final variable, and uniformly continuous in the rest. This gives them similar topological
properties.
In the only variable, f ∈ Inc(Θ, A) has continuity points C(
f). In the final variable, U ∈ Φ has
continuity points C ′ (U ):
Definition. For f ∈ Inc(Θ, A), let C(
f)
:= {θ : f is continuous at θ
}.
For U ∈Φ, let C ′ (U ) = {θ′ : U (θ, a, ·) is continuous at θ′ for all (θ, a)
}.
Equivalently, using uniform continuity of U w.r.t. (θ, a), C ′ (U ) is the set of continuity of U ′ in L1,
where U ′ (θ′) : (θ, a) 7→U(θ, a,θ′
).
Lemma B.1. Both C(
f)
and C ′ (U ) are co-countable, have full measure, and are dense.
There are connections between L1 convergence and pointwise convergence. If fi → f in Inc(Θ, A),
fi (θ) → f (θ) for θ ∈ C(
f). We can also allow the inner variable to converge, giving fi (θi ) → f (θ) for
θi → θ. This property is true for U ∈Φ also.
Lemma B.2. 1. If fi , f ∈ Inc(Θ, A), fi → f = f∞, and θi → θ ∈C(
f), then fi (θi ) → f (θ).
2. If Ui ∈ Φ with Ui(θ, a,θ′
)uniformly continuous in (θ, a) over
(i ,θ′
), and Ui → U := U∞, and
θi → θ and ai → a and θ′i → θ′ ∈C ′ (U ), then Ui(θi , ai ,θ′i
)→U(θ, a,θ′
).
If strategies and payoffs converge, then separating equilibrium payoffs converge:
Lemma B.3. If fi → f in Inc(Θ, A), and Ui →U =U∞ with Ui(θ, a,θ′
)uniformly continuous in (a,θ)
over(i ,θ′
), (Ui ) fi
→U f in L1.
When varying Θ, the following result on the closeness of strategies defined on different Θ will be
useful:
Lemma B.4. If Θi ↗Θ∞, and fi , gi ∈ Inc(Θ∞, A) agree on Θi , then d(
fi , gi)→ 0. In particular if gi = g
is constant then fi → g .
Proof. This is trivial for Θ∞ finite so take Θ∞ = Θ. {θmin,θmax} ∈Θi . Then d(
fi , gi)≤ δ · |A|, where δ is
the width of the largest interval in Θ/Θi .
28
B.4 Topological lemmas: proofs
Proof of Lemma B.1
C(
f)
is co-countable since discontinuities of an increasing function are countable.
Claim. C ′ (U ) is co-countable:
Proof. Let U ′ (θ′) : (θ, a) 7→U(θ, a,θ′
). U ′ is weakly increasing. So is J : θ′ 7→ ∫
U ′ (θ′) (θ, a) d(θ, a).
If U has a discontinuity at(θ0, a0,θ′0
), the function θ′ → U
(θ0, a0,θ′
)must be discontinuous at
θ′0, since U is absolutely continuous in (a,θ). The function is weakly increasing, so must jump up at
θ′0, with a discontinuity of some δ > 0. Then there exists a ball B of area ε around (θ0, a0) such that∣∣U ′ (θ′) (θ, a)−U ′ (θ′) (θ0, a0)∣∣< δ/3 for any θ′. Then for any θ′− < θ′0 < θ′+, U ′ (θ′+)
(θ, a) >U ′ (θ′−)(θ, a)+
δ/3 on B , so∫
B U ′ (θ′+)−∫B U ′ (θ′−)≥ ε ·δ/3, so J
(θ′+
)− J(θ′−
)> ε ·δ/3, so J is discontinuous at θ′0. Since
J is weakly increasing, it can have at most countably many discontinuities.
For finite Θ, C(
f) = C ′ (U ) = Θ and all results are trivial. For Θ = Θ, both C
(f)
and C ′ (U ) must
contain {θmin,θmax}, since {θmin} and {θmax} are open sets. So co-countability implies C(
f)
has full
measure and is dense.
Proof of Lemma B.2
Both parts of the lemma follow from the more general statement:
Lemma. Suppose X and Y are metric measure spaces such that B (x,δ)×B(y,δ
)has positive measure
for x ∈ X , y ∈ Y ,δ > 0. Suppose the functions gi : X ×Y → R are measurable, with gi(x, y
)uniformly
continuous in x over(i , y
). Suppose gi → g = g∞. Suppose xi → x and yi → y, with g continuous at(
x, y). Then gi
(xi , yi
)→ g(x, y
).
Proof. Suppose otherwise. Then by uniform continuity of gi in x, gi(x, yi
)9 g
(x, y
). So, restricting to
a sub-sequence,∣∣gi
(x, yi
)− g(x, y
)∣∣≥ 3ε, for some ε> 0, for all i .
Take δ such that if d(x2, x1
)< 2δ, for any i and y∗,∣∣gi
(x2, y∗)− gi
(x1, y∗)∣∣≤ ε (by uniform conti-
nuity in x), and such that for d(y∗, y
)< δ,∣∣gi
(x, y∗)− gi
(x, y
)∣∣≤ ε (by continuity of gi at(x, y
)).
Take the open cube C = B (x,δ) × B(y,δ
). This has measure M > 0. On C ,
∣∣g i − g∣∣ ≥ ε. Then∥∥gi − g
∥∥L1 ≥ M ·ε, contradicting gi → g .
Proof of Lemma B.3
Claim. (Ui ) f is uniformly continuous in f over i .
Proof. If∥∥ f − g
∥∥L1 ≤ δ2, µ
({θ :
∣∣ f (θ)− g (θ)∣∣≥ δ}) ≤ δ. For any ε > 0, take 0 < δ ≤ ε such that for
|a2 −a1| < δ, |Ui (θ, a2,θ)−Ui (θ, a1,θ)| < ε for any (i ,θ). When ε= 1, this δ is δ1. Then for∥∥ f − g
∥∥L1 ≤
δ2,∫ ∣∣(Ui ) f (θ)− (Ui )g (θ)
∣∣ dθ ≤ ε+δ ·B ≤ ε · (1+B), where B = 1+|A|/k.
This implies∥∥(Ui ) fi
, (Ui ) f
∥∥L1
→ 0 as i →∞. So it is sufficient to show (Ui ) f →U f .
29
Claim. Ui(θ, f (θ) ,θ
)→U(θ, f (θ) ,θ
)for θ ∈C ′ (U )
Proof. Suppose not. Then d(Ui
(θ∗, f (θ∗) ,θ∗
),U
(θ∗, f (θ∗) ,θ∗
)) > 4.ε infinitely often for some θ∗ ∈C ′ (U ).
If Ui > U + 4ε at(θ∗, f (θ∗) ,θ∗
)infinitely often, then for θ′ ∈ [θ∗,θ∗+δ2] for some δ2 >
0, d(U
(θ∗, f (θ∗) ,θ∗
),U
(θ∗, f (θ∗) ,θ′
)) < ε, and so d(Ui
(θ∗, f (θ∗) ,θ′
),U
(θ∗, f (θ∗) ,θ′
)) > 3ε. By
uniform continuity in (θ, a), for d((θ, a) ,
(θ∗, f (θ∗)
)) < δ1, and θ′ ∈ [θ∗,θ∗+δ2] as above,
d(Ui
(θ, a,θ′
),U
(θ, a,θ′
))> ε i.o.. This contradicts Ui →U in L1.
The case of U >Ui +4ε at(θ∗, f (θ∗) ,θ∗
)i.o. proceeds similarly.
So restricting to C ′ (U ), which by Lemma B.1 has full measure, and using dominated convergence
(the spaces are bounded a.e.),∫ ∣∣Ui
(θ, f (θ) ,θ
)−U(θ, f (θ) ,θ
)∣∣ dθ→ 0 as required.
B.5 IC implies weakly increasing above a∗
Lemma B.5. If f satisfies IC on S and f ≥ a∗ on S, then f is weakly increasing on S.
Proof. Suppose θ1,θ2 ∈ S with θ1 < θ2, but f (θ1) > f (θ2). Then f (θ2) ≥ a∗ (θ2) ≥ a∗ (θ1). So
f (θ1) > f (θ2) ≥ a∗ (θ1). So by strict quasi-concavity,(
f (θ2) ,θ1) Âθ1
(f (θ1) ,θ1
). So by monotonicity(
f (θ2) ,θ2)Âθ1
(f (θ1) ,θ1
), contradicting IC.
B.6 Characterization of Sep(U )
For a weakly increasing function to be in Sep(U ), it is enough that is is weakly separating over a dense
subset of Θ:
Lemma B.6. The following are equivalent:
1. f ∈ Sep(U )
2. f ∈ Inc(Θ, A) and is weakly separating over C(
f)
3. f ∈ Inc(Θ, A) and is weakly separating over a dense subset S of Θ.
Proof. Despite the possibility of discontinuities in U and f , if f is weakly separating over T , the func-
tion U f must be uniformly continuous on T , as the supremum of the uniformly continuous collection
of functions{θ 7→U
(θ, f
(θ′
),θ′
),θ′ ∈ T
}. Combined with uniform continuity of U in the first argu-
ment, this implies uniform continuity of(θ,θ′
) 7→U(θ, f
(θ′
),θ′
)on T ×T .
1 ⇒ 2 Take g ∈ Sep(U ) with g = f a.e.. If f is continuous at θ ∉ {θmin,θmax}, then f (θ) = g (θ): oth-
erwise, since g is weakly increasing f 6= g on [θ−ε,θ] or on [θ,θ+ε] for some ε > 0. f = g on
{θmin,θmax} because {θmin} and {θmax} have positive measure. So f = g on C(
f), and so f is
weakly separating over C(
f).
2 ⇒ 3 C(
f)
is dense.
30
3 ⇒ 1 U f is uniformly continuous on S. Let U∗ (θ) := limθ′∈S,θ′→θU f(θ′
). U∗ extends U f from S to Θ
and is continuous.
There exists g (θ) satisfying: Ug (θ) =U∗ (θ) and f(θ′
) ≤ g (θ) for θ′ ∈ S, θ′ ≤ θ and g (θ) ≤ f(θ′
)for θ′ ∈ S, θ ≤ θ′. To achieve this, set g = f on S and outside S there is a solution to the
above equations lying between the values of S above and below θ. Such values exist because
{θmin,θmax} ⊆ S by the metric on Θ.
g = f on C(
f), so g = f a.e..
U∗ (θ) =U(θ, g (θ) ,θ
)≥U (θ, a,θmin) on S and so, by continuity of U∗ and of U in θ, g satisfies
IR on Θ.
U(θ, g
(θ′
),θ′
)is continuous in θ′as U
(θ′, g
(θ′
),θ′
)is and U is uniformly continuous in the first
argument. So since U(θ, g
(θ′
),θ′
)≤U(θ, g (θ) ,θ
)holds on S ×S it must hold on Θ×Θ by con-
tinuity in θ and θ′. So g satisfies IC and is weakly separating.
B.7 Proof of Lemma 1 (Sep(U ) closed)
Since Sep(U ) is a subset of the closure of Sep(U ), it is sufficient to show:
Claim. Sep(U ) is closed.
Proof. Suppose f ′i ∈ Sep(U ), and f ′
i → f . Then taking fi ∈ Sep(U ) with fi = f ′i a.e., we have fi → f .
Then fi (θ) → f (θ) on C(
f)
by Lemma B.2. Since U(θ, fi (θ) ,θ
)≥U (θ, a,θmin) for all a, we must have
U(θ, f (θ) ,θ
)≥U (θ, a,θmin) for all a. So f satisfies IR on C(
f). For any θ,θ′ ∈C
(f), U
(θ, fi
(θ′
),θ′
)≤U
(θ, fi (θ) ,θ
), and by taking the same limits, this holds for f . So f satisfies IC on C
(f). By Lemma B.6,
f ∈ Sep(U ).
B.8 Proof of Proposition 1 (continuity of Sep)
Parts 2 and 3 follow immediately from part 1:
Proposition. Suppose Θi ↗ Θ∞. Suppose Ui ∈ Φ (Θ∞), with Ui → U = U∞ and Ui(θ, a,θ′
)uniformly
continuous in (a,θ) over(i ,θ′
). Suppose fi ∈ Sep(Ui ) and fi ↗ f = f∞. Then f ∈ Sep(U∞).
Proof. If Θ∞ is finite, eventually Θi =Θ∞. A limit point of separating equilibria must converge point-
wise and be separating by continuity of U in a. So assume Θ∞ = Θ.
W.l.o.g., assume fi ∈ Sep(Ui ). Take fi ∈ Inc(Θ, A) with fi = fi |Θi . Then fi → f .
Let S :=C ′ (U )∩C(
f). By Lemma B.1, S is dense. By Lemma B.2, for θ ∈ S, fi (θi ) → f (θ) if θi → θ.
Since Ui → U in L1, for any(θ, a,θ′
)at which U∞ is continuous, there exists a sequence(
θi , ai ,θ′i) → (
θ, a,θ′)
with Ui(θi , ai ,θ′i
) → U∞(θ, a,θ′
). Moreover if θ′ is interior, the sequences θi
can be chosen to be monotonic in either direction. By uniform continuity of Ui in (θ, a) we have:
Ui(θ, a,θ′i
) →U∞(θ, a,θ′
)for some increasing and some decreasing monotonic sequence θ′i → θ′, for
interior θ′.
31
By Lemma B.2, Ui(θi , ai ,θ′i
)→U∞(θ0, a0,θ′0
)for θ′0 ∈C ′ (U∞).
For θ ∈ S, take θi → θ,θi ∈Θi . By IR for Ui , Ui(θi , fi (θi ) ,θi
)≥Ui (θi , a,θmin), and by taking limits,
U∞(θ, f (θ) ,θ
)≥U∞ (θ, a,θmin). So f satisfies IR for U∞.
For θ,θ′ ∈ S, take θi ,θ′i ∈Θi with(θi ,θ′i
)→ (θ,θ′
). By IC for Ui , Ui
(θi , fi (θi ) ,θi
)≥Ui(θi , fi
(θ′i
),θ′i
),
and by taking limits U∞(θ, f (θ) ,θ
)≥U∞(θ, f
(θ′
),[θ′
]), so f satisfies IC for U∞.
By Lemma B.6, f ∈ Sep(U∞).
B.9 Proof of Lemma 2 (a finite equilibrium)
Definition 2 implies χ (θ0) = a∗ (θ0). If 1 uniquely defines a weakly separating equilibrium, that equi-
librium must satisfy IC, so must satisfy definition 2, and uniquely since the condition is stricter. So
only the claims about definition 1 need to be proved:
Claim. Definition 1 defines a unique χ, weakly increasing, with χ≥ a∗, satisfying IC and IR, when Θ is
restricted to {θ0, . . .θi }, for any i .
Proof. This is true for i = 0. Suppose it is true for i −1 (with i ≥ 1), giving χ′ on {θ0 . . .θi−1}.
Let Ai := {(χ (θi−1) ,θi−1
)ºθi−1 (a,θi )}. Then Bi := Ai ∩
{a ≥χ (θi−1)
}.
For a solution to exist for χ (θi ), Bi must be non-empty. For U ∈ ΦB , the set Bi is guaranteed to
be non-empty: it contains amax because (amax,θi ) ¹θi−1 U (a,θmin) for some a by the bound condition,
and (a,θmin) ¹θi−1
(χ (θi−1) ,θi−1
)by IR for θi−1. The maximum of U (θi , a,θi ) on Ai occurs on Bi : if
x ∈ Ai \Bi , then (x,θi ) ¹θi−1
(χ (θi−1) ,θi−1
)∼θi−1 (minBi ,θi ), so by single crossing (x,θi ) ≺θi (minBi ,θi ).
Using the fact that U (θi , a,θi ), strictly quasi-concave in a, Bi is a closed finite interval, and definition
1 defines a unique χ (θi ), with χ (θi ) ≥χ (θi−1).
So definition 1 gives χ uniquely on {θ0, . . .θi }, weakly increasing, with χ=χ′ on {θ0, . . .θi−1}.
χ satisfies IC for (θ,θi ), θ < θi : by definition(χ (θi−1) ,θi−1
)ºθi−1
(χ (θi ) ,θi
). Since χ (θi−1) ≤ χ (θi ),
by single crossing,(χ (θi−1) ,θi−1
) ºθ (χ (θi ) ,θi
)for θ < θi . By IC for χ′,
(χ (θ) ,θ
) ºθ (χ (θi−1) ,θi−1
) ºθ(χ (θi ) ,θi
).
χ satisfies IC for (θi ,θ), θ < θi :(χ (θi−1) ,θi−1
) ∼θi−1 (minBi ,θi ), so by single crossing,(χ (θi−1) ,θi−1
) ¹θi (minBi ,θi ) ¹θi
(χ (θi ) ,θi
). By IC for χ′,
(χ (θi−1) ,θi−1
) ºθi−1
(χ (θ) ,θ
), and by sin-
gle crossing,(χ (θi−1) ,θi−1
)ºθi
(χ (θ) ,θ
), so
(χ (θi ) ,θi
)ºθi
(χ (θ) ,θ
).
So χ satisfies IC. χ also satisfies IR:
Suppose (a,θmin) Âθi
(χ (θi ) ,θi
). Then a∗ (θi ) < χ (θi ) and a < χ (θi ), or else (a,θmin) ¹θi (a,θi ) ¹θi(
χ (θi ) ,θi). We know (a,θmin) Âθi
(χ (θi ) ,θi
)ºθi (amax,θi ), so by the intermediate value theorem there
existsc ∈ [a, amax] with (a,θmin) ∼θi (c,θi ). By single crossing, (a,θmin) ºθi−1 (c,θi ). By IR for θi−1,(χ (θi−1) ,θi−1
) ºθi−1 (a,θmin), so(χ (θi−1) ,θi−1
) ºθi−1 (c,θi ). So Ai contains c, and so(χ (θi ) ,θi
) ºθi
(c,θi ) ∼θi (a,θmin), contradicting the assumption. So IR is satisfied for θi .
32
B.10 Proof of Proposition 2 (Riley equilibrium)
Proof of part 1 (existence)
Let U max (θ) := sup f ∈Sep±(U ) U f (θ). Let A (θ) := {a : U (θ, a,θ) ≥U max (θ)
}. A(θ) is a non-empty closed
interval containing a∗ (θ) and excluding amax. Let g (θ) = max A (θ). Then a∗ ≤ g , and Ug ≥ U f for
f ∈ Sep± (U ).
Claim. g ∈ Sep± (U ).
Proof. For each θ, g (θ) is a limit point of actions f (θ) satisfying separating IR for θ, so g satisfies
separating IR by continuity of U w.r.t. a.
Now take any θ1,θ2. Take a sequence fn ∈ Sep± (U ) with Ug (θ2) = limU fn (θ2). We have
U(θ1, g (θ1) ,θ1
) ≥ U(θ1, fn (θ1) ,θ1
) ≥ U(θ1, fn (θ2) ,θ2
), the second inequality by IC for fn , and taking
limits gives U(θ1, g (θ1) ,θ1
)≥U(θ1, g (θ2) ,θ2
). So g satisfies IC.
Since g ≥ a∗, g is weakly increasing by Lemma B.5. So g ∈ Sep(U ).
Proof of part 2 (uniqueness)
As in the proof of part 1, let U max (θ) := supq∈Sep±(U ) Uq (θ), A (θ) := {a : U (θ, a,θ) ≥U max (θ)
}, and
g (θ) := max A (θ). Also let h (θ) := min A (θ).
Suppose f ∈ Sep± (U ) and g are both maximal, so that U f = Ug . Let S := {θ : f (θ) 6= g (θ)
}. Then
on S, h (θ) and g (θ) are distinct and lie strictly on either side of a∗ (θ). On S, f (θ) = h (θ), and outside
f (θ) = g (θ).
Claim. There is no θ∗ ∈ S with f (θ) ≤ f (θ∗) for θ < θ∗.
Proof. Suppose this were true for θ∗.
For some maximal interval (θ∗∗,θ∗] or [θ∗∗,θ∗], f (θ) = f (θ∗).
This can be split into two intervals I1 and I2, where I1 6⊆ S and ; 6= I2 ⊆ S, because g is weakly
increasing. (If U(θ, a,θ′
)is strictly increasing in θ′, I1 =∅ and I2 =
{θ∗
}.)
For θ < θ∗∗, f (θ) < f (θ∗). IC requires that(
f (θ) ,θ) ºθ
(f (θ∗) ,θ∗
). So for θ < θ∗, f (θ∗) ∉(
h (θ) , g (θ)). So f (θ∗) ≥ g (θ), since f (θ∗) > f (θ).
This holds also for I1. So for θ below I2, f (θ∗) ≥ g (θ).
So(
f (θ) ,θ) ºθ (
a∗ (θ′
),θ′
)for θ′ ∈ I2. So if f
(θ′
)is altered to a∗ (
θ′)
on I2, no lower type wants to
mimic a type in I2, nor does a type in I2 want to mimic any lower type. Moreover because there are
no signaling incentives within I2 (which follows from condition 2 of Φ), no type in I2 wants to mimic
another type in I2.
Then define α (θ) to be equal to f below θ∗∗, equal to a∗ (θ′
)on I2, and on [θ∗,θmax] equal to a
weakly increasing separating equilibrium of [θ∗,θmax] starting at a∗(θ∗). By Proposition 3, this exists.
By considerations above, α (θ∗) ∈ Sep± (U ), and yet Uα (θ∗) >U f (θ∗), a contradiction.
a) Suppose f ∈ Sep(U ). Since f is weakly increasing, f (θ) ≤ f (θ∗) for θ < θ∗, and f = g by the above
argument.
33
b) Suppose f ∈ Sep± (U ).
i. Suppose Θ is finite. Let θ∗ = minS. Again f (θ) ≤ f (θ∗) for θ < θ∗, so f = g .
ii. Suppose Θ is an interval. Take θ1,θ3 ∈ S with θ1 < θ3.(g (θ1) ,θ1
) ºθ1
(g (θ3) ,θ3
)because
g ∈ Sep(U ) and the reverse is true for θ3, so by continuity of u in θ, we have for some
θ2 ∈ (θ1,θ3):(g (θ1) ,θ1
) ∼θ2
(g (θ3) ,θ3
). Since (h (θ1) ,θ1) ∼θ1
(g (θ1) ,θ1
), we have by sin-
gle crossing (h (θ1) ,θ1) ≺θ2
(g (θ1) ,θ1
)and similarly (h (θ3) ,θ3) Âθ2
(g (θ3) ,θ3
). Putting
the inequalities together, we have (h (θ3) ,θ3) Âθ2 (h (θ1) ,θ1). Therefore we must have
h (θ3) > h (θ1) because otherwise single crossing gives (h (θ3) ,θ3) Âθ1 (h (θ1) ,θ1) and so
(h (θ3) ,θ3) Âθ1 (h (θ1) ,θ1), contradicting IC on f . So f must be increasing on S.
Moreover on S, we must have g (θ1) ≤ h (θ2) or else h (θ2) = f (θ2) ∈ (h (θ1) , g (θ1)
)and so
θ1 would want to mimic θ2.
Therefore the non-empty open intervals(h (θ) , g (θ)
)over θ ∈ S are disjoint. Therefore S
must be countable.
B.11 Proof of Lemma 4 (characterizations of R (U ))
1.
(a) Let f = R (U ). Then f ≥ a∗ and satisfies IC. Suppose, contrary to the claim, that α ≥ a∗
satisfies IC, with α (θ∗) < f (θ∗) for some θ∗. Let β (θ) := min({
f (θ) ,α (θ)})
.
Uβ (θ) ≥ U f (θ), so β satisfies IR since f does. Since both f and β satisfy IC,
U(θ,β
(θ′
),θ′
)≤ max({
U f (θ) ,Uα (θ)})=Uβ (θ), so β satisfies IC. So β ∈ Sep± (U ).
But U f 6≥Uβ, since Uβ (θ∗) >U f (θ∗), contradicting f =R (U ).
(b) This is immediate from 1.a), since elements of Sep(U ) satisfy IC.
2. Suppose χ differs from R (U ) first at θ∗. θ∗ must have weakly lower utility under χ, but χ (θ∗)
uniquely maximizes utility subject to separating from lower types, which R (U ) (θ∗) must also
do. So χ (θ∗) =R (U ) (θ∗).
B.12 Proof of Proposition 3 (continuity of R)
Since R (U ) is dominant, it maximizes any integral of payoffs over types over separating equilibria, and
it does so uniquely:
Lemma B.7. If I(
f)
:= ∫U f (θ) dα (θ), with α a measure on Θ, then R (U ) maximizes I on Sep(U ). If α
is absolutely continuous with respect to µΘ, then the maximum is unique up to a.e.(µ)
(and so a.e. (α))
equivalence.
Proof. I (R (U ))− I(
f)= ∫ (
UR(U ) (θ)−U f (θ))
dα (θ). This is weakly positive since the inner term is for
all θ, proving the first part. If the integral is 0, then the function UR(U ) −U f must be 0a.e.(α).
34
Proof of part 2 (continuity inΘ)
Let iΘ : Inc(Θ∞, A) → Inc(Θ, A) be restriction to Θ: iΘ(
f) 7→ f |Θ. Then given a set S of functions f ∈
Inc(Θ, A), i−1Θ (S) is the set of functions in Inc(Θ∞, A) whose restriction to Θ is in S.
If f is a measurable function Θ→ A, let I(
f)
:= ∫U f (θ)dθ. By Lemma B.3, I is continuous.
Let Sep∗ (U ) ⊆ Sep(U ) be the subset of functions f in Sep(U ) with f ≥ a∗U .
Claim. Some member ri of i−1Θi
R(U |Θi
)maximizes I
(f)
over f ∈ i−1Θi
Sep∗ (U |Θi
).
Proof. For i =∞ this follows from Lemma B.7, part 2. For i <∞:
Let r ′i := R
(U |Θi
), and let ri (θ) := r ′
i (θ) on Θi and elsewhere choose ri to maximize U (θ, ·,θ)
subject to ri (θ) ≥ r ′i (θ−), for all θ− ∈ Θi ,θ− ≤ θ. Then ri (θ) ≤ r ′
i
(θ+
)for θ+ ∈ Θi ,θ ≤ θ+, because
r∗i
(θ+
)≥ a∗ (θ+
)≥ a∗ (θ). So ri is increasing and so ri ∈ i−1Θi
R(U |Θi
).
Suppose f ∈ i−1Θi
Sep∗ (U |Θi
). Then f (θ) ≥ f (θ−) ≥ r∗
i (θ−), for Θi 3 θ− ≤ θ. So Uri (θ) ≥U f (θ) and
so ri maximizes θ→U f (θ) on f ∈ i−1Θi
Sep∗ (U |Θi
)and so maximizes I .
Let the maximum be mi = I (ri ). Then i → mi is decreasing because i → i−1Θi
Sep∗ (U |Θi
)is decreas-
ing (w.r.t. set inclusion). In particular mi ≥ m∞.
Take a convergent sub-sequence of ri ; by Proposition 1.2 this must converge to an element of
r∗ ∈ Sep(U ). By Lemma B.3, mi → I (r∗), so I (r∗) ≥ m∞.
By Lemma B.7, I is uniquely maximized (up to a.e. equality) over Sep(U ) with maximum m∞. So
r∗ =R (U ) a.e.. So ri →R (U ) since all convergent sub-sequences converge to this.
Since ri |Θi =R(U |Θi
), Lemma B.4 implies that R
(U |Θi
)↗R (U ).
Proof of part 3 (continuity in U )
Let ri :=R (Ui ), so we need to show ri → r∞.
For finite Θ this is true by using the finite characterization of the Riley equilibrium. This gives
pointwise convergence, which is equivalent to L1 convergence for finite Θ. So assume Θ= Θ.
Restrict to a convergent sub-sequence ri → r∗, suppressing sub-sequence notation. By Proposi-
tion 1.1, r∗ ∈ Sep(U∞).
Let Ii(
f)
:= ∫(Ui ) f (θ) dµ (θ). By Lemma B.3, I∞ (ri ) → I∞ (r∗) and Ii (ri ) → I∞ (r∗).
Claim. Ii (ri ) → I∞ (r∞).
Proof. Define finite approximations Θi to Θ∞ := Θ, such that each finite Θi contains no discontinu-
ities of U∞. Then on Θ j , Ui →U∞, because(a 7→Ui
(θ, a,θ′
)) → (a 7→U∞
(θ, a,θ′
))uniformly for each(
θ,θ′), because Ui
(θ, a,θ′
) →U∞(θ, a,θ′
)pointwise with uniform continuity in a. So using the finite
result, R(U j |Θi
)→R(U∞|Θi
).
As in the proof of part 2, let r ij ∈ i−1
ΘiR
(U j |Θi
)maximize I j
(f). From the above, r i
j → r i∞, and so
I j
(r i
j
)→ I∞
(r i∞
)by Lemma B.3.
Also I j
(r i
j
)↘ I j
(r∞
j
)for each j . That the limit is I j
(r∞
j
)is implied by part 2. If follows that
I j(r j
)= I j
(r∞
j
)→ I∞
(r∞∞
)= I∞ (r∞).
35
So I∞ (r∞) = I∞ (r∗). So r∗ ∈ Sep(U∞) maximizes I∞, so r∗ =R (U∞).
Proof of part 1 (continuity in U and Θ)
By Lemma B.4, it is enough to show that there exists ri ∈ Inc(Θ∞, A) with ri |Θi = R(Ui |Θi
)and ri →
R (U∞).
As in the proof of part 3, let I j(
f)
:= ∫ (U j
)f (θ) dµ (θ), and let r i
j ∈ i−1Θi ,Θ∞R
(U j |Θi
)maximize I j
(f).
By Proposition 1.1, there is an element r∗ of Sep(U ) with a sub-sequence of(r i
i
)converging to r∗.
We have: I j
(r i
j
)→ I∞
(r i∞
)by part 3, and I j
(r i
j
)↘ I j
(r∞
j
)by part 2.
Since Ix(r y
x)
is decreasing in y , fori ≥ j , Ii(r∞
i
) ≤ Ii(r i
i
) ≤ Ii
(r j
i
). Taking the limit i → ∞,
I∞(r∞∞
) ≤ lim Ii(r i
i
) ≤ I∞(r j∞
), where the middle term may be set-valued, and taking limits in j ,
I∞(r∞∞
)≤ lim Ii(r i
i
)≤ I∞(r∞∞
), so lim Ii
(r i
i
)= I∞(r∞∞
)= I (R (U∞)).
By Lemma B.3, for the sub-sequence of(r i
i
)converging to r∗, Ii
(r i
i
)converges to I (r∗). So I (r∗) =
I (R (U∞)), and so r∗ =R (U∞) and ri →R (U∞).
B.13 Proof of Proposition 4 (IC and uniqueness)
Proof of part 1 (uniqueness with initial condition)
f is weakly increasing by Lemma B.5.
Fix δ> 0. For δ< δ, let Uδ
(θ, a,θ′
):=U
(θ+δ, a,θ′
)on
[θmin,θmax − δ
](which includes Θi ).
f ≥R (U ) by Lemma 4.1.
Claim. If θ < θ′ < θ+δ, then f(θ′
)≤ max{
x : Uδ
(θ, x,θ′
)=Uδ
(θ, f (θ) ,θ
)}. I.e. f
(θ′
)is weakly less than
what is needed to separate from θ at f (θ) according to Uδ.
Proof. By IC for f , U(θ′, f
(θ′
),θ′
)≥U(θ′, f (θ) ,θ
).
Therefore, by single crossing, since f is weakly increasing and θ′ ≤ θ + δ, U(θ+δ, f
(θ′
),θ′
) ≥U
(θ+δ, f (θ) ,θ
)Rewriting, Uδ
(θ, f
(θ′
),θ′
)≥Uδ
(θ, f (θ) ,θ
). So f
(θ′
)≤ max{
x : Uδ
(θ, x,θ′
)=Uδ
(θ, f (θ) ,θ
)}.
Claim. f ≤ rδ :=R (Uδ)
Proof. 1. It follows from the previous claim that: If a∗Uδ
(θ) ≤ f (θ) ≤ rδ (θ), then Uδ
(θ, f (θ) ,θ
) ≥Uδ (θ,rδ (θ) ,θ) and so f
(θ′
)≤ max{
x : Uδ
(θ, x,θ′
)=Uδ (θ,rδ (θ) ,θ)}≤ rδ
(θ′
), for θ < θ′ < θ+δ.
2. So if f ≥ a∗Uδ
on [θ1,θ2] and f ≤ rδ at θ1, then f ≤ rδ on [θ1,θ2] follows, by splitting the interval
up into types at most δ apart.
3. Since U and therefore Uδ are continuous, the separating equilibria f and rδ are continuous.
4. rδ ≥ a∗Uδ
. Suppose f (θ∗) > rδ (θ∗). From 2., f ≥ a∗Uδ
must be violated on [θmin,θ∗]. (Since
f (θmin) = a∗ (θmin) ≤ a∗Uδ
(θmin) = rδ (θmin)
5. Take θ < θ∗ minimal such that f ≥ a∗Uδ
on[θ,θ∗
]. By 4. and continuity, f
(θ) = a∗
Uδ
(θ)
and so
f ≤ rδ at θ. By 2., f ≤ rδ on[θ,θ∗
], a contradiction.
36
Uδ→U as δ→ 0, so by Proposition 3.3, rδ =R (Uδ) →R (U ) on[θmin,θmax − δ
].
Since R (U ) ≤ f ≤ R (Uδ) for all δ > 0, f = R (U ) on[θmin,θmax − δ
]. This is true for all δ, so f =
R (U ) on [θmin,θmax), and by continuity of f on Θ.
Proof of part 2 (uniqueness without initial condition)
Let f (θ) :=a θ = θmin
f (θ) θ > θmin
.
On θ > θmin, IC, together with f ≥ a∗, imply that f (θ) ≥ a. Lemma B.5 implies that f is weakly
increasing on this set. So f ≥ a and is weakly increasing. By continuity of U and IC for (θ,θ0), f is
continuous at θmin and so f is continuous, since f is continuous on (θmin,θmax].
f satisfies IC on (θmin,θmax] since f does, and so satisfies IC on (θmin,θmax] by continuity of U and
f . Restricted to A, f satisfies IR at θmin: i.e. a∗(U |A) (θmin) = f (θmin) = a.
Part 1 then implies that U |A ∈ΦSep(
A)
and f =R(U |A
).
B.14 Proof of Proposition 5 (a condition for R (U1) ≤R (U2))
First assume Θ = {θ0, . . .θk } is finite. Suppose R (U1) (θi ) ≤ R (U2) (θi ). Then R(U j
)(θi+1) =
max(s j
i , a∗U j
(θi+1)), where s j
i ≥R(U j
)(θi ) and U j
(θi ,R
(U j
)(θi ) ,θi
)=U j
(θi , s j
i ,θi+1
). Then:
U1 (θi ,R (U1) (θi ) ,θi ) ≥U1 (θi ,R (U2) (θi ) ,θi ) ≥U1(θi , s2
i ,θi+1)
The second inequality above holds by U1 ≤′SC U2. So U1
(θi , s1
i ,θi+1) ≥ U1
(θi , s2
i ,θi+1), so s1
i ≤ s2i .
Also a∗U2
(θi+1) ≥ a∗U1
(θi+1). So R (U1) (θi+1) ≤R (U2) (θi+2).
So by induction, R (U1) ≤R (U2).
Proposition 3.2 extends the result to Θ= Θ.
B.15 Proof of Proposition 6 (supermodular comparative statics)
Proof of part 1 (strategy increasing in parameter)
This is an immediate corollary of Proposition 5.
Proof of part 2 (supermodular value function)
We shall use a result from Topkis (1978):
Theorem. (Topkis (1978)) If u(a j , a− j
)is supermodular, then maxa j u
(a j , a− j
)is supermodular in a− j .
Claim. v is weakly supermodular.
37
Proof. First assume Θ is finite. Take adjacent θ1 < θ2.
Let ZB be the set of z such that θ2 is weakly bound by separation from θ1, and ZU the set of z such
that θ2 is not strictly bound by separation from θ1:
ZB := {z : U (z)
(θ1, f (z) (θ1) ,θ1
)= U (z)(θ1, f (z) (θ2) ,θ2
)}ZU :=
{z : f (z) (θ2) = argmax
a
(U (z) (θ2, a,θ2)
)}On z ∈ ZB and θ ∈ {θ1,θ2}, v (z,θ) = U (z)
(θ, f (z) (θ2) ,θ2
), and v (z,θ) has weakly increasing differ-
ences in (z,θ) on this set since U (z) (θ, a,θ2) has weakly increasing differences in (z,θ) and (z, a) and
f (z) (θ2) is increasing in x (part 1).
On z ∈ XU , utility for θ2 is maxa w (z,θ2, a), where w (z,θ, a) := U (z) (θ, a,θ). w is weakly super-
modular, since U (z)(θ, a,θ′
)is. So by Topkis (1978), maxa w (z,θ, a) is a weakly supermodular function
of (z,θ), and more generally so is maxa≥l w (z,θ, a) for any l .
Take z2 > z1 in ZU and let l := f (z1) (θ1).
v (z2,θ2)− v (z1,θ2) = maxa
w (z2,θ2, a)−maxa
w (z1,θ2, a)
= maxa≥l
w (z2,θ2, a)−maxa≥l
w (z1,θ2, a)
≥ maxa≥l
w (z2,θ1, a)−maxa≥l
w (z1,θ1, a)
= maxa≥l
w (z2,θ1, a)− v (z1,θ1)
≥ v (z2,θ1)− v (z1,θ1)
The first inequality holds by weakly increasing differences of maxa≥l w (z,θ, a). The second holds
because f (z2) (θ1) ≥ l by part 1, so v (z2,θ1) = w(z2,θ1, f (z2) (θ)
)≤ maxa≥l w (z2,θ1, a).
So on {θ1,θ2} weakly increasing differences holds over ZB and ZU , closed subspaces of Z whose
union is Z . Between any two points in zB ∈ ZB , zU ∈ ZU , there is a point in ZB∩ZU , so weakly increasing
differences holds over Z . So weakly increasing differences holds over Θ and Z .
Proposition 3.2 and Lemma B.3 extend the result to Θ= Θ.
Proof of part 3 (value increasing in parameter)
v (z,θmin) is weakly (strictly) increasing in x: in the Riley equilibrium θmin takes an action a maximizing
U (z) (θmin, a,θmin), and this maximum must be weakly (strictly) increasing in z since U (z)(θ, a, θ
)is
weakly (strictly) increasing in z.
Weak supermodularity of v (part 2) then implies the result.
38
B.16 Proof of Lemma 5 (bounds on comparative statics)
Proof of 1 (bound on increase in strategy)
Let dz = z1 − z0. Let Θ′ be the set of θ for which R(U (z0)
)(θ) ≤ amax −α ·dz . Outside this set, the claim
holds trivially. Now restrict to Θ′.Let U
(θ, a,θ′
):= U (z1)
(θ, a +α ·dz ,θ′
)on A′ := [amin, amax] − α · dz , and let A′′ :=
[amin, amax −α ·dz ].
Claim. U ≤′SC U (z0) on A′′.
Proof. Let ∆ = (U (z0)−U
). Then ∆
(θ, a,θ′
) = U (z0)(θ, a +dz ,θ′
)−U (z0 +dz )(θ, a +α ·dz ,θ′
). Then
∆(θ, a2,θ′
)−∆(θ, a1,θ′
)≤ 0 for a1 < a2, and ∆(θ, a,θ′2
)−∆(θ, a,θ′1
)= 0.
Then U ∈ΦSep(
A′′), and:
R(U (z1)
) = R(U
)+α ·dz
≤ R(U |A′′
)+α ·dz
≤ R(U (z0)
)+α ·dz
Proof of part 2 (bound on supermodularity)
Let w (z,θ, a) := U (z) (θ, a,θ).
Claim. Suppose U (z)(θ, a,θ′
)is continuous. For any l , w∗ : (z,θ) 7→ maxa≥l w (z,θ, a) has an increas-
ing difference bound of(βzθ+βzθ′ +α ·βza
).
Proof. First suppose Θ= Θ.
Let a∗ (z,θ) = argmaxa≥l w (z,θ, a). a∗ is continuous, so a∗ (z,θ) is uniformly continuous in θ for
each z. Since w has an increasing difference bound of δ :=βθa +βaθ′ in (θ, a), for θ < θ′:
w(z,θ′, a∗ (
z,θ′))+w
(z,θ, a∗ (z,θ)
)−w(z,θ, a∗ (
z,θ′))−w
(z,θ′, a∗ (z,θ)
)≤δ · (θ′−θ)(
a∗ (z,θ′
)−a∗ (z,θ))
For θ1 < θ2, and 0 ≤ i ≤ n, let θni := i
n ·θ2 + n−in ·θ1. Then:
w∗ (z,θ2)−w∗ (z,θ1) =n−1∑i=0
[w∗ (
z,θni+1
)−w∗ (z,θn
i
)]=
n−1∑i=0
[w
(z,θn
i+1, a∗ (z,θn
i
))−w(z,θn
i , a∗ (z,θn
i
))]+o (1)
Also w has an increasing difference bound of βzθ+βzθ′ in (z,θ). So:
39
w∗ (z2,θ2)+w∗ (z1,θ1)−w∗ (z1,θ2)−w∗ (z2,θ1)
= limn−1∑i=0
[w(z2,θn
i+1, a∗ (z2,θn
i
))−w(z2,θn
i , a∗ (z2,θn
i
))−w
(z1,θn
i+1, a∗ (z1,θn
i
))+w(z1,θn
i , a∗ (z1,θn
i
))]
≤ limn−1∑i=0
[(βzθ+βzθ′
)(θn
i+1 −θni
)(z2 − z1)+βza ·
(a∗ (
z2,θni
)−a∗ (z1,θn
i
))(θn
i+1 −θni
)]= (
βzθ+βzθ′)
(θ2 −θ1) (z2 − z1)+βza · (θ2 −θ1) ·∫ [
a∗ (z2,θ)−a∗ (z1,θ)]
dθ
≤ (βθz +βθ′z +α ·βza
)(θ2 −θ1) (z2 − z1)
This shows the result for Θ= Θ.
For finite Θ the assumption that there is a member of Φ(Θ
)extending each U (z) means the result
still applies.
Claim. The result holds when Θ is finite and U (z)(θ, a,θ′
)is strictly increasing in θ′.
Proof. We can assume U (z)(θ, a,θ′
)is continuous in z by subtracting the function
U (z) (θmin, amin,θmin) of z: this is continuous by the assumed increasing difference bounds.
Then U (z)(θ, a,θ′
)is continuous.
Take two adjacent θ1 < θ2. As in the proof of Proposition 6.2, define:
ZB := {z : U (z)
(θi , f (z) (θi ) ,θi
)= U (z)(θi , f (z) (θi+1) ,θi+1
)}ZU :=
{z : f (z) (θi+1) = argmax
a
(U (z) (θi+1, a,θi+1)
)}1. On z ∈ ZB and θ ∈ {θ1,θ2}, v (z,θ) = U (z)
(θ, f (z) (θ2) ,θ2
).
For z1, z2 ∈ ZB , z1 < z2, take ai := f (zi ) (θ2), so that 0 ≤ a2 −a1 ≤α · (z2 − z1):
v (z2,θ2)+ v (z1,θ1)− v (z2,θ1)− v (z1,θ2)
= U (z2) (θ2, a2,θ2)+U (z1) (θ1, a1,θ2)−U (z2) (θ1, a2,θ2)−U (z1) (θ2, a1,θ2)
= [U (z2) (θ2, a1,θ2)+U (z1) (θ1, a1,θ2)−U (z2) (θ1, a1,θ2)−U (z1) (θ2, a1,θ2)
]+[
U (z2) (θ2, a2,θ2)+U (z2) (θ1, a1,θ2)−U (z2) (θ1, a2,θ2)−U (z2) (θ2, a1,θ2)]
≤ βzθ · (z2 − z1) (θ2 −θ1)+βza · (a2 −a1) (θ2 −θ1)
≤ (βzθ+α ·βza
)(z2 − z1) (θ2 −θ1)
2. Now take z1, z2 ∈ ZU , z1 < z2.
Let l := f (z2) (θ1) ≤ f (z1) (θ2). Define w∗ and a∗ accordingly. Then v (z2,θ1) = w∗ (z2,θ1) and
v (z1,θ1) ≤ w∗ (z1,θ1), so:
40
v (z2,θ1)− v (z1,θ1) ≥ w∗ (z2,θ1)−w∗ (z1,θ1)
v (z2,θ2)− v (z1,θ2) = w∗ (z2,θ2)−w∗ (z1,θ2)
So:
v (z2,θ2)+ v (z1,θ1)− v (z1,θ2)− v (z2,θ1)
≤ w∗ (z2,θ2)+w∗ (z1,θ1)−w∗ (z1,θ2)−w∗ (z2,θ1)
≤ (βzθ+βzθ′ +α ·βza
)(θ2 −θ1) (z2 − z1)
Since U (z)(θ, a,θ′
)is strictly increasing in θ′, f (z) (θ2) > f (z) (θ1). Since f is continuous in z,
δ := max(
f (z) (θ2)− f (z) (θ1))
is attained, so δ> 0. Since f is uniformly continuous in z, for z2−z1 < ε,
f (z2) (θi )− f (z2) (θi ) < δ. Then for z2 − z1 < ε, f (z2) (θ1) < f (z1) (θ1)+δ≤ f (z1) (θ2).
So the property holds when z2 − z1 < ε.
3. Therefore v has an increasing difference bound of βzθ+βzθ′+α·βza on {θ1,θ2}×[z, z +ε] for any
interval [z, z +ε] ∈ Z . So it has the increasing difference bound on {θ1,θ2}×Z . So it has it on Θ×Z .
Claim. The result holds when Θ is finite.
Proof. Define U ′i (z)
(θ, a,θ′
):= U (z)
(θ, a,θ′
)+εiθ′, where εi → 0. The result holds for U ′
i by the previ-
ous claim, and taking limits, for U .23
Claim. The result holds for continuum Θ.
Proof. Take finite approximations Θi ↗Θ= Θ and define: U ′i (z)
(θ, a,θ′
):= ∫
U (z)(θ, a,θ′+ζ) dµi (ζ),
where µi has a truncated N (0,εi ) distribution (say) and where εi → 0. U ′i (z)
(θ, a,θ′
)preserves the
increasing difference bounds and converges in L1 to U (z) for each z. Then Proposition 3.1 gives the
result for U .
B.17 Proof of Proposition 7
B.17.1 Proof of 1
The strategy f and initial distribution ψ∗ result in a measure µΘ×A over types and signals, which pro-
jected to A is µA (S) :=µΘ×A (Θ×S).
Let Sp(a) = ⋂{T :µΘ×A
(T C ×O
)= 0 for some open set O containing a}. I.e. Sp(a) is the set of
types taking action a.
23To ensure U ′i (z) ∈ ΦSep, A may have to be extended and U ′
i
(θ, amax +ε,θ′
)set to U ′
i
(θ, amax,θ′
)+v(θ,ε), where v is strictly supermodular and strictly decreasing in ε. This will guarantee that U ′
i (z) ∈ΦSep.
41
1. No pooling Supposeψ∗ (S) > 0, and Sp(a1)∩S 6=∅. I.e. a1 is in the support of actions taken by types
in a set S of positive measure. (This is equivalent to: for any open set O containing a1,µΘ×A (S ×O) > 0.)
Let r1 := limsupa→a1ρ
(β (a) , a
)be the highest response to an action close to a1.24 Then u f ,β (θ∗) =
v (θ∗, a1,r1) for some type θ∗ in S.
Let ah be the higher solution of v (θ∗, a,r1) = v (θ∗, a1,r1), so that v (θ∗, a,r1) is strictly decreasing
after ah . For some ε, for all a2 ∈ [ah , ah +ε), there exists r (a2) such (a1,r1) ∼θ∗ (a2,r (a2)). r (ah) = r1,
and r is strictly increasing.
All lower types strictly prefer (a1,r1) to (a2,r (a2)). So all lower types prefer their equilibrium ac-
tions to (a1,r1) (weakly) to (a2,r (a2)) (strictly). So for any δ > 0 there exists some some r ′ such that
types above θ∗−δ strictly prefer(a2,r ′) to (a1,r1), and lower types reverse this preference. In the con-
tinuous case, r ′ is found by making type θ∗−δ indifferent.
So by D1, β (a2) ≥ [θ∗−δ] for all δ, so β (a2) ≥ [θ∗]. So ρ(β (a2) , a2
) ≥ ρ ([θ∗] , a2) ≥ ρ ([θ∗] , a1). It
follows that ρ ([θ∗] , a1) ≤ r1 or else ρ(β (a2) , a2
) ≥ ρ ([θ∗] , a1) > r1 for all a2 > ah and type θ∗ would
gain by taking a2 close to ah .
Since this is true for some θ∗ ∈ S for all positive measure S with Sp(a1) ∩ S 6= ∅, taking Sε =[x −ε, x], where x = sup
(Sp(a1)
), and taking a limit ε → 0 gives ρ ([x −ε] , a1) ≤ r1, so by continuity
ρ(sup
(Sp(a1)
), a1
)≤ r1.
We now have the result for finite types: in a pool with highest type θ∗, take S = {θ∗
}; then
ρ ([θ∗] , a1) > ρ (β (a1) , a1
)because [θ∗] >β (a1). In general:
Suppose β (a) is non-degenerate for actions a ∈ AP with positive measure µA (AP ) > 0. Then for
almost all (µA) a1 ∈ AP , β (a1) < sup(Sp(a1)
). So ρ
(β (a1) , a1
) < ρ(sup
(Sp(a1)
), a1
) ≤ r1. This means
taking a1 is not rational for any type, since taking a nearby signal is superior. So a1 is not taken by any
type, contradicting that such a1 have positive measure.
Therefore β (a) is degenerate for almost all (µA) a.
2. Riley equilibrium Here we need to apply one of the extra conditions.
a) Finite types By induction. Suppose the equilibrium has f (θ) = R (U ) (θ) up to θi . Let a :=R (U ) (θi+1), and r := ρ ([θi+1] , a). Since the equilibrium is separating, by the definition of R, type θi+1
weakly prefers (a, r ) to his equilibrium action(s). Since R (U ) is separating, θ1, ...θi weakly prefer their
equilibrium actions to (a, r ).
Taking a+ just above a, type θi+1 is indifferent between (a, r ) and(a+,r+)
, where r+ > r . Then θi+1
weakly prefers(a+,r+)
to his equilibrium action(s), and by single crossing, lower types strictly prefer
their equilibrium actions to(a+,r+)
. So by D1, β(a+)≥ [θi+1].
So θi+1 by deviating can achieve a payoff of at least v(θi+1, a+,ρ
([θi+1] , a+))
, and by continuity
in a, must achieve payoff at least the maximum separating payoff v (θi+1, a, r ), and since the Riley
equilibrium is unique and the equilibrium is separating, we must have f (θi+1) =R (U ) (θi+1).
24Note that a = a1 is allowed inside the limit.
42
b) Continuum of types, U satisfying strong single crossing If U satisfies strong single crossing,
the separating equilibrium is increasing, and so Proposition 4.1 implies f =R (U ) a.e..
c) Continuum of types, ρ ([θ] , a) strictly increasing in a Equilibrium payoff of type θ is uE (θ).
f (θ) places positive measure on A (θ) := {a :β (a) = [θ]
}for almost all θ, on a set Θ−. Choose g (θ) ∈
A (θ) on Θ− such that U E (θ) =U(θ, g (θ) ,θ
). Then g is separating over Θ−.
i. Suppose g (θ) < a∗ (θ) on I ∩Θ−, where I is an interval of positive measure. Take θ∗ ∈ I ∩Θ− and
in the interior of I . Let x = g (θ∗), rx = ρ ([θ∗] , x).
Take y ≥ a∗ (θ∗) satisfying U E (θ∗) =U(θ∗, y,θ∗
). Then y > g (θ) for θ < θ∗; otherwise type θ would
strictly prefer(g (θ∗) ,θ∗
), contradicting IC. Moreover y > g (θ)+ε for θ < θ∗. Let ry = ρ
([θ∗] , y
). By the
assumption on ρ, ry > rx .
Take y− < y close to y . Then a∗ (θ∗) < y , y− > y − ε. Let r− satisfy U E (θ∗) = v(θ∗, y−,r−)
. Taking
y− close enough to y , r− > rx . Then type θ∗ is indifferent to(y−,r−)
, while lower types are lose by
choosing (x,rx ) (by IC) and lose further (by single crossing) by choosing(y−,r−)
. So β(y−) ≥ [θ∗].
Then type θ∗ strictly gains by choosing y−, a contradiction.
ii. Suppose, on the other hand g (θ) ≥ a∗ (θ) on a set that is dense in Θ. g is separating on this set.
Then there exists h (θ) with h ≥ a∗ and g = h on a dense set, with h separating. By Proposition 4.1,
g =R (U ). f gives the same payoff as g , so f =R (U ) a.e. by Proposition 2.2.
B.17.2 A strengthening of IR satisfied by the Riley equilibrium
We know that in a Riley equilibrium, no type strictly prefers any (a,θmin) to equilibrium. This is the IR
condition. We can generalize it. Consider the set NIC(θ′
)of actions a which if taken by θ′ would make
some θ < θ′ want to mimic θ′, when θ takes the Riley equilibrium action:
Definition. Let NIC(θ,θ′
):= {
a :(a,θ′
)Âθ (R (U ) (θ) ,θ)}, and NIC
(θ′
):=⋃
θ<θ′ NIC(θ,θ′
).
With two types, Θ = {θ0,θ1}, to separate from θ0 an action a outside NIC(θ1) must be taken. Pro-
vided this happens, no type strictly prefers (a,θ1) to equilibrium utility. In general, fixing θ′, to separate
from types θ < θ′ an action a outside NIC(θ′
)must be taken. Then no type strictly prefers
(a,θ′
)to
equilibrium utility.
Lemma B.8. For any θ′, if a 6∈ NIC(θ′
), then for any θ∗,
(a,θ′
)¹θ∗ (R (U ) (θ∗) ,θ∗).
Proof. This is immediate for θ∗ < θ′. For θ∗ ≥ θ′:Let S := NIC
(θ′
). This is open, so A \ S is closed.
i. For θ′′ ≥ θ′, on A \ S, U(θ′′, a,θ′′
)is maximized at some x ≥ supS. To see this, suppose it is
maximized at x < supS, and this is strictly better than any x > supS. Take y ∈ S, y > x. If(x,θ′′
) ºθ′′(y,θ′′
), then
(x,θ′′
) Âθ (y,θ′′
)for θ < θ′′ and so x ∈ S. So for y ∈ S, y > x,
(x,θ′′
) ≺θ′′ (y,θ′′
), and taking
limits as y → supS, we have(x,θ′′
)¹θ′′ (supS,θ′′), contradicting the assumption.
ii. Let Θ := {θ ∈Θ : θ ≥ θ′} and A := [
supS, amax].
43
Take f =R (U ) θ ∈Θ∩ [
θmin,θ′)
R(U |Θ,A
)θ ∈Θ
.
f is weakly increasing. f satisfies IC on Θ := Θ ∩ [θmin,θ′
]and Θ. If θ1 ∈ Θ and θ2 ∈ Θ,(
θ1, f (θ1)) ºθ1
(θ′, f
(θ′
))and
(θ′, f
(θ′
)) ºθ′ (θ2, f (θ2)
), so by single crossing
(θ′, f
(θ′
)) ºθ1
(θ2, f (θ2)
)and so
(θ1, f (θ1)
)ºθ1
(θ2, f (θ2)
). Similarly
(θ2, f (θ2)
)ºθ2
(θ1, f (θ1)
). So f satisfies IC.
Since f is also above a∗, by Lemma 4.1, f ≥ R (U ); but f maximizes payoffs subject to weaker
conditions, so f =R (U ).
iii. IR for R(U |Θ,A
)implies
(f (θ∗) ,θ∗
)ºθ∗ (a,θ′
)for a ≥ supS, and so by part i., for all a ∉ S.
B.17.3 Proof of part 2 ( f =R (U ) satisfies D1 for some β)
First we define β. Let S be the set of (θ, a) for which{r : v (θ, a,r ) ≥U f (θ)
}is non-empty, and S A be
the set of a such that (θ, a) ∈ S for some θ. S is closed by continuity of v and U f . On S let r (a,θ) :=min
{r : v (θ, a,r ) ≥U f (θ)
}. This is continuous in θ. On S A , let M (a) := (
argmin)θ r (a,θ). By continuity
of r , this is a closed set. Set β (a) = [max M (a)] on S A , and arbitrarily outside.
This satisfies D1 by construction. It remains to show that β satisfies Bayes’ rule and f is rational
given β. Let the equilibrium separating response be ρ′ (θ) := ρ ([θ] , f (θ)
).
Claim. β(
f (θ))= [θ] (Bayes’ rule)
Proof. By single crossing, and since f is strictly increasing, if θ0 < θ1 < θ2, θ0 strictly prefers(f (θ1) ,ρ′ (θ1)
)to
(f (θ2) ,ρ′ (θ2)
), and so strictly prefers U f (θ0) to
(f (θ2) ,ρ′ (θ2)
), so r
(f (θ2) ,θ0
) >ρ′ (θ2) and the same holds for θ2 > θ0.
Therefore for Θ a continuum, M(
f (θ)) = {θ}. For finite Θ, the above gives {θi } ⊆ M
(f (θi )
) ⊆{θi−1,θi ,θi+1}. But we know from the finite characterization of R (U ) that θi+1 strictly prefers his equi-
librium action to that of θi . So {θi } ⊆ M(
f (θi ))⊆ {θi−1,θi }. In either case, β
(f (θ)
)= [max M
(f (θ)
)]=[θ].
Let o(θ, a
):= (
a,ρ(θ, a
))be the outcome of beliefs θ and signal a.
Claim. If a ∈ NIC(θ′), then β (a) is supported on[θmin,θ′−δ]
for some δ> 0.
Proof. Suppose θ < θ′ and a ∈ NIC(θ,θ′
). Let α
(θ,θ′
):= sup
(NIC
(θ,θ′
)). Then
([θ] , f (θ)
) ∼θ([θ′
],α
(θ,θ′
)). Then (a,r ) ∼θ o
([θ′
],α
(θ,θ′
))for some r < ρ
([θ′
], a
) ≤ ρ([θ′
],α
(θ,θ′
)). By single
crossing, (a,r ) ≺θ′′ o([θ′
],α
(θ,θ′
)) ¹θ′′ ([θ′
], f
(θ′
))for θ′′ ≥ θ′. By continuity of v (θ, a,r ) in θ, this
must also hold for θ′′ ≥ θ′−δ for some δ > 0. So by D1, β (a) is supported on[θmin,θ′−δ]
for some
δ> 0.
Claim. No type has a profitable deviation
Proof. Suppose there is a profitable deviation:([θ] , f (θ)
) ≺θ(β (a) , a
). Define θ′ by: ρ
([θ′
], a
) =ρ
(β (a) , a
). (There exists a solution by continuity of ρ; it is unique by monotonicity of ρ in θ.)
Then([θ] , f (θ)
) ≺θ ([θ′
], a
). So by Lemma B.8, a ∈ NIC
(θ′
). Then by the previous claim β (a) ≤[
θ′−δ]for some δ. This contradicts ρ
([θ′
], a
)= ρ (β (a) , a
).
44
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