Testing Behavioral Hypotheses in Signaling Games · 2019. 12. 2. · Perfect Bayesian Equilibrium (PBE) is a standard solution concept for signaling games. A Sender observes his type,

Testing Behavioral Hypotheses in Signaling Games∗

Adam Dominiak†1 and Dongwoo Lee‡2

1,2 Department of Economics, Virginia Tech

August 24, 2019

Abstract

In this paper, we introduce an equilibrium concept for signaling games, called Focused Hy-

pothesis Testing Equilibrium (HTE). This equilibrium notion incorporates Ortoleva’s (2012)

theory of belief updating on zero-probability events via selecting and updating hypotheses.

Hypotheses are beliefs about strategies. If an equilibrium message is observed, the hypothesis

that the sender plays his equilibrium strategy is selected and updated by Bayes’ rule. However,

if an out-of-equilibrium message is observed, this hypothesis is rejected. Then, a new hypoth-

esis about a strategy that generates the observed message is selected and updated via Bayes’

rule. Each Focused HTE is a Perfect Bayesian Equilibrium (PBE). Conversely, we show that

each PBE can be explained by a Focused HTE, providing a novel justification for PBE beliefs.

Hypotheses facilitate reasoning about off-path beliefs. We impose different behavioral restric-

tions on hypotheses and use stronger equilibrium notions as refinement criteria for PBE. We

compare our refinements with the Intuitive Criterion. Our strongest refinement justifies out-of-

equilibrium beliefs that are immune to the Stiglitz-Mailath critique of the Intuitive Criterion.

Keywords: Signaling games, Perfect Bayesian Equilibrium, Hypothesis Testing Equilibrium, be-lief updating, Bayes’ rule, maximum likelihood updating, out-of-equilibrium beliefs, refinements.JEL Classification: C72, D81, D83

∗The authors are deeply indebted to Hans Haller, George Mailath, Gerelt Tserenjigmid, Matthew Kovach, KevinHe, and the audiences of the 28th International Conference on Game Theory and the 88th Southern Economic Associ-ation (SEA) Meetings, and the seminar participants at Australian National University for their valuable comments andfruitful discussions.†Email: [email protected]‡Email: [email protected]

1

1 Introduction

Signaling games is an important class of dynamic games with incomplete information. They referto interactive situations in which one party uses observable actions of an informed opponent tomake inferences about hidden information. Signaling games have been applied to explain a varietyof economic phenomena including job search (Spence, 1973), advertising (Neslon, 1974; Milgromand Roberts, 1986), dividends (Bhattacharya, 1979; John and Williams, 1985), product quality(Miller and Plott, 1985), warranties (Gal-Or, 1989), limit pricing (Milgrom and Roberts, 1982),elections (Banks, 1990), social norms (Bernheim, 1994), or lobbying (Lohmann, 1995).1

Perfect Bayesian Equilibrium (PBE) is a standard solution concept for signaling games. ASender observes his type, and chooses an optimal message. A Receiver observes the message butnot the type, forms her posterior belief about the Sender’s types, and best responds with an action.As the name indicates, posterior beliefs are derived by Bayes’ rule. However, this updating ruleposes a serious limitation on PBE. Since Bayes’ rule does not specify how beliefs are derived atinformation sets with zero probability, PBE allows for arbitrary and thus multiple off-path beliefs.

In this paper, we elaborate on a solution concept that admits belief updating for all messages,including out-of-equilibrium messages. In a nutshell, beliefs are derived by selecting and updatinghypotheses about strategies. We will argue that hypotheses provide a useful language for reasoningabout beliefs and they facilitate design of different refinement criteria for off-path beliefs.

Our first goal is to introduce an equilibrium concept called the Focused Hypothesis TestingEquilibrium (HTE). This equilibrium notion incorporates a novel theory of belief updating onzero-probability events, the so-called Hypothesis Testing model axiomatized by Ortoleva (2012).The key element of this theory is a set of hypotheses. We call a belief of the Receiver aboutstrategies of the Sender a hypothesis. For each message, the Receiver selects a hypothesis aboutthe Sender’s strategy that in his view generates the message, and updates it via Bayes’ rule.

To illustrate the main idea, consider the following setting of a labor-market game a la Spence(1973). A worker applying for a job has either low (L) or high (H) skill. An employer knows onlythe prior probability distribution over the job applicant’s types, p(H) and p(L). The worker signalswhether he has education or not. Given the signal, the employer forms her posterior belief aboutthe types, and assigns the job applicant to either an executive job or a manual one.2

Suppose that the employer believes that both types pool on education (i.e., education is ac-quired regardless of skills). This belief combined with the prior information defines a hypothesisaccording to which education is chosen by the low-skilled type with probability p(L) and by thehigh-skilled type with probability p(H). If the employer observes education, she may select the

1Riley (2001) provides a comprehensive survey of economic applications of signaling games.2The players’ payoffs are depicted in Figure 1. This game will be analyzed throughout the paper.

2

“pooling” hypothesis, and use it to derive her posterior belief. By applying Bayes’ rule, her poste-rior belief coincides with the prior information about the types.

However, when no education is observed, the employer concludes that her “pooling” hypothesiswas wrong. She selects a new hypothesis about a strategy that generates the observed signal. Forinstance, the employer may believe that both types separate; i.e., education is chosen by the high-skilled type with probability p(H) and no education is chosen by the low-skilled type with p(L).By updating this hypothesis, she infers that no education is signaled by the low-skilled type.

In Focused HTE, the Receiver has a prior over a set of hypotheses (i.e., a second-order prior).Before any information is revealed, the Receiver chooses an initial hypothesis. It is the most likelyhypothesis with respect to her second-order prior. According to this hypothesis, the Sender playshis equilibrium strategy. The Receiver updates this hypothesis via Bayes’ rule to derive her on-the-path beliefs. However, if an out-of-equilibrium message is observed, the initial hypothesis isrejected. Then, the Receiver updates her second-order prior via Bayes’ rule, and selects a newhypothesis. It is the most likely one according to the updated second-order prior. The Receiverupdates the new hypothesis via Bayes’ rule to derive her out-of-equilibrium belief. This updatingprocedure provides a system of posterior beliefs which are well-defined for all information sets. 3

The idea to derive beliefs from hypotheses about strategic behavior is not new. It was in-formally suggested by Kreps and Wilson (1982) to justify beliefs of sequential equilibrium inextensive-form games. In particular, the authors argued that beliefs should be structurally consis-tent. That is, at each information set, each belief should be derived from a single strategy thatgoverned the previous moves via Bayes’ rule. In a later paper, Kreps and Ramey (1987, p.1332)provided the following interpretation of structural consistency:4

“[...] the player who is moving should posit some single strategy combination which, in

his view, has determined moves prior to his information set, and that his beliefs should

be Bayes-consistent with this hypothesis. If the information set is reached with positive

probability in equilibrium, then beliefs are formed using the equilibrium strategy. If,

however, the information set lies off the equilibrium path, then the player must form

some single “alternative hypothesis” as to the strategy governing prior play, such that

under the hypothesis the information set is reached with positive probability.”

As far as we are aware of, Focused HTE is the first solution concept that formally incorporatesstructural consistency into signaling games via the Hypothesis Testing model of Ortoleva (2012).5

3In Ortoleva’s (2012) model, an agent rejects her initial hypothesis and selects a new one if, according to the initialhypothesis, the observed event has probability equal or smaller than a threshold ε ≥ 0. For our equilibrium concept,the threshold is zero. The acronym ”focused” means that the Receiver considers hypotheses that she actually use.

4Kreps and Ramey (1987) showed that sequential-equilibrium beliefs do not need to be structurally consistent insome extensive-form games. In Section 3, we will show that PBE beliefs are structurally consistent in signaling games.

5? applies the Hypothesis Testing model to study optimal persuasion under strategic information design. The

3

Our first result establishes the relationship between Focused HTE and PBE. We show thatFocused HTE and PBE are equivalent solution concepts. In particular, for each out-of-equilibriumbelief of a PBE there exists a hypothesis about a strategy of the Sender that induces the belief. Thisresult proves that PBE beliefs are structurally consistent in the sense of Kreps and Wilson (1982).

Our second goal is use hypotheses as a tool to reduce the number of PBEs (or, Focused HTEs).To this end, we will strengthen our equilibrium concept by imposing two behavioral restrictionson hypotheses. Our first restriction requires hypotheses to be (second-order) rational. A rationalhypothesis is a belief about rational strategies of the Sender (i.e., strategies that best respond torational strategies of the Receiver).6 The corresponding equilibrium notion is called Rational HTE.

We use the Rational HTE supporting a PBE as an argument in favor of the equilibrium and itsoff-path beliefs. That is, a PBE is said to pass the Rational Hypothesis Testing (HT) refinement ifthere exists a Rational HTE that supports the PBE; otherwise, the equilibrium fails the refinement.

We compare our refinement with the well-known Intuitive Criterion of Cho and Kreps (1987).The Intuitive Criterion does not build on any theory of belief updating. Instead, it eliminatesbeliefs that assign strictly positive probabilities to types that cannot benefit from sending an out-of-equilibrium message. In general, the Intuitive Criterion and the Rational HT refinement are notnested.7 However, there are intuitive PBEs that pass the Rational HT refinement.8 If for each off-the-path message of a PBE there is a single type that could benefit from sending the message, thenthe PBE passes the Rational HT refinement. In this case, the Rational HTE induces the IntuitiveCriterion outcome according to which the Receiver infers the “potentially deviating” type.

Our second restriction requires rational hypotheses to be behaviorally consistent. A new hy-pothesis is said to be behaviorally consistent with the initial hypothesis if, after updating it alongthe equilibrium path, it rationalizes the same behavior of the Receiver as the initial hypothesis. Thecorresponding equilibrium notion is called Behaviorally Consistent HTE.

Our main rationale for Behaviorally Consistent HTE is the Stiglitz-Mailath critique of the In-tuitive Criterion (see Cho and Kreps, 1987, p.203 and Mailath, 1988). According to their critique,the Intuitive Criterion may select beliefs that cause inconsistencies in reasoning about behaviorson and off the equilibrium paths. Behaviorally consistent hypotheses rule out such inconsisten-cies. Therefore, we use the existence of Behaviorally Consistent HTE supporting a given PBEas an additional refinement criterion. We derive a condition under which each Rational HTE isBehaviorally Consistent HTE, and solve the educational signaling game of Spence (1973).

Sender can confirm or disconfirm the Receiver’s understanding of a prior. The author explores different ways how theReceiver forms a new prior in light of unexpected decisions of the Sender.

6In Focused HTE, the initial hypothesis is always rational while new hypotheses do not need to be rational.7That is, there is a PBE that passes the Intuitive Criterion but fails the Rational HT refinement and vice versa;

there is a PBE that passes the Rational HT refinement but fails the Intuitive Criterion.8We use the term intuitive PBE as a reference to a PBE that passes the Intuitive Criterion.

4

Finally, we show that our strongest solution concept is consistent with empirical findings.Brandts and Holt (1992) challenged the Intuitive Criterion from an experimental perspective. Theyran a series of experiments to test predictions of the Intuitive Criterion.9 For instance, in one oftheir experiments, a majority of subjects played in line with a PBE that fails the Intuitive Criterion.We show that Behaviorally Consistent HTE can explain the results of Brandts and Holt (1992).

This paper is organized as follows. In Section 2, we recapitulate the standard PBE. In Section3, we formalize the Focused HTE notion, and derive our equivalence result. In Section 4, wedefine the Rational HTE, and introduce our first refinement criterion. In Section 5, we comparethe Rational HT refinement with the Intuitive Criterion. In Section 6, we elucidate the Stiglitz-Mailath critique of the Intuitive Criterion, and define our second refinement based on BehaviorallyConsistent HTE. In Section 7, we explain the experimental findings of Brandts and Holt (1992).In Section 8, we solve the educational signaling game of Spence (1973). In Section 9, we providefinal remarks. Appendix A collects all proofs. In Online Appendix, we compare the Rational HTrefinement with other popular refinement rules studied in the economic literature.

2 Perfect Bayesian Equilibrium

In this section, we recapitulate the Perfect Bayesian Equilibrium (PBE) for signaling games.A signaling game consists of two players, a Sender and a Receiver. Nature draws a type for

the Sender from a finite set of types Θ according to a prior probability distribution p on Θ with afull support (i.e., supp(p) = Θ). The Sender observes his type, and chooses a message m froma finite setM. We denote by s : Θ → M a strategy for the Sender. The Receiver observes themessage, but not the type, chooses an action a from a finite setA, and the game ends. We denote byr :M→A a strategy for the Receiver. Players’ payoffs are given by uS, uR : Θ×M×A → R.The prior information p is commonly known. The class of such signaling games is denoted by G.

Given a message m, we denote by µ(·|m) a posterior belief over the Sender’s types (i.e., aprobability distribution over Θ). We denote by µ := {µ(·|m)}m∈M a family of posterior beliefs.A strategy profile and a family of posterior beliefs are summarized by

(s, r, µ

).

We study PBE in pure strategies. Formally, a (pure) PBE is defined as follows.10

9In this paper, the games implemented by Brandts and Holt (1992) are presented in Figure 1 and Figure 2.10In Appendix B, we show that our main results carry over to PBE in behavioral strategies.

5

Definition 1 (PBE) (s∗, r∗, µ∗) is a Perfect Bayesian Equilibrium if:

(i) s∗(θ) ∈ arg maxm∈M

uS (θ,m, r∗(m)) for each θ ∈ Θ,

(ii) r∗(m) ∈ arg maxa∈A

∑θ∈Θ

µ∗(θ|m)uR(θ,m, a) for each m ∈M,

(iii) µ∗(θ|m) =π(θ,m)

π(Θ,m)for each θ ∈ Θ if π(Θ,m) > 0, and

µ∗(·|m) is an arbitrary probability distribution over Θ if π(Θ,m) = 0,where

π(θ,m) =

{p(θ), if s∗(θ) = m,

0, otherwise.

Conditions (i) and (ii) ensure sequential rationality. Each type best responds to the Receiver’s (op-timal) strategy, and the Receiver best responds to each message with respect to her posterior belief.Condition (iii) specifies how posterior beliefs are determined. For each equilibrium message (i.e.,m ∈ M such that π(Θ,m) > 0), Bayes’ rule is applied. However, for an out-of-equilibrium mes-sage (i.e., m ∈M such that π(Θ,m) = 0), Bayes’ rule cannot be applied and posterior beliefs aredetermined arbitrarily. We denote byM◦ the set of out-of-equilibrium messages.

Below, we derive PBEs for the labor-market game alluded in the Introduction.

Example 1. Consider the signaling game depicted in Figure 1. A worker has either type θL (lowskill) or θH (high skill). Knowing his type, the worker decides on whether to invest in education

Figure 1: Labor-Market Game 1 from Brandts and Holt (1992).

(E) or no education (N ). Given the signal, an employer assigns the worker to either an executive

6

job (e) or a manual job (m). The prior information p about types is given by p(θL) = 1/3 andp(θH) = 2/3. Notice that both worker types prefer the executive job regardless of their educationstatus. Moreover, education is more costly for the low-skilled worker. For the employer, educationis not productive since her payoff is unaffected by the signal. Thus, the employer prefers to matchtype θH with the executive job and type θL with the manual job. There are two pooling PBEs.

PBE-1: In the first pooling PBE, both types signal education, i.e.,

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, µ∗(θL|E) = 1/3 and µ∗(θL|N) ≥ 1/2.

PBE-2: In the second pooling PBE, both types signal no education, i.e.,

s∗(θL) = E∗(θH) = N, r∗(E) = m, r∗(N) = e, µ∗(θL|N) = 1/3 and µ∗(θL|E) ≥ 1/2.

Notice that there are multiple equilibria due to the large amount of out-of-equilibrium beliefs.Our first goal is to elaborate an equilibrium concept that is consistent with all PBEs. Then, we willstrengthen the equilibrium concept to reduce the number of PBEs (and our-of-equilibrium beliefs).

In the next section, we introduce a solution concept that admits belief updating for all messages,including out-of-equilibrium messages.

3 Focused Hypothesis Testing Equilibrium

In this section, we introduce the Focused Hypothesis Testing Equilibrium (HTE), and show thatPBE and Focused HTE are equivalent solution concepts for signaling games.

The main component of Focused HTE is a set of hypotheses. Hypotheses are beliefs overstrategies. Denote by S := {s : Θ → M} the set of (pure) strategies for the Sender. Letβ : S → [0, 1] be a probability measure that represents a belief about the Sender’s strategies. Abelief β over S combined with the prior probability distribution p over Θ is called a hypothesis.

Definition 2 (Hypothesis) A hypothesis π is the probability distribution on Θ×M induced by a

belief β on S and the prior probability distribution p on Θ such that, for every (θ,m) ∈ Θ×M:

π(θ,m) =∑

s∈S s.t.s(θ)=m

β(s)p(θ). (1)

A hypothesis π ascribes probability π(θ,m) to the event “type θ sends message m.” A strategys ∈ S is said to generate message m, if s(θ) = m for some θ ∈ Θ). A hypothesis π is consistentwith m, if π(θ,m) > 0 (i.e., β(s) > 0 for some s ∈ S that generates m. By construction, eachhypothesis is consistent with the prior information about types (i.e., π(θ,M) = p(θ) for all θ ∈ Θ).

7

When β is a degenerate belief (i.e., β(s) = 1 for some s ∈ S and β(s′) = 0 for any s′ 6= s), π iscalled a simple hypothesis. In this case, the Receiver believes that the Sender plays a single strategy(i.e., π(θ,m) = p(θ) if β(s) = 1 for some s ∈ S such that s(θ) = m; otherwise π(θ,m) = 0).

Example 2a. In the game of Figure 1, the employer has four strategies S = {s1, s2, s3, s4}. Thus,a degenerate belief β on S combined with the prior information p defines the simple hypotheses:

1) π1 := {π1(θL, N) = 1/3, π1(θH , E) = 2/3} if β(s1) = 1 and s1 := {s(θL) = N, s(θH) = E},

2) π2 := {π2(θL, E) = 1/3, π2(θH , N) = 2/3} if β(s2) = 1 and s2 := {s(θL) = E, s(θH) = N},

3) π3 := {π3(θL, E) = 1/3, π3(θH , E) = 2/3} if β(s3) = 1 and s3 := {s(θL) = E, s(θH) = E},

4) π4 := {π4(θL, N) = 1/3, π4(θH , N) = 2/3} if β(s4) = 1 and s4 := {s(θL) = N, s(θH) = N}.

According to π1, the employer believes that workers separate; the high-skilled worker signals Ewhile the low-skilled worker signals N . According to π2, the employer believes that workers“reversely” separate; the high-skilled worker signals N while the low-skilled worker signals E.According to π3 (resp., π4), the employer believes that both workers pool on E (resp., on N ).

The employer may believe that workers pool on N with probability β(s4) = λ and that theyseparate with probability β(s1) = 1−λ. Such belief induces the following non-simple hypothesis:

πλ := {π(θL, N) = 1/3, π(θL, E) = 0, π(θH , N) = λ2/3, π(θH , E) = (1− λ)2/3}.

According to this hypothesis, the employer believes that her opponent mixes between s1 and s4.To specify how the Receiver selects and updates hypotheses, we apply the Focused Hypothesis

Testing model axiomatized by Ortoleva (2012). It is a theory of dynamic choice that admits beliefupdating on zero-probability events.Below, we elucidate how this theory works in our setup.

Denote by ∆(Θ×M) the set of all probability measures on Θ×M. Let Π ⊂ ∆(Θ×M) bethe set of all hypotheses associated with a signaling game in G. The Receiver holds a second-order

prior over Π, denoted by ρ, with a finite support (i.e., |supp(ρ)| ∈ N). We assume that ρ induces astrict partial order over supp(ρ). Before any information is revealed, the Receiver selects an initialhypothesis π∗. It is the most likely hypothesis with respect to the second-order prior ρ, i.e.,

{π∗} := arg maxπ∈supp(ρ)

ρ(π). (2)

Upon arrival of a message m, the Receiver conducts a test. If π∗ is consistent with m, she acceptsπ∗, and updates it via Bayes’ rule. However, if π∗ is inconsistent with m (i.e., π∗(Θ,m) = 0), π∗ isrejected. Then, the Receiver updates her second-order prior ρ given m via Bayes’ rule, and selects

8

a new hypothesis π∗∗m . It is the most likely hypothesis according to her updated second-order prior,i.e.,

{π∗∗m } := arg maxπ∈supp(ρ)

ρm(π), where ρm(π) =π(Θ,m)ρ(π)∑

π′∈supp(ρ)

π′(Θ,m)ρ(π′). (3)

The Receiver updates π∗∗m via Bayes’ rule to determine her posterior belief over Θ. Notice thatposterior beliefs are well-defined if for each message m ∈ M, there exists a hypothesis π ∈supp(ρ) that is consistent with m (i.e., π(Θ,m) > 0).

The Receiver is said to hold a focused second-order prior ρ if its support contains hypothesesthat are actually used, i.e.,

suppF (ρ) := {π∗} ∪⋃

m∈M s.t.

π∗(Θ,m)=0

{π∗∗m }, (4)

where π∗ is the most likely hypothesis with respect to ρ and π∗∗m is the most likely hypothesis withrespect to the updated second-order prior ρm given a zero-probability message m according to π∗

(i.e., π∗(Θ,m) = 0). This is the essence of Ortoleva’s Focused Hypothesis Testing model.11

A (pure) Focused HTE consists of a strategy profile (s∗, r∗), a focused second-order prior ρ,and a family of posterior beliefs µ∗ρ = {µ∗ρ(·|m)}m∈M derived via Ortoleva’s updating procedure.

Definition 3 (Focused HTE) (s∗, r∗, ρ, µ∗ρ) is a Focused Hypothesis Testing Equilibrium if:

(i) s∗(θ) ∈ arg maxm∈M

u (θ,m, r∗(m)) for each θ ∈ Θ,

(ii) r∗(m) ∈ arg maxa∈A

∑θ∈Θ

µ∗ρ(θ|m)uR(θ,m, a) for each m ∈M,

(iii) µ∗ρ(θ|m) =π∗(θ,m)

π∗(Θ,m)if π∗(Θ,m) > 0, where {π∗} := arg max

π∈suppF (ρ)

ρ(π), and

π∗(θ,m) =

{p(θ), if s∗(θ) = m,

0, otherwise,

(iv) µ∗ρ(θ|m) =π∗∗m (θ,m)

π∗∗m (Θ,m)if π∗(Θ,m) = 0, where {π∗∗m } := arg max

π∈suppF (ρ)

ρm(π).

11We consider a special case of the Focused Hypothesis Testing model of (see Ortoleva, 2012, Definition 4). Inhis model, an agent rejects her initial belief if the conditional event has a probability equal or smaller than a thresholdε ≥ 0. Such model is said to be minimal if any ε′ < ε leads to different decisions than under ε (see Ortoleva, 2012,Definition 3). In our setup, the initial hypothesis is rejected at information sets with zero probability (i.e., ε = 0). Bythis assumption, our equilibrium notion incorporates, strictly speaking, a Minimal Focused Hypothesis Testing model.One advantage of this model specification is that it admits a unique representation (see Ortoleva, 2012, Proposition 2).

9

Conditions (i) and (ii) ensure sequential rationality (as in PBE). Conditions (iii) and (iv) ensurethat beliefs are well-defined for all messages, including out-of-equilibrium messages. In FocusedHTE, the Receiver best responds to each message with respect to the posterior belief that is derivedfrom a hypothesis about the Sender’s strategy that – in her view – generates the message.

On the equilibrium path, posterior beliefs are derived from the initial hypothesis π∗ according towhich the Sender plays his equilibrium strategy s∗ (i.e., for each m ∈M such that π∗(Θ,m) > 0).Off the equilibrium path, π∗ is rejected (i.e., for each m◦ ∈ M◦ such that π∗(Θ,m◦) = 0). Then,a new hypothesis π∗∗m◦ is selected to derive the out-of-equilibrium belief given m◦. According toπ∗∗m◦ , the Receiver believes that her opponent plays a (non-equilibrium) strategy that generates m◦

(i.e., β(s) > 0 for some s 6= s∗ such that s(θ) = m◦ implying π∗∗(Θ,m◦) > 0).The second-order prior ρ might be interpreted as a probability distribution over a population

of Senders. That is, ρ(π) is the fraction of Senders in the population that behave according toπ ∈ supp(ρ). Such distribution might be purely subjective or based on exogenous information.

The example below derives two pooling Focused HTEs for the labor-market game of Figure 1.

Example 2b. Consider the set of simple hypotheses {π1, π2, π3, π4} presented in Example 2a.There are two Focused HTEs with respect to simple hypotheses.

FHTE-1: In the first Focused HTE both types signal education; i.e.,

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, suppF (ρ) = {π1, π3} such that ρ(π1) < ρ(π3),

µ∗ρ(θL|E) = 1/3 and µ∗ρ(θL|N) = 1.

Initially, the employer selects π3 according to which she believes that both types choose E (i.e.,π∗ = π3). Updating of π3 yields the prior distribution p. When the out-of-equilibrium signal N isobserved, the pooling hypothesis π3 is rejected. The employer selects π1 according to which shebelieves that workers separate (i.e., π∗∗N = π1). By updating π1, the employer infers that N mustbe chosen by the low-skilled type (i.e., µ∗ρ(θL|N) = 1).

FHTE-2: In the second Focused HTE, both worker types signal no education, i.e.,

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, suppF (ρ) = {π2, π4} such that ρ(π2) < ρ(π4)

µ∗(θL|N) = 1/3 and µ∗(θL|E) = 1.

When E is observed, the initial hypothesis π4 is discarded and π2 is selected (i.e., π∗∗E = π2).According to π2, the employer believes that workers “reversely” separate. By updating π2, theemployer infers that E is sent by the high-skilled worker (i.e., µ∗ρ(θL|E) = 1).

Notice that FHTE-1 and FHTE-2 coincide with the two pooling PBEs in which the employerinfers the worker types from the respective out-of-equilibrium messages (see PBE-1 and PBE-2).

10

Our first result demonstrates that each PBE can in fact be explained by a Focused HTE.

Theorem 1 Let (s∗, r∗, µ∗) be a PBE and M◦ be the set of out-of-equilibrium messages. Then,

there exists a Focused HTE, (s∗, r∗, ρ, µ∗ρ), that supports the PBE, i.e.,

(i) µ∗ρ(·|m) = µ∗(·|m) for each equilibrium message m, and

(ii) µ∗ρ(·|m◦) = µ∗(·|m◦) for each out-of-equilibrium message m◦ ∈M◦.

Theorem 1 provides an explanation for PBE beliefs. PBE beliefs might be seen as being derived viaOrtoleva’s updating procedure. For each belief, there exists a hypothesis that induces the belief.12

On the equilibrium path, posterior beliefs are derived from a simple hypothesis according towhich the Sender plays a (pure) equilibrium strategy. However, hypotheses used to derive off-pathbeliefs do not need to be simple. In other words, the Receiver may believe that out-of-equilibriummessage is an outcome of a mixed strategy, even though only pure strategies are played. Theorem1 shows that counter-factual reasoning about mixed behavior allows us to explain all PBE beliefs.

There are two remarks in order.

Remark 1: On the one hand, Theorem 1 is not surprising. Then, we use mixed extension (i.e.,non-simple hypotheses) to explain the plethora of PBEs. On the other hand, such extension isnecessary for the existence of a Focused HTE. If we constrain hypotheses to be simple, there aregames for which a Focused HTE does not exist even though a (pure) PBE exists (see Appendix D).

Remark 2: Theorem 1 shows that PBE beliefs are structurally consistent. Following Kreps andWilson (1982), a belief at an information set is structurally consistent if there exists a single (be-havioral) strategy under which the information set can be reached with a positive probability andfrom which the belief can be derived via Bayes’ rule. Since each hypothesis can be associated witha mixed strategy, and by the Kuhn Theorem, each mixed strategy has a payoff-equivalent behav-ioral strategy, Theorem 1 shows that beliefs of a (pure) PBE are structurally consistent.13 Thereare two important differences to Kreps and Wilson (1982). First, besides structurally consistency,we also require that posterior beliefs are consistent with the prior information about types. Sec-ond, we derive posterior beliefs via the Focused Hypothesis Testing model of Ortoleva (2012). Incontrast, Kreps and Wilson (1982) informally use a sequence of hypotheses with a lexicographicbelief system.14

Finally, it should be remarked that Focused HTE and PBE are equivalent solution concepts.Since each Focused HTE is PBE by Definition 3, Theorem 1 implies the following corollary.

12In Appendix B, we show that Theorem 1 trivially extends to PBE in behavioral strategies (see Theorem 3).13This results holds also true for PBEs in behavioral strategies (see Theorem 3 in Appendix B).14? showed that their model of lexicographic beliefs can be applied to refine Nash Equilibrium in normal-form

games.

11

Corollary 1 PBE and Focused HTE are equivalent solution concepts.

In the next section, we strengthen our equilibrium notion to reduce the number of PBEs (orequivalently, the number of Focused HTEs).

4 Rational Hypothesis Testing Equilibrium

In this section, we introduce a stronger equilibrium notion and suggest it as a refinement for PBE.The idea is to derive out-of-equilibrium beliefs from hypotheses about rational behavior. In

Focused HTE, the initial hypothesis is about (second-order) rational behavior. According to thishypothesis, the Sender plays her (equilibrium) strategy that best responds to the Receiver’s (equi-librium) strategy. However, new hypotheses do not need to be about “best-responding” strategies.Strategies that generate an out-of-equilibrium message must differ from the Sender’s equilibriumstrategy and some of them may be irrational. To narrow down PBE beliefs, we eliminate irrationalstrategies.

From now on, we require that hypotheses are about (second-order) rational strategies, called ra-

tional hypotheses. More precisely, we call a belief about strategies of the Sender that best respondto rational strategies of the Receiver a rational hypothesis.

A strategy r :M→A for the Receiver is rational if r(m) is a best response to some posteriorbelief over Θ, i.e.,

r(m) = a ∈ BR(Θ,m) :=⋃

{µ : µ(Θ|m)=1}

BR(µ,m) (5)

for each m ∈M.15 LetR be the set of rational strategies for the Receiver.A strategy s : Θ→M for the Sender is second-order rational if there exists r ∈ R such that,

s(θ) = m ∈ arg maxm′∈M

uS (θ,m′, r(m′)) , (6)

for each θ ∈ Θ. Denote by B the set of second-order rational strategies for the Sender.A belief β on B together with the prior probability distribution p defines a rational hypothesis.

Definition 4 (Rational Hypothesis) A rational hypothesis is the probability distribution π on Θ×M induced by a belief β on B and the prior probability distribution p on Θ such that, for every

(θ,m) ∈ Θ×M,

π(θ,m) =∑

s∈B s.t.s(θ)=m

β(s)p(θ). (7)

15BR(µ,m) := arg maxa∈A

∑θ∈Θ

µ(θ|m)uR(θ,m, a).

12

A rational hypothesis is called simple, if β is a degenerate belief on B (i.e., β(s) = 1 for s ∈ Band β(s′) = 0 for any s′ 6= s). According to a simple-rational hypothesis π, the Receiver believesthat her opponent chooses a single strategy s that best responds to some of her rational strategies.

A Focused HTE with suppF (ρ) that contains only rational hypotheses is called Rational HTE.

Definition 5 (Rational HTE) (s∗, r∗, ρ, µ∗ρ) is a Rational Hypothesis Testing Equilibrium if (s∗, r∗, ρ, µ∗ρ)

satisfies conditions (i) through (iv) of Definition 3 and suppF (ρ) contains rational hypotheses.

It is immediate that Rational HTE is a more stringent solution than PBE (or equivalently, Fo-cused HTE). Therefore, we will use the Rational HTE that supports a PBE as a refinement criterionfor the PBE.

Remark 3. It should be remark that Rational HTE is stronger than the equilibrium notion sug-gested by Ortoleva (2012, Section IV) to solve the Beer and Quiche game of Cho and Kreps(1987). In line with our equilibrium concept, also Ortoleva (2012) assumed that ε = 0 (i.e., theinitial hypothesis is rejected as soon a zero-probability signal arrives). Besides that, there are twosubstantial differences. First, Ortoleva (2012) used simple hypotheses. Second, simple hypothesesare about first-order rational strategies.16 That is, his hypothesis notion is about a single strategyof the Sender that best responds to some strategy of the Receiver.17 Yet, the Receiver’s strategydoes not need to be rational and off-path beliefs may be derived from strategies that best respondto never-best responses. We rule out such beliefs by requiring second-order rationality.

A given PBE is said to pass the Rational Hypothesis Testing (HT) refinement if there exists aRational HTE supporting the PBE, otherwise the equilibrium fails the refinement.

Definition 6 (Rational HT Refinement) A PBE, (s∗, r∗, µ∗), is said to pass the Rational Hypoth-

esis Testing refinement if there exists a Rational HTE, (s∗, r∗, ρ, µ∗ρ), that supports the PBE. That

is, there exist ρ and a family of posterior beliefs µρ := {µρ(·|m)}m∈M such that

(i) µ∗ρ(·|m) = µ∗(·|m) for each equilibrium message m, and

(ii) µ∗ρ(·|m◦) = µ∗(·|m◦) for each out-of-equilibrium message m◦ ∈M◦.

Fix a PBE, (s∗, r∗, µ∗). The algorithm that verifies the above refinement operates in two steps.In the first step, for each m◦ ∈M◦, we check if there exists a rational hypothesis that is consistentwith the out-of-equilibrium message (i.e., π such that π(Θ,m◦) > 0). Let Πm◦ be the set of such

16Recently, Sun (2019) showed that Ortoleva’s equilibrium notion is a special case of sequential equilibrium thatcan be used as a refinement criterion thereof.

17It can be shown that Ortoleva’s Hypothesis Testing Equilibrium might not exist even if we consider non-simplehypotheses about first-order rational behavior. Therefore, to ensure existence of an “HTE” whenever a PBE exists, wehave to go beyond first-order rationality and allow for non-simple hypotheses as we assumed in Theorem 1.

13

hypotheses. If Πm◦ is a non-empty set, the second step applies. In the second step, we verify ifthere is a rational hypothesis in Πm◦ that induces the out-of-equilibrium belief of the PBE; i.e.,

for some π ∈ Πm◦ ,π(θ,m◦)

π(Θ,m◦)= µ∗(θ|m◦) for each θ ∈ Θ.

Accordingly, a PBE might fail the Rational HT refinement for two reasons. First, there is anout-of-equilibrium message m◦ ∈ M◦ for which Πm◦ is empty.18 Second, Πm◦ is a non-emptyset. However, none of the rational hypotheses induces the PBE belief (i.e., for all π ∈ Πm◦ ,µρ(·|m◦) 6= µ∗(·|m◦)). This means that the out-of-equilibrium belief is inconsistent with the priorinformation about types under the Sender’s (second-order) rational strategies that generate m◦.19

The following example shows that PBE-1 but not PBE-2 passes the Rational HT refinement.

Example 3. Consider again the simple hypotheses defined in Example 2a. Notice that only s1,s3 and s4 are best responses against rational strategies of the employer (i.e., B = {s1, s3, s4}).20

Therefore, the simple hypotheses π1, π3 and π4 are rational.Consider PBE-1. The rational hypothesis π1 is consistent with the out-of-equilibrium message

N . By updating π1 given N , the employer infers that N is sent by the low-skilled worker. Thus,

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, suppF (ρ) = {π1, π3} such that ρ(π1) < ρ(π3),

µ∗ρ(θL|E) = 1/3 and µ∗ρ(θL|N) = 1

is the Rational HTE supporting the pooling PBE with µ (θL|N) = 1.Consider now PBE-2 and the out-of-equilibrium messageE. Notice that π1, π3 and any mixture

thereof are rational hypotheses that are consistent with E. However, none of them induces the PBEbelief µ(θL|E) ≥ 1

2. For each belief β on B such that β(s1) = γ and β(s3) = (1− γ), we have

π(γ) := {π(θL, N) = γ1/3, π(θL, E) = (1− γ)1/3, π(θH , N) = 0, π(θH , E) = 2/3}.

where π(0) = π3 and π(1) = π1. By updating of π(γ) given E, we have

π(γ)(θL, E)

π(γ)(Θ, E)=

(1− γ)1/3

1− γ1/3≤ 1/3 for each γ ∈ [0, 1].

The employer infers that E is more likely to be chosen by the high-skilled worker, and assigns the

18For example, Πm◦ is empty when the out-of-equilibrium messagem◦ is a dominated action for every type θ ∈ Θ.19Example 3 shows that PBE-2 fails the Rational HT refinement for this reason.20More precisely, s1 := {s(θL) = N, s(θH) = E} best responds against r1(E) = m and r1(N) = m; s3 :=

{s(θL) = E, s(θH) = E} best responds against r3(E) = e and r3(N) = m, and s4 := {s(θL) = N, s(θH) = N}best responds against r4(E) = m and r4(N) = e. Moreover, r1, r3 and r4 are rational strategies of the employer.

14

worker to the executive job e instead of m. Thus, PBE-2 fails the Rational HT refinement.

Summing up, we have shown that Rational HTE can be used as a refinement criterion for PBE.In the next section, we compare the Rational HT refinement with the popular Intuitive Criterion.

5 Rational HT Refinement versus Intuitive Criterion

In this section, we compare our refinement with the Intuitive Criterion of Cho and Kreps (1987).Contrary to our approach, the Intuitive Criterion does not build on a theory of belief updating.

It is a payoff-based refinement. The idea is to eliminate out-of-equilibrium beliefs that assign apositive probability to types that have not incentive to deviate from the equilibrium strategy.

Let us briefly recall the Intuitive Criterion. Fix a PBE, (s∗, r∗, µ∗). Consider an out-of-equilibrium messagem◦ ∈M◦. Denote by u∗S(θ) the Sender’s equilibrium payoff when his type isθ ∈ Θ (i.e., u∗S(θ) = uS(θ, s∗(θ), r∗(m))). Let T (m◦) ⊆ Θ be the set of types that cannot improveupon their equilibrium payoff by sending m◦. That is, for all θ ∈ T (m◦):

u∗S(θ) > maxa∈BR(Θ,m◦)

uS(θ,m◦, a), (8)

whereBR(Θ,m◦) :=

⋃{µ : µ(Θ|m◦)=1}

BR(µ,m◦) (9)

is the set of best responses for the Receiver against m◦ with respect to beliefs defined over Θ.21

Denote by I(m◦) := Θ \T (m◦) the set of types that can be better off than their equilibrium payoffby choosing the out-of-equilibrium message m◦. Let

BR(I(m◦),m◦

):=

⋃{µ:µ(I(m◦)|m◦)=1}

BR(µ,m◦) (10)

be the set of best responses for the Receiver against m◦ with respect to beliefs defined over I(m◦).If there exists a type θ′ ∈ Θ such that

u∗S(θ′) < mina∈BR(I(m◦),m◦)

uS(θ′,m◦, a), (11)

the PBE fails the Intuitive Criterion. If the PBE passes the Intuitive Criterion, only the out-

21BR(µ,m◦) := arg maxa∈A

∑θ∈Θ

µ(θ|m◦)uR(θ,m◦, a).

15

of-equilibrium beliefs that assign a zero probability to each type in T (m◦) are admitted (i.e.,µ(θ|m◦) = 0 for all θ ∈ T (m◦)). If I(m◦) is a singleton (i.e., I(m◦) = {θm◦} for some θm◦ ∈ Θ),the Receiver learns the type that could benefit from playing m◦ (i.e., µ(θm◦ |m◦) = 1). A classof signaling games with I(m◦) being a singleton for every m◦ ∈ M◦ is important for economicapplications. In such games, the Intuitive Criterion selects unique out-of-equilibrium beliefs.

Example 4. Consider the PBE with pooling on E (i.e., PBE-1). Given the equilibrium payoff, thelow-skilled type could be better off by sending the out-of-equilibrium message N if he believes toget the executive job. That is, I(N) = {θL} and T (N) = {θH}. As long as the employer believesthat N is sent by the low-skilled worker, there is no type that has an incentive to choose N . There-fore, the PBE passes the Intuitive Criterion yielding the out-of-equilibrium belief µ∗(θL|N) = 1.22

As shown before, this pooling PBE passes also the Rational HT refinement (see Example 3).We begin our comparison by an observation that both refinements are not nested. That is, there

is a PBE that passes the Rational HT refinement but fails the Intuitive Criterion, and vice versa;there is a PBE that passes the Intuitive Criterion but fails the Rational HT refinement.

Observation 1 The Rational HT refinement and the Intuitive Criterion are not nested.

However, there are PBEs that pass the Rational HT refinement whenever they pass the IntuitiveCriterion. We call a PBE that passes the Intuitive Criterion the intuitive PBE.23 If for each out-of-equilibrium message m◦ ∈ M◦ of an intuitive PBE, there exists a single type that could benefitfrom sending m◦, then there exists a Rational HTE supporting the PBE. In this case, the RationalHT refinement selects the same belief as the Intuitive Criterion. According to this belief, theReceiver infers that the out-of-equilibrium message m◦ is sent by the “potentially deviating” type.

Theorem 2 Let (s∗, r∗, µ∗) be a PBE that passes the Intuitive Criterion. If for each out-of-

equilibrium messagem◦ ∈M◦, I(m◦) is a singleton (i.e., I(m◦) = {θm◦} for some θm◦ ∈ Θ), then

there exists a Rational HTE (s∗, r∗, ρ, µ∗ρ) that supports the PBE. In particular, for eachm◦ ∈M◦:

µ∗(θm◦|m◦) = µ∗ρ(θm◦|m◦) = 1. (12)

It should be remarked that Theorem 2 derives a class of intuitive PBEs that can be supportedby a Rational HTE with respect to simple hypotheses. The single-type condition (i.e., I(m◦) =

{θm◦}), guarantees that there exists a (second-order) rational strategy s ∈ B that generates the out-of-equilibrium message m◦ (i.e., s(θm◦) = m◦). If the Receiver believes that the Sender follows

22PBE-2 fails the Intuitive Criterion. Only the high-skilled type could benefit from sending the out-of-equilibriummessage E. Therefore, the employer learns that E is sent by the high-skilled worker (i.e., µ∗(θH |E) = 1) and bestresponds with e. This, however, causes the high-skilled worker to signal E instead of N .

23A PBE that fails the Intuitive Criterion is referred to as the unintuitive PBE.

16

such strategy (i.e., β(s) = 1), her belief combined with the prior information p induces a simple-rational hypothesis π that is consistent with m◦ (i.e, π(θ◦|m◦) = p(θ◦) > 0).24 After updating πgiven m◦, the Receiver infers that the out-of-equilibrium message is sent by θm◦ . For this reason,πm◦ justifies the same out-of-equilibrium belief as the Intuitive Criterion (i.e., µ∗(θm◦ |m◦) = 1).25

As mentioned before, the single-type condition implies that the Intuitive Criterion outcome isunique. However, there may be more than one Rational HTE supporting the intuitive PBE. Toguarantee uniqueness, an auxiliary condition is needed. If, in addition to the single-type condition,each out-of-equilibrium message m◦ ∈ M◦ is a never-best response for each type that cannot bebetter off by sendingm◦ than the equilibrium payoff (i.e., for all θ ∈ T (m◦)), then there is a uniqueRational HTE supporting the intuitive PBE. Formally, the uniqueness condition is stated below.

Corollary 2 Consider an intuitive PBE with I(m◦) being a singleton for each messagem◦ ∈M◦.

If, for each out-of-equilibrium message m◦ ∈M◦, it is true that

m◦ 6= s(θ) ∈ arg maxm∈M

uS(θ,m, r(m)), (13)

for any r ∈ R and each θ ∈ T (m◦), then the Rational HT refinement outcome is unique.

In the special case where m◦ is dominated by another message for each type that cannot improveupon his equilibrium payoff (i.e., θ ∈ T (m◦)), the uniqueness condition (13) is naturally satisfied.

Notice that the Intuitive Criterion has the same limitation as PBE if for some message m◦,I(m◦) contains more than one type (i.e., |I(m◦)| ≥ 1). In this case, the Intuitive Criterion admitsarbitrary beliefs over I(m◦). When I(m◦) = Θ, the Intuitive Criterion does not reduce the set ofbeliefs at all while the Rational HT refinement may reduce them (see Appendix C for an example).For this reason, a variety of stronger refinement concepts have been suggested in the economic lit-erature (see Cho and Kreps (1987), Banks and Sobel (1987), Mailath, Okuno-Fujiwara, and Postle-waite (1993), Eso and Schummer (2009), Fudenberg and He (2018, 2019)). In Online Appendix,we compare the Rational HT refinement with other refinements for PBEs where |I(m◦)| ≥ 1.

6 Behaviorally Consistent Hypotheses

In this section, we recapitulate the Stiglitz-Mailath critique of the Intuitive Criterion. Then, weintroduce an equilibrium concept whose out-of-equilibrium beliefs are immune to their critique.

24This is true for any intuitive PBE with |I(m◦)| ≥ 1 for each m◦ ∈ M◦. In Online Appendix (Lemma A1), weshow that for each out-of-equilibrium message there exists a rational hypothesis that is consistent with the message.

25The converse of Theorem 2 does not work (see PBE-II derived in Section 6). PBE-II satisfies the single-typecondition of Theorem 2. However, while PBE-II passes the Rational HT refinement, it fails the Intuitive Criterion.

17

The idea is to derive out-of-equilibrium beliefs from hypotheses that are “consistent” with theinitial hypothesis in the sense that they both rationalize the same behavior on the equilibrium path.

In Rational HTE, the initial hypothesis rationalizes the Receiver’s behavior on the equilibriumpath while new hypotheses rationalize her behavior off the path. However, the Receiver may selecta new hypothesis which – after updating it along the equilibrium path – will rationalize a differentaction than her equilibrium action. Such hypothesis is said to be behaviorally inconsistent with theinitial hypothesis. Behaviorally inconsistency is problematic. As shown below, it might provide anargument against the new hypothesis along similar lines as the argument of Stiglitz (see Cho andKreps, 1987, p.203) and Mailath (1988) against beliefs admitted by the Intuitive Criterion.

Let us briefly recall the Stiglitz-Mailath critique by means of the following example.

Figure 2: Labor-Market Game 2 in Brandts and Holt (1992)

Example 5a. Consider the labor-market game depicted in Figure 2. There are two pooling PBEs.PBE-I: In the first pooling PBE both worker types signal E; i.e,

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, µ∗ (θL|E) = 1/3 and µ∗ (θL|N) ≥ 1/2.

PBE-II: In the second pooling PBE both worker types signal N ; i.e.

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, µ∗ (θL|N) = 1/3 and µ∗ (θL|E) ≥ 1/2.

Consider PBE-II. The Intuitive Criterion asserts that only the high-skilled worker could benefitby sending the out-of-equilibrium message E. That is, I(E) = {θH}. Therefore, if E is observed,the employer infers that education is chosen by the high-skilled type (i.e., µ(θH |E) = 1). However,

18

given such belief, the employer prefers to assign the worker to the executive job e instead ofmatching him with the manual job m. Therefore, the pooling PBE fails the Intuitive Criterion.26

Suppose the employer reasons further. Since the worker signaling E receives the executivejob, the high-skilled type is strictly better off by signaling E instead of N . If the employer reasonsconsistently, then she should infer that only the low-skilled worker can signal N . Therefore, shewill best respond by matching the worker signaling N with the manual job m. This will in turninduce the low-skilled worker to signal E. This chain of reasoning provides an argument againstthe posterior belief that E is signaled by the high-skilled worker (i.e., µ(θH |E) = 1). As a conse-quence, we might discard the unintuitive PBE even though the belief used against the equilibriumis “implausible” itself. This is the Stiglitz-Mailath critique regarding the Intuitive Criterion.

To eliminate such “implausible” beliefs, we require that new hypotheses are behaviorally con-sistent. Let π∗ be an initial hypothesis. A rational hypothesis is behaviorally consistent with π∗ if– after updating it along the equilibrium path – it rationalizes the same behavior as π∗.

Definition 7 (Behaviorally Consistent Hypothesis) Let π∗ be an initial hypothesis. A rational

hypothesis π is behaviorally consistent with the initial hypothesis π∗ if for each message m ∈ Msuch that π∗(Θ,m) > 0 and for each action a∗ ∈ A such that

a∗ ∈ arg maxa∈A

∑θ∈Θ

π∗(θ,m)

π∗(Θ,m)uR(θ,m, a), (14)

it is true that π(Θ,m) > 0 and

a∗ ∈ arg maxa∈A

∑θ∈Θ

π(θ,m)

π(Θ,m)uR(θ,m, a). (15)

Below, we define the notion of Behaviorally Consistent HTE.

Definition 8 (Behaviorally Consistent Hypothesis Testing Equilibrium) (s∗, r∗, ρ, µ∗ρ) is a Be-

haviorally Consistent Hypothesis Testing Equilibrium if (s∗, r∗, ρ, µ∗ρ) satisfies conditions (i) through

(iv) in Definition 3, and suppF (ρ) contains behaviorally consistent hypotheses.

Behaviorally consistent hypotheses are included in the set of rational hypotheses and, as wewill show below, they ensure that beliefs are immune to the Stiglitz-Mailath critique. For thisreason, we apply Behaviorally Consistent HTE as an additional refinement criterion for PBE. TheBehaviorally Consistent HT refinement is defined in the same way as the Rational HT refinement(see Definition 6). A PBE is said to pass the Behaviorally Consistent HT refinement if there exists aBehaviorally Consistent HTE that supports the PBE; otherwise the equilibrium fails the refinement.

26PBE-I passes the Intuitive Criterion yielding the posterior belief µ(θL|N) = 1.

19

Below, we illustrate the refinement.

Example 5b. We will show that PBE-II can be supported by a Behaviorally Consistent HTE (seeExample 5a). Consider the following simple-rational hypotheses:

1) π′1 := {π1(θL, N) = 1/3, π1(θH , N) = 2/3} if β(s′1) = 1,

2) π′2 := {π2(θL, E) = 1/3, π2(θH , N) = 2/3} if β(s′2) = 1,

where s′1 and s′2 are best responses of the worker against rational strategies r1, r2 ∈ R of theemployer.27

1) s′1 := {s(θL) = N, s(θH) = N} is the best response against r1(E) = m and r1(N) = e,

2) s′2 := {s(θL) = E, s(θH) = N} is the best response against r2(E) = m and r2(N) = m,

Consider the following Rational HTE supporting the pooling PBE-II with µ∗ (θL|N) = 1; i.e.,

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, suppF (ρ) = {π′1, π′2} such that ρ(π′2) < ρ(π′1),

µ∗ρ (θL|N) = 1/3 and µ∗ρ (θL|E) = 1.

By updating the initial hypothesis π′1, the employer infers that signalN is more likely to be cho-sen by the high-skilled type, and therefore assigns the job applicant to the executive job e. Considernow the new hypothesis π′2. According to π′2, the employer believes that workers “reversely” sep-arate; i.e., the high-skilled worker signals N while the low-skilled worker signals E. By updatingπ′2 on the equilibrium path, the employer infers that signal N is chosen by the high-skilled worker.Therefore, π′2 rationalizes the same action as the initial hypothesis π′1, showing that π′2 is behav-iorally consistent with π′1. Hence, the above equilibrium is Behaviorally Consistent HTE, showingthat the PBE-II with µ∗ (θL|N) = 1 passes the Behaviorally Consistent HT refinement.

Because of behavioral consistency, the employer has no reason to deviate from her equilibriumstrategy if she would use π′2 on the equilibrium path. At this stage, the inconsistency in reasoningabout optimal behaviors on and off the equilibrium paths is prevented. The out-of-equilibriumbelief µ(θL|E) = 1 justified by the new hypothesis π′2 is immune to the Stiglitz-Mailath critique.For this reason, we argue that there is no reason to refute the pooling PBE-II with µ(θL|E) = 1.

Below, we derive a condition under which Rational HTE is Behaviorally Consistent HTE.

27Altogether there are four best responses B = {s′1, s′2, s′3, s′4} in this game. The other best responses s′3 and s′4and the simple-rational hypotheses π′3 and π′4 are presented in Example 5c.

20

Proposition 1 Let (s∗, r∗, ρ, µ∗ρ) be a Rational HTE. If the Receiver’s best-response correspon-

dence is single-valued along the equilibrium path, i.e.,

{r∗(m)} = arg maxa∈A

∑θ∈Θ

µ∗ρ(θ|m)uR(θ,m, a) for each m ∈M such that π∗(Θ,m) > 0, (16)

then there exists a Behaviorally Consistent HTE that supports the Rational HTE.

One question remains open. Can we justify an intuitive PBE by a Behaviorally ConsistentHTE? As we have shown before, if a PBE passes the Intuitive Criterion and I(m◦) satisfies thesingle-type condition, then the PBE passes the Rational HT refinement yielding the same out-of-equilibrium belief as the Intuitive Criterion. However, the intuitive PBE does not need to pass theBehaviorally Consistent HT refinement unless an auxiliary condition is satisfied.

Let π be a probability distribution on Θ ×M. Let m be a message and θ be a type such thatπ(θ,m) > 0. Another type θd ∈ Θ signaling m under π (i.e., π(θd,m) > 0) is called a dummy forthe message m, if the Receiver’s best response to m with respect to π remains unchanged underany probability distribution π on Θ ×M such that π(θ,m) = π(θ,m) for all θ ∈ Θ \ {θd} andπ(θd,m) = 0 (i.e., under any π that ascribes zero probability to the event “type θd sends m”).

Definition 9 (Dummy Types) Let (s∗, r∗, µ∗) be a PBE and π be a probability distribution on

Θ×M induced by the equilibrium strategy s∗, i.e., π(θ,m) = p(θ) if, s∗(θ) = m and π(θ,m) = 0,

otherwise. Consider a type θd that sends an equilibrium message m ∈ M, i.e., s∗(θd) = m. Type

θd is called a dummy for m if for any a ∈ A such that

a ∈ arg maxa∈A

∑θ∈Θ

π(θ,m)

π(Θ,m)uR(θ,m, a), (17)

it is true that

a ∈ arg maxa∈A

∑θ∈Θ

π(θ,m)

π(Θ,m)uR(θ,m, a), (18)

for any probability distribution π on Θ×M such that

π(θ,m) =

{π(θ,m), if θ ∈ Θ \ {θd},

0, if θ = θd.

Consider a PBE in which type θd sends an equilibrium message m (i.e., π(θd,m) > 0). If type θd

is a dummy for m, then the equilibrium message does not convey any relevant information aboutθd. That is, if the Receiver considers a probability distribution π that assigns zero probability tothe event “θd sends m,” then π rationalizes the same best response to m as her equilibrium action.The existence of dummy types leads to the following result.

21

Proposition 2 Consider a pooling PBE that passes the Intuitive Criterion. Let m∗ be the pooling

message. If I(m◦) is a singleton for each m◦ ∈ M◦, and θd ∈ I(m◦) is a dummy for the equilib-

rium message m∗, then the intuitive PBE passes the Behaviorally Consistent HT refinement.

The example below illustrates the meaning of a dummy type for an intuitive PBE.

Example 5c. Consider the PBE-I with pooling on E (see Example 5a). Notice that this equilib-rium passes the Intuitive Criterion and it satisfies the single-type condition (i.e., I(N) = {θL}).Therefore, by Theorem 2, PBE-I also passes the Rational HT refinement. Consider the other twosimple-rational hypotheses for the labor-market game depicted in Figure 2:

3) π′3 := {π3(θL, E) = 1/3, π3(θH , E) = 2/3} if β(s′3) = 1,

4) π′4 := {π4(θL, N) = 1/3, π4(θH , E) = 2/3} if β(s′4) = 1.

where s′3 and s′4 are best responses of the worker against rational strategies r3, r4 ∈ R of theemployer:

3) s′3 := {s(θL) = E, s(θH) = E} is the best response against r3(E) = m and r3(N) = e,

4) s′4 := {s(θL) = N, s(θH) = E} is the best response against r4(E) = e and r4(N) = e.

Notice that the low-skilled type is a dummy type for the equilibrium messageE. That is, even if theemployer would believe that the low-skilled worker does not acquire education (i.e. π(θL, E) = 0),it would not change her best response to the equilibrium message E. Therefore, we can find

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, suppF (ρ) = {π′3, π′4} such that ρ(π′4) < ρ(π′3)

µ∗ (θL|E) = 1/3 and µ∗ (θL|N) = 1,

the Behaviorally Consistent HTE-I that supports the pooling PBE-I with µ∗ (θL|N) = 1.

7 Experimental Findings of Brandts and Holt

In this section we show that Behaviorally Consistent HTE is consistent with empirical findings.Brandts and Holt (1992) ran an experiment testing predictions of the Intuitive Criterion for the

labor-market games analyzed in the previous sections (i.e., Figure 1 and Figure 2.) Interestingly,subjects behaved consistently with PBEs that pass the Behaviorally Consistent HT refinement.

22

Consider the labor-market game in Figure 1 with the intuitive PBE-1 and the unintuitive PBE-2.Since PBE-1 but not PBE-2 can be supported by a Behaviorally Consistent HTE,28 our predictioncoincides with the Intuitive Criterion. Brandts and Holt (1992) reported that 102 out of 128 sub-jects’ decisions matched with the intuitive PBE-1 and only 7 decisions matched with the unintuitivePBE-2. This result is consistent with our prediction.29

There is another interesting finding. Brandts and Holt (1992) analyzed the behaviors of Senders.They found some evidence for the new hypothesis π1 of the Behaviorally Consistent HTE support-ing PBE-1 (see footnote 28). According to π1, the low-skilled type signals N and the high-skilledtype signals E. Brandts and Holt (1992) reported that 84 out of 84 high-skilled subjects played E.However, 24 out of 44 low-skilled subjects played the out-of-equilibrium message N .

“This type-dependence is consistent with the out-of-equilibrium beliefs that support

the intuitive [. . . ] equilibrium” (see Brandts and Holt, 1992, p.1357).

A majority of Receivers seemed to believe that N is sent by low-skilled types in accordance withπ1. Then, 17 out of 24 Receivers who observedN responded with the equilibrium actionm. Hence,the reported type-dependence and subjects’ replies to N suggest that π1 is a reasonable hypothesis.

Consider now the game in Figure 2 with the intuitive PBE-I and the unintuitive PBE-II.30 Forthis game, the predictions form our refinement and the Intuitive Criterion are different. Then,PBE-I and PBE-II pass the Behaviorally Consistent HT refinement (see Examples 5b, and 5c).Interestingly, a majority of subjects behaved consistently with the unintuitive PBE-II. As Brandtsand Holt (1992) reported, only 23 out of 144 subjects’ decisions matched with the intuitive PBE-Iwhile 84 out of 144 decisions matched with the unintuitive PBE-II.

As in the previous case, Brandts and Holt (1992) found some evidence for the new hypothesisπ′2 of the Behaviorally Consistent HTE supporting the unintuitive PBE-I. According to π′2, Senders“reversely” separate, i.e., the low-skilled type chooses E while the high-skilled type chooses N .Brandts and Holt (1992) found that 72 out of 99 high-skilled Senders played N while 20 out of 45

low-skilled Senders played E. Notably, a significant number of Receivers believed that Sendersdo “reversely” separate. Then, 24 out of 47 Receivers who observed E responded with m.

In sum, Behaviorally Consistent HTE can explain better the experimental results of Brandts

28 In Example 3, we have shown that PBE-1 is supported by the Rational HTE with the initial hypothesis π3 andthe new hypothesis π1. By updating π1 on the equilibrium path, we have that µρ(θH |E) = 1 and the employer willmatch the job applicant with the executive job e. Therefore, π1 rationalizes the same behavior as π3, showing that π1

is behaviorally consistent with π3. Thus, PBE-1 passes the Behaviorally Consistent HT refinement. Regarding PBE-2,we have shown that it does not even pass the Rational HT refinement.

29This game was implemented in their Treatment 1. The results of this treatment are summarizes in Table 3 (parts(a) and (c)) (see Brandts and Holt, 1992, p.1358).

30This game was implemented in their Treatment 5. Results of this treatment are summarized in Table 4 (parts (a)and (b)) (see Brandts and Holt, 1992, p.1363).

23

and Holt (1992) than the Intuitive Criterion, making our solution concept empirically relevant.31

8 Educational Signaling Game.

In this section, we apply our equilibrium notions to solve the signaling game of Spence (1973).His model is known to have a plethora of pooling PBEs and all of them fail the Intuitive Criterion.We show that some pooling PBEs can be supported by Rational and Behaviorally Consistent HTE.

We consider a finite version of the Spence model. As in our previous games, there is a workerand an employer. The worker has either low (L) or high (H) productivity, depicted by types θL orθH where θL < θH . The prior information p is given by p(θL) = 1− α and p(θH) = α ∈ (0, 1).

The worker knows his type θ, and chooses an education level e from a setM = {e0, e1, . . . , eN}where e0 := 0 < e1 < . . . < eN .32 The worker’s payoff is given by

uS(θ, e, w) = w − e

θfor θ ∈ {θL, θH}, (19)

where w denotes the wage and eθ

is the cost of choosing e by type θ. Education is more costly tothe low-productivity type. It is assumed that the worker can always find a job at wage w = θL.Figure 3 depicts type-dependent indifference curves (the red one for θL and the blue one for θH).

The employer observes the education level e but not the worker’s type, and offers a wage w.The employer’s payoff is given by

uR(θ, e, w) = −(θ − w)2. (20)

This implies that the employer offers a wage that is equal to the average productivity the averageproductivity (i.e., w(e) = E(θ|e) = µ(θH |e)θH +

(1 − µ(θH |e)

)θL).33 Notice that we allow for

A = R+. We denote by E(θ) := αθH +(1− α)θL the average productivity when the employer’s

belief coincides with the prior information about types.We study pooling behavior; i.e., both types choose the same education level e∗. What levels of

education can be accommodated by pooling PBE? In any PBE, the payoff of the low-productivity

31Other studies have tested the Intuitive Criterion. For instance, Banks, Camerer, and Porter (1994) found evidencein favor of intuitive PBE. However, implementing similar games as Banks, Camerer, and Porter (1994), Brandts andHolt (1993) could not find unequivocal support for intuitive PBEs. Instead, Brandts and Holt (1993) replicated similarpatterns of equilibrium behaviors as reported in Brandts and Holt (1992). Over a series of treatments, a majority ofsubjects behaved consistently with an unintuitive PBE that is also consistent with Behaviorally Consistent HTE.

32To simplify our analysis, we assume that M is sufficiently large yet finite. In particular, we assume that Mcontains education levels en := θL(θH − θL) and eN := θH(θH − θL). For an analysis of the Spence model withM = R+, the reader is referred to Fudenberg and Tirole (1991, Chapter 8, p.329).

33Alternatively, we can consider a perfect cometition among many employers. Jeong (2019) studies imperfect com-petition among employers in the context of job market signaling. He investigates how wage offers change dependingon the degree of competition.

24

w

e

θH

θL

E(θ)w − e

θL

w − eθH

e∗ x en eN

E(θ)− e∗

θLE(θ)− e∗

θH

Figure 3: Pooling PBE with education level e∗ and wage w∗ = E(θ).

type from choosing e∗ at wage w∗ = E(θ) must be greater than her payoff from choosing noeducation at the minimum wage w = θL. Formally, the pooling message e∗ must satisfy

E(θ)− e∗

θL≥ θL, or equivalently, e∗ ≤ α(θH − θL)θL︸ ︷︷ ︸

:=x

. (21)

Condition (21) specifies an upper bound x for education levels supportable by a pooling PBE.Let e∗ ≤ x be a pooling message. Notice that neither type has incentive to deviate from

e∗ at wage w∗ = E(θ) as long as the wages paid out-of-equilibrium paths satisfy the followingconditions. For each education level below e∗ (i.e., e < e∗), the offered wage w(e) satisfies

E(θ)− e∗

θL≥ w(e)− e

θL, (22)

and for each education level above e∗ (i.e., e′ > e∗), w(e′) satisfies

E(θ)− e∗

θH≥ w(e′)− e′

θH. (23)

Hence, there is a large amount of PBEs admitting various wage schemes and off-path beliefs.For example, consider the following family of pooling PBEs:

(i) s∗(θL) = s∗(θH) = e∗ such that e∗ ≤ x := α(θH − θL)θL,

(ii) w∗(e) =

E(θ) if e ≥ e∗,

θL if e < e∗,

25

(iii) µ∗(θL|e) =

1− α if e ≥ e∗,

1 if e < e∗.

In this PBE, the employer believes that any education below the equilibrium level e∗ is chosen bythe low-productivity type, and offers the minimum wage. Any education equal to and above e∗

does not convey any information about types and the employer offers the average productivity.34

As mentioned before, each pooling PBE fails the Intuitive Criterion.35 Thus, a natural questionarises as to whether we can defend some PBEs by Rational HTE. We indeed can. We will demon-strate that there is a set of pooling messages that can be supported by a Rational HTE. Moreover,the “size” of the subset depends on the prior information p about types.

Let us first elucidate why a PBE may fail the Rational HT refinement. To this end, consideran education level e ∈ {en, . . . , eN} (see Figure 3).36 Notice that the low-productivity type has noincentive to choose such e even if he is paid the highest wage w = θH . For each e ∈ {en, . . . , eN},the payoff of θL is lower than his payoff for no education at the minimum wage w = θL. Therefore,any strategy where θL chooses e ∈ {en, . . . , eN} is a never-best response.37 This in turn means thatthere does not exist a rational hypothesis according to which θL chooses e ∈ {en, . . . , eN} witha positive probability. For this reason, the above PBE and any other pooling PBE that admits anoff-path belief such that µ(θL|e) > 0 for some e ∈ {en, . . . , eN} fails the Rational HT refinement.

When does a pooling PBE pass the Rational HT refinement? Since only the high-productivitytype can choose e ∈ {en, . . . , eN} as a best response, µ(θH |e) = 1 is the only out-of-equilibriumbelief that can be justified by a rational hypothesis. Given such belief, the employer offers thehighest wage. This implies that the payoff of the high-productivity type from choosing a (pooling)message e∗ at w∗ = E(θ) must greater than his payoff from choosing e ∈ {en, . . . , eN} at w = θH .That is, e∗ must satisfy

E(θ)− e∗

θH≥ θH −

e2

θH, or equivalently, e∗ ≤ (θH − θL)(θL − (1− α)θH)︸ ︷︷ ︸

:=y

. (24)

Condition (24) specifies an upper bound y for education levels that can be supported by a RationalHTE provided y ≥ 0. When y < 0, none of the pooling PBEs can be supported by Rational HTE.

Rational HTE refines PBE in two dimensions. Since y < x, the first dimension is poolingmessages. The second dimension is off-path beliefs (or wages). For each e∗ ≤ y, only the pooling

34Notice that offering E(θ) for a pooling message e∗ and the minimum wage θL for each out-of-equilibrium mes-sage e 6= e∗ rationalized by µ∗(θL|e∗) = 1− α and µ∗(θL|e) = 1 for e 6= e∗ is another family of pooling PBEs.

35For each pooling PBE, there is an out-of-equilibrium message that only type θH could be better off than hisequilibrium payoff. If the employer believes that type θH sends such message, he will pay the highest wage and thentype θH will indeed deviate from the PBE. Therefore, all pooling PBEs fail the Intuitive Criterion in the Spence game.

36The lower and upper bounds of this set are en := θL(θH − θL) and eN := θH(θH − θL).37Even under wage w(e) such that w(e) = θH for e > en and w(e) = θL for e ≤ en, θL will not choose e > en.

26

PBE in which the employer infers the high-productivity type from each e ∈ {en, . . . , eN} (i.e.,µ(θH |e) = 1) passes the Rational HT refinement.38 Below, we provide an example of such PBEs.

Observation 2 The following pooling PBE passes the Rational HT refinement:

(i) s∗(θL) = s∗(θH) = e∗ such that 0 ≤ e∗ ≤ y := (θH − θL),(θL − (1− α)θH

)

(ii) w∗(e) =

θL if e < e∗,

E(θ) if e∗ ≤ e ≤ en,

θH if en < e ≤ eN .

(iii) µ∗(θL|e) =

1 if e < e∗,

1− α if e∗ ≤ e ≤ en,

0 if en < e ≤ eN .

In this refined PBE, the employer believes that any education level below e∗ is chosen by the low-productivity type, and pays the minimum wage. For any education level between e∗ and en, shecannot infer the type, and pays the average productivity. For any education level above en, theemployer believes that it is chosen by the high-productivity type, and thus pays the highest wage.

As mentioned before, a Rational HTE exists if and only if α ≥ θH−θLθH

. In other words, thenumber of refined PBEs is a function of α, the fraction of high-productivity types in the market.The larger the fraction of such types is the more pooling PBE passes the Rational HT refinement.Yet, when α < θH−θL

θH, none of the pooling PBEs can be justified by Rational HTE (see Figure 4).

w

e

θH

θL

E(θ)

E(θ)′

w − eθL

w − eθH

e∗ y x en eN

Figure 4: Refinement of pooling PBE at E(θ); no refinement at E(θ)′

What about Behaviorally Consistent HTE? Any pooling PBE that passes the Rational HT re-finement can in fact be supported a Behaviorally Consistent HTE except the one with e∗ = 0. The

38Interestingly enough, there is an experimental evidence in favor of such equilibria. Especially, Kubler, Muller,and Normann (2008) found pooling behavior at lower education levels in the framework of the Spence model.

27

problem is the out-of-equilibrium message eN . There are many rational hypotheses that assign astrictly positive probability to e∗ = 0 and eN . However, when updating them on the path (i.e.,e∗ = 0), none of them rationalizes the employer’s equilibrium decision to offer w∗ = E(θ).

In sum, our requirement to derive beliefs from a rational hypothesis significantly reduces thenumber of pooling PBEs in the education signaling game of Spence (1973). A Rational HTE,provided it exists, is one in which the employer offers the highest wage for each out-equilibrium-message that only the high-productivity type has incentive to choose (i.e., each e ∈ {en, . . . , eN}).

9 Conclusion

In this paper, we have suggested solution concepts for signaling games. The novelty of our equi-librium notions is the updating procedure that admits belief updating at information sets with zeroprobability. Off-path beliefs are derived from hypotheses about strategic behavior of the Sender.

We have shown that hypotheses provide a useful tool for designing belief-driven refinements.To reduce the number of Perfect Bayesian Equilibria, we elaborated on two refinement criteria.

Our first refinement requires that beliefs are derived from rational hypotheses. This criterionensures that off-path beliefs are justified by (second-order) rationality. Our second refinementis more stringent. It requires that beliefs are derived from behaviorally consistent hypotheses,ensuring that off-path beliefs are immune to the Stiglitz-Mailath critique of the Intuitive Criterion.

Our strongest refinement criterion provides an alternative approach to equilibrium selection.Equilibrium selection based on the Intuitive Criterion has been criticized by many authors in-cluding Mailath (1988), van Damme (1989), Mailath, Okuno-Fujiwara, and Postlewaite (1993) asbeing “implausible” due to inconsistency in reasoning between behaviors on and off the equilib-rium paths. The refinement based on behaviorally consistent hypotheses does not only eliminatesuch inconsistencies but it is also consistent with experimental findings on equilibrium behavior.Therefore, we believe that our strongest refinement is worth further exploration and application.

A Proofs

Proof of Theorem 1. Consider a (pure) PBE, (s∗, r∗, µ∗). LetM◦ be the set of out-of-equilibriummessages and {µ∗(·|m◦)}m◦∈M◦ be the family of posterior beliefs of the PBE. The proof consistsof three steps. In Step 1, we construct a hypothesis that induces the belief on the equilibrium path.In Step 2, for each out-of-equilibrium message m◦ ∈ M◦, we construct a hypothesis that inducesthe out-of-equilibrium belief. In Step 3, we construct a Focused HTE that supports the given PBE.

Step 1: Since s∗ is the equilibrium strategy for the Sender, there exists a hypothesis π∗ that justifies

28

the Receiver’s belief on the path. Consider a degenerate belief β such that β(s∗) = 1 and β(s) = 0

for any s ∈ S \ {s∗} where S is the set of strategies for the Sender. This belief together with theprior probability distribution p on Θ defines the following hypothesis:

π∗(θ,m) =

{p(θ), if s∗(θ) = m,

0, otherwise.(25)

By condition (iii) in Definition 1, π∗ induces the Receiver’s belief on the equilibrium path. Thatis, for any m ∈M such that π∗(Θ,m) > 0, we have

µ∗(θ|m) = µρ(θ|m) :=π∗(θ,m)

π∗(Θ,m)for each θ ∈ Θ. (26)

Step 2: Fix an out-of-equilibrium message m◦ ∈M◦. We will show that there exists a hypothesisπm◦ which – after updating it given m◦ – justifies the out-of-equilibrium belief µ∗(·|m◦). Withoutloss of generality, we assume that there areN types in supp (µ∗(·|m◦)). That is, supp (µ∗(·|m◦)) =

{θ1, . . . , θN}. For each i ∈ {1, . . . , N}, we construct a strategy si for the Sender as follows

si(θ) =

{m◦, if θ = θi,

m ∈M \ {m◦}, otherwise.(27)

Now, define β : S → [0, 1] such that

β(si) =

(µ∗(θi|m◦)p(θi)

)/ (1

µ∗(θ1|m◦)p(θ1)

+ . . .+ µ∗(θN |m◦)p(θN )

)for each i ∈ {1, . . . , N}, (28)

and β(s) = 0 for any s ∈ S \ {s1, . . . , sN}. Then, β and p induce the following hypothesis: Forevery (θ,m) ∈ Θ×M,

πm◦(θ,m) =∑

s∈S s.t.s(θ)=m

β(s)p(θ) =∑

si∈{s1,...,sN} s.t.si(θ)=m

β(si)p(θ). (29)

By updating πm◦ given m◦, we have

µρ(θ|m◦) =πm◦(θ,m

◦)

πm◦(Θ,m◦)for every θ ∈ Θ, (30)

29

where πm◦(θ,m◦) and πm◦(Θ,m◦) are defined as follows:

πm◦(θ,m◦) = µ∗(θ|m◦)

/(1

µ∗(θ1|m◦)p(θ1)

+ . . .+ µ∗(θN |m◦)p(θN )

), (31)

andπm◦(Θ,m

◦) =1

µ∗(θ1|m◦)p(θ1)

+ . . .+ µ∗(θN |m◦)p(θN )

. (32)

By plugging πm◦(θ,m◦) and πm◦(Θ,m◦) from Equations (31) and (32) into Equation (30), we getthe out-of-equilibrium belief µρ(·|m◦) that coincides with the PBE belief µ∗(·|m◦). That is,

µρ(θ|m◦) = µ∗(θ|m◦) for each θ ∈ Θ. (33)

This shows that there exists a hypothesis πm◦ that induces the out-of-equilibrium belief µ∗(·|m◦)of the PBE. Notice that we chose the out-of-equilibrium message m◦ arbitrarily. Therefore, foreach out-of-equilibrium message m◦ ∈ M◦, we can construct a hypothesis πm◦ that will inducethe out-of-equilibrium belief µ∗(·|m◦). Let {πm◦}m◦∈M◦ be the family of such hypotheses.

Step 3: We can suitably choose a second-order prior ρ with suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ suchthat

{π∗} := arg maxπ∈suppF (ρ)

ρ(π), (34)

(i.e., π∗ is the most likely hypothesis) and

{π∗∗m◦} := arg maxπ∈suppF (ρ)

ρm◦(π) for each m◦ ∈M◦. (35)

(i.e., π∗∗m◦ is the most likely hypothesis after updating ρ given m◦ ∈M◦). Therefore, there exists aFocused HTE (s∗, r∗, ρ, µ∗ρ) supporting the PBE (s∗, r∗, µ∗). �

Proof of Observation 1. To prove that both refinement criteria are not nested, we provide twocases. In Case 1, we show a PBE that passes the Rational HT refinement but fails the IntuitiveCriterion. In Case 2, we demonstrate a PBE that fails the former but passes the latter criterion.

Case 1: Recall the PBE with pooling on N (i.e., PBE-II) for the labor-market game in Figure 2:

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, µ∗(θL|N) = 1/3 and µ∗(θL|S) ≥ 1/2.

First, we show that PBE-II passes the Rational HT refinement. Consider the rational hypotheses

30

π′1 and π′2 depicted in Example 5b. Then, the Rational HTE

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, suppF (ρ) = {π′1, π′2} such that ρ(π′2) < ρ(π′1)

µ∗ρ(θL|N) = 1/3 and µ∗ρ(θL|E) = 1.

supports the pooling PBE with µ∗(θL|E) = 1

Now, we show that PBE-II fails the Intuitive Criterion. According to the Intuitive Criterion,θH could be better off than her equilibrium payoff if she sends the out-of-equilibrium message E.That is, I(E) = {θH}. This induces the out-of-equilibrium belief µ(θH |E) = 1. However, whenthe Receiver learns that E is sent by θH , she will choose e instead of m. Given that the Receiverplays e against E, type θH will indeed choose E. Thus, PBE-II does pass the Intuitive Criterion.

Figure 5: An intuitive PBE that fails the Rational HT refinement (x = 2 or x = 4)

Case 2: Consider the labor-market game depicted in Figure 5. Assume that x = 4. There exists apooling PBE in which both types choose N . That is,

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, µ∗ (θL|N) = 2/5 and µ∗ (θL|S) ≤ 1/4.

First, we show that there does not exist a Rational HTE that supports this PBE. The Sender hastwo best-response strategies against rational strategies of the Receiver. That is, B = {s1, s2} where

1) s1 := {s1(θL) = E, s1(θH) = E} is the best response against r(E) = e and r(N) = m,

2) s2 := {s2(θL) = N, s2(θH) = N} is the best response against r(E) = m and r(N) = m.

31

Consider β on B = {s1, s2} such that β(s1) = λ and β(s2) = (1 − λ) where λ ∈ [0, 1]. Each βdefines the following rational hypothesis:

π(λ) = {π(θL, N) = (1−λ)2/5, π(θL, E) = λ2/5, π(θH , N) = (1−λ)3/5, π(θH , E) = λ3/5}.

The family {π(λ)}λ∈[0,1] depicts all rational hypotheses for this game. However, none of themrationalizes the Receiver’s equilibrium best response to the out-of-equilibrium message E. Then,for each λ ∈ [0, 1], updating of π(λ) given E delivers µρ(θL|E) = 2/5. However, given this belief,the Receiver will choose e instead of m. Hence, there does not exist a Rational HTE supportingthe above PBE. Therefore, the PBE fails the Rational HT refinement.

Now, we show that the pooling PBE passes the Intuitive Criterion. According to the IntuitiveCriterion, both types θL and θH could be better off than their equilibrium payoff by choosing theout-of-equilibrium message E. That is, I(E) = {θL, θH}. In this case, the Intuitive Criterionadmits any beliefs over I(E) = {θL, θH} including any µ(θL|E) such that µ(θL|E) ≤ 1/4. Weknow that each player does not have an incentive to deviate from the given equilibrium strategy aslong as µ(θL|E) ≤ 1/4. Therefore, the pooling PBE passes the Intuitive Criterion. �

Proof of Theorem 2. Let (s∗, r∗, µ∗) be a PBE. Consider an out-of-equilibrium message m◦ ∈M◦. Let r∗(m◦) = a∗ be the Receiver’s equilibrium response to m◦. By assumption, there is asingle type that has an incentive to deviate from the equilibrium strategy s∗ by sendingm◦. That is,I(m◦) = {θm◦} for some θm◦ ∈ Θ. Moreover, the PBE passes the Intuitive Criterion. This meansthat a∗ is rationalized by µ∗(θm◦ |m◦) = 1, the posterior belief admitted by the Intuitive Criterion.

Let B be the set of (second-order) rational strategies for the Sender. In Step 1, we show thatthere exists s◦ ∈ B that generates m◦, and construct a rational hypothesis πm◦ that is consistentwith m◦ (i.e., πm◦(θ,m◦) > 0 for some θ ∈ Θ), and show that πm◦ justifies the out-of-equilibriumbelief µ∗(θm◦|m◦) = 1. In Step 2, we construct a Rational HTE supporting the intuitive PBE.

Step 1: By the single-type condition, there exists a best response to some belief over Θ (i.e.,a◦ ∈ BR(Θ,m◦)). By the Intuitive Criterion, we have

u∗S(θ) ≤ uS(θ,m◦, a◦) for θ = θm◦ , (36)

andu∗S(θ) > uS(θ,m◦, a◦) for θ ∈ Θ \ {θm◦}, (37)

where u∗S(θ) is the Sender’s equilibrium payoff when his type is θ ∈ Θ. That is, when the Receiverchooses a◦ in response to m◦, only type θm◦ might deviate from the equilibrium strategy s∗.

32

Now, we construct a rational strategy r◦ ∈ R for the Receiver as follows:

r◦(m) =

{r∗(m), if m 6= m◦,

a◦, if m = m◦.(38)

Since r∗(m) is rational for each m 6= m◦, r◦ is a rational strategy for the Receiver. Then, byconstruction of r◦, only type θm◦ best responds with message m◦ against r◦, i.e.,

m◦ ∈ arg maxm∈M

uS (θ,m, r◦(m)) for θ = θm◦ , (39)

andm◦ /∈ arg max

m∈MuS (θ,m, r◦(m)) for θ ∈ Θ \ {θm◦}. (40)

Notice that the equilibrium strategy s∗(θ) is a best response against r◦ for any θ ∈ Θ \ {θm◦}.Therefore, there exists a best-response strategy s◦ against r◦ that generates message m◦. That is,

s◦ := {s◦(θm◦) = m◦, s◦(θ) = s∗(θ) for any θ ∈ Θ \ {θm◦}} is a best response aginast r◦.

Now, define a belief β of the Receiver overB such that β(s◦) = 1 and β(s) = 0 for any s ∈ B\{s◦}.This belief together with the prior information p induces the simple-rational hypothesis:

πm◦ :={π(θm◦ ,m

◦) = p(θm◦), π(θ, s∗(θ)) = p(θ) for any θ ∈ Θ \ {θm◦}}. (41)

Since π(θm◦ ,m◦) = p(θm◦ > 0, πm◦ is consistent with m◦. By updating πm◦ given m◦, we get

µρ(θm◦|m◦) =πm◦(θm◦ ,m

◦)

πm◦(Θ,m◦)= 1, (42)

showing that πm◦ justifies the out-of-equilibrium belief of the Intuitive Criterion, µ∗(θm◦|m◦) = 1.Since m◦ was chosen arbitrary, we can construct a simple-rational hypothesis for any out-of-

equilibrium message. That is, for any m◦ ∈M◦ there exists a simple-rational hypothesis πm◦ thatjustifies µ∗(θm◦|m◦) = 1. Let {πm◦}m◦∈M◦ be the collection of such simple-rational hypotheses.

Step 2: Now, we construct a simple-rational hypothesis π∗ that rationalizes the Receiver’s on-the-equilibrium-path behavior. Since s∗ best responds to r∗ and r∗ is rational, a simple-rationalhypothesis π∗ exists. More precisely, the degenerate belief β such that β(s∗) = 1 and β(s) = 0 forany s ∈ B \ {s∗} together with the prior information p induces the simple-rational hypothesis π∗:

π∗(θ,m) =

{p(θ), if s∗(θ) = m,

0, otherwise.(43)

33

We can now suitably choose a second-order prior ρ with suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ such that

{π∗} := arg maxπ∈suppF (ρ)

ρ(π) and {π∗∗m◦} := arg maxπ∈suppF (ρ)

ρm◦(π) for each m◦ ∈M◦. (44)

Therefore, there exists a Rational HTE (s∗, r∗, ρ, µ∗ρ) supporting the intuitive PBE, (s∗, r∗, µ∗). �

Proof of Corollary 2. Let (s∗, r∗, µ∗) be a PBE andM◦ be the set of out-of-equilibrium messages.The single-type condition states that for each m◦ ∈ M◦, we have I(m◦) = {θm◦} for someθm◦ ∈ Θ. From the proof of Theorem 2 (Step 1), we know that we can construct a family of simple-rational hypotheses {πm◦}m◦∈M◦; each πm◦ being consistent with m◦ (i.e., πm◦(θm◦ ,m◦) > 0). Itremains to be shown that each πm◦ is the only simple-rational hypothesis consistent with m◦.

By Condition (13), there does not exist a rational strategy r ∈ R against which sending an out-of-equilibrium messagem◦ ∈M◦ is a best response for some θ ∈ T (m◦). Recall, T (m◦) is the setof types that cannot improve upon their equilibrium payoff by choosingm◦. This implies that theredoes not exist a best-response strategy that generates m◦ by some type in T (m◦). Equivalently,there does not exist a rational hypothesis π such that π(θ,m◦) > 0 for each θ ∈ T (m◦) andm◦ ∈M◦. Thus, any rational hypothesis πm◦ that is consistent with m◦ ∈M◦ must satisfy

πm◦(θm◦ ,m◦) = πm◦(Θ,m

◦) = p(θm◦) where {θm◦} = I(m◦). (45)

In other words, by updating any such rational hypothesis πm◦ given m◦, including πm◦ , we have

µρ(θm◦|m◦) =πm◦(θm◦ ,m

◦)

πm◦(Θ,m◦)= 1. (46)

Therefore, the family suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ of the Rational HTE (s∗, r∗, ρ, µ∗ρ) thatwe constructed in the proof of Theorem 2 justify the unique family of off-the-path beliefs (i.e.,µ∗ρ(θm◦|m◦) = 1 for any m◦ ∈M◦). Hence, the Rational HT refinement is unique. �

Proof of Proposition 1. Let (s∗, r∗, ρ, µ∗ρ) be a Rational HTE and M◦ be the set of out-of-equilibrium messages. Let suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ be the set of hypotheses where π∗ isthe initial hypothesis and π∗∗m◦ is the new hypothesis selected for the out-of-equilibrium messagem◦. In Step 1, we show that there exists another rational hypothesis πm◦ that consistent with m◦

and that it is behaviorally consistent with respect to π∗. In Step 2, we show that πm◦ induces thesame out-of-equilibrium belief as π∗∗m◦ . In Step 3, we construct a Behaviorally Consistent HTE.

Step 1: Fix an out-of-equilibrium message m◦ ∈ M◦. Let B = {s1, . . . , sN} be the set of bestresponses for the Sender. Without loss of generality, assume that s1 is the equilibrium strategy

34

s∗ (i.e., s∗ = s1). Let β be the Receiver’s belief over B that induces the rational hypothesis π∗∗m◦ .Recall π∗∗m◦ is consistent with m◦ if β(s) > 0 for some s ∈ B that generates m◦. That is,

π∗∗m◦(θ,m◦) =

∑si∈B s.t.si(θ)=m

◦

β(si)p(θ) > 0. (47)

For the sake of simplicity, we set

β(si) = λi ∈ [0, 1] for each si ∈ {s1, . . . , sN}. (48)

Since β is additive on B, we have that λ1 + . . .+ λN = 1.Now, we construct another belief β : B → [0, 1]. For a parameter ε ∈ (0, 1), define β such that

β(s1) = 1− ε(1− λ1) and β(si) = ελi for each si ∈ {s2, . . . , sN}, (49)

Notice that∑

si∈B β(si) = 1 − ε + (λ1 + . . . + λN)ε = 1. Then, for each ε ∈ (0, 1), β inducesanother rational hypothesis, denoted by πm◦(ε), that is consistent with m◦.

By our assumption, A is a finite set and the Receiver’s best-response correspondence is single-valued on the equilibrium path. Therefore, there is a sufficiently small ε∗ ∈ (0, 1) such that πm◦(ε∗)satisfies the following condition: For any message m such that π∗(Θ,m) > 0, we have

arg maxa∈A

∑θ∗∈Θ

π∗(θ,m)

π∗(Θ,m)uR(θ,m, a) = {r∗(m)} = arg max

a∈A

∑θ∈Θ

πm◦(ε∗)(θ,m)

πm◦(ε∗)(Θ,m)uR(θ,m, a).

That is, πm◦(ε∗) rationalizes the same action along the equilibrium path as π∗ does. This provesthat πm◦(ε∗) is behaviorally consistent with the initial hypothesis π∗.

Step 2: We show that πm◦(ε∗) induces the same out-of-equilibrium belief given m◦ as π∗∗m◦ does.By updating πm◦(ε∗) given m◦, we get

µπ(θ|m◦) =πm◦(ε

∗)(θ,m◦)

πm◦(ε∗)(Θ,m◦)=

∑si∈B\{s1} s.t.si(θ)=m

◦

ελip(θ)

∑θ∈Θ

∑si∈B\{s1} s.t.si(θ)=m

◦

ελip(θ)=

∑si∈B\{s1} s.t.si(θ)=m

◦

λip(θ)

∑θ∈Θ

∑si∈B\{s1} s.t.si(θ)=m

◦

λip(θ), (50)

35

for every θ ∈ Θ. By updating π∗∗m◦ given m◦, we get

µ∗π(θ|m◦) =π∗∗m◦(θ,m

◦)

π∗∗m◦(Θ,m◦)

=

∑si∈B\{s1} s.t.si(θ)=m

◦

λip(θ)

∑θ∈Θ

∑si∈B\{s1} s.t.si(θ)=m

◦

λip(θ), (51)

for every θ ∈ Θ. Thus, we haveµπ(·|m◦) = µ∗π(·|m◦), (52)

showing that πm◦(ε∗) induced the out-of-equilibrium belief of the Rational HTE.

Step 3: Since m◦ was chosen arbitrary, we can construct a behaviorally consistent hypothesisπm◦(ε

∗(m◦)) for each out-of-equilibrium message m◦ ∈ M◦.39 Therefore, by suitable choosing asecond-order prior ρ with suppF (ρ) = {π∗, π∗∗m◦(ε∗(m◦))}m◦∈M◦ such that

{π∗} := arg maxπ∈suppF (ρ)

ρ(π) and {π∗∗m◦(ε∗(m◦))} := arg maxπ∈suppF (ρ)

ρm◦(π) for each m◦ ∈M◦,

we constructed a Behaviorally Consistent HTE that supports the Rational HTE. �

Proof of Proposition 2. Let (s∗, r∗, µ∗) be a pooling PBE. Let m∗ be the pooling message andM◦ be the set of out-of-equilibrium messages. Suppose that the PBE passes the Intuitive Criterionand that it satisfies the single-type condition (i.e., for each m◦ ∈ M◦, I(m◦) = {θm◦} for someθm◦ ∈ Θ). By Theorem 2, we know that there exists a Rational HTE (s∗, r∗, ρ, µ∗ρ) that justifies theIntuitive Criterion outcome. That is, for each m◦ ∈ M◦, there exists a simple-rational hypothesisπm◦ that induces the out-of-equilibrium belief µ∗ρ(θm◦ |m◦) = 1. In other words, we have

πm◦ := {π(θm◦ ,m◦) = p(θm◦), π(θ, s∗(θ)) = p(θ) for any θ ∈ Θ \ {θm◦}} , (53)

where I(m◦) = {θm◦}. In the Rational HTE with pooling on m∗, πm◦ takes the particular form:

(i) πm◦(θ,m∗) = π∗(θ,m∗) = p(θ) for θ ∈ Θ \ {θm◦},

(ii) πm◦(θm◦ ,m◦) = p(θm◦),

(iii) πm◦(θm◦ ,m∗) = 0,

(54)

where π∗ is the initial hypothesis of the Rational HTE supporting the pooling PBE.Now, suppose that θm◦ is a dummy type for the equilibrium message m∗. Thus, for any a ∈ A

39Notice that ε∗(m◦) depends on m◦ ∈M◦.

36

such thata ∈ arg max

a∈A

∑θ∈Θ

π∗(θ,m∗)

π∗(Θ,m∗)uR(θ,m∗, a),

it is true thata ∈ arg max

a∈A

∑θ∈Θ

π(θ,m∗)

π(Θ,m∗)uR(θ,m∗, a), (55)

where π is a probability distribution on Θ×M such that

π(θ,m∗) =

{π∗(θ,m∗), if θ ∈ Θ \ {θm◦},

0, if θ = θm◦ .(56)

Notice that the simple-rational hypothesis πm◦ in (54) satisfies Condition (55) and (56). By Con-dition (55), πm◦ is behaviorally consistent with respect to π∗. Notice that m◦ was arbitrarily cho-sen. Therefore, for each out-of-equilibrium message m◦, we can find a simple-rational hypothesisπm◦ that rationalizes the best responses on and off the equilibrium paths of the Receiver. Let{πm◦}m◦∈M◦ be the family of such behaviorally consistent hypotheses.

Thus, we can choose a second-order prior ρ with suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ such that

{π∗} := arg maxπ∈suppF (ρ)

ρ(π) and {π∗∗m◦} := arg maxπ∈suppF (ρ)

ρm◦(π) for each m◦ ∈M◦. (57)

This shows that there exists a Behaviorally Consistent HTE that supports the pooling PBE. �

Proof of Observation 2. We prove that the following PBEs can be supported by rational HTEs.

(i) s∗(θL) = s∗(θH) = e∗ ∈M such that e0 ≤ e∗ ≤ y := (θH − θL)(θL − (1− α)θH

)

(ii) w∗(e) =

θL if e < e∗,

E(θ) if e∗ ≤ e ≤ en,

θH if en < e ≤ eN .

(iii) µ∗(θL|e) =

1 if e < e∗,

1− α if e∗ ≤ e ≤ en,

0 if en < e ≤ eN .

Fix a pooling message and denote it by e∗i := e∗. We first construct a rational hypothesis π∗0that supports the employer’s best-response strategy to e∗i . Consider the following strategy for theemployer and the posterior beliefs over Θ that rationalize it:

w0(e) =

{E(θ), if e ≥ e∗i ,

θL, if e < e∗i .and µ(θL|e) =

{1− α, if e ≥ e∗i ,

1, if e < e∗i .(58)

Given this wage scheme, both types are better off by choosing e∗. Therefore, s0 := {s0(θL) =

e∗i , s0(θH) = e∗i } is the best-response strategy against w0(e). The degenerate belief β such that

37

β(s0) = 1 and p on Θ induce the simple-rational hypothesis π∗0 , yielding the on-the-equilibriumpath belief µ(θL|e∗i ) = p(θL) = 1− α.

As a next step, we construct a rational hypothesis for each out-of-equilibrium message e 6= e∗i .Notice thatM◦ =M\ {e∗i } is the set of out-of-equilibrium messages. We can limit our attentionto the following partition ofM◦:

P(M◦) ={{e0, . . . , ei−1}︸ ︷︷ ︸

Case 1

, {ei+1, . . . , en}︸ ︷︷ ︸Case 2

, {en+1, . . . , eN}︸ ︷︷ ︸Case 3

}(59)

where en := θL(θH − θL) and eN := θH(θH − θL).We consider the three cases in order.

Case 1. Fix e′ ∈ {e0, . . . , ei−1}. Consider the following strategy w1(e) for the employer and theposterior beliefs that rationalize w1(e):

w1(e) =

w′ if e = e′,

θH , if e = en,

θL, elsewhere,

and µ(θL|e) =

θH−w′θH−θL

, if e = e′,

0, if e = en

1, elsewhere.

(60)

Notice that w′ must satisfy the following two conditions. First, w′ for e′ has to make the low-productivity type better off than her payoff for offering education level 0 at the lowest wage θL;i.e,

w′ − e′

θL> θL −

0

θL, or equivalently, w′ > θL +

e′

θL(61)

Second, w′ for e′ has to make the high-productivity type worse off then her payoff for offering enat the highest wage θH ; i.e.,

w′ − e′

θH< θH −

enθH, or equivalently, w′ < θH +

e′

θH− enθH

(62)

From Conditions (61) and (62), µ(θL|e′) is defined as

en − e′

θH(θH − θL)< µ(θL|e′) :=

θH − w′

θH − θL< 1− e′.

θL(θH − θL)), (63)

Thus, we have µ(θL|e′) that rationalizes w′ satisfying Conditions (61) and (62).Against w1(e), the best-response strategy for the worker is s1 := {s1(θL) = e′ , s1(θH) = en}.

Hence, the degenerate belief β such that β(s1) = 1 induces the simple-rational hypothesis π1(e′),yielding the out-of-equilibrium belief µ(θL|e′) = 1 for each e′ ∈ {e0, . . . , ei−1}.

Case 2. Fix e′ ∈ {ei+1, . . . , en}. Consider the following strategy w2(e) for the employer and the

38

posterior beliefs that rationalize w2(e):

w2(e) =

{θH , if e = e′,

θL, elsewhere,and µ(θL|e) =

{0, if e = e′,

1, elsewhere.(64)

Against w2(e), the best-response strategy for the worker is s2 := {s2(θL) = e′, s2(θH) = e′}. Thedegenerate belief β such that β(s2) = 1 induces the simple-rational hypothesis π2(e′), yielding theoff-the-equilibrium-path belief µ(θL|e′) = p(θL) = 1− α for each e′ ∈ {ei+1, . . . , en}.

Case 3. Fix e′ ∈ {en+1, . . . , eN}. Consider the following strategy w3(e) for the employer and theposterior beliefs that rationalize w3(e):

w3(e) =

{θH , if e = e′,

θL, elsewhere,and µ(θL|e) =

{0, if e = e′,

1, elsewhere.(65)

Against w3(e), the best response for the worker is s3 =: {s3(θL) = e0, s3(θH) = e′}. Hence,β(s3) = 1 induce the simple-rational hypothesis π3(e′), yielding the off-the-equilibrium path be-lief µ(θL|e′) = 0 for each e′ ∈ {en+1, . . . , eN}.

Finally, we can suitably choose a second-order prior ρ such that

suppF (ρ) = {π0, π1(e)e∈{e0,...,ei−1}, π2(e)e∈{ei+1,...,en}, π3(e)e∈{en+1,...,eN}},

{π0} := arg maxπ∈suppF (ρ)

ρ(π) and {π∗∗(e)} := arg maxπ∈suppF (ρ)

ρe(π) for each e ∈M◦,

showing that there exists a Rational HTE (s∗, r∗, ρ, µ∗ρ) supporting the pooling PBE with e∗i . There-fore, each pooling PBE with e∗i such that e0 ≤ e∗i ≤ y passes the Rational HT refinement. �

B PBE and Focused HTE in Behavioral Strategies

In this Appendix, we show that Theorem 1 can be extended to setups with behavioral strategies.That is, if we allow for behavioral strategies, then each PBE can again be supported by a FocusedHTE.

Denote by bS : Θ → ∆(M) a behavioral strategy for the Sender. We denote by bS(m|θ) theprobability that that type θ sends message m. Denote by bR :M→ ∆(A) behavioral strategy forthe Receiver. We denote bR(a|m) the probability that the Receiver chooses action a in response m.

A PBE in behavioral strategies is defined as follows.

39

Definition 10 (b∗S, b∗R, µ

∗) is a Perfect Bayesian Equilibrium in behavioral strategies if:

(i) b∗S(·|θ) ∈ arg maxbS(·;θ)∈∆(M)

uS (θ, bS(·; θ), b∗R) for each θ ∈ Θ,

(ii) b∗R(·;m) ∈ arg maxbR(·;m)∈∆(A)

∑θ∈Θ

µ∗(θ|m)uR(θ,m, bR(·;m)) for each m ∈M,

(iii) µ∗(θ|m) =π(θ,m)

π(Θ,m)for each θ ∈ Θ if π(Θ,m) > 0, and

µ∗(·|m) is an arbitrary probability distribution over Θ if π(Θ,m) = 0,where

π(θ,m) =

{b∗S(m; θ)p(θ), if b∗S(m; θ) > 0,

0, otherwise.

Notice that we have a pure PBE if b∗S and b∗R are degenerate behavioral strategies. That is, for eachθ ∈ Θ, b∗S(m|θ) = 1 for some m ∈M; and for each m ∈M, b∗R(a|m) = 1 for some a ∈ A.

The next result shows that each PBE, (b∗S, b∗R, µ

∗), can be supported by a Focused HTE. Thatis, we can always find a set of hypotheses that induce beliefs of the PBE in behavioral strategies.

Theorem 3 For each PBE (b∗S, b∗R, µ

∗) there exists a Focused HTE in behavioral strategies, (b∗S, b∗R, ρ, µ

∗ρ),

that supports the PBE. That is, there are a second-order prior ρ, a set of hypotheses suppF (ρ) and

a family of posterior beliefs µρ := {µρ(·|m)}m∈M such that

(i) µ∗(·|m) =π∗(·,m)

π∗(Θ,m)= µ∗ρ(·|m) = for each equilibrium message m where

{π∗} := arg maxπ∈suppF (ρ)

ρ(π) andπ∗(θ,m)

π∗(Θ,m)=

b∗S(m; θ)∑θ′∈Θ b

∗S(m; θ′)p(θ′)

,

(ii) µρ(·|m) =π∗∗m (·,m)

π∗∗m (Θ,m)= µ∗ρ(·|m◦) = for each out-of-equilibrium message m◦ ∈M◦ where

{π∗∗m } := arg maxπ∈suppF (ρ)

ρm◦(π).

Proof: Consider a PBE, (b∗S, b∗R, µ

∗), in behavioral strategies. Let M◦ be the set of out-of-equilibrium messages. In fact, we can apply the argument presented in Step 2 in the proof ofTheorem 1 to show that for each m◦ ∈ M◦, there is a hypothesis πm◦ that induces the out-of-equilibrium belief µ∗(·|m◦) of the PBE. Let {π∗∗m◦}m◦∈M◦ the set of such hypothesis.

It remains to construct a hypothesis π∗ that induces the PBE belief on the equilibrium path. Tothis end, we need to find a belief β on S such that for each m ∈ M with

∑θ∈Θ b

∗S(θ|m)p(θ) > 0,

40

it satisfies the following identity: For each θ ∈ Θ,

µ∗(θ|m) =b∗S(θ|m)p(θ)∑

θ′∈Θ

b∗S(θ′|m)p(θ′)=

∑s∈S s.t.s(θ)=m

β(s)p(θ)

∑θ′∈Θ

∑s∈S s.t.s(θ′)=m

β(s)p(θ′)=

π∗(θ,m)

π∗(Θ,m). (66)

Notice β can be seen as a mixed strategy. Since signaling games satisfy perfect recall, by the KuhnTheorem, there exists β∗ that is payoff-equivalent with the equilibrium strategy b∗ (Kuhn, 1953).Thus, β∗ together with the prior information p induces the hypothesis π∗ defined in Equation (66).40

Thus, we can suitably choose a second-order prior ρ with suppF (ρ) = {π∗, π∗∗m◦}m◦∈M◦ suchthat

{π∗} := arg maxπ∈suppF (ρ)

ρ(π), and {π∗∗m◦} := arg maxπ∈suppF (ρ)

ρm◦(π) for each m◦ ∈M◦ (67)

(i.e., π∗ is the initial hypothesis and π∗∗m◦ is the most likely hypothesis after updating ρ givenm◦ ∈M◦). Therefore, there exists a Focused HTE (b∗S, b

∗R, ρ, µ

∗ρ) supporting the PBE (b∗S, b

∗R, µ

∗).�

C Intuitive Criterion and Arbitrary Beliefs.

In this Appendix, we present a PBE with I(m◦) = Θ that passes the Rational HT refinement aswell as the Intuitive Criterion. However, the Intuitive Criterion does not reduce the number of PBEbeliefs while our refinement does.

Consider the game depicted in Figure 5. Assume x = 2. There is a PBE with pooling on N :

s∗(θL) = s∗(θH) = N, r∗(E) = m, r∗(N) = e, µ∗(θL|N) = 2/5 µ∗(θL|L) ≤ 1/2.

This PBE passes the Intuitive Criterion. The Intuitive Criterion asserts that I(E) = {θL, θH}and T (E) = ∅. Therefore any probability distribution over Θ = {θL, θH} is admitted. Hence, eachout-of-equilibrium belief is consistent with the Intuitive Criterion.

We have the same rational hypotheses as defined in Case 2 in the proof of Observation 1. Each

40Kreps and Ramey (1987) not all sequential-equilibrium beliefs are structurally consistent. Kreps and Ramey(1987) showed that sequential-equilibrium beliefs can be justified by a weaker consistency notion, called convexstructural consistency. That is, they showed that each out-of-equilibrium belief in a sequential equilibrium can bejustified by updating a convex combination of probabilities derived from a finite set of (behavioral) strategies.

41

β on B = {s1, s2} such that β(s1) = λ and β(s2) = (1− λ) where λ ∈ [0, 1] defines

π(λ) := {πλ(θL, E) = λ2/5, πλ(θH , E) = λ3/5, πλ(θL, N) = (1−λ)2/5, πλ(θH , N) = (1−λ)3/5}.

Updating π(λ) given E yields the out-of-equilibrium belief µρ(θL|E) = 2/5 for each λ ∈ (0, 1].Hence, the PBE can be supported by a Rational HTE with ρ such that suppF (ρ) = {π(λ), π2} andρ(π(λ)) < ρ(π2). Moreover, the Rational HT refinement selects the unique belief µρ(θL|E) = 2/5.

D Non-Existence of Focused HTE under Simple Hypotheses

In this Appendix, we show that a Focused HTE constrained to simple hypotheses may not existeven though a (pure) PBE exists.

Consider the game depicted in Figure 6. This game has only one PBE with pooling on E:

Figure 6: Non-existence of Focused HTE with simple hypothesis

s∗(θL) = s∗(θH) = E, r∗(E) = e, r∗(N) = m, µ∗(θL|E) = 0.9, and µ∗(θL|N) ∈ [0.475, 0.525].

Recall that a simple hypothesis is defined by a degenerate belief on the set of pure strategies.Therefore, there are four simple hypotheses in this game:

1) π1 := {π1(θL, N) = 0.9, π1(θH , E) = 0.1} if β(s1) = 1 and s1 := {s(θL) = N, s(θH) = E},

2) π2 := {π2(θL, E) = 0.9, π2(θH , N) = 0.1} if β(s2) = 1 and s2 := {s(θL) = E, s(θH) = N},

42

3) π3 := {π3(θL, E) = 0.9, π3(θH , E) = 0.1} if β(s3) = 1 and s3 := {s(θL) = E, s(θH) = E},

4) π4 := {π4(θL, N) = 0.9, π4(θH , N) = 0.1} if β(s4) = 1 and s4 := {s(θL) = N, s(θH) = N}.

Notice that π3 rationalizes the Receiver’s best response to the equilibrium message E. However,none of the other simple hypotheses in {π1, π2, π4} rationalizes the Receiver’s best response to N .Therefore, there does not exist a Focused HTE with simple hypotheses that can explain the poolingPBE.

References

BANKS, J., C. CAMERER, AND D. PORTER (1994): “An Experimental Analysis of Nash Refine-ments in Signaling Games,” Games and Economic Behavior, 6(1), 1–31.

BANKS, J. S. (1990): “A Model of Electoral Competition with Incomplete Information,” Journal

of Economic Theory, 50(2), 309–325.

BANKS, J. S., AND J. SOBEL (1987): “Equilibrium Selection in Signaling Games,” Econometrica,55(3), 647–661.

BERNHEIM, B. D. (1994): “A Theory of Conformity,” Journal of Political Economy, 102(5),841–877.

BHATTACHARYA, S. (1979): “Imperfect Information, Dividend Policy, and the Bird in the HandFallacy,” Bell Journal of Economics, 10(1), 259–270.

BRANDTS, J., AND C. A. HOLT (1992): “An Experimental Test of Equilibrium Dominance inSignaling Games,” American Economic Review, 82(5), 1350–1365.

BRANDTS, J., AND C. A. HOLT (1993): “Adjustment Patterns and Equilibrium Selection in Ex-perimental Signaling Sames,” International Journal of Game Theory, 22(3), 279–302.

CHO, I.-K., AND D. M. KREPS (1987): “Signaling Games and Stable Equilibria,” Quarterly

Journal of Economics, 102(2), 179–221.

ESO, P., AND J. SCHUMMER (2009): “Credible Deviations from Signaling Equilibria,” Interna-

tional Journal of Game Theory, 38(3), 411–430.

FUDENBERG, D., AND K. HE (2018): “Learning and Type Compatibility in Signaling Games,”Econometrica, 86(4), 1215–1255.

(2019): “Payoff Information and Learning in Signalling Games,” Working Paper.

43

FUDENBERG, D., AND J. TIROLE (1991): Game Theory. MIT Press.

GAL-OR, E. (1989): “Warranties as a Signal of Quality,” Canadian Journal of Economics, pp.50–61.

JEONG, D. (2019): “Job Market Signaling with Imperfect Competition among Employers,” Inter-

national Journal of Game Theory, pp. 1–29.

JOHN, K., AND J. WILLIAMS (1985): “Dividends, Dilution, and Taxes: A Signalling Equilib-rium,” Journal of Finance, 40(4), 1053–1070.

KREPS, D. M., AND G. RAMEY (1987): “Structural Consistency, Consistency, and SequentialRationality,” Econometrica, 55(6), 1331–1348.

KREPS, D. M., AND R. WILSON (1982): “Sequential Equilibria,” Econometrica, 50(4), 863–894.

KUBLER, D., W. MULLER, AND H.-T. NORMANN (2008): “Job-Market Signaling and Screening:An Experimental Comparison,” Games and Economic Behavior, 64(1), 219–236.

KUHN, H. W. (1953): “Extensive Games and the Problem of Information,” in Contributions to the

Theory of Games II, ed. by H. W. Kuhn, and A. W. Tucker, pp. 193–216. Princeton UniversityPress.

LOHMANN, S. (1995): “Information, Access, and Contributions: A Signaling Model of Lobby-ing,” Public Choice, 85(3-4), 267–284.

MAILATH, G. J. (1988): “A Reformulation of a Criticism of the Intuitive Criterion and ForwardInduction,” Working Paper.

MAILATH, G. J., M. OKUNO-FUJIWARA, AND A. POSTLEWAITE (1993): “Belief-Based Refine-ments in Signalling Games,” Journal of Economic Theory, 60(2), 241 – 276.

MILGROM, P., AND J. ROBERTS (1982): “Limit Pricing and Entry under Incomplete Information:An Equilibrium Analysis,” Econometrica, 50(2), 443–459.

(1986): “Price and Advertising Signals of Product Quality,” Journal of Political Economy,94(4), 796–821.

MILLER, R. M., AND C. R. PLOTT (1985): “Product Quality Signaling in Experimental Markets,”Econometrica, pp. 837–872.

NESLON, P. (1974): “Advertising as Information,” Journal of Political Economy, 82(4), 729?754.

44

ORTOLEVA, P. (2012): “Modeling the Change of Paradigm: Non-Bayesian Reactions to Unex-pected News,” American Economic Review, 102(6), 2410–36.

RILEY, J. G. (2001): “Silver Signals: Twenty-Five Years of Screening and Signaling,” Journal of

Economic literature, 39(2), 432–478.

SPENCE, M. (1973): “Job Market Signaling,” Quarterly Journal of Economics, 87(3), 355–374.

SUN, L. (2019): “Hypothesis Testing Equilibrium in Signalling Games,” Mathematical Social

Sciences, 100, 29–34.

VAN DAMME, E. (1989): “Stable Equilibria and Forward Induction,” Journal of Economic Theory,48(2), 476 – 496.

45