Informative tests in signaling environments

Weksler, Ran; Zik, Boaz Article Informative tests in signaling environments Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Weksler, Ran; Zik, Boaz (2022) : Informative tests in signaling environments, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 3, pp. 977-1006, https://doi.org/10.3982/TE4461 This Version is available at: https://hdl.handle.net/10419/296377 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 17 (2022), 977–1006 1555-7561/20220977 Informative tests in signaling environments Ran Weksler Department of Economics, University of Haifa Boaz Zik Institute for Microeconomics, University of Bonn We study a receiver’s learning problem of choosing an informative test in a signaling environment. Each test induces a signaling subgame. Thus, in addition to its direct effect on the receiver’s information, a test has an indirect effect through the sender’s signaling strategy. We show that the informativeness of signaling in the equilibrium that a test induces depends on the relative informativeness of the test’s high and low grades. Consequently, we find that the receiver’s preference relation over tests needs not comply with Blackwell’s (1951) order. Our findings may shed light on phenomena such as grade inflation and information coarsening. Keywords. Signaling games, information design, strategic learning, strategic information transmission. JEL classification. D82, D83, C72. 1. Introduction When decision-makers gather information, they need to decide what kind of information to learn, i.e., which test to choose from their set of available tests.1Often, the decision-maker (henceforth the receiver) makes the learning decision in a signaling environment where an informed agent (henceforth the sender) can take observable costly actions. For example, when public certifiers, such as safety and environmental organizations, test how firms perform in a particular area, firms can signal by spending money on unproductive channels such as advertising or donations. When job market recruiters test potential candidates to evaluate their competence for the job, candidates can signal by indicating they would accept a low wage offer for their initial employment period. When academic institutions test students to identify their qualities, students can signal by participating in extracurricular activities. In this paper, we analyze the receiver’s preferences over tests in such a signaling environment. Ran Weksler: [email protected] Boaz Zik: [email protected] We are grateful to two anonymous referees for their helpful feedback and valuable comments. We also thank Elchanan Ben-Porath, Daniel Fershtman, Alex Gershkov, Benny Moldovanu, and Roland Strausz. Boaz Zik gratefully acknowledges funding by the German Research Foundation (DFG) through CRC TR 224 (Project B01). 1Our notion of a test is identical to the notion of an experiment in Blackwell (1951). ©2022 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4461 978 Weksler and Zik Theoretical Economics 17 (2022) We consider the following model. There is a sender who is fully informed about the state of the world, which is either low or high. There is a receiver who is initially uninformed about the state. The receiver wants to choose an action that matches the state. In the first stage of the game, the receiver chooses an informative test from a given set of feasible tests. The second stage of the game is a signaling stage ` alaSpence (1973), except for the following two features. First, the sender’s cost of signaling does not depend on the state;2we thus simply say that the sender chooses a signaling cost. Second, the sender observes the test choice of the receiver but not the realization of the test (henceforth grade) before choosing his signaling cost. In the last stage of the game, after observing the test’s grade and the sender’s signaling cost, the receiver chooses an action. The sender’s payoff is equal to the receiver’s action minus his signaling cost. In our model, since the sender’s cost of signaling does not depend on the state, if the receiver does not observe a grade of an informative test about the state, then in the unique perfect sequential equilibrium Grossman and Perry (1986), the sender chooses not to signal, and the receiver does not learn any information.3However, if the receiver privately observes a grade of an informative test about the sender’s type, then the different types of the sender face different subjective probabilities regarding the test’s grades. Therefore, although the sender’s payoff is independent of his type, his expected payoff depends on his type. In this case, there exists a unique equilibrium that satisfies the D1 criterion (Bank and Sobel (1987), and Cho and Kreps (1987)); see Daley and Green (2014). In this equilibrium, the low and high types of the sender may choose different signaling costs;4i.e., the equilibrium signaling strategy of the sender discloses information about the state. Therefore, in our model, the receiver’s test choice affects the information that the receiver obtains in equilibrium in two ways. The first is through the test’s intrinsic informativeness level. The second is through the informativeness level of the signaling strategy in the equilibrium that the test induces. We analyze how these two effects influence the receiver’s preference relation over tests. We start by characterizing the unique equilibrium that satisfies the D1 criterion for a given test. Depending on the prior on the state, two kinds of equilibria can emerge: one is the fully pooling equilibrium, where both types of the sender choose not to signal; the other is a semi-pooling equilibrium in which the high type selects one signaling cost while the low type mixes between selecting this cost and not signaling at all. Additionally, we find that in the latter equilibrium, the belief that the receiver develops after observing the pooling signaling cost is the belief that utilizes the test optimally in the sense that it maximizes the difference between the expected posteriors of the high type 2If the sender’s signaling costs depend on the state and satisfy the single-crossing condition, then the case where the receiver is uninformed corresponds to Spence’s (1973) signaling model. In Spence’s model the fully separating equilibrium, in which the state is perfectly revealed, is the unique equilibrium that satisfies stability-based refinements. Therefore, the receiver’s optimal learning strategy is not to learn information independently. 3In this degenerate environment, all the sender’s types share the same preferences over pairs of receiver’s beliefs and actions. Therefore, multiple equilibria satisfy the D1 criterion; in each of them, the sender’s types are indifferent between the equilibrium actions. These equilibria are ranked in terms of efficiency. Grossman and Perry’s refinement selects the efficient fully pooling equilibrium. 4The high (low) type corresponds to the realization of the high (low) state. Theoretical Economics 17 (2022) Informative tests in signaling 979 and the low type. This belief determines the informativeness level of the sender’s signaling strategy in equilibrium: the higher this belief is, the more separation there is in equilibrium, and the more informative the sender’s equilibrium signaling strategy is. We identify the main property that determines the informativeness level of the sender’s signaling strategy in the equilibrium that a test induces. Call a high (low) grade a grade that the high (low) type receives with a higher probability than the low (high) type. Our novel insight is that what matters is not the informativeness level of the test per se, but rather the relative informativeness of the test’s high and low grades. Specifically, we argue that the more informative the test’s low grades are relative to its high grades, the higher is the belief that the signaling cost induces, and the more informative the sender’s signaling is. The intuition for this insight is that a test whose low grades are more informative than its high grades is essentially better at identifying the low type. Intuitively, such a test is used optimally when the interim belief is high, i.e., when the belief assigns a low probability to the event that the sender’s type is low, as it manifests the test’s relative advantage. We present a series of formal results that establish this insight and study its implications for the receiver’s preference relation over tests. In Lemma 4, we identify conditions under which a test that is derived from another test by increasing the informativeness of its low (high) grades induces more (less) informative signaling in equilibrium. A direct implication of this lemma is that a more informative test in the sense of Blackwell (1951) does not necessarily induce a more informative signaling strategy than a less informative test.5We present a sufficient condition for a test to be both intrinsically more informative and more informative in terms of the signaling strategy it induces than another test, and we show that the receiver prefers the former test to the latter independently of the prior. We then show that because more informative tests do not necessarily induce more informative signaling strategies, the receiver’s preference relation over tests does not comply with Blackwell’s partial order. Specifically, there are cases where the receiver prefers a less informative test to a more informative test independently of the prior. We next analyze the receiver’s preference relation over pairs of symmetric tests,6one whose low grades are more informative than its high grades and one whose high grades are more informative than its low grades. We show that in the absence of signaling, the receiver’s preferences over these tests depend on the prior. However, when the sender can signal, we show that since the former test induces a more informative equilibrium signaling strategy than the latter, the receiver always prefers the former test to the latter, independently of the prior. Our model corresponds to a learning problem in signaling environments with two key features. First, the sender’s signaling costs are independent of the property that is 5In the paper, we say that a test/signal is more informative than another test/signal if and only if it dominates it in the sense of Blackwell (1951). 6Two tests are symmetric if, for every grade, the probability of the low (high) type to observe the grade under the first test is equal to the probability of the high (low) type to observe this grade under the second test. 980 Weksler and Zik Theoretical Economics 17 (2022) the subject of interest. Second, the receiver can commit to the test.7The applications we mention at the beginning of this section seem to include these features, and our results may help shed light on several phenomena in the context of these applications.8For example, grade inflation in schools, where the vast majority of the students receive grades A and B, and only a small fraction of the students receive lower marks, is criticized for not providing sufficient information to the market about the students’ qualities.9How- ever, our results show that because a grade inflation policy may include low grades that are more informative than high grades, such a policy may provide more information to the market than other grading policies that are intrinsically more informative. Our results may also help explain why public certifiers deliberately coarsen the information they reveal to the public Harbaugh and Rasmusen (2018) or why employers delegate their hiring decisions to human resources companies which may be relatively less well equipped to assess the quality of job candidates.10 The explanation we offer is that the receiver may intentionally choose a less informative test because it induces a more informative signaling strategy. Related literature In Spence’s (1973) seminal paper about signaling, as in many subsequent signaling papers, the sender’s signaling costs satisfy a single-crossing condition, such that the unique equilibrium under stability-based refinements is fully separating where the receiver gains full information. Daley and Green (2014)andFrankel and Kartik (2019) consider signaling environments with information loss. Daley and Green (2014)study a signaling environment ` a la Spence, where the receiver observes an exogenous test. They show that if the receiver’s test is sufficiently informative, then the unique equilibrium that satisfies the D1 criterion involves some pooling.11 Frankel and Kartik (2019) consider a signaling environment where a sender’s type is two-dimensional, such that 7Additionally, if the realization of the grade of the test were public, then the signaling of the sender should take place before the realization of the grade. Note that in our model in which the receiver privately observes the grade, there is no restriction on the order of the signaling action and the realization of the grade. 8These applications include a certifier and a firm, schools and students, and an employer and a potential candidate. In these applications, it seems natural that the receiver can commit to the test but not to her action. Moreover, in many cases, it is natural to presume that the signaling cost of the sender is independent of the property that is the subject of interest. That is, that the monetary loss of a firm on advertising or donation does not depend on its quality, that a candidate’s disutility from a lower salary is not related to his competence level, and that the cost of participation in extracurricular activities is independent of a student’s academic level. 9The phenomenon of grade inflation is well documented. Several papers present models that predict grade inflation as the outcome of strategic interaction between competing universities; see, e.g., Chan, Hao, and Suen (2007), Popov and Bernhardt (2013), and Boleslavsky and Cotton (2015). 10We thank an anonymous referee for suggesting this example. 11Alós-Ferrer and Prat (2012) study a similar question. They find that in the presence of a test, pooling equilibria can satisfy the intuitive criterion Cho and Kreps (1987). Other papers analyze signaling environments that are different from Spence (1973) where the receiver is exposed to additional information; see, e.g., Weiss (1983) and Feltovich, Harbaugh, and To (2002). Theoretical Economics 17 (2022) Informative tests in signaling 981 one dimension corresponds to the intrinsic value of the sender and the other dimension corresponds to his ability to signal. They show that, generically, there is some loss of information in equilibrium. Ball (2021) studies an information design problem of choosing a scoring rule, a mapping from the sender’s action to a distribution of scores, in a multi-feature extension of Frankel and Kartik’s (2019)model. 12 Specifically, he shows that a less informative scoring rule may induce a more informative sender’s signaling strategy and so the receiver can obtain more information in equilibrium by coarsening the scoring rule. Bonatti and Cisternas (2020) study a scoring rule design problem in a dynamic monopolistic screening setting and find a qualitatively similar result. In this paper, we study the information design problem of choosing an informative test in a variation of Daley and Green’s (2014) model and find a related qualitative result: a less informative test may induce a more informative signaling strategy and, thus, may ultimately lead to more information revelation in equilibrium. However, the mechanisms by which scoring rules and tests affect the sender’s signaling strategy are different. A scoring rule affects the sender’s signaling strategy by altering the way the receiver observes the chosen signaling cost, whereas a test affects the sender’s signaling strategy by modifying the sender’s types preferences over the receiver’s interim belief, i.e., her belief following the signaling action. Specifically, in tests, unlike in scoring, different sender’s types face different expected posterior beliefs for the same signaling cost. This property enables meaningful signaling in environments where the sender’s payoff is state-independent. Our work joins other papers that deal with environments where a receiver’s test choice affects the strategic behavior of an informed sender. Rosar (2017)andHarbaugh and Rasmusen (2018) both study a receiver’s optimal test choice in environments where the sender can decide on whether or not to participate in the test. The information that the receiver obtains from a test also depends on the way it affects the sender’s participation strategy. Rosar (2017) considers an environment with a risk-averse sender who is partially informed about whether his quality is high or low. Harbaugh and Rasmusen (2018) consider an environment where the sender is fully informed about his quality and incurs an exogenous fixed cost from participating in the test. Both papers find that the receiver’s optimal test uses coarse grading to increase participation. Boleslavsky and Kim (2021) study the problem of Bayesian persuasion, where a sender chooses a test to affect a receiver’s behavior, in environments where the underlying state is generated by another agent’s unobservable effort. The sender’s test affects the receiver’s information in equilibrium further by affecting the strategic behavior of the agent. Hence, the sender’s optimal test also depends on how the test incentivizes the agent’s effort in the equilibrium it induces. 12Frankel and Kartik (2021) consider the case where the receiver can commit to her action following each of the sender’s signaling costs in Frankel and Kartik’s (2019) setting. They show that the receiver finds it optimal to commit to taking a more moderate action than her best response following each of the sender’s signaling costs. Whitmeyer (2021) compares the receiver’s payoff from the optimal scoring rule and her maximal payoff when she can commit to her actions and characterizes conditions under which the optimal scoring rule yields the same payoff for the receiver as in the commitment case. 982 Weksler and Zik Theoretical Economics 17 (2022) The rest of the paper is organized as follows. In Section 2,wepresentthemodel.In Section 3, we consider the possible equilibria in the sequential subgame that a test induces. In Section 4, we show how the properties of a test affect the informativeness level of the sender’s signaling strategy in equilibrium and study the implications of this result on the receiver’s preferences over tests. Section 5is devoted to a discussion. Section 6 concludes. Most of the proofs are relegated to the Appendix. 2. The model 2.1 The environment There is a sender (he) who either has a low value (henceforth the low type)orahigh value (henceforth the high type). The sender’s type is high with a prior probability of μ0∈(0, 1). For simplicity, we assume that the values of the low type and the high type are 0 and 1, respectively, and identify the sender’s type with its value.13 The sender knows his type, ω∈{0, 1}, and can signal by taking a costly observable action c∈R+(henceforth signaling cost). There is a receiver (she) who chooses a test π:{0, 1}→S from a finite set of feasible tests R⊆, and an action, a∈R.14 We assume that is the set of all partially informative tests; i.e., the set of all tests that are informative but not fully informative. The receiver’s payoff as a function of the sender’s type and her action is UR(ω,a)=− (ω−a)2. The sender’s payoff is the receiver’s action minus his signaling cost, US(ω,a,c)=a−c. Note that the sender’s payoff does not depend directly on his type. The receiver and both of the sender’s types act to maximize their expected payoffs. 2.2 Time line The game consists of three periods. In the first period, the receiver chooses a test π∈R. In the second period, the sender observes the receiver’s test choice and decides on a signaling cost c∈R+. The receiver observes the sender’s signaling cost and forms an interim belief about the sender’s type, denoted by μπ(c)∈[0, 1]. In the third period, the receiver observes the test’s grade s∈S, forms a posterior belief about the sender’s type, and takes an action a∈R. 2.3 Notations To simplify the exposition, we introduce a series of notations. Since the type space is binary, we identify a belief μ∈{0, 1}with a number μ∈[0, 1]that corresponds to the probability that the belief μassigns to the event that the sender’s type is 1, which is also the expected value given the belief. We denote by posπ(s,μ)∈[0, 1]the posterior belief given a test π∈, an initial belief μ∈[0, 1],andagrades∈Sthat is derived using Bayes’s 13All our results hold for arbitrary values of the high type and the low type hand lwith h>l. 14We assume without loss of generality that all tests have the same grade set S, as we can always define S to be the union over all the possible grades in the set of feasible tests. We assume that Ris finite to ensure the existence of an optimal test. Theoretical Economics 17 (2022) Informative tests in signaling 983 rule whenever possible,15 i.e., posπ(s,μ):=μ·π(s|1) (1−μ)·π(s|0)+μ·π(s|1). We denote by VR π(μ)the expected payoff of the receiver from a test π∈and an initial belief μwhen she chooses her action to be equal to the expected value of the sender given her posterior belief, i.e., VR π(μ):=− μ· s∈S π(s|1)1−posπ(s,μ)2+(1−μ)· s∈S π(s|0)−posπ(s,μ)2. We denote by Vω π(μ)the expected value of the receiver’s posterior belief from the perspective of the sender of type ω,ω∈{0, 1},fromatestπ∈and an initial belief μ, i.e., Vω π(μ):= s∈S π(s|ω)·posπ(s,μ). We denote by Uω π(μ,c)the expected payoff of the sender of type ω,ω∈{0, 1},fromatest π∈, an initial belief μ, and a signaling cost cwhen the receiver chooses her action to be equal to the expected value of the sender given her posterior belief, i.e., Uω π(μ,c)≡Vω π(μ)−c. 2.4 Strategies and equilibrium We now define the players’ strategies and the equilibrium concept we use. Definition 1. A strategy for the sender is a mapping σ:{0, 1}×R→R+that assigns to each pair of type ω∈{0, 1}and test π∈Ra probability distribution over the possible signaling costs. We let σπ(c|ω)denote the probability that σ(π,ω)assigns to the signaling cost c, and supp(σ(π)) ≡supp(σ(π,0 ))∪supp(σ(π,1 )). Definition 2. The first-period strategy of the receiver is a choice of test π∈R.The third-period strategy of the receiver is a mapping a:R×S×R+→Rthat assigns to each tuple of test π∈R,grades∈S, and signaling cost c∈R+a probability distribution over the receiver’s possible actions. A strategy for the receiver is a pair {π,a(·,·,·)}. Every test π∈Rinduces a signaling subgame. Our equilibrium concept requires that the sender’s strategy, the receiver’s third-period strategy, and the receiver’s interim 15The only cases where posπ(s,μπ(c)) cannot be obtained using Bayes’ rule is when π(s|1)=0(π(s|0)= 0),π(s|0)>0(π(s|1)>0), and μ=1(μ=0). That is, cases where the receiver observes a grade that only one of the types can get when the receiver’s initial belief is that the sender’s type is the other type with certainty. In this case, the only reasonable conclusion is that the initial belief is incorrect, i.e., posπ(s,1 )=0 (posπ(s,0 )=1). 984 Weksler and Zik Theoretical Economics 17 (2022) beliefs must form a perfect Bayesian equilibrium (henceforth PBE) that satisfies the D1 criterion (Bank and Sobel (1987), Cho and Kreps (1987)) in every subgame that is induced by any test π∈R. Specifically, since the receiver’s loss function is quadratic, then, in equilibrium, the unique action that the receiver selects is equal to the sender’s expected value given her posterior belief from observing the signaling cost and the test’s realized grade, which is derived using Bayes’ rule. The receiver’s first-period strategy must choose a test π∗∈Rthat induces a subgame whose equilibrium outcome maximizes the receiver’s expected payoff. Definition 3. We say that a pair of strategies {σ∗(·,·),{π∗,a∗(·,·,·)}} and a system of interim beliefs {μπ(c)}π∈R,c∈R+form an equilibrium if and only if the following conditions hold: (i) If c∈supp(σ∗(π)), then the receiver’s interim belief μπ(c)is obtained from μ0 using Bayes’ rule, i.e., μπ(c)=μ0·σπ(c|1) μ0·σπ(c|1)+(1−μ0)·σπ(c|0). If c/∈supp(σ∗(π)), then the interim belief μπ(c)must satisfy the D1 criterion, as defined below. (ii) For every π∈,s∈S,andc∈R+we have that a∗(π,s,c)=posπs,μπ(c). (iii) If c∈supp(σ∗(π,ω)), then c∈argmax c∈R+ Uω πμπc,c. (iv) The test π∗satisfies π∗∈argmax π∈R c∈supp(σ∗(π))μ0·σ∗ π(c|1)+(1−μ0)·σ∗ π(c|0)·VR πμπ(c). In condition (iv) of the definition we implicitly assume that |supp(σ∗(π))|<∞.This assumption is valid because, as we show in the following section, for every π∈in any D1 equilibrium, it holds that |supp(σ∗(π))|<∞. The D1 criterion that is mentioned in condition (i) of the definition is defined as follows. Definition 4. Consider a PBE of the subgame that is induced by some π∈with equilibrium strategies {σ∗(π,·),a∗(π,··)}. For each ω∈{0, 1}, consider some cω∈ supp(σ∗(π,ω)). For every c/∈supp(σ∗(π)) and ω∈{0, 1}, define Bω(c):={μ|Uω π(μ,c)> Uω π(μπ(cω),cω)}. The D1 criterion requires that if Bω(c)⊂Bω(c), then μπ(c)=ω. The intuition behind the widely used D1 criterion is the following. Given an equilibrium, once the receiver observes a signaling cost that is not an element of the support of Theoretical Economics 17 (2022) Informative tests in signaling 991 with grades {L,H}and probability functions π(L|1)=γπ (L|0)=1 π(H|1)=1−γπ (H|0)=0. 4.2 Symmetric tests In this subsection, we want to further establish the point that the receiver tends to prefer tests whose low grades are more informative to tests whose high grades are more informative. To do so, we compare the receiver’s preferences over pairs of symmetric tests, one whose low grades are more informative than its high grades and one whose high grades are more informative than its low grades. We show that in the absence of signaling, the receiver prefers the latter test to the former when the prior is smaller than 1 2and the former test to the latter when the prior is greater than 1 2. However, in a signaling environment, the receiver prefers the former test to the latter, independently of the prior. In our analysis, we concentrate on a subset of tests that consist of tests that are convex combinations of binary tests. We use the following definitions and notations. Definition 6. Tests πand πare symmetric if for every s∈S,wehavethatπ(s|1)= π(s|0)and π(s|0)=π(s|1).Wedenoteby ˆπthe symmetric test of π. Definition 7. We say that a binary test is L-informative (H-informative)ifπ(h|1)> π(l|0)(π(h|1)<π (l|0)) and that it is N-informative if π(h|1)=π(l|0). We now introduce the notion of a convex combination of binary tests: given a set of binary tests {π1,,πk}and a set of real numbers {p1,,pk}such that pi>0 for every i∈{1, ,k}and k i=1pi=1, we denote by ⊕k i=1piπithe test such that with probability pi, the receiver observes a realization of the test πi. Definition 8. We say that a test k i=1piπiis L-informative (H-informative)ifπiis either L-informative (H-informative) or N-informative for every i∈{1, ,k},andthere exists j∈{1, ,k}for which the test πjis L-informative (H-informative). We say that the test πis N-informative if πiis N-informative for every i∈{1, ,k}.Wedenotethe union over all tests that are L-informative, N-informative, and H-informative by d. We are now ready to formulate a lemma that describes a connection between L- informative/H-informative tests and their dividing beliefs. Lemma 6. If π∈dis an L-informative (H-informative) test, then μ∗ π>1 2(μ∗ π<1 2), and if π∈dis an N-informative test, then μ∗ π=1 2. We prove the lemma by directly using Lemma 3. Intuitively, since an L-informative (H-informative) test has an advantage at identifying the low (high) type, the high type sets the initial probability of the low (high) type to be lower than 1 2to utilize the test optimally. 992 Weksler and Zik Theoretical Economics 17 (2022) It is easy to see that if a test is L-informative, then its symmetric test is H-informative and vice versa. In the following proposition, we compare the receiver’s preferences over such pairs of L-informative and H-informative symmetric tests in environments with and without signaling. Proposition 4. Consider an L-informative test πand its symmetric (H-informative) test ˆπ. •In an environment without signaling, if μ0<1 2,then ˆπRπ,andifμ0>1 2,then πRˆπ. •In an environment with signaling, for every μ0∈(0, 1), we have that πRˆπ. Proposition 4shows that in an environment without signaling, the receiver prefers the H-informative (L-informative) test if the prior is lower (greater) than 1 2; i.e., the receiver’s preference over symmetric tests is symmetric. The intuition for this result is the same intuition that accompanies Lemma 6. An H-informative (L-informative) test is better at identifying the high (low) type. Therefore, it provides more information to the receiver when the initial probability of the high (low) type is low. However, in a signaling environment, the symmetry breaks, i.e., the receiver prefers the L-informative test independently of the prior. The intuition for this result is that although the tests are symmetric in terms of the direct information they provide, they are not symmetric in terms of the information they provide through the signaling channel, where the L- informative test has an advantage over the H-informative test. Specifically, consider a low prior in which without the signaling stage, the receiver prefers the H-informative test to the L-informative test. The key point is that when there is signaling, under the L-informative test, the signaling cost effectively moves the prior to a prior that is greater than 1 2, in which the L-informative test is better, and the action of no signaling, which perfectly reveals that the sender’s type is low, is selected with a higher probability than under the H-informative test. To further explain this last point, we present a sketch of the proof of the second part of the proposition. Let μ0∈(0, 1). Assume first that μ0≤μ∗ ˆπ. Consider the equilibrium under πand ˆπ. Since πand ˆπare symmetric, we get that conditional on observing a positive signaling cost, the equilibrium payoff of the receiver is the same, i.e., VR ˆπ(μ∗ ˆπ)=VR π(μ∗ π). When the receiver observes a signaling cost of 0, then she learns that the sender is of type 0 with certainty. Now, since μ∗ ˆπ<μ ∗ π, the ex ante probability that the receiver would observe the signaling cost 0 is greater under πthan under ˆπ. Therefore, the expected payoff of the receiver is greater under πthan under ˆπ. The proof in the case when μ∗ ˆπ<μ 0<μ ∗ πrelies on a similar but a more subtle argument. If 1 2<μ ∗ π≤μ0, then under both ˆπand π, we get a fully pooling equilibrium, and the receiver’s payoff is VR ˆπ(μ0)and VR π(μ0), respectively. Since 1 2<μ 0,wehavethatVR ˆπ(μ0)<VR π(μ0). 5. Discussion 5.1 The receiver’s commitment power A natural question that arises in our model is whether the receiver benefits from her ability to commit to the test. It is straightforward to see that this is indeed the case. A more Theoretical Economics 17 (2022) Informative tests in signaling 993 informative test induces more information for every signaling strategy of the sender. Therefore, if the receiver cannot commit to the test, then, in equilibrium, the receiver chooses a more informative test over a less informative test. Hence, the property that in our setting the receiver’s preference relation does not comply with Blackwell’s partial order implies that the receiver strictly benefits from her commitment power. 5.2 Signaling costs Another natural question that arises concerns the connection between the receiver’s test choice and the expected signaling cost. At first glance, this connection is negative; i.e., a higher informativeness level of a test leads to a lower expected signaling cost. The intuition for this connection is the following. As the test becomes more informative, the high type chooses to differentiate itself more through the test and less through costly signaling. This insight is established in Daley and Green (2014), who analyze the effect of the informativeness level of binary symmetric tests on the expected signaling cost in equilibrium. However, our results show that it could be the case that a less informative test’s dividing belief is smaller than the prior, while the more informative test’s dividing belief is higher than the prior. In such a case, the less informative test induces a fully pooling equilibrium with a signaling cost of 0, while the more informative test induces a separating equilibrium with a positive signaling cost. That is, our results show that in some cases, the negative connection between the test’s informativeness level and the expected signaling cost breaks. 6. Conclusion We have studied the receiver’s learning problem in a signaling model ` alaSpencewith type-independent signaling costs. Our main insight is that tests whose low grades are more informative than their high grades induce more information to be conveyed through the channel of the sender’s signaling costs. We established this insight through a series of formal results. We showed that because of this property, the receiver tends to prefer tests whose low grades are more informative. Appendix Proof of Lemma 1Consider the difference in the expected posteriors of the high type and the low type as a function of the initial belief: V1 π(μ)−V0 π(μ)= s∈Sπ(s|1)−π(s|0)·μ·π(s|1) (1−μ)·π(s|0)+μ·π(s|1). Define Rπ(s):=π(s|1) π(s|0)and rewriting we get V1 π(μ)−V0 π(μ)= s∈S π(s|0)Rπ(s)−1·μ·Rπ(s) 1−μ+μ·Rπ(s). 994 Weksler and Zik Theoretical Economics 17 (2022) Now posπ(s,μ)=μ·Rπ(s) 1−μ+μ·Rπ(s) and ∂2posπ(s,μ) ∂μ2=2·1−Rπ(s)·Rπ(s) 1−μ+μ·Rπ(s)3 so we get that if ∞>R π(s)>1, then ∂2posπ(s,μ) ∂μ2<0, i.e., posπ(s,μ)is a strictly concave function of μ; if 0 <R π(s)<1, then ∂2posπ(s,μ) ∂μ2>0, i.e., posπ(s,μ)is a strictly convex function of μ. Therefore, for every grade s∈Sthat satisfies that Rπ(s)= 1, we get that (Rπ(s)−1)· (μ·Rπ(s)) (1−μ+μ·Rπ(s)) is a strictly concave function. Since π∈is informative, we get that there exists s∈Sfor which Rπ(s)= 1. Therefore, we get that V1 π(μ)−V0 π(μ)is a sum of concave and strictly concave functions, and, therefore, it is a strictly concave function. Since V1 π(μ)−V0 π(μ)is strictly concave and defined on a closed interval, we get that it has a unique maximizer. Proof of Proposition 1Given a test π∈, our model is a special case of Daley and Green’s (2014) model in which the signaling cost is type-independent. Therefore, our characterization of equilibria in the second-period subgame stems from their characterization (see Proposition 3.8 in Daley and Green (2014)). Proof of Lemma 2Assume by contradiction that the lemma is not true; i.e., there exists atestπ∈such that μπ= μ∗ π. Assume first that there exists π∈such that μπ<μ ∗ π. Consider a prior ˆμwith μπ<ˆμ<μ ∗ π. From Proposition 1,wehavethatforsuchaprior the unique equilibrium is fully pooling without any signaling. The payoff of the low type in this equilibrium is V0 π(ˆμ). Define ˆ c:=V0 π(μ∗ π)−V0 π(ˆμ)>0, and consider a deviation to the signaling cost ˆ c. The receiver observes the deviation and forms a belief regarding the type of sender that deviated. Assume that the belief of the receiver given this deviation is μ∗ π. Given this belief, we can see that the expected payoff of the low type from such a deviation is V0 πμ∗ π−ˆ c=V0 π(ˆμ). It follows that given this belief, the low type is indifferent between its equilibrium payoff and its payoff from this deviation. Let us compute now the expected payoff of the high type from this deviation given the same belief μ∗ π: V1 πμ∗ π−ˆ c=V1 πμ∗ π−V0 πμ∗ π−V0 π(ˆμ). Theoretical Economics 17 (2022) Informative tests in signaling 995 It follows that the difference between the payoff of the high type given this deviation (when the belief is μ∗ π) and its equilibrium payoff is V1 πμ∗ π−V0 πμ∗ π−V0 π(ˆμ)−V1 π(ˆμ)=V1 πμ∗ π−V0 πμ∗ π−V1 π(ˆμ)−V0 π(ˆμ). From the definition of μ∗ πas the unique belief that maximizes the difference V1 π(μ)− V0 π(μ)and the last equation, we get that the expected payoff of the high type given this deviation when the belief is μ∗ πis higher than its equilibrium payoff. Now, from the fact that the expected payoff from this deviation is strictly increasing in the belief of the receiver given the deviation, we get that the set of beliefs that satisfy that the payoff of the high type is strictly larger than its equilibrium payoff is a superset of the set of beliefs that satisfy that the payoff of the low type is strictly larger than its equilibrium payoff. From the definition of the D1 criterion, we get that the belief after such a deviation must be that the type that deviated is the high type. Clearly, under this belief, this deviation is a profitable one at least for the high type. It follows that the equilibrium we considered is not an equilibrium in contradiction to Proposition 1. Assume now that there exists a test π∈such that μπ>μ ∗ π. Consider a prior ˆμwith μπ>ˆμ>μ ∗ π. From Proposition 1, we have that under this prior the unique equilibrium is the following: The high type chooses the signaling cost c:=V0 π(μπ)and the low type mixes between the same signaling cost cand zero, such that the equilibrium interim belief after the receiver observes the signaling cost cis μπ. Consider now a deviation to the signaling cost ˆ c:=V0 π(μ∗ π). Notice that because μπ>μ ∗ π,wehavethatV0 π(μπ)> V0 π(μ∗ π), it follows that c>ˆ c. Assume that the belief of the receiver after observing a deviation to the signaling cost ˆ cis μ∗ π. From the definition of ˆ cit is clear that the payoff of the low type after this deviation and given the interim belief μ∗ πis zero, which is equal to its equilibrium payoff. The payoff of the high type from this deviation given the belief μ∗ πis V1 πμ∗ π−ˆ c=V1 πμ∗ π−V0 πμ∗ π. Notice that the equilibrium payoff of the high type is V1 π(μπ)−c=V1 π(μπ)−V0 π(μπ). From the definition of μ∗ πas the unique belief that maximizes the difference V1 π(μ)− V0 π(μ), we get that the deviation given the interim belief μ∗ πstrictly improves the high type’s payoff relative to its equilibrium payoff. Now, from the fact that the expected payoff from this deviation is strictly increasing in the belief of the receiver given the deviation, we get that the set of beliefs such that the payoff of the high type is strictly larger than its equilibrium payoff is a superset of the set of beliefs such that the payoff of the low type is strictly larger than its equilibrium payoff. From the definition of the D1 criterion, we get that the belief after such a deviation must be that the type that deviated is the high type. Clearly, under this belief, this deviation is a profitable one for both types. It follows that the equilibrium we considered is not an equilibrium in contradiction to Proposition 1. 996 Weksler and Zik Theoretical Economics 17 (2022) Proof of Lemma 3To analyze whether some belief μis greater than, less than, or equal to μ∗ π, we develop the expression ∂[V1 π(μ)−V0 π(μ)] ∂μ . First, we develop the expression ∂posπ(s,μ) ∂μ =π(s|1)(1−μ)π(s|0)+μπ(s|1)−π(s|1)−π(s|0)π(s|1) (1−μ)π(s|0)+μπ(s|1)2. Rearranging the expression, we get that ∂posπ(s,μ) ∂μ =π(s|1)π(s|0) (1−μ)π(s|0)+μπ(s|1)2. After further development, we get that ∂posπ(s,μ) ∂μ =1 μ(1−μ)posπ(s,μ)−posπ(s,μ)2 and so ∂V1 π(μ)−V0 π(μ) ∂μ = s∈Sπ(s|1)−π(s|0)·1 μ(1−μ)·posπ(s,μ)−posπ(s,μ)2. Given μ, we want to see whether  s∈Sπ(s|1)−π(s|0)·1 μ(1−μ)·posπ(s,μ)−posπ(s,μ)20, which is equivalent to  s∈S π(s|0)·1 2−posπ(s,μ)2 s∈S π(s|1)·1 2−posπ(s,μ)2 . Proof of Lemma 4We prove part (i) of the lemma; part (ii) can be proved analogously. Consider μ=μ∗ π>1 2. From Lemma 3,wehavethat  s∈supp(π) π(s|0)·1 2−posπ(s,μ)2 = s∈supp(π) π(s|1)·1 2−posπ(s,μ)2 .(1) We can write this equation in an equivalent way:  s∈S− ππ(s|0)−π(s|1)·1 2−posπ(s,μ)2 = s∈S+ ππ(s|1)−π(s|0)·1 2−posπ(s,μ)2 .(2) Theoretical Economics 17 (2022) Informative tests in signaling 997 We now use the conditions in the lemma to show that  s∈S− ππ(s|0)−π(s|1)·1 2−posπ(s,μ)2 < s∈S+ ππ(s|1)−π(s|0)·1 2−posπ(s,μ)2 .(3) This will end the proof, because according to Lemma 3, if the right-hand side is bigger than the left-hand side, then μ∗ π<μ=μ∗ π. Becausewehavethat ˜πl(π)=˜πl(π)and that ˜πβ(π)=˜πβ(π), we can deduce that the left-hand side of (2) is equal to the left-hand side of (3), so it is sufficient to show that  s∈S+ ππ(s|1)−π(s|0)·1 2−posπ(s,μ)2 < s∈S+ ππ(s|1)−π(s|0)·1 2−posπ(s,μ)2 .(4) From the fact that ˜πh(π)B˜πh(π), we can deduce that there exists a set of tests {ˆπs}s∈S+ π such that the test ˜πh(π)is equivalent to a test that is defined according to the next sequential procedure: first the test ˜πh(π)is activated and then after every possible result s∈S+ π,thetest ˆπsis activated. This equivalence allows as to write (4) in the following equivalent way:  s∈S+ ππ(s|1)−π(s|0)·1 2−posπ(s,μ)2 < s∈S+ π ˆ s∈Sˆπsπ(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0)·1 2−pos(s,ˆ s,μ)2 .(5) We prove that (5) is true case by case. That is, for every s∈S+ π,weprovethat π(s|1)−π(s|0)·1 2−pos(s,μ)2 < ˆ s∈Sˆπsπ(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0)·1 2−pos(s,ˆ s,μ)2 .(6) To see why this is true, first notice that clearly we have that π(s|1)−π(s|0)= ˆ s∈Sˆπsπ(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0).(7) 998 Weksler and Zik Theoretical Economics 17 (2022) It follows that 1= ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0) π(s|1)−π(s|0).(8) From this it follows that we can write (6) in the following equivalent way: 1 2−pos(s,μ)2 < ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0) π(s|1)−π(s|0)·1 2−pos(s,ˆ s,μ)2 .(9) Now notice that because we have that ˜πβ(π)=˜πβ(π)and because s∈S+ π,itmust be that pos(s,μ)>μ>1 2and also for every ˆ s∈Sˆπswe have that pos(s,ˆ s,μ)>μ>1 2. Additionally, notice that the function (1 2−x)2is concave and increasing when x>1 2. Last, notice that by construction the distribution of posteriors that corresponds to {pos(s,ˆ s,μ)}ˆ s∈Sˆπsis a mean preserving spread of pos(s,μ); that is, we have that pos(s,μ)= ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0) π(s|1)+π(s|0)·pos(s,ˆ s,μ). (10) From the concavity of the function (1 2−x)2, we get that 1 2−pos(s,μ)2 < ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0) π(s|1)+π(s|0)·1 2−pos(s,ˆ s,μ)2 . (11) In the last part of the proof we will establish that  ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0) π(s|1)+π(s|0)·1 2−pos(s,ˆ s,μ)2 < ˆ s∈Sˆπs π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0) π(s|1)−π(s|0)·1 2−pos(s,ˆ s,μ)2 . (12) To do this we will prove the next lemma. Lemma 7. For every ˆ s∈Sˆπs, •if pos(s,ˆ s,μ)>pos(s,μ),then(π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0)) (π(s|1)−π(s|0)) >(π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0)) (π(s|1)+π(s|0)) ; •if pos(s,ˆ s,μ)<pos(s,μ),then(π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0)) (π(s|1)−π(s|0)) <(π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0)) (π(s|1)+π(s|0)) . Proof. We will prove the first claim of the lemma; the proof of the second claim is analogous. We start with rewriting the inequality in the lemma: π(s|1)ˆπs(ˆ s|1)−π(s|0)ˆπs(ˆ s|0) π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0)>π(s|1)−π(s|0) π(s|1)+π(s|0). (13) Theoretical Economics 17 (2022) Informative tests in signaling 999 Rewriting again we get π(s|1)ˆπs(ˆ s|1) π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0)−π(s|0)ˆπs(ˆ s|0) π(s|1)ˆπs(ˆ s|1)+π(s|0)ˆπs(ˆ s|0) >π(s|1) π(s|1)+π(s|0)−π(s|0) π(s|1)+π(s|0), (14) and again, poss,ˆ s,1 2−1−poss,ˆ s,1 2>poss,1 2−1−poss,1 2. (15) This is true if and only if poss,ˆ s,1 2>poss,1 2. (16) This is clearly true because we have that pos(s,ˆ s,μ)>pos(s,μ). Combining this lemma and the fact that the function (1 2−x)2is increasing when x> 1 2, we get that (12) is true. To see this, notice that both the left-hand side and the righthand side of (12) are convex combinations over the same numbers. From the lemma and the fact that the function (1 2−x)2is increasing when x>1 2, we can deduce that the right-hand side puts larger weights on large numbers, i.e., (1 2−pos(s,ˆ s,μ))2when pos(s,ˆ s,μ)>pos(s,μ)>1 2, and lower weights on small numbers, i.e., (1 2−pos(s,ˆ s,μ))2 when 1 2<pos(s,ˆ s,μ)<pos(s,μ). This ends the proof. Proof of Proposition 2We first partition the set of possible priors (0, 1)into three regions: (0, μ∗ π],(μ∗ π,μ∗ π),[μ∗ π,1 ). If μ0∈(μ∗ π,1 ], then we have, according to Proposition 1and Lemma 2,thatunder both tests the unique equilibrium is the fully pooling equilibrium. It follows that under such priors, the fact that the sender can signal plays no role and, therefore, the receiver would prefer the more informative test π.19 If μ0∈[μ∗ π,μ∗ π), then we have that under the test πthe unique equilibrium is a fully pooling equilibrium, while under the test πthe unique equilibrium is a semipooling equilibrium. That is, under the test πthe receiver first gets an informative signal through the signaling channel and then she gets another signal directly from the test. Clearly, the effective signal that the receiver observes through this procedure is more informative than πand we know that πis more informative than π. The proof follows from the transitivity of the Blackwell relation. If μ0∈(0, μ∗ π), then we have that under both tests the unique equilibrium is a semipooling equilibrium. We first want to show that the informative signal that the receiver observes through the signaling strategy is more informative under the test πthan under the test π.Denotebysigπand sigπthe test that corresponds to the informative signal 19Recall that when we say that a test/signal is more informative than another test/signal, we mean that in the sense of Blackwell (1951). 1000 Weksler and Zik Theoretical Economics 17 (2022) that the receiver gets through the signaling strategy under πand π, respectively. After the signaling stage under πthe prior is spread into two posteriors 0 and μ∗ πwhile under the test πthe prior is spread into the posteriors 0 and μ∗ π. Because we have that μ∗ π> μ∗ π, it follows that the posteriors distribution under πis a mean preserving spread of the posteriors distribution under π. Now, because we are in a binary state environment, it follows that sigπBsigπ. The effective signal that the receiver observes in equilibrium under the test π(π)is the result of a procedure in which first the receiver observes sigπ (sigπ)and then for each posterior (interim belief) the test π(π)is activated. Note that we have that πBπand also that sigπBsigπ. It follows that both the first stage and the second stage of the procedure are more informative under πthan under πand, therefore, the receiver prefers πalso under priors in the set (0, μ∗ π). Proof of Lemma 5Applying the condition of Lemma 3to binary tests, we get the following result. Let πbe a binary test, and let μ∈[0, 1]be an initial distribution, μ∗ πμif and only if  1 2−posπ(l,μ) posπ(h,μ)−1 2 . Now if π(h|1)=1, then posπ(l,μ)=0, which implies that the left-hand side is equal to 1 2. Since πis partially informative (as we assume in the model), we get that posπ(h,μ)<1 for every μ<1andposπ(h,μ)=1onlyifμ=1. Therefore, we get that the left-hand side is equal to the right-hand side if and only if μ=1, and so μ∗ π=1. If π(l|0)=1, then posπ(h,μ)=1, which implies that the right-hand side is equal to 1 2. Since πis partially informative (as we assume in the model), we get that posπ(l,μ)>0 for every μ>0andposπ(h,μ)=0 if and only if μ=0. Therefore, we get that the left-hand side is equal to the right-hand side if and only if μ=0, and so μ∗ π=0. To complete the proof we show that if π∈does not include a fully informative grade, i.e., a grade that some type receives with probability 0, then μ∗ π∈(0, 1).Theargument is the following. If a grade s∈Sis partially informative, i.e., both type 0 and type 1 receive it with positive probabilities, then posπ(s,0 )=0andposπ(s,1 )=1. Since both grades in the support of πare partially informative, we get that V1 π(0)=V0 π(0)=0and V1 π(1)=V0 π(1)=0. Hence,wegetthatthefunctionV1 π(μ)−V0 π(μ)is strictly concave with V1 π(0)−V0 π(0)=V1 π(1)−V0 π(1)=0. Therefore, μ∗ π=argmaxμ∈[0,1]V1 π(μ)−V0 π(μ)∈ (0, 1). Proof of Lemma 6Again, from Lemma 3we have that for binary test πwe have that μ∗ πμif and only if  1 2−posπ(l,μ) posπ(h,μ)−1 2 . Atestπhas a μ∗ π1 2if and only if 1 2−1−π(h|1) 1−π(h|1)+π(l|0)π(h|1) π(h|1)+1−π(l|0)−1 2,