Perceptions of Correlation, Ambiguity and Communication Gilat Levy and Ronny Razin, LSE1 Abstract: We suggest a new framework to analyse how individuals update their beliefs following group communication. Our model is based on two main assumptions. First, while individuals know their own information source, they have ambiguity about the joint distribution of information sources in the group. Second, this ambiguity is bounded by the perception that individuals have on the level of correlation across information sources. We show how communication in the presence of exogenous ambiguity over information sources induces endogenous ambiguity over the state of the world, and fully characterize the set of possible inferences that individuals can make for di¤erent perceptions of correlation. Our comparative measure of the perception of correlation restricts the set of inferences in a meaningful way; speci…cally, a lower perception of correlation is associated with a lower ambiguity over the state of the world. We study the behavioral implications of the model to risky and cautious shifts in groups, jury deliberation, and common-value auctions. 1 Introduction In many social interactions, individuals who are exposed to di¤erent sources of information also communicate their information or beliefs to each other. Often the web of the di¤erent information sources is quite complex, rendering the task of updating beliefs rather di¢ cult. A simple strategy in these situations is to treat all sources of information as if they were independent.2 This strategy always leads to a unique prescription of how to interpret di¤erent bits of information and is the subject of a recent literature that analyses the behavioral implications of “correlation neglect”.3 1 We thank Eddie Dekel, Andrew Ellis, Je¤ Ely, Erik Eyster, Tristan Gagnon-Bartsch, Daniel Krahmer, Francesco Nava, Matthew Rabin, Alp Simsek and Joel Sobel for helpful comments. We also thank seminar participants at the Queen Mary Theory workshop, PSE, ESSET 2015, Manchester University, University of Bonn, LBS, Ecole Polytechnique, Bristol University, University of St. Andrews, Rotterdam, and the UCL-LSE Theory workshop. 2 Indeed this is the approach taken in the Machine Learning literature (Russel and Norvig 2003). This literature deals with categorization and provides a heuristic to aggregate information across di¤erent attributes under the assumption that the attributes are independent. 3 Ortoleva and Snowberg (2015) analyze the e¤ect of correlation neglect on the polarisation of beliefs. De Marzo et al (2003) and Gagnon-Bartsch and Rabin (2015) study how it a¤ects the di¤usion of information in social networks. Glaeser and Sunstein (2009) and Levy and Razin (2015a, 2015b) explore 1 Treating information sources as independent is clearly an over-simpli…cation: A more careful strategy might be to take into account at least some of the di¤erent possible correlations between sources of information. In this paper we suggest a framework to model these kinds of strategies and to compare how di¤erent perceptions of correlation might imply di¤erent behavior. Intuitively, the perception of correlation is related to con…dence, and in two di¤erent ways. As already explored in the literature, neglecting the correlation across information sources may make beliefs more extreme. Individuals who believe that their information sources are independent will tend to have beliefs with lower variance.4 The second channel by which perception of correlation might induce con…dence is through a Knightian notion of uncertainty. While there is only one way to interpret information if it is independent, there might be many ways for the information to be correlated. Therefore, someone who believes that information sources are independent will be more con…dent in this Knightian sense, than someone who entertains di¤erent possibilities for how the information held by himself and others should be interpreted. Our framework captures this idea that the less an individual perceives correlation, the tighter is the set of possible interpretations of the information he considers. We consider a simple environment in which individuals receive signals according to some joint information structure. Individuals …rst view their private signals and update their posteriors about the state of the world. They then communicate these posteriors truthfully to each other. The model is based on two main assumptions. First, that individuals know their own marginal information structure, but face ambiguity about the joint information structure. Second, that such ambiguity is restricted by individuals’perception of correlation in the joint distribution of information. We analyze what di¤erent perceptions of correlation imply to individuals’post-communication beliefs and behaviour. We characterise the perception of correlation by a single parameter, a; which bounds the degree of pointwise mutual information between information sources. Speci…cally, consider two individuals, 1 and 2, each receiving a signal, s and s0 ; respectively. Let q(s; s0 j!) denote the joint distribution of the signals conditional on a state !; and q 1 (sj!) and q 2 (s0 j!) denote the marginal distributions of s and s0 conditional on !: The pointwise the implications for group decision making in political applications. Recent experimental evidence is in Eyster and Weizsacker (2011), Kallir and Sonsino (2009) and Enke and Zimmermann (2013). Madarazs (2012) takes the opposite approach and assumes projection bias which implies an over-estimation of correlation. 4 This arises -with standard information structures such as the normal distribution- in Ortoleva and Snowberg (2015) and Glaeser and Sunstein (2009). However, Sobel (2014) shows that correlation neglect is not a necessary condition for extreme beliefs. 2 0 q(s;s j!) mutual information is de…ned in the Information Theory literature as ln q1 (sj!)q 2 (s0 j!) : We assume that individuals’ambiguity over joint information structures is restricted to those q(s;s0 j!) 1 for some parameter 0 < a 1: The higher is a; the lower for which a q1 (sj!)q 2 (s0 j!) a is the perceived correlation. As a benchmark, we analyze the resulting beliefs under full correlation neglect (FCN), a = 1; which implies a perception of (conditional) independence across information sources. Our …rst result is conceptual, formalizing the intuition above about the relation between perceptions of correlation and Knightian uncertainty. We show that (exogenous) ambiguity about joint information structures induces endogenous ambiguity about the state of the world. Speci…cally, we show that, pre-communication, individuals have a unique posterior about the state of the world but that, after communication, they have a set of possible posteriors. Therefore, in our model communication creates ambiguity about the state of the world.5 We fully characterise the set of beliefs that arises from communication and show that this is a convex set that depends on the parameter a in a simple way. In particular, the set of beliefs is strictly monotone in a, i.e., the higher is a; the smaller is the set in a set-inclusion sense. Therefore, individuals with di¤erent perceptions of correlation will develop di¤erent levels of ambiguity when interacting in groups. We show that with su¢ cient ambiguity aversion, this implies that individuals with lower perception of correlation will exhibit a more risky behaviour. Thus our measure of perception of correlation provides a meaningful way to compare beliefs and behaviour across individuals. Our analysis unveils a trade-o¤ in terms of group size. Speci…cally, we show that the set of possible beliefs for a perception a is spanned around the (unique) belief of an individual with FCN and that as the FCN belief converges to be degenerate, the set of beliefs for a > 0 shrinks and converges to the FCN belief. Larger groups may indeed be associated with an FCN belief that is close to being degenerate. As a result, while larger groups can possibly aggregate more information, the actual behaviour of its members might (wrongly) converge to treating information as independent. We analyze the implications of the above to several group decision making applications. We start with risky and cautious shifts, that were …rst explored in Stoner (1968).6 These shifts relate to the di¤erent behaviour of individuals before and after they engage in group communication. In a simple model of investment, we show that (i) Cautious shifts and small groups: individuals with a high perception of correlation (low a) will always exhibit a cautious shift in small groups. (ii) Divergence: individuals who pre-deliberation 5 There is recent experimental evidence showing the relation between ambiguity and group communication. See Levati et al (2014) and Keller and Sarin (2007). 6 For a recent analysis, see Eliaz el al (2006). 3 behave in similar ways, diverge in their actions following communication. (iii) Large group polarisation e¤ect: As the group size grows large, depending on the state of the world, substantial cautious and risky shifts will occur. Thus, the e¤ect of group size can be non-monotonic and individuals with high perception of correlation may switch from a cautious to a risky shift when the group becomes larger. Another application we study is jury deliberation. Our model …ts this application well: Juries comprise of groups of individuals who take decisions after deliberation, which is an explicit part of the process. The jurors are exposed to the same evidence and so correlation and its perception are important features. Truthful communication during deliberations is prescribed by law and con…rmed in experiments on jury decisions.7 Our analysis contrasts then with the canonical models of Austen-Smith and Banks (1996) and Feddersen and Pesendorfer (1998) that have all modelled juries with no deliberation stage and assuming conditionally independent private signals.8 We …nd a simple condition characterising when convictions and acquittals arise. Acquittals for example are more likely to occur when the group is small, the burden of proof is high, and the perception of correlation is high (low a). In this case jurors become less con…dent after deliberation and thus more averse to conviction. We also analyze the normative implications of our model in two extreme cases: When the true information structure satis…es (conditional) independence, large juries adjudicate e¢ ciently for all perceptions of correlation, while when the true information structure satis…es full correlation, large juries induce conviction shifts when the burden of proof is high. The experimental literature on juries has either followed the canonical model closely as in Guarnaschelli, McKelvey and Palfrey (2000) and Goeree and Yariv (2011) or followed the jury protocol as much as possible as in Schkade, Sunstein and Kahneman (2000). In the …rst set of experiments, private information was generated independently, participants were told that it is so, and large juries made correct decisions. In Schkade et al (2000) in contrast, the same information was publicly provided to the participants, and a severity shift (towards harsher punishments) was identi…ed. Our results can shed light then on both types of experiments. Finally, we show how our framework can be extended -from a communication model- to other environments. We analyze a simple model of common-value auctions when individuals have ambiguity about the joint information structure. We show that our framework a¤ects bidding strategies in non-trivial ways: the winner’s curse is mitigated for some types who become pessimistic and submit too low bids, but is exacerbated for other 7 See Goeree and Yariv (2011) and Guarnaschelli et al (2000). Coughlan (2000) and Austen-Smith and Feddersen (2006) allow for deliberation but keep the conditional independence of signals. 8 4 types. We show that the support of bids contains that of the standard case of independent information. We also show that this result is qualitatively di¤erent to the equilibrium outcome in two alternative models. In a model with standard exogenous ambiguity about the state of the world, the support of bids is a downward shift of the one in the standard model. In a cursed equilibrium (Eyster and Rabin 2005), the support of bids is contained within the support of the standard model. Finally we show that a small perception of correlation induces lower revenues to the seller compared with the standard model. We proceed as follows. The next Section discusses the related literature. Section 3 presents the model and the FCN benchmark. The main result, Theorem 1, characterizing the set of rationalisable beliefs with the bounds on correlation, is presented in Section 4. In Section 5 we consider the implication of Theorem 1 to behaviour in an investment model, to choice shifts in groups, and to jury deliberations. The extension to auctions is provided in Section 6, where we also brie‡y discuss repeated communication in networks. All proofs that are not in the text are relegated to an Appendix. 2 Related Literature Our paper is related to Sobel (2014) who asks whether one can rationalise a given set of data (pre and post-deliberation actions) by …nding some information structure that would rationalise this data. Sobel (2014) shows that by having the freedom to choose information structures, a modeler can rationalise almost any data. Given the data of predeliberation beliefs, we similarly span a set of rationalisable beliefs. Our focus is di¤erent though. First, in our approach it is the economic agent, rather than the modeler, who spans the possible information structures. Second, our analysis rests on a restriction of the set of possible information structures using the notion of pointwise mutual information. We show that this implies meaningful constraints on the set of post-deliberation beliefs that can be rationalised.9 The literature on group decision-making studies di¤erent types of polarisation arising from deliberation. A recent literature focuses on the divergence of beliefs arising when a group of individuals are exposed to the same information. Sethi and Yildiz (2012) consider a simple group communication model in which individuals truthfully exchange posterior beliefs. They assume that individuals know the information structures of others. Their focus is on polarisation that arises from non-common priors. In contrast, in our model even if players share all the information that they have (including for example non-common priors, their signals, and their marginal information structures), uncertainty 9 In these two ways we also di¤er from Bergemann, Brooks and Morris (2015) who study the set of achievable utilities when considering, as modelers, the set of di¤erent feasible information structures. 5 remains with regard to the joint information structure. Acemoglu, Chernozhukov and Yildiz (forthcoming) analyze a model in which players are uncertain about the informativeness of their (public) signals and also have di¤erent priors. They show that substantial asymptotic disagreement may arise even if the two players have vanishing uncertainties about the distribution of the signals. Baliga, Hanani and Klibano¤ (2013) characterise environments (sets of priors and “neutral”signals) in which ambiguity aversion can lead to polarisation. Borgers, Hernando-Veciana and Krahmer (2013) have shown that such polarisation can arise in a fully rational environment when signals are complements. Andreoni and Mylovanov (2012) derive divergence as a result of multidimensional information and di¤erences in tastes. In our analysis such divergence arises as a result of di¤erent perceptions of correlation, which translate into di¤erent, endogenous, levels of ambiguity. Another literature focuses on the e¤ect of deliberation on the strength (variance) of beliefs. This kind of polarisation arises for example in models that assume correlation neglect, such as Glaeser and Sunstein (2009), Levy and Razin (2015a,2015b) and Ortoleva and Snowberg (2015). Sobel (2014) shows that a similar result can also arise for rational players. In our model, this type of polarisation is likely to arise in large groups when players’beliefs converge to be as if they have full correlation neglect. Our framework nests correlation neglect but is qualitatively di¤erent. Our focus is on the relation between perceptions of correlation and the level of ambiguity, with correlation neglect being the extreme case which results in no ambiguity. Due to our focus on ambiguity we get di¤erent results. We can show for example that, under max-min behaviour, an individual with a lower perception of correlation will always invest more in a risky asset, compared to an individual with a higher perception of correlation. In contrast, when one compares an individual with correlation neglect to a rational individual with no ambiguity, the former will have more extreme beliefs but sometimes in the risky and sometimes in the cautious direction. Our approach is also related to recent papers in the behavioral economics and bounded rationality literature. First, our analysis complements Ellis and Piccione (2015) who use an axiomatic approach to represent decision makers a¤ected by the complexity of correlations among the consequences of feasible actions. Second, in the naïve social learning literature (Bohren 2014, Eyster and Rabin 2010, 2014, Gagnon-Bartsch and Rabin 2015), individuals interpret, often incorrectly, the actions of predecessors as truthful indications of their signals, and typically do not take into account that others may have learnt from others’ signals. In our model individuals observe the true posterior of their peers, but may not necessarily account for all possible correlations in the underlying information structures.10 A key additional di¤erence is that we focus on ambiguity. 10 In the analogy-based equilibrium concept of Jehiel (2005), individuals bundle game nodes into classes, 6 3 The model and preliminary results A group of individuals, N = f1; 2; :::; ng; share a common prior p(!) (full support) about a state of the world ! 2 ; where is …nite (the model can be easily extended to the case of non-common priors). Players also receive some private information governed by a joint information structure. The players know their own marginal information structure but face ambiguity about what the joint information structure is. We assume that after updating their own beliefs based on their private information they communicate their posteriors truthfully. We are interested in characterizing the rationalisable postcommunication beliefs, as a function of individuals’perception of the correlation in the joint information structure. 3.1 Information and pre-communication beliefs A joint information structure is given by ( ni=1 S i ; q(s; !)) where S i is a …nite set of possible private signals for individual i and q(s; !) is a joint probability on ni=1 S i . Let joint be the set of all such joint information structures. For each such information structure : and any !; let q(sj!) q(s;!) p(!) An individual or marginal information structure is denoted by (S i ; q i (s; !)) where S i is a …nite set of possible signals and q i (s; !) is a joint probability on S i . For each q i (s;!) i ind such information structure and any !; let q (sj!) . Let be the set of all p(!) such information structures that are marginals of information structures in the set joint . That is, it includes all (S i ; q i (s; !)) such that there exists a ( ni=1 S i ; q(s; !)) 2 joint where P i q i (s; !) = s i 2 n ; !). j q(s; s j=1;j6=i S Pre-communication, individuals know their own individual information structure and observe their signal. Our …rst key assumption is that individuals have ambiguity over the feasible set joint ; that is, they consider all joint information structures as de…ned above. This ambiguity will generate a set of beliefs about the state of the world, but only following communication, as we will show. We assume that whenever individuals update their beliefs on the state of the world, they will consider each joint information structure, one by one, and generate a belief on the state of the world for joint information structures that are “consistent”with their obchoose best-responses to their analogy-based expectations, and expectations correctly represent the average behavior in every class. In the cursed-equilibrium concept of Eyster and Rabin (2005), individuals fail to fully acknowledge the correlation between others’ strategy and their type/information. These types of reasoning can be adopted to yield patterns of information processing that are related to correlation neglect, but are di¤erent from our analysis as again they are concerned with the relation between players’ behaviour and their information. 7 servations. The next Lemma shows that the set of possible pre-communication posteriors is a singleton. Lemma 0: Pre-communication, each player has a unique posterior. Given an obseri 0 vation of some s0 2 S i ; individual i updates his belief to q i (!js0 ) = P p(!)q (s j!) : i 0 0 v2 i p(v)q (s jv) Proof: Individual i observes s 2 S and considers all joint information structures in which have a marginal information structure that accords with his own. That is, P all ( nj=1 S^j ; q^(s; !)) for which s i 2 j6=i S^j q^(s0 ; s i j!) = q i (s0 j!) for all !: For any such joint information structure ( n S^j ; q^(s; !)); we generate the posterior belief about state joint ! as q^i (!js0 ) = P P v2 ^j s i 2 j6=i S P j=1 p(!)^ q (s0 ;s ^j s i 2 j6=i S i j!) p(v)^ q (s0 ;s i jv) i 0 = P p(!)q (s j!) i 0 v2 p(v)q (s jv) for all !: Lemma 0 shows that the ambiguity about the joint information structures, the set joint , does not necessarily imply ambiguity over the state of the world. Pre-communication, individuals know their own individual information structure, observe a signal and compute their posterior belief. Their knowledge of the feasible set of joint information structures has no implications for Bayesian updating or in other words each joint information structure they consider will lead them to the same belief on the state of the world. As they have no information on others’signals, the law of iterated expectations implies that each joint information structure which is consistent with their own marginal information structure will generate the same belief. With some abuse of notation we will sometimes suppress s and denote this posterior belief by q i (!): 3.2 Communication and rationalisable beliefs At the communication stage, individuals truthfully reveal their posterior beliefs, q i (!) for any i; to each other. Let q = (q i (:))i2N denote the vector of group posterior beliefs.11 We assume that in this stage, the individual forgets her own marginal information structure, and thus the post-communication knowledge consists of q and the ambiguity over joint . This assumption implies that the set of rationalisable beliefs will not depend on the particulars of the information structure, which is a useful property for situations in which the modeler does not have that kind of information. Note that there are a few alternative modelling approaches, such as assuming that individuals truthfully exchange their signals and marginal distributions. Such alternative approaches would lead to qualitatively similar results as long as individuals have ambiguity about the joint information structures (in our auction application we use such an alternative approach and let players maintain their knowledge of the marginal information structure). 11 For expositional purposes we focus only on posteriors which are full support. 8 Individuals rationalise the posteriors they observe in order to update their beliefs about the state of the world. In other words, they take each joint information structure in their set, one by one, see if it is consistent with the vector of posteriors (in the sense we de…ne below) and if so, generate a belief on the state of the world. The following de…nition formalizes the process by which individuals rationalise their observation of q. Let s =(s1 ; :::; sn ). Below we denote by q^j (sj ; !) the marginal distribution of (sj ; !) derived from the joint distribution q^(s; !): De…nition 1: A joint information structure ( nj=1 S^j ; q^(s; !)) rationalises a belief i (:) q^j (s ;!) given q; if there is (s1 ; :::; sn ) 2 ni=j S^j such that (i) q j (!) = Pr(!jsj ) = P q^jj (sj ;v) , v2 8j 2 N; (ii) i (!) = Pr(!js1 ; :::; sn ) = P q^(s1 ;:::;sn ;!) : ^(s1 ;:::;sn ;v) v2 q In this case we say that i (!) is rationalisable, given q; by ( nj=1 S^j ; q^(s; !)). Note that rationalisability is an ex-post approach and therefore individuals do not take into consideration whether the event they had conditioned upon is likely to have happened. We extend the approach to ex-interim rationalisations in Section 6.1. Proposition 1 in Sobel (2014) shows that rationalisability alone does not restrict the feasible set of post-deliberation beliefs. That is, for any vector of (interior) individual posteriors, any group belief can be rationalized. To give an example, consider an environment with two individuals and two states of the world, 0 and 1, an equal prior, and two signals s and s . We can construct information structures which rationalize strong beliefs of individuals that the state is 1 pre-communication, but certain beliefs that the state is 0 post-communication. Below, in each cell, we describe the joint probability of receiving the two signals conditional on the state of the world: !=0 s s 0 s 0 s 0 0 1 0 2 0 + 0 !=1 s s 0 s 1 s 1 1 2 1 The marginal probability of receiving s in state 0 is 0 > 0 and in state 1 is 1 > 0; for each player: Thus for each player i the probability that the state is 1 after receiving an s signal is q 1 = q 2 = 1 +1 0 ; which we can set as close to 1 as we wish by setting 0 in…nitely low compared with 1 : Upon exchanging beliefs q 1 and q 2 , individuals may perceive that each had received s and the above to be the prevailing information structure: However this implies that post-communication, they believe that the probability that the state is 1 is zero. In what follows we make our second key assumption, that individuals restrict attention to joint information structures with a bounded level of correlation. 9 3.3 Bounds on correlation To de…ne bounds on the correlation between information structures, we use the local notion of pointwise mutual information (PMI). Let f (x1 ; :::; xn ) be a joint probability distribution of random variables x~1 ; :::; x~n ; with marginal distributions fi (:): The pointwise n) mutual information (PMI) at (x1 ; :::; xn ) is ln[ f (x1i;:::;x ]. PMI was suggested by Church fi (:) and Hanks (1991) and is used in information theory and text categorization or coding, to understand how much information one word or symbol provides about the other, or to measure the co-occurance of words or symbols. For example, for two variables, it can also be written as f (x1 ; x2 ) ] = h(x1 ) h(x1 jx2 ) ln[ i fi (:) where h(x1 ) = log2 Pr(X1 = x1 ) is the self information (entropy) of x1 and h(x1 jx2 ) is the conditional information. Summing over the PMI’s, we can derive the well known measure of mutual information, P P M I(X1 ; X2 ) = x1 2X1 x2 2X2 f (x1 ; x2 ) ln[ f (xi f1i;x(:)2 ) ] = H(X1 ) H(X1 jX2 ); which can be shown to be always non-negative as it equals the amount of uncertainty about X1 which is removed by knowing X2 : We can also express mutual information by using the KullbackLeibler divergence between the joint distribution and the product of the marginals: M I(X1 ; X2 ) = DKL (f (x1 ; x2 )jf (x1 )f (x2 )); and it can therefore capture how far from independence individuals believe their information structures are. For our purposes, the local concept of the PMI is a more suitable concept than the MI, as we are looking at ex-post rationalisations given some set of signals.12 The concept of the PMI is closely related to standard measures of correlation and specifically it implies a bound on the concordance between information structures. The most common measure of concordance is Spearman’s rank correlation coe¢ cient or Spearman’s , a nonparametric measure of statistical dependence between two variables.13 It assesses how well the relationship between the variables can be described using a monotonic function. A perfect Spearman correlation of +1 or -1 occurs when each of the variables is a perfect monotonic function of the other. In the Appendix we show (Proposition A1) that there is a 0 < < 1 such that any joint information structure with bounded PMI’s has a Spearman’s in [ ; ]. This also implies that we can put bounds directly on the copula.14 12 The PMI therefore does not distinguish between rare or frequent events. See Schmid and Schmidt (2007) for an extension of the coe¢ cient to more than two variables. 14 One alternative speci…cation would be to make the bounds assumption directly on the copula, so 13 10 Let ePMI be the exponent of the PMI, i.e., assumption: f (x1 ;:::;xn ) . i fi (:) We now make the following Assumption A1: Individual j only considers joint information structures ( ni=1 S i ; q(s; !)) 2 joint ; such that the ePMI, at any state ! and for any vector of signals s = (s1 ; :::; sn ); is bounded from below by aj and from above by a1j where 0 < aj 1; i.e., aj q(s1 ; :::; sn j!) i2N q(si j!) 1 ; 8s 2 aj n i i=1 S and 8! 2 : Thus individuals have some “capacity” level a for their perception for correlation.15 The assumption allows us to have a one-parameter characterization of an individual’s perception of correlation across information structures. Each individual j is characterized by a number aj ; so that he considers only joint information structures for which at any point (that is for any vector of signals and at any state), the ePMI is no smaller than aj and no larger than a1j .16 Note that a joint information structure which satis…es independence 1 ;:::;sn j!) would have q(si2N = 1 at any point; whenever a joint information structure does not q(si j!) satisfy independence then the ePMI is less than 1 for some (s1 ; :::; sn ; !); and is greater than 1 for some (s01 ; :::; s0n ; ! 0 ). Let C(ai ; q) denote the set of beliefs i (!) that are rationalisable (as in De…nition 1) by an information structure that satis…es A1 with ai , given the observation of q: Note that as ai > 0; the example of the information structure given above cannot rationalize a belief in C(ai ; q): We next prove some preliminary results and then fully characterize C(ai ; q) in Theorem 1 in the next Section. 3.4 A benchmark: Full correlation neglect Consider …rst an individual j with full correlation neglect ( FCN), that is, aj = 1: The following characterises the set of rationalisable beliefs for this individual. Proposition 1 (Sobel 2014, Proposition 5): After observing q, the set of beliefs that are rationalisable for individual j satisfying FCN (aj = 1) is a singleton. The unique that C(u;v) is bounded, where C(u; v) is the copula of u; v: This is implied by our bounds below on uv RR the PMI. Nelsen (2006) also shows that Spearman = 12 [C(u; v) uv]dudv; this implies that the distance between the relevant copula and the product copula (independence) is closely related to notions of concordance. 15 This methodology is similar to the rational inattention model in Sims (2003) which allows individuals to consider models of the world with …nite Shannon capacity. 16 All the results can be easily generalized if instead of the upper bound a1 we use some …nite b 1. 11 posterior is F CN 1 p(!)n (!) = P v2 1 1 p(v)n Q q i (!) i Q i q (v) 1 i Proof: Consider for brevity a group of two individuals. We show that contemplating any information structures (S i ; q i (s; !)); (S j ; q j (s; !)) in ind (by FCN this will span all the joint information structures that satisfy independence), i will update to F CN (!). Suppose i believes that the information structures are (S i ; q i (s; !)); (S j ; q j (s; !)) and that the two individuals had observed (si ; sj ): He would then, using Bayes rule, update to p(!)q i (si j!)q j (sj j!) P : p(v)q i (si jv)q j (sj jv) v But this is equivalent to the result stated as: = q i (!)q j (!) P q i (v)q j (v) =( ) p(!) p(v) v i j P p(!)q (si j!)q P (sj j!) v P v p(v)q i (si jv)dv P v v p(v)q j (sj jv)dv p(v)q i (si jv)q j (sj jv)dv p(v)q i (si jv)dv P v p(v)q j (sj jv)dv p(!)q i (si j!)q j (sj j!) = P : p(v)q i (si jv)q j (sj jv)d! v Under the assumption of (conditional) independence, posteriors are su¢ cient statistics and no knowledge of the information structure is needed. In the next section we generalize this result to allow individuals to perceive some correlation in their information sources, which is the focus of our paper. Still, the benchmark case is useful for our analysis below. It is always included in the set of rationalisable beliefs, and as we will show, in large groups F CN (!) approximates the limit beliefs in many standard environments.17 3.5 A useful result and an example We conclude this Section by proving a useful result that will allow us to characterise the set C(ai ; q): The result states that any belief which is in C(ai ; q); can be rationalized by an information structure that consists of only two possible signals for each group member. 17 In Levy and Razin (2015c) we also use this benchmark to analyse correlation neglect in repeated information exchange and show how the limit beliefs relate to F CN (!) in a simple manner. 12 Proposition 2: Let i (:) 2 C(ai ; q). Then there exists an information structure (S 0 ; q 0 ) with S 0 = fs ; s gn which rationalises i (:) and satis…es A1. This result arises from the ex post nature of the rationalisation: what is important is the vector of signals that an individual perceives to be the one that was attained by the group members, and all other vectors of signals could be bundled together in terms of their joint probability. We show in the proof that such a construct will still satisfy A1. The following example shows how we can use Proposition 2 to characterise the set C(ai ; q). Example 1: Consider an environment with two individuals and two states of the world, 0 and 1; and a uniform prior. We now consider the post-communication beliefs of player i; that is, the set C(ai ; q). By Proposition 2 any such belief in C(ai ; q) can be rationalised by an information structure with two signals s and s . Therefore, a general joint information structure that rationalises a belief in C(ai ; q) is given by the following parameterization: !=0 s s 0 2 s 0 !=1 s s 1 2 s 1 s 1 0 0 0 1 0 1 2 0 + 0 s 1 1 1 1 1 1 1 2 1 + 1 where 10 ; 20 are the marginal probabilities of s in state 0 for players 1 and 2 respectively and 11 ; 21 are the marginal probabilities of s in state 1 for these players respectively. For 1 0 example, an ePMI constraint on (s ; s ; ! = 1) for player i is ai : Note that 1 2 ai 0 0 the eight ePMI constraints for player i imply that all entries in the above matrices have to be strictly positive. Suppose wlog that players believe they had received (s ; s ) and note that rationalisability implies that individuals consider information structures which satisfy q i (!js ) = i i 1 2 1 i i = q (!) for i 2 f1; 2g given q = (q (!); q (!)): 1+ 0 To check for the boundaries of C(ai ; q) let us try to look for the belief that minimizes the probability of ! = 1; e.g., wlog we try and minimize 1 +1 0 : Observe that at the 1 minimum, some ^ 1^+^ ; it has to be that for any !, at least one ePMI constraint has to 0 bind. To see why, suppose that this is not true in the ! = 1 matrix. Then we can change the information structure that lead to this minimum belief to ^1 2 1 1 1 1 1 " ^1 + " 1 ^1 + " 2 1 + ^1 " for some " so that each entry is still positive. This new joint information structure is still rationalisable as the marginals did not change, and for a small enough " the 13 ePMI constraints will still be satis…ed. But we have decreased the beliefs that ! = 1 to ^1 " 1 1 < ^ 1^+^ ; a contradiction to the minimum belief being ^ 1^+^ . Thus, the boundaries ^ 1 "+^ 0 0 0 i of the beliefs in the set will depend on a ; and to …nd its boundaries, we need to consider which of the constraints is binding. 4 Perceptions of correlation and ambiguity The following Theorem characterises C(ai ; q); the set of post-communication rationalisable beliefs of player i that are derived from information structures that satisfy A1 for some ai . Theorem 1: A belief i (!) is in C(ai ; q) i¤ ! i (!) = P v2 1 p(!)n 1 v Q q j (!) Q j2N 1 p(v)n 1 q j (v) j2N for a vector = ( ! )!2 satisfying ! 2 [ai ; a1i ] for all !: Thus the minimum (maximum) belief on state ! is derived when ! = ai ( a1i ) and for all other v; v = a1i (ai ): Moreover, the set C(ai ; q) is compact and convex. The Theorem shows that ambiguity over information structures generates a set of beliefs whenever ai < 1, which is a natural and simple extension of F CN (!): When ai = 1; C(ai ; q) becomes a singleton, coinciding with F CN (!). Communication creates ambiguity over the state of the world: Individuals -who start with a unique belief when on their own- end up with a set of beliefs following communication. Each joint information structure that is consistent with the beliefs of others will generate a di¤erent probability over the states of the world and individuals cannot further re…ne this set of beliefs. While naturally ambiguity over the set of joint information structures can only decrease following communication, ambiguity in the beliefs over the state of the world increases. Thus our model can be used to underpin the sets of prior beliefs in models of ambiguity aversion. 4.1 Perceptions of correlation and beliefs We now provide some implications of the Theorem which will prove useful in the applications to behaviour in Section 5. We …rst de…ne a measure of perception of correlation and show what it implies in terms of beliefs. We then show how the FCN benchmark approximates beliefs in large groups. 14 First, the characterization in the Theorem allows us to formalise a comparative notion of perception of correlation and connect it to di¤erent notions of con…dence: De…nition 1: An individual i has a lower perception of correlation than individual j i¤ ai > aj : De…nition 2: An individual i has more con…dence than individual j if C(ai ; q) C(aj ; q): De…nition 3: An individual has stronger beliefs post-communication if the variance of each posterior in C(ai ; q) is lower than the variance of q i (!): De…nition 2 highlights a novel relation between correlation and con…dence, in particular between perceptions of correlation and the scale of Knightian ambiguity, as derived from the Theorem: Corollary 1: If individual i has a lower perception of correlation than individual j then for any q; individual i has more con…dence than individual j: We show in Section 5.1 what this implies for behaviour. Speci…cally, in a simple investment model, we show that an individual with a lower perception of correlation will invest more in a risky asset compared to an individual with a higher perception of correlation. De…nition 3 formalises the notion of con…dence explored in the literature on “correlation neglect”. As individuals will have sets of beliefs our de…nition will involve incomplete comparisons but our model nests such a notion as well; to see this, consider for example the case of two states of the world: Corollary 2: For any q; if an individual j has stronger beliefs post-communication, so does individual i who has a lower perception of correlation:18 Second, we illustrate how the FCN benchmark can anchor the set of beliefs. If F CN (!) is close to being degenerate on some !, this implies that C(ai ; q) collapses to F CN (! 0 ) for all ! 0 , as long as ai is bounded. Thus, F CN (!) can anchor beliefs if it becomes degenerate. Intuitively, the larger is the group the more informative the FCN posterior might be. Suppose just for simplicity that all individuals have the same bound a; and consider a 18 To see this, note that in the simple binary model (and equal prior), the variance of a belief q is q(1 q): Thus variance of beliefs is maximized at a half. If q i > 0:5; then variance decreases for all beliefs ai i2N q j if the minimum beliefs in the set has lower variance, i.e., ai > q i : Thus, for any vector qj + 1 (1 q j ) i2N ai i2N of beliefs, this is more likely to arise when ai is higher and closer to 1. If q i < 0:5; then we have to look at the maximum belief in the set that will satisfy more vectors of beliefs if ai is closer to 1. 1 ai 15 j 1 i2N q ai j i i2N q +a i2N (1 qj ) < q i : Again, this will occur for sequence of such bounds that may depend on the size of the group, (an )1 n=1 : Assume that for all sizes n; this sequence satis…es 0 < < an 1 for some : Let qn be the sequence of vectors of beliefs of the n individuals in the group and let Fn CN be the relevant full correlation neglect sequence of posteriors given these beliefs. Let Cn = C(an ; qn ): Corollary 3: If Fn CN ! (!) where (!) is a degenerate belief, then for any sequence in (!) 2 Cn : i n (!) ! (!) Beliefs in large groups can therefore converge to those resulting from full correlation neglect. Alternatively, if an is not bounded, the result can be generalized to the case in which the rate of convergence of Fn CN to a degenerate distribution is faster than the rate of convergence of an to zero. Note that even if information is conditionally independent, this does not guarantee that Fn CN converges to be degenerate; it is in fact straightforward to construct such information structures with n ! 1 in which the Fn CN either converges or not to be degenerate. As a result, we can have both ambiguous and unambiguous sets of beliefs with large groups.19 Conversely, if Fn CN converges to be degenerate, it does not imply that it converges to the truth. We discuss such normative issues in Section 5. 4.2 Proof of Theorem 1 In Step 1 we prove that all rationalisable beliefs by information structures that satisfy A1 must belong to the set C(ai ; q) as characterized in the Theorem. In Step 2 we construct a family of information structures that spans all beliefs in the set as characterized in the result. In Step 3 we prove the convexity of C(ai ; q). Note that due to the nature of the ePMI constraints, one cannot simply take a convex combination of information structures. To prove convexity we therefore need to use our construction to …nd a new information structure to rationalise any convex combination of beliefs in C(ai ; q). In what follows we consider a generic parameter a (i.e., we suppress the superscript i): We assume that individuals rationalize the set of posteriors they observe by believing that all have received the signal s . For any v 2 ; let v = Pr(all receive s jv) and let i v = Pr(i receives s jv): 19 Suppose that information structures are independent and each individual receives an informative sigCN nal with precision that is …xed. This indeed will lead F to converge to be degenerate with probability n one. Another information structure that maintains independence however is that each individual receives CN an informative signal with precision that decreases in n: If the precision decreases fast enough, then F n would be uninformative in the limit. 16 Step 1: Any ~(!) 2 C(a; q) satis…es Y a p(!)n 1 X 1 1 a p(v)n 1 Y i2N a q i (v)+ p(!)n 1 i2N v2 ;v6=! Y X ~(!) q i (!) i2N Note that ~(!) = X p(!) p(v) 1 a p(v)n 1 Y q i (!) i2N q i (v)+ a1 1 p(!)n 1 i2N v2 ;v6=! ! Y 1 1 a p(!)n 1 q i (!) Y : q i (!) i2N : By the ePMI for (s ,s :::; s ; v); for any v; we have: v +p(!) ! v6=! p(!)a 1 a X Y i p(v) i ! Y p(!) ! X p(v) i v i v6=! p(!) a1 a v X Y i p(v) Y i v : i v6=! v6=! i ! By rationalisability, for any two states ! and z; we have: p(!) i! p(z) iz ; q i (z) = X q i (!) = X p(v) iv p(v) v2 i z Thus in any rationalisable belief, Y a p(!)n 1 1 a X q i (!) i 1 p(v)n 1 a p(!)n 1 1 a X v6=! 1 p(v)n 1 Y q i (v) i v6=! Y Y i a p(!)n 1 i Y = p(!) q (z) p(z) q i (!) p(!) ! X p(v) i i !: 1 1 a p(!)n 1 a v X p(!) p(v) X v Y i 1 p(v)n 1 q i (!) Y q i (v) , i v6=! v6=! q i (!) , v2 i q i (!) q i (v)+ i v 1 1 a p(!)n 1 ! + p(!) ! X v6=! v6=! a p(v)n 1 Y Y q i (!) i q i (v)+ a1 1 p(!)n 1 i Step 2: For any ~(!) that satis…es the characterization in Theorem 1, there exists an information structure that satisfy A1 and rationalizes this belief. Take any vector ( ! )!2 that satis…es a ! ~(!) = P v2 ! 1 p(!)n 1 v Q 1 a for any ! and consider the belief q j (!) Q j2N 1 p(v)n 1 q j (v) : j2N Using this vector ( ! )!2 we now construct an information structure that will satisfy all ePMI constraints and the rationalisability constraints, and will rationalise the belief ~(!). 17 Y i : q i (!) Let ! = ! Q i ! i (!) = " qp(!) : This implies that this information structure Q 1 i ! and let i2N ! p(!) generates the belief as desired as ~(!) = P ! p(v) v v2 p(!) Note that P i ! p(v) i i v v2 q (!) = P i v2 i q (v) = P p(!)n 1 v q j (!) Q j2N 1 p(v)n 1 q j (v) : j2N = q (!) which implies that the posterior beliefs of all v2 individuals are rationalized. We now specify the joint distribution over signals, making sure that all the ePMI constraints are satis…ed. For all states, set the joint probability of each event in which two or more players receive s in some state, but not when all players receive s , to satisfy independence. For example, the probability that all m individuals in the set J and only Q j Q i these individuals receive s in state !, for 1 < m < n; is (1 ! ! ). Thus for all j2J i2N=J these cases the ePMI constraints are satis…ed. At any state, we then need to verify the ePMI constraints in the following events: when one player exactly had received s ; or when all received s . Let us focus on some state !. Consider …rst the event in which only one player had received s : P r(si = s ; all others receive s j!) = " q i (!) p(!) " ! q i (!) P Q ( p(!) J N=fig j2J Q j ! k ! )) (1 k2N=J[fig jJj 1 The ePMI is: i i (!) " qp(!) ! (!) " qp(!) ( P i (!) " qp(!) 1 = ! Q k2N=i Q Q (1 (1 k6=i Q j ! k ! )) (1 k2N=J[fig k (!) " qp(!) ) k6=i P ( J k (!) ) (" qp(!) Q J N=fig jJj 1 j2J Q N=fig jJj 1 j2J Q j ! k ! )) (1 k2N=J[fig k (!) " qp(!) ) !"!0 1; as for all j; j! goes to 0 with ": Thus, the ePMI can be made smaller than than a; if " is small enough. Consider now the event that all players had received s in state !: Pr(all received signal s j!) = (1 (n " q i (!) ) p(!) 1)(" (1 q i (!) p(!) " P Q q i (!) )( p(!) J N=fig j2J jJj 2 ! " j ! and greater Q k ! )) (1 k2N=J[fig q i (!) P Q ( p(!) J N=fig j2J jJj 1 18 1 a j ! Q (1 k2N=J[fig k ! ))); where here we subtract all the events in which two or more received s (but at most n 1); and the n 1 events in which just one player had received s which we had described above. The ePMI is: P Q j Q P Q j Q q i (!) q i (!) q i (!) q i (!) k (1 " p(!) ) (1 " p(!) )( (1 ! J N=fig jJj 2 j2J (1 " 1 ( = P Q J N=fig jJj 2 j2J Q (1 " k6=i ! "!0 1: j ! Q k ! )) (1 ! )) (n 1)(" k2N=J[fig (n 1)( k2N=J[fig Q q i (!) ) p(!) q i (!) p(!) (1 " k6=i n 1 !" " p(!) ( J N=fig jJj 1 j2J q k (!) ) p(!) Q ( k2N " q k (!) ) p(!) ! p(!) q k (!) ) p(!) (1 " q i (!) P ( J p(!) q i (!) ) p(!) Q (1 " k6=i Q N=fig jJj 1 j2J q k (!) ) p(!) j ! ! Q Step 3: C(a; q) is compact and convex. Compactness comes from steps 1 and 2. To prove convexity consider two beliefs and 0 that are in C(ai ; q). Note that Steps 1 and 2 imply a belief (:) is in C(a; q) if and only if for any v; ! 2 we have, (!) = (v) q i (!) i2N ! p(!)n v Q 1 q i (v) : i2N p(v)n or that all likelihood ratios satisfy, Q 1 Q q i (!) i2N (!) (v) n 1 p(!) a2 Q i q (v) i2N p(v)n q i (!) i2N 1 p(!)n 1 Q : q i (v) a2 (1) i2N 1 p(v)n 1 To prove convexity we show that we can …nd a vector with elements between a and a1 that spans + (1 ) 0 : It will be enough to show that + (1 ) 0 has likelihood ratios in the bounds in (1). Note that ; 0 satisfy Q q i (!) i2N n 1 p(!) a2 Q i q (v) (!) (v) i2N p(v)n Q Q q i (!) i2N i2N 1 p(!)n 1 p(!)n 1 2 Q Q ; a 2 q i (v) q i (v) a i2N 1 p(v)n we have that: (!) + (1 (v) + (1 0 ) (!) ) 0 (v) i2N 1 p(v)n (v) + (1 (v) + (1 Q q i (!) 0 0 (!) 1 p(!)n 1 Q ; q i (v) a2 0 (v) i2N 1 p(v)n Q q i (!) i2N Q 1 q i (!) i2N ) (v) 1 p(!)n 1 1 p(!)n 1 Q Q = ; q i (v) q i (v) ) 0 (v) a2 a2 i2N p(v)n 19 q i (!) i2N i2N 1 p(v)n 1 (1 k2N=J[fig which again can be made smaller than a and larger than a1 for low enough ". Thus all constraints in state ! can be satis…ed. Therefore we have proved step 2. Q (1 k ! k2N=J[fig k ! ))) and similarly that, (!) + (1 (v) + (1 ) 0 (!) ) 0 (v) Q q i (!) i2N n 1 p(!) a2 Q i q (v) i2N p(v)n So there must exist 5 that spans + (1 : 1 ) 0: Perceptions of correlation and behaviour We now illustrate the behavioral implications of Theorem 1. We consider two related applications that have been studied extensively in the literature. We …rst consider the phenomena of risky and cautious shifts in groups (Stoner 1968). We then explore the implications of our model to jury decision making. Both applications are based on a simple model in which the actions of individuals are monotone in their beliefs. To consider how perceptions of correlation a¤ect behaviour, we have to take a stand on how individuals take actions in the face of ambiguity. We assume for simplicity that individuals have ambiguity aversion and concretely, that when individuals are faced with ambiguity, they use the max-min preferences as in Gilboa and Schmeidler (1989). Note that given the convexity of the set of beliefs C(ai ; q), we can use other attitudes towards ambiguity to generate similar results. 5.1 Risk taking and group shifts We consider a simple investment model. To …x ideas, assume a binary model with two states of the world, ! 2 f0; 1g. Assume that there is a safe asset which provides the same returns L > 0 at any state, and a risky asset which provides 0 at state 0 and H > L in state 1. Individuals have one unit of income to invest which they can split across these two assets. Assume a standard concave utility V (:) of wealth, so that an individual would invest a higher share in the risky asset the higher are her beliefs that the state is 1. Pre-communication they would then base their investment decision on q i (1); while post-communication they would base their investment decision on min i (1) in the set C(ai ; q). We denote a higher investment in the risky asset as more risky behaviour, as for the outside observer, such a behaviour is associated with more risk taking. Comparison of individuals’behaviour: Given our measure of perception of correlation, and Theorem 1, it is then straightforward to show: 20 Lemma 1: Following communication, an individual i who has a lower perception of correlation than an individual j will invest more in the risky asset compared with j. Our measure of the perception of correlation has therefore behavioral implications in a standard setup of risk taking. It is easy to see why this arises. By the strict inclusion property in Theorem 1, the individual with a lower perception of correlation will have a higher minimum belief in his set C(ai ; q). Note that “standard”results in the literature on correlation neglect are typically of the form that individuals with more correlation neglect will take more extreme decisions, but depending on the state of the world these could be either on the risky or on the cautious side.20 The result above applies to any state of the world and any set of signals, and arises from the reduced ambiguity that comes with lower perception of correlation. Remark 1 (Identi…cation): Note that di¤erent perceptions of correlation can be identi…ed in choice data, as long as there is choice data on behaviour of individuals in the face of ambiguity. When faced with exogenous ambiguity, the behaviour of individuals can indicate their attitudes towards ambiguity. Once individuals have the same attitudes towards ambiguity, we can identify their perceptions of correlation from their behaviour as indicated in Lemma 1. Choice shifts in groups: We now take the investment example further to explore “group shifts”as a function of group size. Group shifts, …rst described by Stoner (1968), compare between individual behaviour and group behaviour, or between individual behaviour before group deliberation and after. Do individuals’ beliefs and behaviour become more cautious or more risky after group deliberation? A cautious (risky) shift implies that an individual will invest less (more) in the risky asset following group communication compared with her behaviour before group communication. We will show that: (i) Cautious shifts and small groups: Individuals with a high enough perception of correlation always experience a cautious shift; (ii) Divergence: Individuals’ behaviour -even when similar pre-communication- may substantially diverge following exposure to the same information, and …nally, that (iii) Large groups: group size matters so that in large groups substantial cautious or risky shifts will occur. We therefore highlight the e¤ect of group size, which can be non-monotonic. The following result shows how communication -through the ambiguity it creates- can always cause a cautious shift: 20 In the papers of Glaeser and Sunstein (2009) or Ortoleva and Snowberg (2015), individuals believe that a set of signals is independently drawn rather than correlated. This implies that they have more extreme beliefs -be it high or low beliefs- compared to an individual who realizes that these signals are correlated. 21 Lemma 2: Fix n and q, there is a cautious shift. > 0 such that if ai < then i will experience a The intuition is simple: if the bound is low enough, even when an individual observes that others have “optimistic”beliefs that the state is 1, she could imagine an information structure that has su¢ cient correlation and implies that beliefs should be pessimistic. The proof follows from Theorem 1. The result highlights the di¤erence between the endogenous ambiguity in our model and standard exogenous ambiguity. Here ambiguity arises only after communication and so the cautious shift above arises due to the communication in the group and due to su¢ cient ambiguity about information sources. In contrast, in a standard model with ambiguity about the state of the world, such ambiguity will a¤ect the individual both pre and post communication. Note that heterogeneity in perceptions of correlation might also explain a form of group divergence as studied in Lord et al (1979). They show that individuals who observe the same information can diverge in their beliefs or actions. In our case, small groups and di¤erent perceptions of correlation can create such a divergence, in which some will move to riskier behaviour, while some will move to invest in safer assets, after observing the same information. For example, consider the case in which they all start with the same private posterior beliefs and hence the same level of pre-communication investment. Suppose that these beliefs are conducive for investment so that q i (1) = q > 12 . Individuals with low perception of correlation, for example with a = 1; will experience a risky shift given the observation of others’ positive beliefs, while those with high enough perception of correlation will exhibit a cautious shift in line with Lemma 2. What happens when groups become large? From the analysis above we know that in large groups individuals may have beliefs that are close to the FCN beliefs. To illustrate, the following …gure describes the behaviour of individuals with di¤erent perceptions of correlation, as a function of group size. Consider again the case when all individuals start with the same beliefs q i (1) = q > 12 ; but di¤er in their perceptions of the correlation. 22 While individuals with high perception of correlation (low a) will decrease their investment in a small group, when the group becomes large their beliefs will converge to the FCN belief. Here, larger groups increase the FCN belief, implying that beliefs attain the highest possible lower bound. As a result, large groups induce a risky shift for all individuals. More generally, the …gure above illustrates the non-monotonic e¤ect of group size for individuals with low a. 5.2 Group shifts in jury decisions We now use our model to study decision making in Juries. In many ways our model …ts well with the Jury application. Juries are groups of individuals who take decisions after deliberation, which is an explicit part of the process. The jurors are exposed to the same evidence and proceedings and so correlation in their information and its perceptions are important features. Finally, our assumption of truthful communication during deliberations is con…rmed in experiments on juries.21 We adopt the canonical model of juries to our framework. There are n jurors who decide to acquit, A, or convict, C, a defendant. There are two states of the world, one in which the defendant is guilty, G, and another in which she is innocent, I. Assume that the prior about these two states is uniform and that it is common knowledge. As in our model, a vector of beliefs, qn = (qni ; :::; qnn ), denotes the posterior private belief of each juror that the defendant is guilty, attained after observing their signals from the trial. During deliberation the Jurors share their beliefs qni truthfully. Each Juror has a 21 Goeree and Yariv (2011) and Guarnaschelli et al (2000) have shown that when the option to deliberate (or to conduct a straw poll before decisions) is o¤ered, individuals overwhelmingly tend to be truthful, even when there are con‡icting preferences. 23 perception of correlation parameter ai < 1, satis…es A1 with this bound, and updates his beliefs to the set C(ai ; qn ): Each Juror is denoted by a cuto¤ i 2 (0:5; 1) such that she prefers to convict i¤ her posterior belief that the defendant is guilty is above the cuto¤ i : We assume that to convict, a unanimous vote for conviction must be reached. As all information is shared, there are no “pivotal” considerations, and we can simply assume that each juror votes sincerely.22 Given Theorem 1 and ambiguity aversion, individual i ai q j will then vote to convict i¤ ai qj + 1 j n (1 qj ) > i . j n ai j n Note that the model di¤ers from the investment model of the previous Section as now each juror makes a binary decision by using her cuto¤, and the …nal decision is an aggregation of individual views. In what follows, given the vector qn , let q^n be the number such that 1 q^nq^n is the geometric j average of f (1 qnqj ) gj2N ; i.e., n q^n 1 q^n =( j 1 qn j2N (1 q j ) ) n : n Lemma 3: (i) The decisive juror is juror i 2 N which maximizes (ai )2 (1 i 2 N: (ii) The jury convicts (acquits) i¤ ( (ii) As n grows large, if q^n 1 q^n > q^n n 1 ) > i 2 1 q^n (a ) 1 > 1 ( 1 q^nq^n 6 i i ) for all i i < 1) the jury only convicts (acquits). The result gives a simple characterisation of jury behaviour: the information held by the group indicating that the defendant is guilty (summarized by the geometric mean q^n ) has to be strong compared to the adjusted cuto¤ of the decisive juror. Such juror is determined not only by her taste parameter ; but also by her perception of correlation a. The lower is the perception of correlation (high a); the more “persuasive”is the information held by the group, while a higher perception of correlation will make the individual more cautious. When the group becomes large, F CN (G) n = ( 1 q^nq^ )n n ( 1 q^nq^ )n +1 becomes degenerate at either 0 or n 1 and overwhelms both the taste parameters and the perception of correlation. We now consider the normative implications of our model to jury decision making for large groups and shed light on the experimental evidence. While large groups can potentially aggregate more information, they will also behave according to the FCN benchmark. We consider two extreme cases, when the joint information structure satis…es (conditional) independence and when it satis…es full correlation. Consider …rst the case in which the true sources of information are conditionally independent. For example assume that conditional on the state of the world, an individual 22 The analysis can also then be easily extended to other voting rules. 24 gets a signal i (signaling state I) or g (signaling state G) which are correct with probability p > 21 and independently drawn. In this case, by Lemma 3(ii) and the law of large numbers, large juries take the correct decision for all perceptions of correlation. This is in contrast with the predictions of Feddersen and Pesendorfer (1998), but in line with the experiments of Goeree and Yariv (2011) and Guarnaschelli et al (2000). These experiments allow for deliberation and …nd that larger juries are more accurate in their decisions. Moreover, they use conditionally independent information sources and inform the participants that this is the case. Consider now the case in which the true sources of information are correlated. For simplicity suppose that there is one draw of the signal above and all observe the same draw and form a posterior q i = q. One criterion for e¢ ciency, given the requirement of unanimity, might be that the jury would convict if q > maxi2N i > 12 : In our model, by Lemma 3(ii), they convict whenever q > 21 and acquit otherwise. The experiments reported in Schkade et al (2000) show a severity shift that arises when jurors deliberate so that they tend to award higher punitive damages. In Schkade et al (2000) there is little control over both the information sources of participants and what they might believe about them: Participants are only shown the same videos of evidence and are not given any additional information. Our model generates such shifts for large groups; for the case described above, some individuals with i > q > 12 would acquit pre-deliberation, while all would convict post-deliberation.23 The trade-o¤ we alluded to above can allow us to …nd the optimal size of juries as a function of the true correlation embedded in the joint information structure. The example of independence described above preserves the e¢ ciency of large groups in our model, as in such a case, large groups have more information and behave as if they assume independence. The example of full correlation described above implies that a larger group has no bene…ts in terms of additional information and only damages e¢ ciency as individuals exhibit FCN-type of behaviour: A unique juror may be better in this case. 6 Discussion and Extensions We now discuss a few extensions of the model. We …rst consider common-value auctions and then discuss a possible extension to repeated communication in networks. 23 When q < 21 ; all individuals would acquit pre-deliberation as well as post-deliberation as 25 i > 12 : 6.1 Common-value auctions We now take a …rst step of extending our approach from a communication model into a strategic game. In particular, we consider the e¤ect of di¤erent perceptions of correlation on behaviour in common value auctions. The common value auction is a good application to consider because of the “winner’s curse”which arises when individuals need to condition their valuation on the event of winning the auction, and hence on the information content of others’signals in this case. We present here a simple example of a …rst price-auction with two possible valuations and two signals. We compare the equilibria in our model to those in the canonical case (when signals are conditionally independent and individuals are aware of it). We show that the support of equilibrium bids is increasing, set-wise, in the perception of correlation. Therefore, compared to the canonical model, the lowest bid is lower and the highest bid is higher. Moreover, the winner’s curse is mitigated for the low type and sometimes exacerbated for the high type. The results di¤er from the case of exogenous ambiguity about the state of the world, and from a cursed-equilibrium analysis (Eyster and Rabin 2005). Consider then a …rst-price common values auction with two bidders, two possible common valuations v = 0 and v = 1, referring to a low state L and a high state H respectively, and a uniform prior. Each individual can receive one of two signals: l or h: We assume that the true information structure is symmetric among the players and across the states, and that the (marginal) probability of receiving the signal l in state L; or the signal h in state h; is q > 12 : To put some structure on the equilibrium, we impose that individuals know the true marginal probability distribution generating both their signals.24 Thus, individuals perceive the following family of information structures: !=L l l L h q L h q 1 L 2q + L !=H l h l H 1 q H h 1 q 2q 1 + H H so that L and H are the parameters on which there is ambiguity. To simplify we assume that both players have the same a: Individuals have therefore ambiguity over all information structures as above which also satisfy the ePMI constraints for some a; e.g., q 1 1 L L a ; a q(1 ; and so on. We focus on a su¢ ciently close to 1. q2 a q) a 24 In our main result, Theorem 1, individuals did not know the marginal information structures, which had led to a characterization that did not depend on any parameters besides a: Here we assume that individuals know the marginals; and thus choosing an information structure subject to the ePMI constraints will depend on the speci…c details of the marginal distributions. 26 An equilibrium is denoted by a pair of bidding strategies for the two players, (b1 (:); b2 (:)); and a symmetric equilibrium has b1 (:) = b2 (:) b(:): We consider again max-min behaviour. Speci…cally, in equilibrium, given an observed signal, a bidding strategy maximizes the utility of the individual under the worst case scenario (that is, when nature chooses the worst information structure from the set described above). The equilibrium comprises of a unique bid for the type with the low signal, denoted by ba (l); and a probability distribution Fa (b) describing the bids of the type with the high signal, over [ba (l); ba (h)] for some maximum bid ba (h). In the Appendix we show: Proposition 3: (i) The minimum bid ba (l) increases in a and the maximum bid ba (h) decreases in a. (ii) The expected payment of the high type increases in a for low q and decreases in a for high q. (iii) The seller’s revenue increases in a: Thus, a higher perception of correlation increases the maximum bid and decreases the minimum bid. Moreover, for high q, the expected payments of the high type increases when we move from a = 1 to a smaller a: The di¤erence in the reactions of the low and high type to a decrease in a follow from the di¤erent beliefs they focus on in equilibrium, as we show below. Therefore, ambiguity over information structures implies that the winner’s curse a¤ects the distribution of bids in a non-trivial way. To see how this comes about, let us consider the bidder with the low signal. At the minimum equilibrium bid, he would win only when the other bidder receives the low signal. Thus for any b; he “chooses” H and L to (recall that v = 1; v = 0): min ( L; H = min L; H H ba (l)) L+ H ba (l)); L ( ba (l)) + H (1 L + H )( which implies that L (the probability to receive two l signals in state L) is set at its maximum (and hence greater than the independent benchmark of q 2 ); and H is set at the minimum possible (lower than the independent benchmark of (1 q)2 ), subject to the ePMI constraints. Denote this worst case information structure for type l by l = f lL ; lH g:25 Given that there is no rent for the low type, the equilibrium bid will then be equal to the expected valuation given two l signals under l : ba (l) = E l (vjl; l) = l H l L + l H < (1 (1 q)2 = E(vjl; l) q)2 + q 2 Thus, the bid of the low type is lower compared with the canonical case, implying that his expected utility (computed for the true information structure) will be higher. 25 We show in the appendix that l = fq 2 + ( 1 a a )(1 27 q)2 ; a(1 q)2 g: What about the high type? suppose that the high type bids just above ba (l): He wins only if the other individual had received l; and analogously to the above, we show in the appendix that the worst case scenario for this bid will constitute the belief that the signals (h; l) are most likely to indicate that the state is L: This implies to minimize L and maximize H ; subject to the ePMI constraints. This induces some belief h = f hL ; hH g; which is the polar belief to l :26 The maximum bid in the support of the high type, ba (h); will be associated with an expected valuation that depends only on the h signal, that is, E(vjh) = q for all a; as winning with the highest equilibrium bid provides no information; however, compared with the independent case, the utility of the type h from the minimum bid in his support will be lower (as he considers a worse event). As the utility has to be equal for all bids, this implies that the maximum bid ba (h) will be higher compared with the canonical case. Speci…cally, we show that: ba (h) = Pr(ljh)E l (vjl; l) + Pr(hjh)E h (vjh; h); and h Fa (b) = h Pr h (ljh)(b E l (vjl; l)) ; Pr h (hjh)(E h (vjh; h) b) where Pr h (ljh) describes the probability of receiving an l signal conditional on having an h signal under the information structure h : In the appendix we show that the expected payment of the high type (as manifested also in the maximum bid above) puts more weight on E h (vjh; h) and less on E l (vjl; l) the higher is q: For all information structures, a high q implies that upon receiving h; the individual is more likely to face a rival player with h as well. Moreover, as h minimizes E (vjl; h); it therefore maximizes E (vjh; h) and more so the lower is a: This implies that a lower a and a higher q will combine to increase the expected bid of the high type. Thus, the winner’s curse is exacerbated for this type and his expected utility will therefore be lower for high q compared with the canonical case. Overall however, taking into consideration also the lower bid of the low type, the seller’s pro…ts decrease with a lower a for all q: We make three remarks about the above characterization. First note that the characterization is not equivalent to a model in which the individuals simply start with some unique worst case beliefs, as each type (l or h) uses a di¤erent worst case belief to justify their best response ( l and h respectively). Second, our characterisation di¤ers from that of a model in which there is some exogenous ambiguity on the state of the world, through the prior for example.27 In this case, We show in the appendix that h = fq 2 (1 a)(1 q)2 ; a1 (1 q)2 g; and that such belief is the worst case scenario for all bids of h greater than the bid of l. 27 Bose et al (2006) analyse optimal auctions under exogenous ambiguity on the prior. 26 28 again, both types would see the worst case scenario as the lowest prior on the state. Thus both the minimum and the maximum bids would be reduced. We show this formally in the appendix (Proposition A2). Finally the characterisation di¤ers from a cursed-equilibrium characterisation à la Eyster and Rabin (2005). In their framework, individuals do not realize that others’strategies relate to others’information. The cursed equilibrium applied to the auction model presented here will yield a minimum bid which is higher than in the canonical case and a maximum bid which is lower than in the canonical case, in contrast with our results.28 6.2 Networks and social learning The assumptions set up in our model are quite reasonable in the context of repeated information exchange and information di¤usion in networks. First, as is the case in most of the literature, for such environments it is common to consider that individuals are not strategic (see the survey in Jackson 2011). Second, it is reasonable to assume that individuals are not aware of the information structures of others. Finally, as information ‡ows in the network, individuals may be unaware that they are exposed to the same information they have been exposed to in the past, and thus correlation neglect is likely to arise. It is therefore reasonable to assume that individuals are characterized by some perception or possibly misperception of correlation. In Levy and Razin (2015c) we use the FCN benchmark and show that the limit beliefs in the network are easily computable (when the matrix describing the network is irreducable and aperiodic).29 The characterization in Theorem 1 can potentially be applied to repeated interaction environments in which individuals also perceive some correlation in their information sources. Speci…cally, a recent literature in development economics has highlighted both the importance of social networks, and the importance of ambiguity aversion in inducing cautious decisions in the case of technology adoption.30 Following from our analysis, ambiguity over the joint information structure may induce cautious behaviour in the presence of social networks. Still, if initial beliefs are su¢ ciently 28 An l type individual who is cursed will mistakenly use E(vjl) > E(vjl; l) at least with some probability to evaluate winning against another low type, whereas an h type who is cursed will mistakenly use E(vjh) < E(vjh; h) at least with some probability to evaluate winning against another h type. 29 We highlight in Levy and Razin (2015c) two testable di¤erences between our results and results that would follow from using the DeGroot heuristic (DeGroot, 1974). The DeGroot heuristic (see De Marzo et al 2003, Golub and Jackson 2010) involves averaging of beliefs and would thus prevent beliefs from becoming polarized (not in the convex hull of the initial beliefs). Moreover, if all in society share the same beliefs, these would not change. The FCN allows for polarisation in beliefs and for a dynamic consensus (as long as beliefs are not degenerate). 30 See Conley and Udry (2004), Bryan (2013) and Engle-Warnick et al (2011). 29 optimistic and the network su¢ ciently connected, our result about large groups can point to some environments in which this cautious behaviour is mitigated. 7 Conclusion We suggest a new framework to analyse individuals’perceptions of correlation and their e¤ect on behaviour. Our model is built on two main premises, one that individuals have ambiguity about information structures and the second that they have bounds on the level of correlation between these information structures. 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Public Disagreement, American Economic Journal: Micro, 4(3), 57-95. [48] Sims, C. (2003), Implications of Rational Inattention, Journal of Monetary Economics, Volume 50, Issue 3, 665–690. [49] Sobel, J. (2014). On the relationship between individual and group decisions, Theoretical Economics 9, 163–185. [50] Stoner, J. A. F. (1968). Risky and cautious shifts in group decisions: the in‡uence of widely held values. Journal of Experimental Social Psychology: 442-459. 33 8 Appendix Proposition A1: Assume that there are two individuals, i and j. There is a 0 < < 1 such that any joint information structure that satis…es A1 has a Spearman’s (Kendal’s ) in [ ; ]: Proof of Proposition A1: The bounds on the ePMI imply that there is an " such q(si ;sj j!) that qi (si j!)q "; 1 + "]: This implies that jq(si ; sj j!) q i (si j!)q j (sj j!)j j (s j!) 2 [1 j "q i (si j!)q j (sj j!): Summing up over all (si ; sj ) and given x; y we get that jQ(x; yj!) Qi (xj!)Qj (yj!)j "Qi (xj!)Qj (yj!) ": This implies that the distance between the copula of any such information structure to the product copula is bounded by ". Among all such information structures, take the supremum according to the highest copula. That information structure has a Spearman’s (Kendall’s ) that is strictly smaller than 1 (See Theorem 5.9.6 and Theorem 5.1.3 in Nelsen 2006). Among all such information structures, take the in…mum according to the lowest copula. That information structure has a Spearman’s (or Kendall’s ) that is strictly larger than -1 (See Theorem 5.9.6 and Theorem 5.1.3 in Nelsen 2006). By Theorem 5.1.9 in Nelsen (2006), any other information structure will have a Spearman’s (Kendall’s ) in between the two copulas above. Proof of Proposition 2: Assume that an information structure (S; q) rationalises (:): Without loss of generality relabel signals so that the vector of signals that rationalises i (!) is (s ; s :::; s ) so that i (!) = q(!js ; s :::; s ). In addition we have that the following rationalizability and ePMI constraints are satis…ed, i 8i 2 N and 8! 2 ; q i (!) = q i (!js ) 8s =(s1 ; :::; sn ) 2 i2N S i and 8! 2 ; aj q(sj!) i i2N q (si j!) 1 for all j 2 N aj Construct the new information structure (S 0 ; q 0 ) by keeping the same distribution over signals as in (S; q); while keeping the label s and bundling all possible signals s 6= s under one signal s : In particular, 8! 2 ; q 0 (s ; :::; s j!) = q(s ; :::; s j!) P q 0 (s ; s ; :::; s j!) = q(s; s ; :::; s j!); s2S 1 =fs g and so on. Note that (S 0 ; q 0 ) rationalises i (:) by de…nition. It remains to show that the ePMI constraints hold for (S 0 ; q 0 ) so that it satis…es A1. Note …rst that the ePMI constraint for (s ; :::; s ) holds by de…nition of (S 0 ; q 0 ). Consider any other pro…le of signals s 2fs ; s gn : The ePMI constraint for s can be expressed in terms of the information 34 Pl j=1 cj 0 j=1 cj where cj = q(sj j!) for some sj = (sj1 ; :::; sjn ) 2 m2N S m where Q i j we sum over all sj that compose s; and c0j = q (si j!): But as the original ePMI i2N P l cj 1 j Plj=1 for all j: Thus the ePMI constraints hold, this also implies that a aj 0 structure (S; q) as Pl 0 c j=1 j 0 constraints are satis…ed also for (S ; q ): Proof of Lemma 1: The proof follows from the construction in Theorem 1, Corollary 1 and maxmin preferences. These imply that an individual i who has a lower perception of correlation than an individual j; will choose to invest according to a higher belief about state 1 and hence will invest more in risky asset. Proof of Lemma 2: The proof follows from the construction in Theorem 1. Fix n and q, as ai goes to zero, the set C(ai ; q) converges to span all possible beliefs. Therefore there is a > 0 such that if ai < ; i will have a minimum belief that is lower than his q i and hence will experience a cautious shift. Proof of Lemma 3: (i)+(ii): To convict we need ai ai j qnj j 1 j qn + ai j (1 (ai )2 i (a ) qnj ) i > i 1 j i 21 i qnj ) (1 > i for all i , > ( 1 qnj q^n n ) . for all i , q^n i (iii) As n increase we have that (ai )2 1 i is …xed but ( 1 q^nq^n )n converges either to 0 or to in…nity, implying that only convictions arise in the former case or only acquittals arise in the latter case. Proof of Proposition 3: (i) We …rst characterize the equilibria, show that the maximum bid decreases in a; and that the minimum bid decreases in a; for a close to 1. Consider a low type. For any ba (l), this type’s expected utility is perceived as min ( L; H = min L; H L + L (v H )( L L+ ba (l)) + v+ H H (v H L+ ba (l)) v ba (l)) H which (assuming ba (l) > v) is resolved by setting L to be the highest possible value L and H to be the lowest possible value, given the ePMI constraints, which are: a q2 q 1 1 1 1 1 1 L L H ; a q(1 ; a 1 (12q+ ; a (1 Hq)2 ; a 1q(1q q)H and a 2q 1+ : a q) a q)2 a a a q2 a 35 Note that these can be re-written as: 1 2 1 aq 2 q ;q q(1 q) q aq(1 q); a(1 q)2 + 2q 1 L L L a a 1 2 (1 q) + 2q 1 and a 1 a(1 q)2 (1 q)2 ; 1 q a1 q(1 q) 1 q aq(1 q), aq 2 + 1 2q H H a 1 2 q + 1 2q: H a Note that for a < 1; a(1 q)2 > aq 2 + 1 2q and for a close to one, a(1 q)2 > 1 2 1 q a1 q(1 q). Moreover, for a < 1; a1 (1 q)2 q + 1 2q and for a close to one a 1 1 2 (1 q) 1 q aq(1 q): In addition for a < 1; a (1 q)2 + 2q 1 a1 q 2 and for a a close to one, a1 (1 q)2 + 2q 1 q aq(1 q): Finally, for a < 1 a(1 q)2 + 2q 1 > aq 2 and for a close to one, a(1 q)2 + 2q 1 > q a1 q(1 q): Therefore, for a close to one, the ePMI constraints for H become: q)2 a(1 and the ePMI constraints for a(1 L H 1 (1 a q)2 L 1 (1 a q)2 + 2q are: q)2 + 2q 1 1 Therefore for a su¢ ciently close to 1; the solution is lH = a(1 q)2 ; lL = a1 (1 q)2 + 2q 1: Denote this by l : Note that the low type cannot have any rent as in the standard model, and thus we set ba (l) = E l (vjl; l) = a(1 a(1 q)2 q)2 + a1 (1 q)2 + 2q 1 < (1 (1 q)2 q)2 + q 2 Taking a derivative of E l (vjl; l) with respect to a; it is straightforward to see that it is positive. Thus the bid of the low type increases with a: We will establish later that this type will not want to use any other bid given the behaviour of the high type. Now let us consider the high type. Wlog we can consider a mixed strategy with support on [ba (l); ba (h)]; as bidding less than ba (l) will provide a zero utility. First let us consider a bid just above ba (l) which allows the individual to win against the low type only. We then need to solve the following, for some information structure = f L ; H g: min Pr(ljh)(E (vjh; l) L; H = min (q L; H L )(v ba (l)) ba (l)) + (1 q H )(v ba (l)); which, as ba (l) > v; yields the need to maximize H and to minimize L : The solution is hH = a1 (1 q)2 ; hL = a(1 q)2 + 2q 1 for a close to one. We denote this information structure by h : Note that this bid provides a utility of Pr h (ljh)(E h (vjh; l) E l (vjl; l)); and that lL + lH = hL + hH : 36 We now consider the highest bid in the support, ba (h): Such bid implies winning for sure and thus unambiguous gain of E(vjh): To be indi¤erent, this bid has to satisfy E(vjh) ba (h) = Pr(ljh)(E h (vjh; l) E l (vjl; l)) h Thus: ba (h) = Pr(hjh)E h (vjh; h) + Pr(ljh)E l (vjl; l) h h = 2q 1+ H + (1 = 2q 1+ H + = 2q 1 1 + (1 a = 2q 1 + (1 l L l L + q)2 + q)2 ( 1 a l H H) L l L l H + l H l H q)2 1 + a(1 q)2 a ) 1)(1 q)2 + q 2 a(1 2 q) + 2q 1 (1 a a+ ( a1 + a a(1 q)2 2 1) a Note that the derivative of a1 a + ( 1 +a 1)(1 ; evaluated at a = 1; is 2q(2q 2 2q+1 < 0: q)2 +q 2 a Thus the maximum bid decreases in a. We now continue to characterize the equilibrium distribution. Let us consider the worst case scenario in terms of utility for some distribution F (b) with density f (b): The expected utility is Z f (b)[Pr(ljh)(E (vjh; l) b) + Pr(hjh)F (b)(E (vjh; h) b)]db Zb f (b)[E (vjh) b (1 F (b)) Pr(hjh)(E (vjh; h) b)]db = b To choose the information structure to minimize utility, we maximise Pr(hjh)(E (vjh; h) = (2q 1)(v v) + ( b) L (v b) + H (v b)) and the solution is therefore, for all b in [v; v]; to maximize H and to minimize which is de…ned as h : F (b) is simply charaterized by using the indi¤erence condition and so: Pr(ljh)(E h (vjh; l) h = Pr(ljh)(E h (vjh; l) h b) + Pr(hjh)F (b)(E h (vjh; h) h ba (l)) 37 b) L; implying that Fa (b) = Pr h (ljh)(b ba (l)) : Pr h (hjh)(E h (vjh; h) b) We complete the equilibrium characterization by showing that given the strategy of the high type, the low type will not deviate. For the low type, bidding any b above ba (l); we choose the belief to minimize expected utility: min Pr(ljl)(E (vjl; l) b) + Pr(hjl)Fa (b)(E (vjl; h) H; L = min (v L; H b)( L (1 Fa (b)) + Fa (b)q) + (v b)( H (1 b) Fa (b)) + Fa (b)(1 As Fa (b) 1; we have again l : Pr h (ljh)(b This gives us a utility of Pr l (ljl)(E l (vjl; l) b)+Pr l (hjl) Pr h (hjh)(E b) = Pr h (ljh) (E l (vjl;h) Pr l (hjl)(b E l (vjl; l))( Pr h (hjh) (E h (vjh;h) 2q 1+ a1 (1 q)2 h+ h L H l + l L H l + l ) L H Pr l (ljl) b) Pr l (hjl) Pr l (ljl) h L+ h H > 1 2 h (vjh;h) b) (E l (vjl; h) ): Note that E l (vjl; h) = = E h (vjh; h); for a su¢ ciently close to 1, and that = Pr l (hjl) ; as lL + lH = the low type does not deviate. 1 ( b) E l (vjl;l)) Pr h (ljh) Pr h (hjh) = q)) 1 1 q a(1 q)2 l l 1 L H h h L H h+ h L H < for all a: Thus the utility is negative and We now proceed to prove (ii) and (iii). Expected payment to seller, , is given by the linear combination of receiving the bid of the low type (when both are l), the expected bid of the high type (when only one is h), and the maximum bid of the two h types: = Pr(l; l)E l [vjl; l] + 2 Pr(l; h)Ea [bi (h)] + Pr(h; h)Ea [max ba (h)] For expositional purposes we write this as 1 = y + (1 2 ) Z x+(1 )y bf (b)db + y 2 Z x+(1 )y b2f (b)F (b)db; y where: = Pr(ljl), according to the true (independent) information structure, = Pr h (ljl) according to the belief of the high bidder, h ; x = E h [vjh; h]; y = E l [vjl; l] = ba (l): We therefore also have ba (h) = x + (1 )y; Fa (b) = 1 xb yb and fa (b) = 1 (x1 b)2 (x y). We start by some preliminary results: Fact 1 @x = @a @y <0 @a 38 < Proof of Fact 1: Note that x = E h (vjh; h) = @x @a @y @a = = 2q 1+ a1 (1 q)2 q 2 (1 a)(1 q)2 + a1 (1 q)2 2q 1+ a1 (1 q)2 @ ( )= 2 @a q (1 a)(1 q)2 + a1 (1 q)2 2 a(1 q) @ = a (q @a 2q 1+ a1 (1 q)2 +a(1 q)2 1)2 Fact 2 = (1 (a2 q 2 2aq 4q a+2q 2 +2 (a2 q 2 2a2 q+a2 +2aq a+q 2 2q+1)2 2aq 4q a+2q 2 +2 : 2a2 q+a2 +2aq a+q 2 2q+1)2 <0 @ =0 @a ja=1 Proof of Fact 2: = Pr h (ljl) = hL + @ @a ja=1 1)2 a (q q) 1 2 @(a+ a ) @a h H = a1 (1 ja=1 q)2 + a(1 1 ) a2 ja=1 2 = (1 q)2 + 2q q) (1 1 = 0: Fact 3 The bid of the low type is increasing in a: Proof of Fact 3: Follows from Fact 1. Fact 4 (i) E[ba (h)] = x(1 + 1 ln(1 )) y 1 ln(1 ): (ii) At a = 1; the expected bid of the high type increases in a for low q and decreases in a for high q: Proof of Fact 4: R b) (i) Note that (x bb)2 db = x 1 b (x b ln (x b) + x ln (x b)) = x (b xx) ln(x ; therefore, b R x+(1 )y b x + ln(1 ): db = (1 )(x y) y (b x)2 Hence R x+(1 )y b E[ba (h)] = 1 (x y) y db = x(1 + 1 ln(1 )) y 1 ln(1 ): (b x)2 @y @ = 0 and @x = @a ; @E[[email protected] (h)] ja=1 = @x (1 + 2 1 @a ja=1 @a ja=1 @a ja=1 ja=1 2(1 q)2 2q+1 + 2 1 2(1 ln(1 2(1 q)2 2q + 1)): q)2 +2q 1 2(1 q)2 2q+1 > 12 ; the expression (1 + 2 1 2(1 ln(1 2(1 q)2 2q q)2 +2q 1 (ii) As @x (1 @a ja=1 ln(1 ))ja=1 = For q + 1)) is strictly increasing, negative for q < q and positive for q > q for some q 2 (0:5; 1): As @x < 0; @a ja=1 we are done. Fact 5 (i) E[max bia (h)] = (x i=1;2 y) 2(( 1 )2 ln (1 ) 1 )+x (ii) The expectation of the maximal bid when both are high types increases in a for low q and decreases in a for high q: Proof of Fact 5: 39 (i) R b(b y) db (x b)3 1 (2b2 ln (x b) + 2x2 ln (x b) 4bx + 2(x b)2 R x+(1 )y b(b y) 4bx+2by xy+3x2 : Therefore, db = 2 y 2(x b) (x b)3 = ln (x b) Hence E[maxi=1;2 bia (h)] = 2( 1 )2 (x (2x 2y 3x +2y ) ) 2(1 )2 (x y) y) 1 R x+(1 y 2 )y b(b y) db (x b)3 1 = xy + 3x2 2by ln (1 2( 1 ) )2 (x 4bx ln (x (2x 2y 3x +2y ) : 2(1 )2 (x y) y) (ln (1 )+ = (x y) 2(( ) ln (1 )+ ) + x: @y (ii) Recalling that @x = @a and that @@a = 0 we have, @a i @E[maxi=1;2 ba (h)] = @x ( 4( 1 )2 ln(1 ) 4 1 + 1); and @a @a i @E[maxi=1;2 ba (h)] q) 2 q) = @x ( 4( q22q(1 ) ln(1 ) 4 q22q(1 + 1) @a @a +(1 q)2 +(1 q)2 ja=1 q) 2 q) For the expression ( 4( q22q(1 ) ln(1 2q(1 q)) 4 q22q(1 +1) there is a q 2 (0:5; 1) +(1 q)2 +(1 q)2 such that the expression is negative for q < q and positive for q > q: As @x < 0; we are @a done. Given the above facts we can write the pro…t function as: = 1 y + (1 2 1 + )x + 1 (x y) (1 2 = x( + 2 (x ( y) ln(1 1 ln (1 1 ) ) ln(1 ( 2x ) 2y ) + 1) + y(1 2 3x + 2y ) 2 1 ( ln (1 ) + 1)) Taking a derivative with respect to a; recalling that (d =da)ja=1 = 0 and that we get, @ @x = ( @a ja=1 @a @x ( = @a 1+2 ( 1 ln (1 @x @a = @y @a ) + 1)) 1) > 0: Common value auction with exogenous ambiguity We now assume that players have ambiguity over the prior; they believe the probability the state is v is in [ 12 "; 21 + "]: The information structure is common knowledge and is !=L sl sh sl q2 q(1 q) sh q(1 q) 1 2q + q 2 !=H sl sh sl (1 q)2 q(1 q) sh q(1 q) 2q 1 + (1 q)2 Proposition A2: Compared to the standard model with no ambiguity, the suppurt of bids in the auction with ambiguity over the prior is strictly lower. In particular, the bid 40 b)) = of the lower type is lower and both the maximum and minimum bids of the high type are lower. Proof of Proposition A2: As above let us conjecture the minimum bid b" (l) for the low type and a distribution for h on [b" (l); b" (h)]: Let us …rst compute the bid of the low type. We aim to minimize for any b the utility from winning upon the other receiving the low signal, in other words: min Pr(ljl)(E" (vjl; l) min1 (1 b" (l)) p2[ 12 "; 12 +"] " = p2[ 21 p)q 2 ( b" (l)) + p(1 "; 2 +"] Assuming b" (l) > v = 0 we then have p = would be 0 in equilibrium, we then set b" (l) = E" (vjl; l) = 1 2 q)2 (1 b" (l)) ": Given that the rent for the low type ( 12 ")(1 q)2 ( 12 + ")q 2 + ( 21 ")(1 q)2 ; and thus the bid is lower than in the case in which there is a unique prior 21 : Consider now the strategy of the high type. If he uses some F (b); we have: Z = Zb f (b)[Pr(ljh)(E" (vjh; l) b) + Pr(hjh)F (b)(E" (vjh; h) " f (b)[E" (vjh) b (1 b)]db " F (b)) Pr(hjh)(E" (vjh; h) b)]db " b To choose the information structure to minimize utility, we min E" (vjh) p (1 F (b)) Pr(hjh)(E" (vjh; h) b) " The above equals pq(1 (1 F (b))q) min + (1 p pq + (1 p)(1 q) pq 2 + (1 F (b)) pq + (1 p)(1 p)(1 q)2 b q) For any F (b) 1 of the other bidder, this decreases in p and hence we choose the lowest p = 21 ": Let us look at b" (h) = Pr" (ljh)E" (vjl; l) + Pr" (hjh)E" (vjh; h): Note that E" (vjh; h) < 2 +(1 p)(1 q)2 E(vjh; h); E" (vjl; l) < E(vjl; l); and that Pr" (hjh) < Pr(hjh) = pqpq+(1 : We therep)(1 q) Pr" (ljh) b E" (vjl;l) : fore have a uniformly lower value. It is straightforward to show that F" (b) = Pr " (hjh) E" (vjh;h) b To complete the equilibrium characterization note that the low type will not deviate for a bid b if min(Pr(ljl)(E" (vjl; l) " " b) + Pr(hjl)F" (b)(E" (vjl; h) " 41 b)) < 0 Note that Pr" (ljl)(E" (vjl; l) b)+Pr" (hjl)F" (b)(E" (vjl; h) b)) = q) F (b)(p b)) increases in p for all F" (b) b) + p(1 q(1 q)+(1 p)q " 1 lowest p = 2 ": Given the same belief p = 21 "; we have that Pr(ljl)(E" (vjl; l) " = (b as E" (vjl;h) b E" (vjh;h) b b) Pr" (ljh) b E" (vjl; l) (E" (vjl; h) " Pr" (hjh) E" (vjh; h) b Pr" (ljh) E" (vjl; h) b )<0 E" (vjl; l)( Pr(ljl) + Pr(hjl) " " Pr" (hjh) E" (vjh; h) b = Pr(ljl)(E" (vjl; l) " 1 and thus again we choose the b) + Pr(hjl)F" (b)(E" (vjl; h) " p(1 q)2 +(1 p)q 2 p(1 q)2 ( p(1 q)+(1 p)q p(1 q)2 +(1 p)q 2 b) + Pr(hjl) Pr" (ljl) Pr" (ljh) > Pr by the Pr" (hjl) " (hjh) Pr" (ljh) b E" (vjl;l) F (b) which Pr" (hjh) E" (vjh;h) b < 1 and b) MLRP which is satis…ed here. Finally note is computed in the canonical case with no that F" (b) = ambiguity, so F (b) …rst order stochastically dominates F" (b) for the same support: This implies that the seller’s revnues would be lower. 42

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# Perceptions of Correlation, Ambiguity and Communication