Bayesian Heuristics for Group Decisions

Ba y esian Heuristics for Group Decisions M. Amin Rahimian & Ali Jadbabaie ? ? Institute for Data, Systems, and Society (IDSS), Massach usetts Institute of T ec hnology (MIT), Cambridge, MA 02139, USA. Emails: rahimian , jadbabai @mit.edu W e prop ose a mo del of inference and heuristic decision making in groups that is ro oted in the Bay es rule but av oids the complexities of rational inference in partially observed environmen ts with incomplete information. According to our mo del, the group members b ehav e rationally at the initiation of their interac- tions with eac h other; how ever, in the ensuing decision ep ochs, they rely on a heuristic that replicates their exp eriences from the ﬁrst stage. Subsequently , the agents use their time-one Bay esian up date and repeat it for all future time-steps; hence, up dating their actions using a so-called Bayesian heuristic . This model a voids the complexities of fully rational inference and also pro vides a b eha vioral and normative foundation for non-Ba y esian updating. It is also consistent with a dual-pro cess psyc hological theory of decision mak- ing, where a controlled (conscious/slo w) system dev elops the Ba y esian heuristic at the b eginning, and an automatic (unconscious/fast) system tak es ov er the task of heuristic decision making in the sequel. W e sp ecialize this mo del to a group decision scenario where priv ate observ ations are received at the b eginning, and agen ts aim to take the best action giv en the aggregate observ ations of all group mem b ers. W e presen t the implications of the c hoices of signal structure and action space for suc h agents. W e show that for a wide class of distributions from the exp onential family the Bay esian heuristics take the form of an aﬃne up date in the self and neigh b oring actions. F urthermore, if the priors are non-informative (and p ossibly improper), then these action up dates become a linear combination. W e inv estigate the requirements on the mo deling parameters for the action up dates to constitute a conv ex com bination as in the DeGroot mo del. The results rev eal the nature of assumptions that are implicit in the DeGroot updating and highligh ts the fragility and restrictions of suc h assumptions; in particular, w e show that for a linear action up date to constitute a conv ex combination the precision or accuracy of priv ate observ ations should b e balanced among all neigh b oring agen ts, requiring a notion of social harmony or homogeneit y in their observ ational abilities. F ollowing the DeGro ot model, agents reach a consensus asymptotically . W e deriv e the requirements on the signal structure and net work topology such that the consensus action aggregates information eﬃcien tly . This in volv es additional restrictions on the signal likelihoo ds and netw ork structure. In the particular case that all agents observ e the same n umber of i.i.d. samples from the same distribution, then eﬃciency arise in degree-regular balanced structures, where all no des listen to and hear from the same n umber of neigh b ors. W e next shift attention to a ﬁnite state model, in whic h agents take actions ov er the probabilit y simplex; th us revealing their beliefs to each other. W e show that the Bay esian heuristics, in this case, prescrib e a log-linear update rule, where each agen t’s belief is set prop ortionally to the pro duct of her o wn and neigh- b oring beliefs. W e analyze the evolution of beliefs under this rule and sho w that agents reac h a consensus. The consensus b elief is supported ov er the maximizers of a w eighted sum of the log -lik eliho ods of the initial observ ations. Since the w eights of the signal lik eliho o ds coincide with the net work cen tralities of their respec- tiv e agen ts, these w eigh ts can b e equalized in degree-regular and balanced top ologies, where all no des ha ve the same in and out degrees. Therefore, in suc h highly symmetric structures the supp ort of the consensus b elief coincides with the maximum likelihoo d estimators (MLE) of the truth state; and here again, balanced regular structures demonstrate a measure of eﬃciency . Nev ertheless, the asymptotic beliefs systematically reject the less probable alternatives in spite of the limited initial data, and in contrast with the optimal (Ba yesian) belief of an observ er with complete information of the en vironment and priv ate signals. The latter would assign probabilities proportionally to the lik eliho o d of every state, without rejecting any of the p ossible alternatives. The asymptotic rejection of less probable alternatives indicates a case of group p olar- ization, i.e. o verconﬁdence in the group aggregate that emerges as a result of the group interactions. Unlike the linear action updates and the DeGro ot mo del whic h entail a host of knife-edge conditions on the signal structure and model parameters, we observ e that the b elief updates are unw eighted; not only they eﬀectiv ely in ternalize the heterogeneity of the priv ate observ ations, but also they compensate for the individual priors. Thence, we are led to the conclusion that m ultiplicative b elief up dates, when applicable provide a relativ ely robust description of the decision making behavior. Key wor ds : opinion dynamics; so cial learning; Ba yesian learning; non-Ba yesian learning; rational learning; observ ational learning; statistical learning; distributed learning; distributed hypothesis testing; distributed detection; DeGro ot model; linear regression; conjugate priors; exp onen tial families MSC2000 subje ct classiﬁc ation : Primary: 91B06; secondary: 91A35, 62C10 OR/MS subje ct classiﬁc ation : Primary: Games/group decisions: V oting/committees; secondary: Organizational studies: Decision making Eﬀectiv eness/p erformance Information; Netw orks/graphs: Heuristics History : This v ersion is dated September 12th, 2016. A preliminary v ersion w as presented in NBER-NSF Seminar on Bay esian Inference in Econometrics and Statistics on April 30, 2016. 1 Rahimian and Jadbabaie: Bayesian Heuristics for Group De cisions 2 1. In tro duction. Daniel Kahneman in his highly acclaimed w ork, “ Thinking, F ast and Slow ”, p oin ts out that the prop er wa y to elicit information from a group is not through a public discus- sion but rather conﬁden tially collecting each person’s judgment [ 52 , Chapter 23]. Indeed, decision making among groups of individuals exhibit many singularities and imp ortan t ineﬃciencies that lead to Kahneman’s noted advice. As a team con v erges on a decision expressing doubts ab out the wisdom of the consensus choice is suppressed; subsequently teams of decision mak ers are aﬄicted with gr oupthink as they app ear to reach a consensus. 1 The mechanisms of uncritical optimism, o v erconﬁdence, and the illusions of v alidity in group in teractions also lead to gr oup p olarization , making the individuals more amenable to w ard extreme opinions [ 92 ]. In more abstract terms, agen ts in a so cial net work exc hange opinions to beneﬁt from eac h other’s exp erience and information when making decisions ab out adoption of technologies, purc hasing pro ducts, v oting in elections and so on. The problem of so cial learning is to characterize and understand such interactions and it is a classical fo cus of researc h in micro economics [ 45 , Chapter 8], [ 39 , Chapter 5], [ 18 ]. Research on formation and ev olution of beliefs in so cial netw orks and subsequen t shaping of the individual and group b eha viors ha ve attracted m uch attention amongst div erse comm unities in engineering [ 58 , 97 ], statistics [ 66 ], economics [ 47 , 65 ], and so ciology [ 82 ]. An enhanced understanding of decision making and learning in so cial netw orks sheds light on the role of individuals in shaping public opinion and ho w they inﬂuence eﬃciency of information transmissions. These in turn help us improv e our predictions ab out group b eha vior and provide guidelines for designing eﬀectiv e so cial and organizational policies. 1.1. Rational social learning. The rational approac h adv o cates application of Ba y es rule to the en tire sequence of observ ations successively at every step. How ever, such rep eated applications of Ba y es rule in netw orks b ecome v ery complex, esp ecially if the agents are una ware of the global net w ork structure; and as they use their lo cal data to mak e inferences ab out all p ossible contin- gencies that can lead to their observ ations. While some analytical prop erties of rational learning is deduced and studies in the literature [ 32 , 1 , 68 , 66 , 42 ], their tractable mo deling and analysis remains an imp ortan t problem in netw ork economics and contin ues to attract attention. Some of the earliest results addressing the problem of social learning are due to Banerjee (1992) [ 9 ], and Bikhchandani et al. (1998) [ 13 ] who consider a complete graph structure where the agent’s observ ations are public information and also ordered in time, suc h that each agen t has access to the observ ations of all the past agen ts. These assumptions help analyze and explain the in terplay b et ween public and priv ate information leading to fashion, fads, herds etc. Later results by Gale and Kariv [ 32 ] relax some of these assumptions by considering the agents that make sim ultane- ous observ ations of only their neighbors rather than the whole netw ork, but the computational complexities limit the analysis to netw orks with only t wo or three agen ts. In more recen t results, Mueller-F rank [ 68 ] provides a framew ork of rational learning that is analytically amenable. Mossel, Sly and T amuz [ 66 ] analyze the problem of estimating a binary state of the world from a single initial priv ate signal that is independent and iden tically distributed amongst the agen ts conditioned on the true state; and they show that b y rep eatedly observing eac h other’s b est estimates of the unkno wn as the size of the netw ork increases with high probability Ba y esian agents asymptotically learn the true state. Hence, the agen ts are able to combine their initial priv ate observ ations and learn the truth. F urther results b y Kanoria and T amuz [ 53 ] pro vide eﬃcient algorithms to calculate eac h agen t’s estimate in the case of tree net works. On the one hand, the prop erties of rational learning mo dels are diﬃcult to analyze b ey ond some simple asymptotic facts such as con v ergence. On the other hand, these mo dels make unrealistic 1 Gar Klein proposes a famous metho d of pr oj e ct pr emortem to o vercome the groupthink through an exercise: imagining that the the planned decision was failed in implementation and writing a brief rep ort of the failure [ 57 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group De cisions 3 assumptions ab out the complexit y and amoun t of computations that agents p erform b efore com- mitting to a decision. T o a v oid these shortcomings, an alternativ e “ non-Bayesian ” approach relies on simple and in tuitiv e heuristics that are descriptiv e of ho w agen ts aggregate the reports of their neigh b ors before coming up with a decision. 1.2. Heuristic decision making. Heuristics are used widely in the literature to mo del social in teractions and decision making [ 34 , 35 , 36 ]. They provide tractable to ols to analyze b oundedly rational b ehavior and oﬀer insights ab out decision making under uncertaint y . Hegselmann and Krause [ 43 ] inv estigate v arious w ays of av eraging to mo del opinion dynamics and compare their p erformance for computations and analysis. Using suc h heuristics one can a void the complexities of fully rational inference, and their suitabilit y are also v eriﬁed in experimental studies b y Grimm and Mengel [ 40 ] and Chandrasekhar, Larreguy and Xandri [ 19 ]. The study of such heuristics started in 1974 with the seminal work of DeGro ot [ 20 ] in linear opinion p o oling, where agen ts up date their opinions to a conv ex combination of their neigh b ors’ beliefs and the co eﬃcien ts corresp ond to the lev el of conﬁdence that each agent puts in each of her neighbors. More rece n tly , Jadbabaie, Molavi and T ahbaz-Salehi [ 47 , 48 , 65 ] consider a v ariation of this mo del for streaming observ ations, where in addition to the neighboring b eliefs the agents also receive priv ate signals. Despite their widespread applications, theoretical and axiomatic foundations of so cial inferences using heuristics and non- Ba y esian up dates ha v e receiv ed limited attention and only recently [ 65 , 69 ], and a comprehensiv e theory of non-Bay esian learning that reconciles the rational and b oundedly rational approac hes with the widely used heuristics remains in demand. The main goal of our pap er is to address this gap and to do so from a b eha vioral p ersp ectiv e. A dual pro cess theory for the psyc hology of mind and its op eration identiﬁes tw o systems of thinking [ 24 ]: one that is fast, in tuitiv e, non-delib erativ e, habitual and automatic (system one); and a second one that is slow, atten tive, eﬀortful, delib erativ e, and conscious (system tw o). 1 Ma jor adv ances in b eha vioral economics are due to incorp oration of this dual pro cess theory and the subsequen t mo dels of b ounded rationality [ 51 ]. Reliance on heuristics for decision making is a distinctiv e feature of system one that av oids the computational burdens of a rational ev aluation; system t w o on the other hand, is b ound to deliberate on the options based on the av ailable infor- mation b efore making recommendations. The interpla y b et ween these tw o systems and ho w they shap e the individual decisions is of paramoun t imp ortance [ 17 ]. Tv ersky and Kahneman argue that h umans hav e limited time and brainp ow er, therefore they rely on simple rules of thum b, i.e. heuristics, to help them make judgmen ts. How ever, the use of these heuristics causes p eople to mak e predictable errors and sub jects them to v arious biases [ 96 ]. Hence, it is imp ortant to understand the nature and prop erties of heuristic decision making and its consequences to individual and organizational c hoice behavior. This premise underlies many of the recen t adv ances in b eha vioral economics [ 93 ], and it motiv ates our work as well. 1.3. Our con tribution. In this work w e are concerned with the op erations of system one: we aim to study heuristics for information aggregation in group decision scenarios when the relev an t information is disp ersed among many individuals. In such situations, individuals in the group are sub jected to informational (but not strategic) externalities. By the same tok en, the heuristics that are developed for decision making in such situations are also aimed at information aggregation. In our mo del as the agent exp eriences with her en vironment her initial resp onse would engage 1 While man y decision science applications fo cus on developing dual process theories of cognition and decision mak- ing (cf. [ 25 , 63 ] and the references therein); other researchers identify m ultiple neural systems that derive decision making and action selection: ranging from reﬂexive and fast (Pa vlovian) resp onses to deliberative and pro cedural (learned) ones; and these systems are in turn supported b y sev eral motoric, p erceptual, situation-categorization and motiv ational routines which together comprise the decision making systems [ 80 , Chap yter 6]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 4 her system t wo: she rationally ev aluates the reports of her neigh b ors and use them to mak e a decision. How ever, after her initial experience and by engaging in rep eated in teractions with other group members her system one takes o v er her decision pro cesses, implemen ting a heuristic that imitates her (rational/Ba y esian) inferences from her initial exp erience; hence a v oiding the burden of additional cognitiv e pro cessing in the ensuing in teractions with her neighbors. On the one hand, our mo del of inference based on Ba yesian (initial) heuristics is motiv ated by the real-world b eha vior of p eople induced by their system one and reﬂected in their spur-of-the- momen t decisions and impromptu b ehavior: basing decisions only on the immediately observed actions and disregarding the history of the observ ed actions or the p ossibilit y of correlations among diﬀeren t observ ations; i.e. “ what you se e is al l ther e is ” [ 52 ]. On the other hand, Bay esian heuristics oﬀer a b oundedly rational approac h to mo del decision making o ver so cial netw orks. The latter is in the sense of the word as coined by Herb ert A. Simon: to incorporate mo diﬁcations that lead to substan tial simpliﬁcations in the original choice problem [ 87 ]. 1 This in con trast with the Ba yesian approac h which is not only unrealistic in the amount of cognitive burden that it imp oses on the agen ts, but also is often computationally in tractable and complex to analyze. Our main contribution is to oﬀer a normative framework for heuristic decision making, b y relying on the time-one Ba y esian up date and using it for all future decision ep o c hs. This mo del oﬀers a b eha vioral foundation for non-Ba yesian up dating that is compatible with the dual-pro cess psycho- logical theory of decision making. In Section 2 , we describ e the mathematical details of our mo del; in particular, we explain the mathematical steps for deriving the so-called Bay esian heuristics in a given decision scenario. Sp eciﬁc cases of Bay esian heuristics that w e explore in the following sections are the log-linear (multiplicativ e) up dating of b eliefs ov er the probability simplex, and the linear (weigh ted arithmetic av erage) up dating of actions ov er the Euclidean space. In Section 3 , we sp ecialize our group decision mo del to a setting in v olving exp onen tial family of distributions for b oth signal likelihoo ds and agents’ beliefs. The agents aim to estimate the expected v alues of the suﬃcien t statistics for their signal structures. W e show that the Bay esian heuristics in this case are aﬃne rules in the self and neigh b oring actions, and w e giv e explicit expressions for their co eﬃ- cien ts. Subsequen tly , we pro vide conditions under whic h these action up dates constitute a conv ex com bination as in the DeGroot model, with actions conv erging to a consensus in the latter case. W e also inv estigate the eﬃciency of the consensus action in aggregating the initial observ ations of all agen ts across the net work. Next in Section 4 , we discuss a situation where agen ts exc hange beliefs ab out a truth state that can takes one of the ﬁnitely many p ossibilities. The Bay esian heuristics in this case set the up dated b eliefs prop ortional to the pro duct of self and neighboring b eliefs, as rep orted in every decision ep o ch. W e inv estigate the evolution of b eliefs under the prescrib ed up date rules and compare the asymptotic b eliefs with that of a Bay esian agent with direct access to all the initial observ ations; thus characterizing the ineﬃciencies of the asymptotic b eliefs. W e summarize our ﬁndings from the analysis of linear action and log-linear b elief up dates in Section 5 , where we reiterate the assumptions that are implicit in the adoption of p opular aggregation heuristics such as the DeGroot mo del; moreov er, w e discuss the ineﬃciencies that arise as a result of their application. Such heuristics allow us to aggregate the information in our environmen t, and pro vide for desirable asymptotic properties such as consensus; ho wev er, this consensus often fails as an eﬃcient group aggregate for the individuals’ priv ate data. W e pro vide the mathematical pro ofs and the relev an t details for many of the results in the appendices at end of the paper. 1 Simon advocates “b ounded rationality” as compatible with the information access and the computational capacities that are actually p ossessed b y the agents in their environmen ts. Most importantly he proposes the use of so-called “ satisﬁcing ” heuristrics; i.e. to search for alternatives that exceed some “ aspir ation levels ” by satisfying a set of minimal acceptabilit y criteria [ 88 , 89 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 5 2. The mo del. Consider a group of n agents that are lab eled b y [ n ] and in teract according to a digraph G = ([ n ] , E ). 1 The neighborho o d of agent i is the set of all agen ts whom she observes including herself, and it is denoted b y N i = { j ∈ [ n ]; ( j, i ) ∈ E } ∪ { i } ; every no de ha ving a self-loop: ( i, i ) ∈ E for all i . W e refer to the cardinality of N i as the degree of no de i and denote it by deg ( i ). There is a state θ ∈ Θ that is unknown to the agents and it is chosen arbitrarily b y nature from an underlying state space Θ, which is measurable b y a σ -ﬁnite measure G θ ( · ). F or example if a space (Θ or S ) is a coun table set, then we can tak e its σ -ﬁnite measure ( G θ or G s ) to b e the counting measure, denoted b y K ( · ); and if the space is a subset of R k with p ositive Leb esgue measure, then w e can tak e its σ -ﬁnite measure to b e the Leb esgue measure on R k , denoted b y Λ k ( · ). Asso ciated with each agen t i , S i is a measurable space called the signal space of i , and given θ , L i ( · | θ ) is a probabilit y measure on S i , which is referred to as the signal structur e of agen t i . F urthermore, (Ω , F , P θ ) is a probability triplet, where Ω = S 1 × . . . × S n is a pro duct space, and F is a prop erly deﬁned sigma ﬁeld ov er Ω. The probability measure on Ω is P θ ( · ) which assigns probabilities consisten tly with the signal structures L i ( · | θ ) , i ∈ [ n ]; and in suc h a w ay that with θ ﬁxed, the random v ariables s i , i ∈ [ n ] taking v alues in S i , are indep endent. These random v ariables represen t the priv ate signals that agen ts i ∈ [ n ] observ e at time 0. Note that the priv ate signals are indep enden t across the agents. The exp ectation op erator E θ {·} represents in tegration with resp ect to P θ ( dω ), ω ∈ Ω. 2.1. Beliefs, actions and rew ards. An agents’ b elief ab out the unknown allo ws her to mak e decisions even as the outcome is dep endent on the unknown v alue θ . These b eliefs ab out the unkno wn state are probability distributions o ver Θ. Ev en b efore any observ ations are made, ev ery agen t i ∈ [ n ] holds a prior b elief V i ( · ) ∈ ∆Θ; this represents her sub jective biases ab out the p ossible v alues of θ . F or eac h time instan t t , let M i,t ( · ) b e the (random) probabilit y distribution o v er Θ, represen ting the opinion or b elief at time t of agen t i ab out the realized v alue of θ . Moreov er, let the asso ciated exp ectation op erator be E i,t {·} , representing in tegration with resp ect to M i,t ( dθ ). W e assume that all agents share the common kno wledge of signal structures L i ( ·| ˆ θ ) , ∀ ˆ θ ∈ Θ, their priors V i ( · ), and their corresp onding sample spaces S i and Θ for all i ∈ [ n ]. 2 Let t ∈ N 0 denote the time index; at t = 0 the v alues θ ∈ Θ follo w ed by s i ∈ S i of s i are realized and the latter is observ ed priv ately b y eac h agen t i for all i ∈ [ n ]. Asso ciated with ev ery agent i is an action space A i that represents all the c hoices av ailable to her at every point of time t ∈ N 0 , and a utilit y u i ( · , · ) : A i × Θ → R which in exp ectation represen ts her v on Neumann-Morgenstern pref- erences regarding lotteries with indep endent draws from A i and/or Θ. These utilities are additive o v er time corresp onding to successiv e indep endent dra ws. The utility functions and action spaces are common knowledge amongst the agents. Subsequen tly , at every time t ∈ N 0 eac h agent i ∈ [ n ] c ho oses an action a i,t ∈ A i and is rew arded u i ( a i,t , θ ). 1 Some notations: Throughout the pap er, R is the set of real num b ers, N denotes the set of all natural num bers, and N 0 := N ∪ { 0 } . F or n ∈ N a ﬁxed integer the set of integers { 1 , 2 , . . . , n } is denoted by [ n ], while an y other set is represented by a capital Greek or calligraphic letter. F or a measurable set X we use ∆ X to denote the set of all probabilit y distributions ov er the set X . F urthermore, any random v ariable is denoted in b oldface letter, vectors are represen ted in lo wercase letters and with a bar o ver them, measures are denoted by upp er case Greek or calligraphic Latin letters, and matrices are denoted in upp er case Latin letters. F or a matrix A , its spectral radius ρ ( A ) is the largest magnitude of all its eigenv alues. 2 The signal structures L i ( ·| ˆ θ ) , ∀ ˆ θ ∈ Θ and the priors V i ( · ), as well as the corresp onding sample spaces S i and Θ are common knowledge amongst the agents for all i ∈ [ n ]. The assumption of common kno wledge in the case of fully rational (Bay esian) agents implies that given the same observ ations of one another’s b eliefs or priv ate signals distinct agen ts would make iden tical inferences; in the sense that starting form the same belief about the unknown θ , their up dated b eliefs given the same observ ations would b e the same; in Aumann’s words, rational agen ts cannot agree to disagree [ 5 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 6 2.2. Aggregation heuristics. Giv en s i , agent i forms an initial Bay esian opinion M i, 0 ( · ) ab out the v alue of θ and c ho oses her action a i, 0 ← - arg max a i ∈A i R Θ u i ( a i , ˆ θ ) M i, 0 ( dθ ), maximizing her exp ected reward. Here for a set A , we use the notation a ← - A to denote an arbitrary choice from the elements of A that is assigned to a . Not b eing notiﬁed of the actual realized v alue for u i ( a i, 0 , θ ), she then observ es the actions that her neigh b ors hav e taken. Giv en her extended set of observ ations { a j, 0 , j ∈ N i } at time t = 1, she reﬁnes her opinion into M i, 1 ( · ) and mak es a second, and p ossibly diﬀeren t, mov e a i, 1 according to: a i, 1 ← - arg max a i ∈A i Z Θ u i ( a i , θ ) M i, 1 ( dθ ) , (1) maximizing her expected pa y oﬀ conditional on everything that she has observ ed thus far; i.e. maximizing E i, 1 { u i ( a i , θ ) } = E θ { u i ( a i, 1 , θ ) | s i , a j, 0 : j ∈ N i } = R Θ u i ( a i , θ ) M i, 1 ( dθ ). Subsequently , she is granted her net rew ard of u i ( a i, 0 , θ ) + u i ( a i, 1 , θ ) from her past tw o pla ys. F ollo wing realization of rew ards for their ﬁrst t wo plays, in any subsequen t time instance t > 1 eac h agen t i ∈ [ n ] observ es the preceding actions of her neighbors a j,t − 1 : j ∈ N i and tak es an option a i,t out of the set A i . Of particular signiﬁcance in our description of the b ehavior of agents in the succeeding time p eriods t > 1, is the relation: f i ( a j, 0 : j ∈ N i ) := a i, 1 ← - arg max a i ∈A i E i, 1 { u i ( a i , θ ) } (2) deriv ed in ( 1 ), whic h given the observ ations of agent i at time t = 0, sp eciﬁes her (Ba y esian) pa y-oﬀ maximizing action for time t = 1. Once the format of the mapping f i ( · ) is obtained, it is then used as a heuristic for decision making in every future ep o ch. The agents up date their action b y choosing: a i,t = f i ( a j,t − 1 : j ∈ N i ) , ∀ t > 1. W e refer to the mappings f i : Q j ∈N i A j → A i th us obtained, as Bayesian heuristics . 1 , 2 , 3 1 The heuristics th us obtained suﬀer from same fallacies of snap judgmen ts that are asso ciated with the recommen- dations of system one in “Thinking, F ast and Slo w”; ﬂaw ed judgmen ts that rely on simplistic in terpretations: “ what you se e is al l ther e is ”, in Kahneman’s elegant w ords [ 52 ]. Indeed, the use of the initial Bay esian update for future decision ep o c hs en tails a certain lev el of naivet y on the part of the decision mak er: she has to either assume that the structure of her neighbors’ reports ha ve not departed from their initial format, or that they are not b eing inﬂuenced bac k b y her own or other group members and can thus b e regarded as indep endent sources of information. Such naiv ety in disregarding the history of interactions has b een highlighted in our earlier w orks on Bay esian learning without recall [ 75 ], where we in terpret the use of time-one Bay esian update for future decision epo chs, as a rational but memoryless behavior: by regarding their observ ations as b eing direct consequences of inferences that are made based on the initial priors, the agen ts reject an y possibility of a past history b ey ond their immediate observ ations. 2 Similar and related forms of naiv ety ha ve been suggested in the literature. Eyster and Rabin [ 26 , 27 ] prop ose the autarkic mo del of naive inference, where play ers at each generation observe their predecessors but naiv ely think that an y predecessor’s action relies solely on that play er’s priv ate information, thus ignoring the p ossibilit y that successive generations are learning from each other. Bala and Goy al [ 7 ]study another form of naivet y and bounded-rational b eha vior b y considering a v ariation of observ ational learning in which agents observe the action and pa y-oﬀs of their neigh b ors and make rational inferences ab out the action/pay-oﬀ corresp ondences, based on their observ ations of the neigh b oring actions; ho wev er, they ignore the fact that their neigh b ors are themselves learning and trying to maximize their o wn pay-oﬀs. Levy and Razin look at a particularly relev an t cognitive bias called correlation neglect, whic h makes individuals regard the sources of their information as independent [ 60 ]; they analyze its implications to diﬀusion of information, and fo cus in particular, on the voting b ehavior. 3 Cognitiv e and psychological roots of the Bay esian heuristics as aggregation rules can b e traced to Anderson’s seminal theory of information in tegration, dev elop ed throughout 1970s and 1980s [ 3 ]. Accordingly , a so-called “ value function ” assigns psychological v alues to eac h of the stim uli and these psychological v alues are then com bined in to a single psyc hological (and later an observ able) response through what is called the “ inte gr ation function ”. A fundamental assumption is that v aluation can b e represen ted at a higher (molar) lev el as a v alue on the response dimension for eac h stim ulus, as w ell as a weigh t representing the salience of this stimulus in the ov erall resp onse. These v aluations and w eights are themselv es the result of in tegration processes in the low er (molecular) lev el. A t the heart of information in tegration theory is the “ c o gnitive algebr a ” whic h describes the rules b y whic h the v alues and weigh ts of stimuli are in tegrated in to an o verall resp onse [ 4 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 7 3. Aﬃne action up dates, linear up dating and DeGro ot learning. In this section we explore the essen tial mo deling features that lead to a linear structure in the Ba y esian Heuristics. W e present a general scenario that inv olv es the exp onen tial family of distributions and leads to linear action up dates. T o describ e the signal structures, w e consider a measurable sample space S with a σ -ﬁnite measure G s ( · ), and a parametrized class of sampling functions {L ( ·| θ ; σ i ) ∈ ∆ S : σ i > 0 and δ i > 0 } b elonging to the k -dimensional exponential family as follo ws: ` ( s | θ ; σ i , δ i ) := d L ( ·| θ ; σ i , δ i ) d G s = σ i     Λ k ( ξ ( ds )) G s ( ds )     τ ( σ i ξ ( s ) , δ i ) e σ i η ( θ ) T ξ ( s ) − δ i γ ( η ( θ )) , (3) where ξ ( s ) : S → R k is a measurable function acting as a suﬃcien t statistic for the random samples, η : Θ → R k is a mapping from the parameter space Θ to R k , τ : R k × (0 , + ∞ ) → (0 , + ∞ ) is a positive w eigh ting function, and γ ( η ( θ )) := 1 δ i ln Z s ∈S σ i     Λ k ( ξ ( ds )) G s ( ds )     τ ( σ i ξ ( s ) , δ i ) e σ i η ( θ ) T ξ ( s ) G s ( ds ) , is a normalization factor that is constant when θ is ﬁxed, ev en though δ i > 0 and σ i > 0 v ary . This normalization constant for eac h θ is uniquely determined by the functions η ( · ), ξ ( · ) and τ ( · ). The parameter space Θ and the mapping η ( · ) are suc h that the range space Ω θ := { η ( θ ) : θ ∈ Θ } is an open subset of the natural parameter space Ω η := { η ∈ R k : R s ∈S | Λ k ( ξ ( ds )) / G s ( ds ) | τ ( ξ ( s ) , 1 ) e η T ξ ( s ) G s ( ds ) < ∞} . In ( 3 ), σ i > 0 and δ i > 0 for each i are scal- ing factors that determine the quality or informativeness of the random sample s i with regard to the unknown θ : ﬁxing either one of the tw o factors σ i or δ i , the v alue of the other one increases with the increasing informativeness of the observed v alue ξ ( s i ). The following conjugate family of priors 1 are associated with the lik eliho o d structure ( 3 ). This family is determined uniquely by the transformation and normalization functions: η ( · ) and γ ( · ), and it is parametrized through a pair of parameters ( α, β ), α ∈ R k and β > 0: F γ ,η :=  V ( θ ; α, β ) ∈ ∆Θ , α ∈ R k , β i > 0 : ν ( θ ; α, β ) := d V ( · ; α, β ) d G θ =     Λ k ( η ( dθ )) G θ ( dθ )     e η ( θ ) T α − β γ ( η ( θ )) κ ( α, β ) , κ ( α, β ) := Z θ ∈ Θ     Λ k ( η ( dθ )) G θ ( dθ )     e η ( θ ) T α − β γ ( η ( θ )) G θ ( dθ ) < ∞  . F urthermore, we assume that agen ts take actions in R k , and that they aim for a minimum v ariance estimation of the regression function or conditional exp ectation (giv en θ ) of the suﬃcien t statistic ξ ( s i ). Hence, we endow every agent i ∈ [ n ] with the quadratic utilit y u i ( a, θ ) = − ( a − m i,θ ) T ( a − m i,θ ), ∀ a ∈ A i = R k , where m i,θ := E i,θ { ξ ( s i ) } := R s ∈S ξ ( s ) L ( ds | θ ; σ i , δ i ) ∈ R k . Our main result in this section prescrib es a scenario in which each agent starts from a prior b elief V ( · ; α i , β i ) b elonging to F γ ,η and she observes a ﬁxed num b er n i of i.i.d. samples from the distribution L ( · | θ ; σ i , δ i ). The agents then rep eatedly comm unicate their actions aimed at minimum v ariance estimation of m i,θ . These settings are formalized under the following assumption that we term the Exp onential F amily Signal-Utility Structur e . 1 Consider a parameter space Θ, a sample space S , and a sampling distribution L ( ·| θ ) ∈ ∆ S , θ ∈ Θ. Supp ose that s is a random v ariable whic h is distributed according to L ( ·| θ ) for an y θ . A family F ⊂ ∆Θ is a conjugate family for L ( ·| θ ), if starting from any prior distribution V ( · ) ∈ F and for an y signal s ∈ S , the p osterior distribution given the observ ation s = s b elongs to F . Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 8 Assumption 1 (Exp onen tial family signal-utility structure) . (i) Every agent i ∈ [ n ] observes n i i.i.d. private samples s i,p , p ∈ [ n i ] fr om the c ommon sample sp ac e S and that the r andom samples ar e distribute d ac c or ding to the law L ( ·| θ ; σ i , δ i ) given by ( 3 ) as a memb er of the k -dimensional exp onential family. (ii) Every agent starts fr om a c onjugate prior V i ( · ) = V ( · ; α i , β i ) ∈ F γ ,η , for al l i ∈ [ n ] . (iii) Every agent cho oses actions a ∈ A i = R k and b e ars the quadr atic utility u i ( a, θ ) = − ( a − m i,θ ) T ( a − m i,θ ) , wher e m i,θ := E i,θ { ξ ( s i ) } := R s ∈S ξ ( s ) L ( ds | θ ; σ i , δ i ) ∈ R k . The Ba yesian he uristics f i ( · ) , i ∈ [ n ] under the settings prescrib ed by the exp onential family signal-utilit y structure (Assumption 1 ) are linear functions of the neighboring actions with speciﬁed co eﬃcien ts that dep end only on the lik eliho od structure parameters: n i , σ i and δ i as well as the prior parameters: α i and β i , for all i ∈ [ n ]. 1 Theorem 1 (Aﬃne action up dates) . Under the exp onential family signal-utility structur e sp e ciﬁe d in Assumption 1 , the Bayesian heuristics describing the action up date of every agent i ∈ [ n ] ar e given by: a i,t = f i ( a j,t − 1 : j ∈ N i ) = P j ∈N i T ij a j,t − 1 +  i , wher e for al l i , j ∈ [ n ] the c onstants T ij and δ i ar e as fol lows: T ij = δ i σ j ( n j + δ − 1 j β j ) σ i ( β i + P p ∈N i n p δ p ) ,  i = − δ i σ i ( β i + P p ∈N i n p δ p ) X j ∈N i \{ i } α j . The action proﬁle at time t is the concatenation of all actions in a column vector: a t = ( a T 1 ,t , . . . , a T n,t ) T . The matrix T with entries T ij , i, j ∈ [ n ] given in Theorem 1 is called the so cial inﬂuence matrix. The constan t terms  i in this theorem app ear as the rational agen ts attempt to comp ensate for the prior biases of their neighbors when making inferences about the observ ations in their neighborho o d; we denote  = (  T 1 , . . . ,  T n ) T and refer to it as the vector of neighborho o d biases. The evolution of action proﬁles under conditions of Theorem 1 can b e sp eciﬁed as follows: a t +1 = ( T ⊗ I k ) a t +  , where I k is the k × k identit y matrix and ( T ⊗ I k ) is a Kroneck er pro duct. Subsequen tly , the evolution of action proﬁles o ver time follo ws a non-homogeneous p ositiv e linear discrete-time dynamics, cf. [ 28 ]. If the sp ectral radius of T is strictly less than unity: ρ ( T ) < 1, then I − T is non-singular; there is a unique equilibrium action proﬁle giv en b y a e = (( I − T ) − 1 ⊗ I k )  and lim t →∞ a t = a e . If unit y is an eigenv alue of T , then there may b e no equilibrium action proﬁles or an inﬁnity of them. If ρ ( T ) > 1, then the linear discrete-time dynamics is unstable and the action proﬁles ma y gro w unbounded in their magnitude, cf. [ 49 ]. Example 1 (Gaussian Signals with Gaussian Beliefs). Mossel and T am uz [ 67 ] consider the case where the initial priv ate signals as well as the unknown states are normally distributed and the agents all hav e full knowledge of the netw ork structure. They sho w that by iteratively observing their neigh b ors’ mean estimates and up dating their b eliefs using Bay es rule all agents con v erge to the same b elief. The limiting belief is the same as what a Ba y esian agent with direct access to ev eryb o dy’s priv ate signals w ould ha ve hold; and furthermore, the b elief up dates at each step can b e computed eﬃciently and con v ergence o ccurs in a num b er of steps that is b ounded in the netw ork size and its diameter. These results how ever assume complete knowledge of the net w ork structure by all the agen ts. Here, we consider the linear action up dates in the Gaussian setting. Let Θ = R b e the param- eter space asso ciated with the unknown parameter θ ∈ Θ. Supp ose that each agent i ∈ [ n ] holds 1 Some of the non-Bay esian up date rules hav e the prop erty that they resemble the replication of a ﬁrst step of a Ba yesian up date from a common prior. F or instance, DeMarzo, V a yanos and Zwieb el [ 21 ] interpret the w eights in the DeGroot mo del as those assigned initially by rational agents to the noisy opinions of their neighbors based on their p erceiv ed precision. Ho wev er, by repeatedly applying the same w eights o ver and o ver again, the agen ts ignore the need to up date these weigh ts and to account for rep etitions in their information sources (the so-called p ersuasion bias); as one of our main ob jectives, we formalize this setup as a Bay esian heuristic. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 9 on to a Gaussian prior b elief with mean α i β − 1 i and v ariance β − 1 i ; here, γ ( θ ) = θ 2 / 2 and η ( θ ) = θ . F urther supp ose that eac h agen t observes an indep enden t priv ate Gaussian signal s i with mean θ and v ariance σ − 1 i = δ − 1 i , for all i ∈ [ n ]; hence, ξ ( s i ) = s i and τ ( σ i ξ ( s i ) , δ i ) = τ ( σ i s i , σ i ) = (2 π /σ i ) − 1 / 2 exp( σ i s 2 i / 2). After observing their priv ate signals all engage in rep eated comm unica- tions with their neighbors. Finally , we assume that each agents is trying to estimate the mean m i,θ = E i,θ { s i } of her priv ate signal with as little v ariance as p ossible. Under the prescrib ed set- ting, Theorem 1 applies and the Ba y esian heuristic up date rules are aﬃne with the co eﬃcien ts as sp eciﬁed in the theorem with n i = 1 and σ i = δ i for all i . In particular, if α i = (0 , . . . , 0) ∈ R k and β i → 0 for all i , then  i = 0 for all i and the co eﬃcien ts T ij = σ j / P p ∈N i σ p > 0 sp ecify a conv ex com bination: P j ∈N i T ij = 1 for all i . Example 2 (Poisson signals with gamma beliefs). As the second example, supp ose that eac h agen t observ e n i i.i.d. P oisson signals s i,p : p ∈ [ n i ] with mean δ i θ , so that Θ = A i = (0 , + ∞ ) for all i ∈ [ n ]. Moreov er, w e take each agen t’s prior to b e a Gamma distribution with parameters α i > 0 and β i > 0, denoted Gamma( α i , β i ): ν i ( θ ) := d V i d Λ 1 = β i α i Γ( α i ) θ α i − 1 e − β i θ , for all θ ∈ (0 , ∞ ) and each i ∈ [ n ]. Note that here η ( θ ) = log θ , γ ( η ( θ )) = exp( η ( θ )) = θ , κ ( α i , β i ) = Γ( α i ) β i − α i , m i,θ = δ i θ , ξ ( s i,p ) = s i,p , σ i = 1 and τ ( σ i ξ ( s i,p ) , δ i ) = δ s i,p i / ( s i,p !), for all i, p . This setting corresp onds also to a case of Poisson observers with common rate θ and individual exp osures δ i , i ∈ [ n ], cf. [ 33 , p. 54]. The p osterior distribution ov er Θ after observing of the sum of n i P oisson mean δ i θ samples is again a Gamma distribution with up dated (random) parameters P n i p =1 s i,p + α i and n i δ i + β i , [ 33 , pp. 52–53]. Using a quadratic utility − ( a − δ i θ ) 2 , the expected pa y-oﬀ at time zero is maximized b y the δ i -scaled mean of the p osterior Gamma belief distribution [ 33 , p. 587]: a i, 0 = δ i ( P n i p =1 s i,p + α i ) / ( n i δ i + β i ). Given the neighboring information P n j p =1 s j,p = ( n j + β j δ − 1 j ) a j, 0 − α j , ∀ j ∈ N i , agent i can reﬁne her belief in to a Gamma distribution with parameters α i + P j ∈N i [( n j + β j δ − 1 j ) a j, 0 − α j ] and β i + P j ∈N i n j δ j . The subsequent optimal action at time 1 and the resultant Ba y esian heuristics are as claimed in Theorem 1 with σ i = 1 for all i ∈ [ n ]. Here if w e let α i , β i → 0 and δ i = δ > 0 for all i , then  i = 0 for all i and the co eﬃcien ts T ij = n j / P p ∈N i n p > 0 again sp ecify a conv ex combination: P j ∈N i T ij = 1 for all i as in the DeGro ot mo del. In the following t w o subsections, we shall further explore this corresp ondence with the DeGro ot up dates and the implied asymptotic consensus amongst the agen ts. 3.1. Linear up dating and conv ergence. In general, the constant terms  i in Theorem 1 dep end on the neighboring prior parameters α j , j ∈ N i \ { i } and can b e non-zero. Accumulation of constant terms ov er time when ρ ( T ) ≥ 1 prev en ts the action proﬁles from conv erging to any ﬁnite v alues or may cause them to oscillate indeﬁnitely (dep ending up on the mo del parameters). Ho w ever, if the prior parameters are v anishingly small, then the aﬃne action up dates in Theorem 1 reduce to linear update and  i = 0. This requiremen t on the prior parameters is captured by our next assumption. Assumption 2 (Non-informativ e priors) . F or a memb er V ( · ; α, β ) of the c onjugate family F γ ,η we denote the limit lim α i ,β i → 0 V ( · ; α, β ) by V ∅ ( · ) and r efer to it as the non-informative (and impr op er, if V ∅ ( · ) 6∈ F γ ,η ) prior. 1 A l l agents start fr om a c ommon non-informative prior: V i ( · ) = V ∅ ( · ) , ∀ i . 1 Conjugate priors oﬀer a technique for deriving the prior distributions based on the sample distribution (likelihoo d structures). How ever, in lac k of any prior information it is imp ossible to justify their application on an y sub jective basis or to determine their associated parameters for an y agent. Subsequently , the use of non-informative priors is suggested b y Ba y esian analysts and v arious tec hniques for selecting non-informative priors is explored in the literature [ 55 ]. Amongst the many prop osed techniques for selecting non-informativ e priors, Jeﬀery’s metho d sets its choice Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 10 As the name suggest non-informative priors do not inform the agen t’s action at time 0 and the optimal action is completely determined b y the observ ed signal s i,p : p ∈ [ n i ] and its likelihoo d structure, parameterized by σ i and δ i . If we let α i , β i → 0 in the expressions of T ij and  i from Theorem 1 , then the aﬃne action up dates reduce to linear com binations and the preceding corollary is immediate. Corollar y 1 (Linear up dating) . Under the exp onential family signal-utility structur e (Assumption 1 ) with non-informative priors (Assumption 2 ); the Bayesian heuristics describ e e ach up date d action a i,t as a line ar c ombination of the neighb oring actions a j,t − 1 , j ∈ N i : a i,t = P j ∈N i T ij a j,t − 1 , wher e T ij = δ i σ j n j / ( σ i P p ∈N i n p δ p ) . The action proﬁles under Corollary 1 ev olv e as a homogeneous positive linear discrete-time system: a t +1 = ( T ⊗ I k ) a t and if the sp ectral radius of T is strictly less than unity , then lim t →∞ a t = 0. F or a strongly connected social net work with T ii > 0 for all i the Perron-F robenius theory [ 85 , Theorems 1.5 and 1.7] implies that T has a simple p ositive real eigenv alue equal to ρ ( T ). Moreo ver, the left and right eigenspaces asso ciated with ρ ( T ) are b oth one-dimensional with the corresp onding eigen v ectors l = ( l 1 , . . . , l n ) T and r = ( r 1 , . . . , r n ) T , uniquely satisfying k l k 2 = k r k 2 = 1, l i > 0, r i > 0, ∀ i and P n i =1 l i r i = 1. The magnitude of any other eigen v alue of T is strictly less than ρ ( T ). If ρ ( T ) = 1, then lim t →∞ a t = lim t →∞ ( T t ⊗ I k ) a 0 = ( r l T ⊗ I k ) a 0 ; in particular, the asymptotic action proﬁle may not represent a consensus although every action con v erges to some p oint within the con v ex hull of the initial actions { a i, 0 , i ∈ [ n ] } . If ρ ( T ) > 1, then the linear discrete-time dynamics is unstable and the action proﬁles may increase or decrease without b ound; pushing the decision outcome to extremes, w e can asso ciate ρ ( T ) > 1 to cases of p olarizing group interactions. 3.2. DeGro ot up dates, consensus and eﬃciency . In order for the linear action updates in Corollary 1 to constitute a con vex combination as in the DeGro ot mo del, 1 w e need to in tro duce some additional restrictions on the lik eliho od structure of the priv ate signals. Assumption 3 (Lo cally balanced likelihoo ds) . The likeliho o d structur es given in ( 3 ) ar e c al le d lo c al ly b alanc e d if for al l i ∈ [ n ] , ( δ i /σ i ) = ( P j ∈N i δ j n j ) / P j ∈N i σ j n j . Assumption 3 signiﬁes a lo cal balance prop ert y for the tw o exp onential family parameters σ i and δ i and across every neighborho o d in the net w ork. In particular, we need for the likelihoo d structures of ev ery agen t i and her neighborho o d to satisfy: δ i P j ∈N i σ j n j = σ i P j ∈N i δ j n j . Since parameters σ i and δ i are b oth measures of accuracy or precision for priv ate signals of agen t i , the balance condition in Assumption 3 imply that the signal precisions are spread evenly ov er the agen ts; i.e. the quality of observ ations ob ey a rule of so cial balance such that no agent is in a prop ortional to the square root of Fisher’s information measure of the likelihoo d structure [ 81 , Section 3.5.3], while Laplace’s classical principle of insuﬃcien t reason fa vors equiprobability leading to priors whic h are uniform ov er the parameter space. 1 The use of linear av eraging rules for mo deling opinion dynamics has a long history in mathematical so ciology and so cial psychology [ 30 ]; their origins can b e traced to F renc h’s seminal work on“A F ormal Theory of So cial P ow er” [ 29 ]. This was follo wed up by Harary’s inv estigation of the mathematical prop erties of the av eraging mo del, including the consensus criteria, and its relations to Marko v chain theory [ 41 ]. This mo del was later generalized to b elief exc hange dynamics and popularized by DeGroot’s seminal w ork [ 20 ] on linear opinion po ols. In engineering literature, the possibility to ac hieve consensus in a distributed fashion (through lo cal in teractions and information exc hanges b et ween neighbors) is v ery desirable in a v ariet y of applications such as load balancing [ 6 ], distributed detection and estimation [ 15 , 95 , 54 ], tracking [ 56 ], sensor netw orks and data fusion [ 98 , 99 ], as w ell as distributed con trol and rob otics net w orks [ 46 , 64 ]. Early w orks on developmen t of consensus algorithms originated in 1980s with the works of Tsitsiklis et.al [ 94 ] who prop ose a weigh ted a verage protocol based on a linear iterative approach for achieving consensus: eac h node rep eatedly up dates its v alue as a w eighted linear com bination of its o wn v alue and those received b y its neighbors. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 11 p osition of superiority to ev eryone else. Indeed, ﬁxing δ i = δ for all i , the latter condition reduces to a harmonic property for the parameters σ i , when view ed as a function of their resp ective no des (cf. [ 61 , Section 2.1] for the deﬁnition and prop erties of harmonic functions): σ i = X j ∈N i n j σ j P k ∈N i n k , δ i = δ, ∀ i. (4) Ho w ever, in a strongly conceded social netw ork ( 4 ) cannot hold true unless σ i is a constant: σ i = σ for all i . Similarly , when σ i = σ is a constant, then under Assumption 3 δ i is spread as a harmonic function o v er the net w ork no des, and therefore can only take a constant v alue δ i = δ for all i , cf. [ 61 , Section 2.1, Maxim um Principle]. In particular, ﬁxing either of the parameters σ i or δ i for all agen ts and under the local balance condition in Assumption 3 , it follo ws that the other parameter should b e also ﬁxed across the netw ork; hence, the ratio σ i /δ i will b e a constan t for all i . Later when we consider the eﬃciency of consensus action we introduce a strengthening of Assumption 3 , called globally balanced likelihoo d (cf. Assumption 4 ), where the ratio δ i /σ i should b e a constant for all agen ts across the net work. Examples 1 and 2 ab o ve provide tw o scenarios in which the preceding balancedness conditions may b e satisﬁed: (i) ha ving σ i = δ i for all i , as was the case with the Gaussian signals in Example 1, erasures that the likelihoo ds are globally balanced; (ii) all agents receiving i.i.d. signals from a common distribution in Examples 2 (Poisson signals with the common rate θ and common exp osure δ ) mak es a case for likelihoo ds b eing lo cally balanced. Theorem 2 (DeGro ot up dating and consensus) . Under the exp onential family signal- utility structur e (Assumption 1 ), with non-informative priors (Assumption 2 ) and lo c al ly b alanc e d likeliho o ds (Assumption 3 ); the up date d action a i,t is a c onvex c ombination of the neighb oring actions a j,t − 1 , j ∈ N i : a i,t = P j ∈N i T ij a j,t − 1 , P j ∈N i T ij = 1 for al l i . Henc e, in a str ongly c onne cte d so cial network the action pr oﬁles c onver ge to a c onsensus, and the c onsensus value is a c onvex c ombination of the initial actions a i, 0 : i ∈ [ n ] . In ligh t of Theorem 2 , it is of interest to kno w if the consensus action agrees with the min- im um v ariance un biased estimator of m i,θ giv en all the observ ations of every agent across the net w ork, i.e. whether the Bay esian heuristics eﬃciently aggregate all the information amongst the net w orked agents. Our next result addresses this question. F or that to hold w e need to introduce a strengthening of Assumption 3 : Assumption 4 (Globally balanced likelihoo ds) . The likeliho o d structur es given in ( 3 ) ar e c al le d glob al ly b alanc e d if for al l i ∈ [ n ] and some c ommon c onstant C > 0 , δ i /σ i = C . In particular, under Assumption 4 , σ i δ j = σ j δ i for all i, j , and it follows that the lo cal balance of lik eliho o ds is automatically satisﬁed. W e call the consensus action eﬃcient if it coincides with the minimum v ariance un biased estimator of m i,θ for all i and giv en all the observ ations of ev ery agen t across the netw ork. Our next result indicates that global balance is a necessary condition for the agents to reach consensus on a globally optimal (eﬃcient) action. T o pro ceed, let the net w ork graph structure b e encoded b y its adjacency matrix A deﬁned as [ A ] ij = 1 ⇐ ⇒ ( j, i ) ∈ E , and [ A ] ij = 0 otherwise. T o express the conditions for eﬃciency of consensus, we need to consider the set of all agents who listen to the b eliefs of a given agent j ; w e denote this set of agents by N out j := { i ∈ [ n ] : [ I + A ] ij = 1 } and refer to them as the out-neighborho o d of agent j . This is in con trast to her neigh b orho o d N j , which is the set of all agen ts whom she listens to. Both sets N j and N out j include agen t j as a mem b er. Theorem 3 ((In-)Eﬃciency of consensus) . Under the exp onential family signal-utility structur e (Assumption 1 ) and with non-informative priors (Assumption 2 ); in a str ongly c onne cte d so cial network the agents achieve c onsensus at an eﬃcient action if, and only if, the likeliho o ds ar e glob al ly b alanc e d and P p ∈N out j n p δ p = P p ∈N i n p δ p , for al l i and j . The eﬃcient c onsensus action is then given by a ? = P n j =1  δ j n j a j, 0 / P n p =1 n p δ p  . Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 12 F ollo wing our discussing of Assumption 3 and equation ( 4 ), we p ointed out that if either of the tw o parameters σ i and δ i that characterize the exp onen tial family distribution of ( 3 ) are held ﬁxed amongst the agents, then the harmonicity condition required for the lo cal balancedness of the likelihoo ds implies that the other parameter is also ﬁxed for all the agents. Therefore the lo cal balancedness in many familiar cases (see Example 2 ) restricts the agents to observing i.i.d. signals: allo wing heterogeneity only in the sample sizes, but not in the distribution of each sample. This sp ecial case is treated in our next corollary , where w e also provide the simpler forms of the Ba y esian heuristics and their linearit y co eﬃcien ts in the i.i.d. case: Corollar y 2 (DeGro ot learning with i.i.d. samples) . Supp ose that e ach agent i ∈ [ n ] observes n i i.i.d. samples b elonging to the same exp onential family signal-utility structur e (Assump- tion 1 with σ i = σ and δ i = δ for al l i ). If the agents have non-informative priors (Assump- tion 2 ) and the so cial network is str ongly c onne cte d so cial network, then fol lowing the Bayesian heuristics agents up date their action ac c or ding to the line ar c ombination: a i,t = P j ∈N i T ij a j,t − 1 , wher e T ij = n j / P p ∈N i n p , and r e ach a c onsensus. The c onsensus action is eﬃcient if, and only if, P p ∈N out j n p = P p ∈N i n p for al l i and j , and the eﬃcient c onsensus action is given by a ? = P n j =1  a j, 0 n j / P n p =1 n p  . It is notable that the consensus v alue pinpointed b y Theorem 2 do es not necessarily agree with the MVUE of m i,θ giv en all the priv ate signals of all agents across the net w ork; in other words, b y following Bay esian heuristics agen ts ma y not aggregate all the initial data eﬃcien tly . As a simple example, consider the exponential family signal-utilit y structure with non-informativ e priors (Assumptions 1 and 2 ) and supp ose that ev ery agent observ es an i.i.d. sample from a common distribution L ( ·| θ ; 1 , 1). In this case, the action up dates pro ceed by simple iterativ e a v eraging: a i,t = (1 / |N i | ) P j ∈N i a j,t − 1 for all i ∈ [ n ] and any t ∈ N . F or an undirected graph G it is well-kno wn that the asymptotic consensus action following simple iterative av eraging is the degree-weigh ted a v erage P n i =1 ( deg ( i ) / |E | ) a i, 0 , cf. [ 38 , Section I I.C]; and the consensus action is diﬀerent form the global MVUE a ? = (1 /n ) P n i =1 a i, 0 unless the so cial net w ork is a regular graph in which case, deg ( i ) = d is ﬁxed for all i , and |E | = n.d . Remark 1 (Efficiency of Balanced Regular Structures). In general, if we assume that all agen ts receiv e the same n umber of i.i.d. samples from the same distribution, then the condi- tion for eﬃciency of consensus, P p ∈N out j n p = P p ∈N i n p , is satisﬁed for balanced regular structures. In suc h highly symmetric structures, the num b er of outgoing and incoming links are the same for ev ery no de and equal to ﬁxed n umber d . Our results shed light on the deviations from the globally eﬃcient actions, when consensus is b eing achiev ed through the Ba y esian heuristics. This ineﬃciency of Bay esian heuristics in globally aggregating the observ ations can b e attributed to the agents’ naivet y in inferring the sources of their information, and their inabilit y to interpret the actions of their neighbors rationally , [ 38 ]; in particular, the more central agents tend to inﬂuence the asymptotic outcomes unfairly . This sensitivit y to so cial structure is also due to the failure of agents to correct for the rep etitions in the sources of the their information: agen t i may receive multiple copies that are all inﬂuenced by the same observ ations from a far wa y agent; ho wev er, she fails to correct for these rep etition in the sources of her observ ations, leading to the co-called p ersuasion bias, [ 21 ]. 1 1 Sob el [ 90 ] provides a theoretical framework to study the interpla y b et w een rationality and group decisions and p oin ts out the subsequen t ineﬃciencies in information aggregation. The seminal w ork of Janis [ 50 ] provides v arious examples in volving the American foreign p olicy in the mid-tw entieth where the desire for harmony or conformit y in the group ha ve resulted in bad group decisions, a phenomenon that he coins groupthink . V arious other works hav e lo ok ed at the c hoice shift tow ard more extreme options [ 23 , 91 ] and group p olarization [ 44 , 83 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 13 4. Log-Linear b elief up dates. When the state space Θ is ﬁnite, the action space is the probabilit y simplex and the agen ts b ear a quadratic utilit y that measures the distance b et ween their action and the p oint mass on the true state, the communication structure b et ween the agents is rich enough for them to reveal their b eliefs at every time p erio d. The Ba y esian heuristics in this case lead to a log-linear up dating of b eliefs similar to what w e analyze in [ 73 , 75 ] under the Ba y esian without recall mo del; the chief question of in terest to the latter works is whether the agen ts, after being exp osed to sequence of priv ate observ ations and while comm unicating with eac h other, can learn the truth using the Ba y esian without recall update rules. 1 The learning framework of [ 73 , 74 , 75 , 78 ] in which agents hav e access to an stream of new observ ations is in con trast with the group decision model of this pap er; the diﬀerence b eing in the fact that here the agen ts hav e a single initial observ ation and engage in group decision making to come up with the b est decision that aggregates their individual priv ate data with those of the other group mem b ers. Consider an en vironment where the state space is a ﬁnite set of cardinality m and agents take actions ov er the ( m − 1)-simplex of probability measures while trying to minimize their distance to a p oin t mass on the true state. Sp eciﬁcally , let Θ = { θ 1 , . . . , θ m } , A i = ∆Θ and for any probability measure M ( · ) ∈ A i with probabilit y mass function µ ( · ) := d M /d K , supp ose that u i ( M , θ j ) = − (1 − µ ( θ j )) 2 − m X k =1 , k 6 = j µ ( θ k ) 2 . Subsequen tly , M i,t = arg max M∈ ∆Θ E i,t { u i ( M , θ ) } , and the agents proceed by truthfully announc- ing their b eliefs to eac h other at every time step. In particular, if w e denote the b elief probabilit y mass functions ν i ( · ) := d V i /d K and µ i,t ( · ) := d M i,t /d K for all t , then we can follo w the steps of [ 77 ] to derive the Ba yesian heuristic f i in ( 2 ) b y replicating the time-one Bay esian b elief up date for all future time-steps: µ i,t ( ˆ θ ) = µ i,t − 1 ( ˆ θ )  Q j ∈N i \{ i } µ j,t − 1 ( ˆ θ ) ν j ( ˆ θ )  P ˜ θ ∈ Θ µ i,t − 1 ( ˜ θ )  Q j ∈N i \{ i } µ j,t − 1 ( ˜ θ ) ν j ( ˜ θ )  , (5) for all ˆ θ ∈ Θ and at an y t > 1. 2 , 3 1 Naiv ety of agents in these cases imp edes their ability to learn; except in simple so cial structures suc h as cycles or ro oted trees (cf. [ 74 ]). Rahimian, Shahramp our and Jadbabaie [ 78 ] show that learning in so cial netw ork with complex neigh b orhoo d structures can b e achiev ed if agents c ho ose a neighbor randomly at every round and restrict their b elief up date to the selected neighbor each time. In the literature on theory of learning in games [ 31 ], log-linear learning refers to a class of randomized strategies, where the probability of eac h action is prop ortional to an exp onential of the diﬀerence betw een the utility of taking that action and the utilit y of the optimal c hoice. Such randomized strategies com bine in a log-linear manner [ 14 ], and they ha ve desirable con vergence prop erties: under prop er conditions, it can b e sho wn that the limiting (stationary) distribution of action proﬁles is supported o ver the Nash equilibria of the game [ 62 ]. 2 In writing ( 5 ), every time agent i regards each of her neighbors j ∈ N i as having started from some prior b elief ν j ( · ) and arriv ed at their currently rep orted b elief µ j,t − 1 ( · ) up on observing their priv ate signals, hence rejecting any p ossibilit y of a past history , or learning and correlation betw een their neigh b ors. Such a rule is of course not the optim um Bay esian up date of agen t i at any step t > 1, b ecause the agen t is not taking into accoun t the complete observ ed history of b eliefs and is instead, basing her inference entirely on the initial signals and the immediately observ ed beliefs. 3 It is notable that the Ba yesian heuristic in ( 5 ) has a log-linear structure. Geometric a veraging and logarithmic opinion p ools ha v e a long history in Bay esian analysis and b eha vioral decision models [ 37 , 84 ] and they can b e also justiﬁed under speciﬁc b eha vioral assumptions [ 65 ]. The are also quite p opular as a non-Bay esian up date rule in engineering literature for addressing problems suc h as distributed detection and estimation [ 86 , 79 , 71 , 59 , 8 ]. In [ 8 ] the authors use Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 14 4.1. An algebra of b eliefs. Both the linear action up dates studied in the previous c hapter as well as the w eigh ted ma jority up date rules that arise in the binary case and are studied in [ 76 , 75 ] hav e a familiar algebraic structure. It is instructive to develop similar structural prop erties for b elief up dates in ( 5 ) and o v er the space ∆Θ, i.e. the p oints of the standard ( m − 1)-simplex. Giv en t wo beliefs µ 1 ( · ) and µ 2 ( · ) o v er Θ we denote their “addition” as µ 1 ⊕ µ 2 ( ˆ θ ) = µ 1 ( ˆ θ ) µ 2 ( ˆ θ ) P ˆ θ ∈ Θ µ 1 ( ˆ θ ) µ 2 ( ˆ θ ) . Indeed, let ∆Θ o denote the ( m − 1)-simplex of probability measure ov er Θ after all the edges are excluded; ∆Θ o endo w ed with the ⊕ op eration, constitutes a group (in the algebraic sense of the w ord). It is easy to v erify that the uniform distribution ¯ µ ( ˆ θ ) = 1 / | Θ | acts as the identit y elemen t for the group; in the sense that ¯ µ ⊕ µ = µ for all µ ∈ ∆Θ o , and given an y such µ w e can uniquely iden tify its in verse as follo ws: µ inv ( ˆ θ ) = 1 /µ ( ˆ θ ) P ˜ θ ∈ Θ 1 /µ ( ˜ θ ) . Moreo v er, the group op eration ⊕ is comm utativ e and we can thus endow the ab elian group (∆Θ o , ⊕ ) with a subtraction op eration: µ 1  µ 2 ( ˆ θ ) = µ 1 ⊕ µ inv 2 ( ˆ θ ) = µ 1 ( ˆ θ ) /µ 2 ( ˆ θ ) P ˜ θ ∈ Θ µ 1 ( ˜ θ ) /µ 2 ( ˜ θ ) . W e are no w in a p osition to rewrite the Ba yesian heuristic for b elief updates in terms of the group op erations ⊕ and  o ver the simplex in terior: µ i,t = ⊕ j ∈N i µ j,t − 1  j ∈N i \{ i } ν j . The ab ov e b elief up date has a structure similar to the linear action up dates studied in ( 1 ): the agen ts incorp orate the b eliefs of their neighbors while comp ensating for the neigh b oring priors to isolate the observ ational parts of the neigh b ors’ rep orts. A key diﬀerence b et ween the action and b elief up dates is in the fact that action up dates studied in Section 3 are w eigh ted in accordance with the observ ational ability of each neighbor, whereas the b elief up dates are not. Indeed, the qualit y of signals are already in ternalized in the reported b eliefs of each neigh b or; therefore there is no need to re-w eigh t the rep orted beliefs when aggregating them. a logarithmic opinion p ool to combine the estimated posterior probability distributions in a Ba yesian consensus ﬁlter; and sho w that as a result: the sum of Kullbac kLeibler divergences b etw een the consensual probability distribution and the local posterior probabilit y distributions is minimized. Minimizing the sum of KullbackLeibler divergences as a wa y to globally aggregate lo cally measured probability distributions is prop osed in [ 11 , 10 ] where the corresp onding minimizer is dubbed the KullbackLeibler a verage. Similar interpretations of the log-linear update are oﬀered in [ 70 ] as a gradient step for minimizing either the KullbackLeibler distance to the true distribution, or in [ 72 ] as a p osterior incorp oration of the most recent observ ations, such that the sum of Kullbac kLeibler distance to the lo cal priors is minimized; indeed, the Ba yes’ rule itself has a pro duct form and the Ba yesian posterior can b e c haracterized as the solution of an optimization problem inv olving the KullbackLeibler divergence to the prior distribution and sub jected to the observ ed data [ 100 ]. In the past, w e ha ve inv estigated the implications of suc h log-linear b ehavior and properties of con vergence and learning when agents are exposed to a stream of priv ate observ ation [ 73 , 74 , 77 , 78 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 15 Giv en the ab elian group structure w e can further consider the “p o w ers” of eac h elemen t µ 2 = µ ⊕ µ and so on; in general for each inetger n and an y b elief µ ∈ ∆Θ o , let the n -th p o wer of µ b e denoted b y n  µ := µ n , deﬁned as follo ws: 1 µ n ( ˆ θ ) = µ n ( ˆ θ ) P ˜ θ ∈ Θ µ n ( ˜ θ ) . Using the ⊕ and  notations, as w ell as the adjacency matrix A w e get: µ i,t +1 = ⊕ j ∈N i µ j,t  j ∈N i \{ i } ν j = ⊕ j ∈ [ n ] ([ I + A ] ij  µ j,t )  j ∈ [ n ] ([ A ] ij  ν j ) . (9) With some abuse of notation, we can concatenate the netw ork b eliefs at every time t in to a column v ector µ t = ( µ 1 ,t , . . . , µ n,t ) T and similarly for the priors ν = ( ν 1 , . . . , ν n ) T ; th us ( 9 ) can b e written in the v ectorized format b y using the matrix notation as follo ws: µ t = { ( I + A )  µ t − 1 }  { A  ν } (10) Iterating o v er t and in the common matrix notation w e obtain: µ t = n ( I + A ) t  µ 0 o  n ( P t τ =0 ( I + A ) τ A )  ν o . (11) The abov e is k ey to understanding the ev olution of b eliefs under the Bay esian heuristics in ( 5 ), as w e will explore next. In particular, when all agents ha ve uniform priors ν j = ¯ µ for all j , then ( 10 ) and ( 11 ) simplify as follo ws: µ t = ( I + A )  µ t − 1 = ( I + A ) t  µ 0 . This assumption of a common uniform prior is the counterpart of Assumption 1 (non-informative priors) in Subsection 3.1 , which pa v ed the w ay for transition from aﬃne action up dates into linear ones. In the case of b eliefs o v er a ﬁnite state space Θ, the uniform prior ¯ µ is non-informative. If all agents start form common uniform priors, the b elief update in ( 5 ) simpliﬁes as follows: µ i,t ( ˆ θ ) = Q j ∈N i µ j,t − 1 ( ˆ θ ) P ˜ θ ∈ Θ Q j ∈N i µ j,t − 1 ( ˜ θ ) . (12) Our main fo cus in the next section is to understand ho w the individual beliefs ev olv e under ( 5 ), or ( 12 ) which is a spacial case of ( 5 ). The gist of our analysis is encapsulated in the group theoretic iterations: µ t = ( I + A ) t  µ 0 , derived ab o v e for the common uniform priors case. In particular, our understanding of the increasing matrix p o w ers ( I + A ) t pla ys a key role. When the net w ork graph G is strongly connected, the matrix I + A is primitive. The P erron-F rob enius theory [ 85 , Theorems 1.5 and 1.7] implies that I + A has a simple positive real eigenv alue equal to its sp ectral radius ρ ( I + A ) = 1 + ρ , where we adopt the shorthand notation ρ := ρ ( A ). Moreo ver, the left and righ t eigenspaces asso ciated with this eigenv alue are b oth one-dimensional and the corresp onding eigen v ectors can b e taken such that they b oth ha v e strictly p ositiv e entries. The magnitude of an y other eigenv alue of I + A is strictly less than 1 + ρ . Hence, the eigenv alues of I + A denoted b y λ i ( I + A ), i ∈ [ n ], can b e ordered in their magnitudes as follows: | λ n ( I + A ) | ≤ | λ n − 1 ( I + A ) | ≤ 1 This notation extends to all real num bers n ∈ R , and it is easy to verify that the following distributive prop erties are satisﬁed: n  ( µ 1 ⊕ µ 2 ) = ( n  µ 1 ) ⊕ ( n  µ 2 ) , ( m + n )  µ 1 = ( m  µ 1 ) ⊕ ( n  µ 1 ) , ( m.n )  µ 1 = m  ( n  µ 1 ) , for all m, n ∈ R and µ 1 , µ 2 ∈ ∆Θ o . Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 16 . . . < λ 1 ( I + A ) = 1 + ρ . Subsequently , we can emplo y the eigendecomp osition of ( I + A ) to analyze the b eha vior of ( I + A ) t +1 . Sp eciﬁcally , we can tak e a set of bi-orthonormal vectors l i , r i as the left and right eigen vectors corresp onding to the i th eigenv alue of I + A , satisfying: k l i k 2 = k r i k 2 = 1, l T i r i = 1 for all i and l T i r j = 0, i 6 = j ; in particular, the left eigenspace asso ciated with ρ is one-dimensional with the corresp onding eigen v ector l 1 = α = ( α 1 , . . . , α n ) T , uniquely satisfying P n i =1 α i = 1, α i > 0, ∀ i ∈ [ n ], and α T A = ( ρ + 1) α T . The entry α i is called the cen trality of agent i and as the name suggests, it measures ho w central is the lo cation of agent in the netw ork. W e can no w use the sp ectral represen tation of A to write [ 49 , Section 6]: ( I + A ) t = (1 + ρ ) t r 1 α T + n X i =2 ( λ i ( I + A ) / (1 + ρ )) t r i l T i ! . (13) Asymptotically , we get that all eigen v alues other than the Perron-F robenius eigenv alue 1 + ρ are sub dominan t; hence, ( I + A ) t → (1 + ρ ) t r 1 α T and µ t = (1 + ρ ) t r 1 α T  µ 0 as t → ∞ ; the latter holds true for the common uniform priors case and also in general, as w e shall see next. 4.2. Becoming certain ab out the group aggregate. W e b egin our inv estigation of the ev olution of b eliefs under ( 12 ) b y considering the optimal resp onse (b elief ) of an age n ts who has b een given access to the set of all priv ate observ ations across the netw ork; indeed, such a resp onse can b e ac hiev e d in practice if one follows Kahneman’s advice and collect each individual’s infor- mation priv ately b efore combining them or allowing the individuals to engage in public discussions [ 52 , Chapter 23]. Starting from the uniform prior and after observing everybo dy’s priv ate data our aggregate b elief about the truth state is given b y the following implemen tation of the Bay es rule: µ ? ( ˆ θ ) = Q j ∈N i ` j ( s j | ˆ θ ) P ˜ θ ∈ Θ Q j ∈N i ` j ( s j | ˜ θ ) . (14) Our next theorem describ es the asymptotic outcome of the group decision pro cess when the agents rep ort their b eliefs and follow the Bay esian heuristic ( 12 ) to aggregate them. The outcome indi- cated in Theorem 4 departs from the global optimum µ ? in tw o ma jor resp ects. Firstly , the agen ts reach consensus on a b elief that is supp orted ov er Θ ♦ := arg max ˜ θ ∈ Θ P n i =1 α i log( ` i ( s i | ˜ θ )), as opp osed to the global (netw ork-wide) lik eliho o d maximizer Θ ? := arg max ˜ θ ∈ Θ µ ? ( ˜ θ ) = arg max ˜ θ ∈ Θ P n i =1 log( ` i ( s i | ˜ θ )); note that the signal log -lik eliho o ds in the case of Θ ♦ are weigh ted b y the centralities, α i , of their resp ectiv e no des. Secondly , the consensus b elief is concentrated uniformly o v er Θ ♦ , its supp ort do es not include the entire state space Θ and those states which score low er on the centralit y-weigh ted likelihoo d scale are asymptotically rejected as a candidate for the truth state; in particular, if { θ ♦ } = Θ ♦ is a singlton, then the agen ts eﬀectively b ecome certain ab out the truth state of θ ♦ , in spite of their ess en tially b ounded aggregate information and in contrast with the rational (optimal) b elief µ ? that is giv en b y the Bay es rule in ( 14 ) and do not discredit or reject an y of the less probable states. This unw arranted certaint y in the face of limited aggregate data is a manifestation of the group p olarization eﬀect that derive the agen t to more extreme b eliefs, rejecting the possibility of an y alternatives outside of Θ ♦ . Theorem 4 (Certain t y ab out the group aggregate) . F ol lowing the Bayesian heuristic b elief up dates in ( 5 ) , lim t →∞ µ i,t ( ˜ θ ) = 1 / | Θ ♦ | for al l i ∈ [ n ] and any ˜ θ ∈ Θ ? , wher e Θ ♦ := arg max ˜ θ ∈ Θ P n i =1 α i log( ` i ( s i | ˜ θ )) . In p articular, if the sum of signal log -likeliho o ds weighte d by no de c entr alities is uniquely maximize d by θ ♦ , i.e. { θ ♦ } = Θ ♦ , then lim t →∞ µ i,t ( θ ♦ ) = 1 almost sur ely for al l i ∈ [ n ] . Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 17 Remark 2 (Efficiency of balanced regular networks). The fact that log-likelihoo ds in Θ ♦ are w eigh ted b y the no de centralities is a source of ineﬃciency for the asymptotic outcome of the group decision process. This ineﬃciency is w arded oﬀ in especially symmetric typologies, where in and out degrees of all no des in the netw ork are the same. In these so-called balanced regular digraphs, there is a ﬁxed integer d suc h that all agen ts receiv e reports from exactly d agen ts, and also send their rep orts to some other d agen ts; d -regular graphs are a sp ecial case, since all links are bidirectional and eac h agent sends her rep orts to and receive rep orts from the same d agents. In such structures α = (1 /n ) 1 so that Θ ? = Θ ♦ and the supp ort of the consensus b elief iden tiﬁes the global maxim um likelihoo d estimator (MLE); i.e. the maximum likelihoo d estimator of the unkno wn θ , giv en the entire set of observ ations from all agents in the net work. 5. Conclusions. W e prop ose the Bay esian heuristics framework to address the problem of information aggregation and decision making in groups. Our mo del is consistent with the dual pro cess theory of mind with one system developing the heuristics through delib eration and slow pro cessing, and another sys tem adopting the heuristics for fast and automatic decision making: once the time-one Ba y esian up date is developed, it is used as a heuristic for all future decision ep ochs. On the one hand, this mo del oﬀers a b eha vioral foundation for non-Ba yesian up dating; in particular, linear action up dates and log-linear b elief up dates. On the other hand, its deviation from the rational c hoice theory captures common fallacies of snap-judgments and history neglect that are observed in real life. Our b eha vioral metho d also complements the axiomatic approac hes which in v estigate the structure of b elief aggregation rules and require them to satisfy speciﬁc axioms such as lab el neutralit y and imp erfect recall, as w ell as indep endence or separabilit y for log-linear and linear rules, resp ectiv ely [ 65 ]. W e sho w ed that under a natural quadratic utilit y and for a wide class of distributions from the exp onen tial family the Bay esian heuristics corresp ond to a minimum v ariance Ba y es estimation with a kno wn linear structure. If the agen ts ha v e non-informativ e priors, and their signal structures satisfy certain homogeneity conditions, then these action updates constitute a conv ex combination as in the DeGro ot mo del, where agen ts reac h consensus on a p oin t in the conv ex hull of their initial actions. In case of b elief up dates (when agents communicate their b eliefs), we sho wed that the agen ts up date their b eliefs prop ortionally to the pro duct of the self and neighboring b eliefs. Subsequen tly , their b eliefs conv erge to a consensus supp orted ov er a maximum likelihoo d set, where the signal lik eliho ods are weigh ted by the cen tralities of their resp ective agents. Our results indicate certain deviations from the globally eﬃcient outcomes, when consensus is b eing achiev ed through the Ba y esian heuristics. This ineﬃciency of Bay esian heuristics in globally aggregating the observ ations is attributed to the agents’ naivet y in inferring the sources of their information, whic h makes them vulnerable to structural netw ork inﬂuences: the share of cen trally lo cated agents in shaping the asymptotic outcome is more than what is w arranted b y the qualit y of their data. Another source of ineﬃciency is in the group p olarization that arise as a result of rep eated group interactions; in case of b elief up dates, this is manifested in the structure of the (asymptotic) consensus b eliefs. The latter assigns zero probabilit y to an y alternativ e that scores lo w er than the maximum in the w eighted lik eliho o ds scale: the agen ts reject the p ossibility of less probable alternatives with certain t y , in spite of their limited initial data. This ov erconﬁdence in the group aggregate and shift tow ard more extreme b eliefs is a key indicator of group polarization and is demonstrated v ery w ell b y the asymptotic outcome of the group decision process. W e pinp oin t some key diﬀerences b etw een the action and b elief updates (linear and log -linear, resp ectiv ely): the former are weigh ted up dates, whereas the latter are unw eighted symmetric up dates. Accordingly , an agen t w eighs each neigh b or’s action diﬀerently and in accordance with the quality of their priv ate signals (whic h are inferred from the actions). On the other hand, when comm unicating their b eliefs the quality of each neigh b or’s signal is already internalized in their Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 18 rep orted beliefs; hence, when incorp orating her neighboring b eliefs, an agen t regards the rep orted b eliefs of all her neigh b ors equally and symmetrically . Moreov er, in the case of linear action up dates the initial biases are ampliﬁed and accumulated in every iteration. Hence, the interactions of biased agen ts are very muc h dominated b y their prior b eliefs rather than their observ ations. This issue can push their choices to extremes, dep ending on the aggregate v alue of their initial biases. Therefore, if the Ba yesian heuristics are to aggregate information from the observed actions satisfactorily , then it is necessary for the agents to b e unbiased, i.e. they should hold non-informative priors ab out the state of the world and base their actions entirely on their observ ations. In contrast, when agen ts exc hange b eliefs with each other the multiplicativ e b elief up date can aggregate the observ ations, irresp ectiv e of the prior b eliefs. The latter are asymptotically canceled; hence, multiplicativ e b elief up dates are robust to the inﬂuence of priors. The Ba yesian heuristics approac h is strongly motiv ated b y the behavioral pro cesses that underlie h uman decision making. These pro cesses often deviate from the predictions of the rational c hoice theory , and our inv estigation of the Ba y esian heuristics highlights b oth the mechanisms for suc h deviations and their ramiﬁcations. In our ongoing research, w e expand this b eha vioral approac h by incorp orating additional cognitive biases such as inatten tiveness, and inv estigate ho w the decision pro cesses are aﬀected. On the one hand, the obtained insights highlight the v alue of educating the public ab out b eneﬁts of rational decision making and unbiased judgmen t, and how to av oid common cognitiv e errors when making decisions. On the other hand, by inv estigating the eﬀect of cognitiv e biases, w e can improv e the practice of social and organizational p olicies, suc h that new designs can accommo date commonly observ ed biases, and work w ell in spite of them. App endix A: Pro of of Theorem 1 . If agen t i starts from a prior belief V i ( · ) = V ( · ; α i , β i ) ∈ F γ ,η , then w e can use the Ba yes rule to verify that, cf. [ 81 , Prop osition 3.3.13], the Radon-Nikodym deriv ativ e of the Bay esian p osterior of agen t i after observing n i samples s i,p ∈ S , p ∈ [ n i ], with lik eliho od ( 3 ) is ν ( · ; α i + σ i P n i p =1 ξ ( s i,p ) , β i + n i δ i ), and in particular the Bay esian posterior at time zero b elongs to the conjugate family F γ ,η : M i, 0 ( · ) = V ( · ; α i + σ i P n i p =1 ξ ( s i,p ) , β i + n i δ i ). Sub ject to the quadratic utilit y u i ( a, θ ) = − ( a − m i,θ ) T ( a − m i,θ ), the exp ected pay-oﬀ at any time time t is maximized is b y c ho osing [ 12 , Lemma 1.4.1]: a i,t = E i,t { m i,θ } := Z θ ∈ Θ m i,θ M i,t ( dθ ) , whic h coincides with her minimum v ariance un biased estimator (Bay es estimate) for m i,θ . The mem b ers of the conjugate family F γ ,η satisfy the following linearit y property of the Ba yes estimates that is k ey to our deriv ations. 1 Lemma 1 (Prop osition 3.3.14 of [ 81 ]) . L et ζ ∈ R k b e a p ar ameter and supp ose that the p ar ameter sp ac e Ω ζ is an op en set in R k . Supp ose further that ζ ∈ Ω ζ has the prior distribution W ( · ; α, β ) with density κ 0 ( α, β ) e ζ T α − β γ ( ζ ) w.r.t. Λ k wher e κ 0 ( α, β ) is the normalization c onstant. If s 0 ∈ S 0 ⊂ R k is a r andom signal with distribution D ( · ; ζ ) and density τ 0 ( s ) e ζ t s − γ 0 ( ζ ) w.r.t. Λ k , then Z ζ ∈ Ω ζ Z s 0 ∈S 0 s D ( ds ; ζ ) W ( dζ ; α , β ) = α β . 1 In fact, such an aﬃne mapping from the observ ations to the Bay es estimate c haracterizes the conjugate family F γ ,η and ev ery mem ber of this family can be uniquely identiﬁed from the constan ts of the aﬃne transform [ 22 ]. Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 19 Hence for an y V ( · ; α, β ) ∈ F γ ,η w e can write Z θ ∈ Θ m i,θ V ( dθ ; α, β ) = Z θ ∈ Θ Z s ∈S ξ ( s ) L ( ds | θ ; σ, δ ) V ( dθ ; α, β ) = Z θ ∈ Θ     Λ k ( η ( dθ )) G θ ( dθ )     e η ( θ ) T α − β γ ( η ( θ )) κ ( α, β ) G θ ( dθ ) × Z s ∈S ξ ( s ) σ     Λ k ( ξ ( ds )) G s ( ds )     τ ( σ ξ ( s ) , δ ) e σ η ( θ ) T ξ ( s ) − δ γ ( η ( θ )) G s ( ds ) = Z ζ ∈ Ω θ     Λ k ( η ( dθ )) G θ ( dθ )     e η ( θ ) T α − β γ ( η ( θ )) κ ( α, β ) G θ ( dθ ) × Z s ∈S ξ ( s ) σ     Λ k ( ξ ( ds )) G s ( ds )     τ ( σ ξ ( s ) , δ ) e σ η ( θ ) T ξ ( s ) − δ γ ( η ( θ )) G s ( ds ) = Z ζ ∈ Ω η e ζ T α − β δ γ 0 ( ζ ) κ ( α, β ) Λ k ( dζ ) Z s 0 ∈S 0 s 0 τ 0 ( s 0 ) σ e ζ T s 0 − γ 0 ( ζ ) Λ k ( ds 0 ) = αδ σ β , (15) where in the p en ultimate equality we hav e employ ed the following change of v ariables: ζ = η ( θ ), s 0 = σ ξ ( s ), γ 0 ( ζ ) = δ γ ( ζ ), τ 0 ( s 0 ) = τ ( s 0 , δ ); and the last equalit y is a direct application of Lemma 1 . In particular, given M i, 0 ( · ) = V ( · ; α i + σ i P n i p =1 ξ ( s i,p ) , β i + n i δ i ), the exp ectation maximizing action at time zero coincides with: a i, 0 = P n i p =1 ξ ( s i,p ) + σ − 1 i α i n i + δ − 1 i β i . (16) Subsequen tly , follo wing her observ ations of a j, 0 , j ∈ N i and from her knowledge of her neighbor’s priors and signal likelihoo d structure, agen t i infers the observed v alues of P n j p =1 ξ ( s j,p ) for all her neigh b ors. Hence, w e get n j X p =1 ξ ( s j,p ) = ( n j + δ − 1 j β j ) a j, 0 − σ − 1 j α j , ∀ j ∈ N i . (17) The observ ations of agent i are therefore augmented b y the set of indep enden t samples from her neigh b ors: { P n j p =1 ξ ( s j,p ) : j ∈ N i } , and her reﬁned b elief at time 1 is again a member of the conjugate family F γ ,η and is giv e b y: M i, 1 ( · ) = V ( · ; α i + X j ∈N i σ j n j X p =1 ξ ( s j,p ) , β i + X j ∈N i n j δ j ) . W e can again inv oke the linearity of the Bay es estimate for the conjugate family F γ ,η and the subsequen t result in ( 15 ), to get that the exp ected pay-oﬀ maximizing action at time 1 is given by: a i, 1 = δ i  α i + P j ∈N i σ j P n j p =1 ξ ( s j,p )  σ i  β i + P j ∈N i n j δ j  . (18) Finally , we can use ( 17 ) to replace for the neigh b oring signals and derive the expression of the action up date of agent i at time 1 in terms of her own and the neighboring actions a j, 0 , j ∈ N i ; leading to the expression of linear Ba y esian heuristics as claimed in Theorem 1 .  Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 20 App endix B: Pro of of Theorem 2 . The balancedness of likelihoo ds (Assumption 3 ) ensures that the co eﬃcien ts of the linear con bination from Corollary 1 sum to one: P j ∈N i T ij = 1, for all i ; th us forming a con vex combination as in the DeGro ot mo del. Subsequen tly , the agen ts b egin b y setting a i, 0 = P n i p =1 ξ ( s i,p ) /n i according to ( 16 ), and at every t > 1 they up date their actions according to a t = T a t − 1 , where a t = ( a 1 ,t , . . . , a n,t ) T and T is the n × n matrix whose i, j -th en try is T ij . Next note from the analysis of con v ergence for DeGro ot mo del, cf. [ 38 , Prop orition 1], that for a strongly connected netw ork G if it is ap erio dic (meaning that one is the greatest common divisor of the lengths of all its circles; and it is the case for us, since the diagonal entries of T are all non-zero), then lim τ →∞ T τ = 1 s T , where s := ( s 1 , . . . , s n ) T is the unique left eigen v ector asso ciated with the unit eigenv alue of T and satisfying P n i =1 s i = 1, s i > 0, ∀ i . Hence, starting from non-informativ e priors agents follow the DeGro ot up date and if G is also strongly connected, then they reac h a consensus at s T a 0 = P n i =1 s i ( P n i p =1 ξ ( s i,p ) /n i ).  App endix C: Pro of of Theorem 3 . W e b egin by a lemma that determines the so-called global MVUE for eac h i , i.e. the MVUE of m i,θ giv en all the observ ations of all agen ts across the net w ork. Lemma 2 (Global MVUE) . Under the exp onential family signal-utility structur e (Assump- tion 1 ), the (glob al) MVUE of m i,θ given the entir e set of observations of al l the agents acr oss the network is given by: a ? i = δ i  α i + P n j =1 σ j P n j p =1 ξ ( s j,p )  σ i  β i + P n j =1 n j δ j  . (19) If we further imp ose non-informative priors (Assumption 2 ), then the glob al MVUE for e ach i c an b e r ewritten as a ? i = δ i  P n j =1 σ j P n j p =1 ξ ( s j,p )  σ i  P n j =1 n j δ j  = δ i σ i n X j =1 σ j n j P n p =1 n p δ p a j, 0 . (20) This lemma can b e pro ved easily b y follo wing the same steps that lead to ( 18 ) to get ( 19 ); making the necessary substitutions under Assumption 2 yields ( 20 ). F rom ( 20 ), it is immediately clear that if some consensus action is to b e the eﬃcien t estimator (global MVUE) for all agents i ∈ [ n ], then w e need δ i σ j = σ i δ j for all i, j ; hence, the global balance is indeed a necessary condition. Under this condition, the lo cal balance of likelihoo ds (Assumption 3 ) is automatically satisﬁed and given non- informativ e priors Theorem 2 guarantees conv ergence to a consensus in strongly connected so cial net w ork. Moreov er, we can rewrite ( 20 ) as a ? i = a ? = ( P n j =1 δ j n j a j, 0 ) / P n p =1 n p δ p , for all i . Hence, if the consensus action ( s T a 0 in the pro of of Theorem 2 , Appendix B ) is to b e eﬃcien t then we need s i = δ i n i / P n j =1 n j δ j for all i ; s = ( s 1 , . . . , s n ) being the unique normalized left eigenv ector asso ciated with the unit eigenv alue of T : s T T = s T , as deﬁned in App endix B . Using δ i σ j = σ i δ j , w e can also rewrite the co eﬃcien ts T ij of the DeGro ot update in Theorem 2 as T ij = δ j n j / ( P p ∈N i n p δ p ). Therefore, by expanding the eigenv ector condition s T T = s T w e obtain that in order for the consensus action s T a 0 to agree with the eﬃcient consensus a ? , it is necessary and suﬃcient to hav e that for all j n X i =1 s i T ij = n X i =1 δ i n i P n j =1 δ j n j ! δ j n j [ I + A ] ij P p ∈N i n p δ p = s j = δ j n j P n j =1 δ j n j , (21) Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 21 or equiv alen tly , X i : j ∈N i δ i n i P p ∈N i n p δ p = X i ∈N out j δ i n i P p ∈N i n p δ p = 1 , (22) for all j . Under the global balance condition (Assumption 4 ), δ i σ j = δ j σ i , the weigh ts T ij = δ j n j / ( P p ∈N i n p δ p ) as given ab o ve, corresp ond to transition probabilities of a no de-weigh ted ran- dom walk on the so cial netw ork graph, cf. [ 16 , Section 5]; where each no de i ∈ [ n ] is weigh ted b y w i = n i δ i . Such a random walk is a sp ecial case of the more common type of random walks on weigh ted graphs where the edge weigh ts determine the jump probabilities; indeed, if for any edge ( i, j ) ∈ E w e set its weigh t equal to w i,j = w i w j then the random w alk on the edge-w eigh te d graph reduces to a random walk on the no de-weigh ted graph with no de w eights w i , i ∈ [ n ]. If the so cial netw ork graph is undirected and connected (so that w i,j = w j,i for all i, j ), then the edge- w eigh ted (whence also the no de-weigh ted) random walks are time-rev ersible and their stationary distributions ( s 1 , . . . , s n ) T can b e calculated in closed form as follo ws [ 2 , Section 3.2]: s i = P j ∈N i w i,j P n i =1 P j ∈N i w i,j . (23) In a no de-w eighted random w alk we can replace w i,j = w i w j for all j ∈ N i and ( 23 ) simpliﬁes in to s i = w i P j ∈N i w j P n i =1 w i P j ∈N i w j . Similarly to ( 21 ), the consensus action will b e eﬃcien t if and only if s i = w i P j ∈N i w j P n i =1 w i P j ∈N i w j = w i P n k =1 w k , ∀ i, or equiv alen tly: ( n X k =1 w k ) X j ∈N i w j = n X i =1 ( w i X j ∈N i w j ) , ∀ i, whic h holds true only if P j ∈N i w j is a common constant that is the same for all agents, i.e. P j ∈N i w j = P j ∈N i δ j n j = C 0 > 0 for all i ∈ [ n ]. Next replacing in ( 22 ) yields that, in fact, C 0 = P i ∈N out j δ i n i for all j , completing the proof for the conditions of eﬃciency .  App endix D: Pro of of Theorem 4 . W e address the more general case of ( 5 ), whic h includes ( 12 ) as a special case (with common uniform priors). F or an y pair of states ˆ θ and ˇ θ , deﬁne the log ratio of b eliefs, lik eliho o ds, and priors as follo ws: φ i,t ( ˆ θ , ˇ θ ) := log  µ i,t ( ˆ θ ) / µ i,t ( ˇ θ )  , λ i ( ˆ θ , ˇ θ ) := log  ` i ( s i | ˆ θ ) /` i ( s i | ˇ θ )  , γ i ( ˆ θ , ˇ θ ) := log  ν i ( ˆ θ ) /ν i ( ˇ θ )  , By concatenating the log-ratio statistics of the n net work ed agen ts, w e obtain the following three v ectorizations for the log-ratio statistics: φ t ( ˆ θ , ˇ θ ) := ( φ 1 ,t ( ˆ θ , ˇ θ ) , . . . , φ n,t ( ˆ θ , ˇ θ )) , λ ( ˆ θ , ˇ θ ) := ( λ 1 ( ˆ θ , ˇ θ ) , . . . , λ n ( ˆ θ , ˇ θ )) , γ ( ˆ θ , ˇ θ ) := ( γ 1 ( ˆ θ , ˇ θ ) , . . . , γ n ( ˆ θ , ˇ θ )) . Rahimian and Jadbabaie: Bayesian Heuristics for Group Decisions 22 Using the ab ov e notation, w e can rewrite the log-linear belief up dates of ( 5 ) in a linearized v ector format as sho wn b elo w: φ t +1 ( ˆ θ , ˇ θ ) =( I + A ) φ t ( ˆ θ , ˇ θ ) − Aγ ( ˆ θ , ˇ θ ) =( I + A ) t +1 φ 0 ( ˆ θ , ˇ θ ) − t X τ =0 ( I + A ) τ Aγ ( ˆ θ , ˇ θ ) =( I + A ) t +1  λ ( ˆ θ , ˇ θ ) + γ ( ˆ θ , ˇ θ )  − t X τ =0 ( I + A ) τ Aγ ( ˆ θ , ˇ θ ) =( I + A ) t +1 λ ( ˆ θ , ˇ θ ) + ( I + A ) t +1 − t X τ =0 ( I + A ) τ A ! γ ( ˆ θ , ˇ θ ) . Next w e use the sp ectral decomposition in ( 13 ) to obtain: 1 φ t +1 ( ˆ θ , ˇ θ ) = (1 + ρ ) t +1 r 1 Λ ( ˆ θ , ˇ θ ) + ((1 + ρ ) t +1 − t X τ =0 (1 + ρ ) τ ρ ) r 1 β ( ˆ θ , ˇ θ ) + o ((1 + ρ ) t +1 ) = (1 + ρ ) t +1 r 1 Λ ( ˆ θ , ˇ θ ) + (1 − t X τ =0 (1 + ρ ) τ − t − 1 ρ ) r 1 β ( ˆ θ , ˇ θ ) + o (1) ! → (1 + ρ ) t +1 r 1 Λ ( ˆ θ , ˇ θ ) , (24) where w e adopt the following notations for the global log lik eliho od and prior ratio statistics: β ( ˆ θ , ˇ θ ) := α T γ ( ˆ θ , ˇ θ ) and Λ ( ˆ θ , ˇ θ ) := α T λ ( ˆ θ , ˇ θ ); furthermore, in calculation of the limit in the last step of ( 24 ) w e use the geometric summation iden tit y P ∞ τ =0 ρ (1 + ρ ) τ − 1 = 1. T o pro ceed denote Λ ( ˆ θ ) := P n i =1 α i ` i ( s i | ˆ θ ) so that Λ ( ˆ θ , ˇ θ ) = Λ ( ˆ θ ) − Λ ( ˇ θ ). Since Θ ♦ consists of the set of all maximizers of Λ ( ˆ θ ), w e hav e that Λ ( ˆ θ , ˇ θ ) < 0 whenever ˇ θ ∈ Θ ♦ and ˆ θ 6∈ Θ ♦ . Next recall from ( 24 ) that φ t +1 ( ˆ θ , ˇ θ ) → (1 + ρ ) t +1 r 1 Λ ( ˆ θ , ˇ θ ) and r 1 is the right P erron-F rob enius eigenv ector with all p ositive en tries; hence, for all ˜ θ ∈ Θ ♦ and any ˆ θ , φ i,t ( ˆ θ , ˜ θ ) → −∞ if ˆ θ 6∈ Θ ♦ and φ i,t ( ˆ θ , ˜ θ ) = 0 whenev er ˆ θ ∈ Θ ♦ ; or equiv alently , µ i,t ( ˆ θ ) / µ i,t ( ˜ θ ) → 0 for all ˆ θ 6∈ Θ ♦ , while µ i,t ( ˆ θ ) = µ i,t ( ˜ θ ) for an y ˆ θ ∈ Θ ♦ . The latter together with the fact that P ˜ θ ∈ Θ µ i,t ( ˜ θ ) = 1 for all t implies that with probability one: lim t →∞ µ i,t ( ˜ θ ) = 1 / | Θ ♦ | , ∀ ˜ θ ∈ Θ ♦ and lim t →∞ µ i,t ( ˜ θ ) = 0 , ∀ ˜ θ 6∈ Θ ♦ as claimed in the Theorem. In the special case that Θ ♦ is a singleton, { θ ♦ } = Θ ♦ , we get that lim t →∞ µ i,t ( θ ♦ ) = 1 almost surely for all i ∈ [ n ]. Ac knowledgmen ts. This w ork was supported by AR O MURI W911NF-12-1-0509. References [1] Acemoglu D, Dahleh MA, Lob el I, Ozdaglar A (2011) Ba y esian learning in so cial net works. 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Bayesian Heuristics for Group Decisions

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