A Bayesian Simulation Model of Group Deliberation and Polarization

Transcription

1 A Bayesian Simulaion Model of Group Deliberaion and Polarizaion Olsson, Erik J Published in: Bayesian Argumenaion 2013 Link o publicaion Ciaion for published version (APA): Olsson, E. J. (2013). A Bayesian Simulaion Model of Group Deliberaion and Polarizaion. In F. Zenker (Ed.), Bayesian Argumenaion (Vol. Synhese Library). Springer. General righs Copyrigh and moral righs for he publicaions made accessible in he public poral are reained by he auhors and/or oher copyrigh owners and i is a condiion of accessing publicaions ha users recognise and abide by he legal requiremens associaed wih hese righs. Users may download and prin one copy of any publicaion from he public poral for he purpose of privae sudy or research. You may no furher disribue he maerial or use i for any profi-making aciviy or commercial gain You may freely disribue he URL idenifying he publicaion in he public poral Take down policy If you believe ha his documen breaches copyrigh please conac us providing deails, and we will remove access o he work immediaely and invesigae your claim. L UNDUNI VERS I TY PO Box L und

2 A Bayesian Simulaion Model of Group Deliberaion and Polarizaion Erik J. Olsson Olsson, E. J. (2013). A Bayesian Simulaion Model of Group Deliberaion and Polarizaion. In Zenker, F. (Ed.) Bayesian Argumenaion, Synhese Library. Springer Verlag. Absrac: The paper describes a simulaion environmen for episemic ineracion based on a Bayesian model called Lapua. An inerpreaion of he model is proposed under which he exchanges aking place beween inquirers are argumenaive. The model, under his inerpreaion, is seen o survive he polarizaion es: if iniially disposed o judge along he same lines inquirers in Lapua will adop a more exreme posiion in he same direcion as he effec of group deliberaion, jus like members of real argumenaive bodies. Our model allows us o sudy wha happens o muual rus in he polarizaion process. We observe ha inquirers become increasingly rusing which creaes a snowball effec. We also sudy condiions under which inquirers will diverge and adop conrary posiions. To he exen ha Bayesian reasoning is normaively correc, he boom line is ha polarizaion and divergence are no necessarily he resul of mere irraional group hink bu ha even ideally raional inquirers will predicably polarize or diverge under realisic condiions. The concluding secion commens on he relaion beween he presen model and he influenial and empirically robus Persuasive Argumen Theory (PAT), and i is argued ha he former is essenially subsumable under he laer. 1. Inroducion There has been a lo of experimenal work in social psychology of group deliberaion and some sriking resuls as well (for an overview of early work, see Isenberg 1986). However, here does no seem o be much work focusing on compuer simulaion of deliberaive processes aking he role of argumenaion seriously. 1 This is unlike many oher areas sudying complex sysems, including economics, where simulaion models abound. I is easy o undersand why hey are found o be so useful: he speed a which a compuer simulaion can be carried ou should be compared wih he someimes many monhs required for meiculously planning and execuing a 1 For an overview of exac models of opinion dynamics see Hegselmann and Krause (2006). See also Zollman (2007). 1

3 conrolled experimen; and, moreover, compuer simulaions allow for precise conrol for parameers ha can be exremely difficul o conrol for in real experimens. This increased speed and conrol is gained, obviously, a he expense of realism because simulaion models need o be idealized in order o be compuaionally workable. Laboraory experimenaion and compuer simulaion are herefore complemenary aciviies. This paper conribues o he sudy of simulaion models of group deliberaion wih he aim of expanding he mehodological oolki available o researchers sudying argumenaion in a social seing. The model, called Lapua, allows for sudying no only he dynamics of belief bu also of rus, including muual rus among inquirers. Lapua was developed by Saffan Angere and he auhor, wih Angere being he main originaor and also he programmer behind he simulaion environmen wih he same name. The plan of he paper is as follows. In secion 2, I describe Lapua as a simulaion framework of episemic ineracion, posponing he descripion of he underlying Bayesian model unil secion 3. In secion 4, an inerpreaion is proposed according o which inquirers in Lapua exchange novel argumens on a common issue. I proceed, in secion 5, o es wheher his model of deliberaion exhibis polarizaion effecs. Condiions under which inquirers diverge are also sudied. In he concluding secion, I commen on he relaion beween he presen model and he influenial Persuasive Argumen Theory (PAT). 2. The Lapua simulaion framework I will choose o inroduce Lapua as a simulaion framework leaving he deails of he underlying model for he nex secion. Social neworks are represened and depiced in he program as direced graphs in which he nodes represen inquirers and he links represen communicaion channels (figure 1). 2

4 Figure 1: The social nework of Sherlock Holmes as represened in Lapua. A number of parameers can be se for each inquirer. The iniial degree of belief is he inquirer s iniial credence in proposiion p. Inquiry accuracy is he reliabiliy of he inquirer s own inquiries. The inquiry chance is he probabiliy ha he inquirer will conduc an inquiry. The inquiry rus is he inquirer s degree of rus in her own inquiries. Likewise, here are a number of parameers for each link. The lisen rus is he recipiens rus in he sender. The hreshold of asserion is he degree of confidence in a proposiion ( p or no-p ) required for he sender o submi a corresponding message o he recipien(s). Wheher a message will hen be submied depends on he lisen chance. For insances, if he hreshold is se a.90, his means ha he sender needs o believe p (no-p) o a degree.90 in order for her o send a posiive (negaive) message in he nework. Running Lapua can mean o consruc a nework, such as ha in figure 1, assign iniial values o he inquirer and link parameers, and hen click on a run buon. Wha happens hen is ha Lapua runs hrough a series of seps, each sep represening a chance for an inquirer o conduc an inquiry, o communicae (send, lisen) o he oher inquirers o which she is hooked up, or o do boh. Afer each sep, Lapua will updae he whole nework according o he informaion received by he inquirers in accordance wih he Bayesian model wih which we shall soon become acquained. Thus, a new degree of belief is compued for each inquirer based on he old degree of belief and he new informaion received hrough inquiry and/or lisening o oher inquirers. Lapua also updaes he inquiry rus and lisen rus parameers in accordance wih Bayesian principles. Lapua oupus no jus wha happens o he individual inquirers during simulaion, bu also collecs some saisical daa. Thus, error dela is he difference beween he iniial and final average degrees of belief in he proposiion p, which is assumed rue by convenion. Given error dela, we can compue he veriisic value (V-value) in he sense of Goldman (1999) for a nework evoluion according o he following simple rule: V-value = -error dela. This means ha an error dela of equals a V-value of Angere (forhcoming), Olsson (2011) and Olsson and Vallinder (in press) discuss various applicaions of Lapua relaing o Goldman s veriisic social episemology. See Vallinder and Olsson (in press b) for a furher philosophical applicaion of Lapua. Lapua also allows is user o specify various feaures or desideraa of neworks a an absrac level. The program can hen randomly generae a large number of neworks of differen sizes having hose feaures, leing hem evolve while collecing various saisics. This is done in Lapua s bach window (figure 2), he perhaps mos powerful feaure of he program. 3

5 Figure 2: The bach window in Lapua. In he bach window, various probabiliy disribuions can be seleced for he several inquirer and link parameers. For insance, he fla disribuion for saring belief indicaes ha Lapua, when selecing he iniial credences in p for a generaed nework, will rea all possible credences as being equally likely o be realized. The selecion of a normal disribuion for Inquiry accuracy, cenered around 0.75 means ha Lapua, when selecing he inquiry accuracy for he inquirers in he generaed neworks, will have a preference for assigning an accuracy of 0.75 and surrounding values. The populaion feaure allows he specificaion of he lower and upper sizes of he neworks o be examined. In his case, Lapua is insruced o generae and sudy neworks having 2 o 20 inquirers. Link chance specifies he densiy of he neworks o be sudied. A link chance of 0.25 indicaes a 25 percen chance ha wo inquirers will be conneced by a direced communicaion link. In Figure 2, he number of rials has been se o 1,000, meaning ha Lapua will generae and sudy 1,000 neworks in accordance wih he saisical crieria specified in he bach window. Finally, he number of seps per rial has been se o 100, indicaing ha he focus is inquirer ineracion over a longer period of ime. 3. The underlying Bayesian model I is ime o elucidae he model underlying he simulaion environmen. This secion follows he exposiion in Angere (forhcoming), excep in one main respec: unlike Angere, I will describe he model in a way ha does no presuppose any more specific inerpreaion of he exchanges 4

6 aking place beween inquirers. Formally, we can ake a social nework S o be a se of inquirers, ogeher wih a binary relaion R on, which we call he nework srucure. This means ha, absracly speaking, a social nework is a direced graph. Following Bayesian radiion, he episemic sae of a person α a ime is assumed o be given by a credence funcion C α : L [0,1]. L can be aken o be a classical proposiional language, and C α is assumed o fulfill he sandard axioms of a probabiliy measure. For he purposes of his paper, le us confine ourselves o he case where inquiry is aimed a discovering wheher a single proposiion p is rue or false. Every inquirer will hen have a credence C α (p) in p, which is a real number beween 0 and 1, for every momen. In our model, here are wo fundamenally differen ways for he inquirers o receive new informaion: inquiry and communicaion. Inquiry can here be aken o include any kind of mehod of alering a credence funcion which does no base iself on informaion given by ohers in he nework. Paradigmaic cases of inquiry include observaion, experimen and aking advice from persons ouside he social nework. No all paricipans' approaches o inquiry are he same, and hey end o vary boh in heir degree of aciviy and heir effeciveness. We say ha a resul of inquiry is posiive if i suppors p, and negaive if i suppors no-p. Le S iα + be he proposiion α's inquiry gives a posiive resul a ime, S iα be he proposiion α's inquiry gives a negaive a, and S iα = S iα + S iα he proposiion ha α's inquiry gives some resul a, posiive or negaive. We represen he paricipans' properies qua inquirers by wo probabiliies: he chance P(S iα ) ha, a any momen, α receives a resul from her inquiries, and he chance P(S iα + S iα p) ha, when such a resul is obained, i is he righ one. To simplify maers, we assume ha he chance ha inquiry gives an appropriae resul does no depend on wheher p is rue or false. P(S iα ) will be referred o as α's aciviy, and P(S iα + S iα p) as her apiude. An inquirer wihou ineres in p would generally have a low aciviy value, while one very ineresed in p, bu engaging in inquiry using fauly mehods would have a high aciviy value bu an apiude close o 0.5, or even below ha. In he laer case, he resuls of her inquiry would acually be negaively correlaed wih he ruh. As a simplificaion, we will assume α's aciviy and apiude o be consan over ime, so we will generally wrie hem wihou he ime index. Jus as inquiry represens he flow of informaion ino he nework, communicaion deals wih how his informaion is disseminaed. Analogously o he inquiry noaion we define S + βα = df β sends a posiive message o α a S βα = df β sends a negaive message o α a S βα = df β sends a posiive or a negaive message o α a 5

7 This srengh of a link βα is hen represenable as a probabiliy P(S βα ) being he chance ha β sends some message, wheher posiive or negaive, o α. Given ha β communicaes wih α, wha does she say? And wha makes her say i? We will leave he firs quesion for he nex secion. The second quesion can be answered by referring o a propery of he link βα ha we will call is hreshold of asserion: a value T βα beween 0 and 1, such ha If T βα > 0.5, β sends a posiive message o α only if C β (p) T βα, and a negaive message only if C β (p) 1 T βα ; If T βα < 0.5, β sends a posiive message o α only if C β (p) T βα, and a negaive message only if C β (p) 1 T βα ; and If T βα = 0.5, β sends a posiive or a negaive message o α independenly of wha she believes, which is modeled by leing her pick wha o say randomly. So far we have described how he inquirers in a social nework engage in inquiry and communicaion, bu we have said nohing abou how hey reac o he resuls of hese pracices. The purpose of he following consideraions is o provide enlighenmen in his regard. We define he reliabiliy of α's source σ as R σα= df P(S + σα S σα p) = P(S σα S σα p) This definiion presupposes ha he probabiliy ha any source sends a posiive message, if p is he case, is equal o he probabiliy ha i sends a negaive message, if no-p is he case. This source symmery simplifies our calculaions, alhough i can be relaxed if we encouner cases where i does no provide a reasonable approximaion. For a discussion, see Olsson (2011). I follows a once ha he reliabiliy of α s inquiry is idenical o her apiude. For oher sources, i is an absracion based on hose sources' performances as indicaions of ruh. In general, an inquirer has no direc access o his value, bu his does no sop her from forming beliefs abou i. Since he number of possible values for he chance R σα is infinie, we need o represen α's credence as a densiy funcion insead of a regular probabiliy disribuion. Thus, for each inquirer α, each source σ, and each ime, we define a funcion τ σα : [0,1] [0,1], called α's rus funcion for σ a, such ha b a C α (a R σα b) = τ σα (ρ)dρ 6

8 for a, b in [0,1]. τ σα (ρ) hen gives he credence densiy a, and we can obain he acual credence ha α has in proposiions abou he reliabiliy of her sources by inegraing his funcion. We will also have use for he expression 1 τ σα, represening α's credence densiy for proposiions abou σ no being reliable, which we will refer o as τ σα. I is reasonable o hink ha an inquirer's credences abou chances should influence her credences abou he oucomes of hese chances. The way his should be done is generally known as he principal principle (Lewis 1980). I says ha if α knows ha he chance ha an even e will happen is, hen her credence in e should be exacly. Applied o our case, his means ha he following principle (PP) mus hold: C α (S + σα S σα R σα = ρ p) = ρ C α (S σα S σα R σα = ρ p) = ρ for all, i.e. α's credence in σ giving a posiive repor, given ha he source gives any repor a all, ha σ's reliabiliy is, and ha p acually is he case, should be. We also have use for an independence posulae. While no sricly necessary, such a posulae will simplify calculaions and modeling considerably. The independence assumpion we use here will be referred o as communicaion independence (CI): C α (p S σα R σα = ρ) = C α (p)c α (S σα )R σα (p) Communicaion independence implies ha wheher σ says anyhing is independen of wheher p is acually rue as well as of σ s reliabiliy. Given (PP) and (CI) we can now define he following expression for α s credence in σ s reliabiliy (see Angere, forhcoming, for he derivaion): 1 0 C α (S + σα p) = C α (S σα ) ρτ σα (ρ)dρ The inegral in his expression is he expeced value τ σα of he rus funcion τ σα, whence (*) C α (S + σα p) = C α (S σα ) τ σα Similarly, 7

9 (**) C α (S + σα p) = C α (S σα ) τ σα We can now derive he crucial expressions C α (p S + σα ) and C α (p S σα ), he credence an inquirer should place in p a given ha she receives a posiive or a negaive message, respecively, from a single source σ: C α (p S + σα ) = C α (p) τ σα C α (p) τ σα +C α ( p) τ σα C α (p S σα ) = C α (p) τ σα C α (p) τ σα +C α ( p) τ σα where τ σα is he expeced value of he rus funcion τ σα. By he Bayesian requiremen of condiionalizaion, we mus have C +1 α = C α (p S + σα ), whenever σ is he only source giving informaion o α a. This means ha hese formulae compleely deermine how α should updae her credence in such a case. No only α's credence in p should be updaed, however. Equally imporan is for α o keep rack of how much o rus her sources. A source ha generally gives very unlikely repors is unlikely o be veridical, and an inquirer should adjus her rus funcion in ligh of his. I urns ou ha our model already deermines how o do his bu we will no go ino he deails here. A full accoun can be found in Angere (forhcoming). Suffice i o menion he following consequence of our model: Even if an inquirer happens o be a perfec inquirer insofar as her inquiry always gives he righ resul, a fairly low sabiliy of her faih in inquiry, ogeher wih her prior judgmen ha p is unlikely, may conspire o make her disrus her own inquiry. This, in urn, may give rise o a vicious circle in which she becomes more and more convinced ha p is false and ha her inquiry is negaively correlaed wih he ruh. The presen model gives rise o a number of qualiaive updaing rules in he case of one message received. We say an inquirer russ a given source if he inquirer s credence in he reliabiliy of he source is greaer han 0.5; disruss he source if i is less han 0.5; and neiher russ nor disruss he source oherwise. We say ha a message ha p (no-p) was surprising o an inquirer if, prior o receiving he message, he inquirer s credence in p (no-p) was less han 0.5; expeced if i was greaer han 0.5; and neiher surprising nor expeced oherwise. In able 1, a +-sign in he lef componen of a pair (_, _) means ha he inquirer s curren belief is reinforced (i.e. her credence in he conclusion is srenghened if above 0.5 and weakened if below 0.5). A - sign means ha he inquirer s curren belief is weakened (i.e. her credence in he conclusion is weakened if above 0.5 and srenghened if below 0.5), whereas 0 means ha he inquirer s credence in he conclusion is lef unchanged. A +-sign in he righ componen of a pair (_, _) 8

10 signifies ha he juror s rus in he source (i.e. credence in is reliabiliy) is srenghened, a - sign ha i is weakened, and 0 ha i is lef unchanged. Table 1 shows how updaing on he informaion from one source affecs an inquirer under various circumsances. Message expeced Neiher nor Message surprising Source rused (+, +) (a) (+, 0) (b) (-, -) (c) Neiher nor (0, +) (d) (0, 0) (e) (0, -) (f) Source disrused (-, +) (g) (-, 0) (h) (+, -) (i) Table 1: Single message updaing in Lapua (for credences sricly beween 0 and 1). Suppose, for example, ha inquirer α s prior credence in p is 0.7. Now α receives a posiive message, i.e. a message in suppor of p, from β, who we assume o be rused by α. Since he message is expeced and he source is rused, we have he siuaion described in cell (a) in able 1. Accordingly, α will reac by raising boh her degree of belief in p and her degree of rus in β. If, by conras, he message sen by β is negaive, we have he siuaion depiced in cell (c), so ha α will respond by lowering boh her degree of belief in p and her rus in β. 2 Le α be he se of all sources from which α receives informaion a. Our Bayesian framework requires ha credences be updaed by means of condiionalizaion: (Cond) C +1 α (p) = C α (p S σα ), where he conjuncion runs over all σ in α. S σα is he message ha α receives from σ a, i.e., + eiher S σα or S σα. The righ hand side of (Cond) can be very hard o assess in he absence of furher assumpions. We can simplify he siuaion considerably by assuming source independence (SI): C α ( S σα p) = C α (S σα p) C α ( S σα p) = C α (S σα p) 2 For proofs of he resuls summarized in Table 1 see Vallinder and Olsson (in press a)), which is a deailed sudy of he dynamics of rus in he Lapua model. 9

11 Source independence saes ha he informaion coming from he sources is independen condiional on he ruh as well as on he falsiy of p. This is he sandard Bayesian way of capuring he idea ha here is no direc influence beween he sources, e.g., ha hey have no conspired o give a cerain message (see, for insance, chaper 2 in Olsson 2005). Given source independence, we can relaively easily compue he lef-hand side of (Cond) by relying on Bayes heorem ogeher wih he heorem of oal probabiliy. See he appendix o his paper for an example of how he machinery works and of he imporan role played by he assumpion of source independence in he updaing of credences. The boom line is ha, given he complexiy of he subjec maer, Lapua is a simple and workable model once we assume source independence. Bu ha, of course, is a echnical moivaion and no a philosophical one. The quesion is wheher an inerpreaion of Lapua can be found under which source independence is rue or a leas highly plausible. This is he issue o which we now urn. 4. Inerpreing Lapua In order o be informaive, an inerpreaion of Lapua should say somehing more precise abou wha kind of messages inquirers receive from inquiry and from he oher inquirers. On wha I will call he opinion disclosure inerpreaion of he model he posiive messages are simply messages o he effec ha p is he case, and he negaive messages ha no-p is he case. We le S + iα be he proposiion α's inquiry signaled ha p is he case a ime, S iα be he proposiion α's inquiry signaled ha no-p is he case a, and S iα = S + iα S iα he proposiion ha α's inquiry signaled eiher ha p or ha no-p is he case a. Similarly, S + βα = df β disclosed her opinion ha p o α a S βα = df β disclosed her opinion ha no-p o α a S βα = df β disclosed her opinion ha p or ha no-p o α a Thus wha happens, a a given poin in a social nework evoluion, is ha one or more inquirers receive messages o he effec ha p (no-p) is he case from heir own inquiries and/or from he oher inquirers. Social nework ineracion on his inerpreaion consiss largely in repeaed disclosure of opinions. The opinions are disclosed only o hose oher inquirers wih whom he inquirer can communicae. The inquirers hen updae heir credence in p a each round by condiionalizaion in he manner described above. This was he original inerpreaion of Lapua as laid ou in Angere (forhcoming) and Olsson (2011). Under i, Lapua can be used, a leas in principle, for sudying he mere exposure effec in 10

12 social psychology, he claim being ha mere exposure o oher group members posiions on some issue can move a given member s credence in similar direcions (for an overview, see Isenberg 1986, pp ). However, here is a problem wih his inerpreaion which needs o be menioned. Suppose an inquirer is repeaedly exposing oher inquirers o her opinion wihou her receiving any new informaion from inquiry in he meanime. Suppose, for example, ha she repeaedly informs he ohers ha her opinion is ha p is rue in consecuive seps of he deliberaion. As Lapua is buil, his will ypically lead he oher inquirers o repeaedly updae heir credence in p in a posiive direcion and o adop an ever increasing rus in he discloser. While his effec may be of lile saisical significance in he end, i is cerainly counerinuiive. There is anoher inerpreaion which does no have his problem. On his inerpreaion wha are exchanged among he inquirers are no opinions bu argumens. More precisely, inquirers exchange argumens for or agains he proposiion p. Since Lapua does no represen he srucure of argumens, his inerpreaion is in some need of jusificaion. Our saring poin will be he assumpion ha deliberaion, as sudied here, is cooperaive, much in he sense of Grice s maxims for cooperaive communicaion (Grice 1975). Thus, we assume ha inquirers adhere o he Maxims of Qualiy and of Relaion. The former saes ha one should no convey wha is believed o be false or unjusified. According o he laer, one should make conribuions ha are relevan. Giving an invalid argumen or an argumen wih false premises would be in violaion of he Maxim of Qualiy. Cooperaive communicaion requires ha all argumens be sound, i.e. valid and based on rue premises a leas in he eyes of he proponen. In his paper, we will ake he inquirers compeence in his regard for graned. The inernal srucure of argumens is imporan if he argumens presened can fail o be sound. The receiver can hen deermine wheher he argumen is valid and based on rue premises by idenifying he argumen srucure, including he premises and he mode of inference (deducive, inducive ec.). Bu if all argumens presened are sound, as we have assumed hem o be, hen i is less obvious ha argumen srucure is of saisical imporance. The assumpion of soundness can herefore be used o moivae viewing argumens as black boxes wihou any inernal srucure. Wha is imporan in an argumen, from his perspecive, is wheher i is a pro or a con argumen vis-à-vis he issue a sake. From his perspecive, he Lapua model makes sense as a simplified and idealized model of argumenaion. We have ye o explain why he problem of repeiion does no arise under his inerpreaion. The key idea is o hink of he argumens ha are pu forward by inquirers in Lapua as novel argumens, i.e. argumens ha have no been advanced earlier in he deliberaion process. Hence, if an inquirer repeaedly argues ha p his should no be inerpreed as he inquirer repeaing he same argumen for p, bu as her advancing a series of 11

13 novel argumens o ha conclusion. If so, he fac ha he inquirers on he receiving side will repeaedly updae heir credence in he conclusion and heir rus in he proponen is no unreasonable. On he conrary, i is wha one would expec should happen. The assumpion of novely can be jusified as follows. Kaplan (1977) found ha, if argumens are presened ha he individual group member is already aware of, a shif in his or her posiion will no occur as a resul of he discussion. The saing of he argumen will be seen as an irrelevan deliberaive conribuion. Vinograd and Bernsein (1978) repor similar findings. In oher words, giving an argumen which has already been aken ino accoun violaes he Maxim of Relaion, which we have assumed ha he inquirers adhere o. If he nework is fully conneced so ha every argumen is presened o everyone, here will be common knowledge abou which argumens have already been presened. Hence, only novel argumens will be advanced. (If he nework is no fully conneced, we adop he same assumpion ha all argumens presened are novel as a useful idealizaion.) We say ha an argumen is posiive if is conclusion is p, and ha i is negaive if is conclusion is no-p. Puing ogeher wha was said above, he proposal is ha we ake S + iα o mean α's inquiry produced a novel posiive argumen a ime, S iα o mean α's inquiry produced a novel negaive argumen a, and S iα = S + iα S iα o mean ha α's inquiry produced some novel argumen, wheher posiive or negaive, a. Similarly, S + βα = df β presened a novel posiive argumen o α a S βα = df β presened a novel negaive argumen o α a S βα = df β presened a novel negaive or a novel posiive argumen o α a The following is a consequence of Lapua under he argumenaion inerpreaion: If T βα > 0.5, β presens a posiive argumen o α only if C β (p) T βα, and a negaive argumen only if C β (p) 1 T βα. Thus, if he hreshold of asserion exceeds 0.5, hen he inquirer will presen an argumen, wheher i be posiive or negaive, if her confidence in he conclusion exceeds he hreshold. A hreshold of asserion exceeding 0.5 capures a sense in which deliberaing agens are sincere. While his is surely he normal case, he model is general enough o allow for inquirers o be insincere, in he following sense: 12

14 If T βα < 0.5, β uers a posiive argumen o α only if C β (p) T βα, and negaive argumen only if C β (p) 1 T βα. In oher words, a hreshold of asserion below 0.5 is inerpreed as a liar hreshold : he inquirer will give an argumen for p only if her degree of belief in p is sufficienly low; and an argumen for no-p only if her degree of belief in no-p is sufficienly low. Seing he hreshold of asserion o a number below 0.5 can be used o model a kind of sraegic communicaion, e.g., lying or acing as he devil s advocae, in he sense of giving an argumen for p (no-p) while personally believing p (no-p) o be false. Finally, if T βα = 0.5, β can uer a posiive or a negaive argumen o α independenly of wha she believes, which is modeled by leing her pick wha o say randomly. 3 The source independence assumpion saes ha inquirers rea oher inquirers as giving independen informaion. Wheher or no we choose he opinion disclosure or he argumenaion inerpreaion of Lapua, assuming source independence has he effec of disconnecing inquirers from realiy afer a few deliberaive rounds. The reason is ha inquirers, when updaing heir credences in p, will ake ino accoun no only he resul of heir own inquiries bu also he informaion coming from oher inquirers, wheher ha informaion is inerpreed as disclosed opinions or novel argumens. This will lead o he credences of inquirers becoming, wih ime, increasingly dependen. Afer a while, posiive (negaive) repors coming from oher inquirers canno be aken anymore as independen indicaions ha p (no-p) is rue, and ye he lisening inquirers in Lapua will rea hem as such. No reason has been presened, however, indicaing ha source independence sysemaically disors simulaion resuls in any paricular direcion. And arguably, source independence is psychologically realisic as a defaul sraegy: real inquirers have a endency o assume source independence in he absence of concree reasons o hink ha sources are no independen. Keeping in mind he considerable simplifying effecs source independence has on he enire model, we are herefore jusified in acceping i as a highly useful idealizaion. 5. Do Bayesian inquirers polarize? 3 The original idea behind Lapua was o simulae communicaion based on inquiry. A possible drawback wih he argumenaion inerpreaion is ha i decouples inquiry from communicaion. The exisence of argumens is no brough in relaion o he resul of inquiry, and wheher or no an inquirer possesses an argumen is no represened in he model. We have been experimening wih a version of he program in which communicaion is possible only if new inquiry has aken place. Preliminary simulaions sugges ha his modificaion does no have any significan saisical effec on simulaion oucome. 13

15 In early work in social psychology i was observed ha group decisions are someimes riskier han he previous privae decisions of he group's members. 4 This observaion paved he way for numerous sudies showing ha risky shif is a pervasive phenomenon bu also ha on cerain decisions groups are acually more cauious ha heir members. Boh risky and cauious shifs are special cases of a group-induced aiude polarizaion (e.g. Moscovici and Zavalloni, 1969). Group polarizaion is said o occur when an iniial endency of individual group members oward a given direcion is enhanced following group discussion (Isengren, 1986, p. 1141) so ha members of a deliberaing group predicably move oward a more exreme poin in he direcion indicaed by he members predeliberaion endencies (Sunsein, 2002, p. 176, ialics removed). Thus, a group of moderaely profeminis women will be more srongly profeminis following group discussion (Myers, 1975). Given ha [g]roup polarizaion is among he mos robus paerns found in deliberaing bodies (Sunsein, 2002, p. 177), we can use polarizaion as a es of empirical adequacy ha any reasonably realisic model of group deliberaion should saisfy. In his secion we es wheher inquirers in Lapua polarize under wha would appear o be normal circumsances characerized by (i) some prior rus in he reliabiliy of he ohers, (ii) an inclinaion o give argumens only if he conclusion is perceived o be more likely o be rue han false, and (iii) an admission o alk in he absence of a high degree of credence in he conclusion. Figure 3 shows he exac parameer seings in he bach window of Lapua. Figure 3 4 Isengren (1986) credis an unpublished maser hesis by James Soner wih his discovery (Soner 1991). 14

16 I was assumed ha he inquirers engage in a closed room debae wihou underaking any inquiry whils deliberaing. Hence, he inquiry chance parameer was se o 0 and he link chance o 1, making every announcemen public wihin he group. The hreshold of asserion was aken o be normally disribued around The social rus parameer (credence in he reliabiliy of ohers) was assumed o be normally disribued in he area above 0.5. Finally, he iniial degree of belief (credence) in p was aken o be posiive and normally disribued jus above 0.5. Lapua was hen insruced o generae 1,000 neworks ( rials ) saisfying hese consrains, allowing each nework o evolve 10 seps. The resul is depiced in figure 4. Figure 4 The lower diagram of figure 4 shows he evoluion of he average credence in p over ime. As we see, afer a view seps he average credence in p converged o a value slighly below 1. The upper diagram of figure 4 shows he number of inquirers per final credence in p afer 10 deliberaive rounds. Virually all inquirers ended up assigning p a credence close o 1. These observaions confirm our predicion: inquirers in Lapua polarize in he sense ha if every inquirer is iniially inclined o believe p, however cauiously, hey will sill believe p afer deliberaion, only much more srongly. The effec is he same muais muandis, if he inquirers iniially favor no-p raher han p, in which case hey will end up believing no-p more srongly. We recall ha inquirers in Lapua updae heir degree of rus in he oher inquirers dynamically, alhough we have no deailed he mechanisms behind rus in his paper (see 15

17 Probabiliy Angere forhcoming). Inuiively, we would expec polarizaion wih regard o he proposiion a sake o be accompanied by increased muual rus among he inquiring agens. This is indeed wha happens in Lapua. This effec was sudied for a small nework of only wo inquirers under circumsances similar o hose in figure 3. More precisely, communicaion chances for inquirer 1 and inquirer 2 were se o 0.94 and 0.88, respecively; and he hreshold for he links ougoing from inquirer 1 and ougoing from inquirer 2 were se o 0.66 and 0.67, respecively. Figure 5 shows he resul. 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 Inq 1 DB Inq 1 rus Inq 2 DB Inq 2 rus 0,2 0, Figure 5 The horizonal axis shows ime. The verical axis displays he relevan credences. We see ha as he inquirers polarize wih regard o heir credence, or degree of belief (DB), in p, hey become increasingly more rusing vis-à-vis each oher. These resuls are easily explained given wha we know abou he underlying Bayesian model. If he inquirers are iniially inclined owards p and some have a hreshold of asserion allowing hem o communicae, he laer will give novel argumens in favor of p. These argumens will be aken ino accoun by he lisening inquirers in he manner previously described, leading hem o adjus heir credence in p, as well as heir rus in he source, upwards (see cell (a) in able 1). Wih ime, an increasing number of inquirers will find heir credence in p exceed heir hreshold of asserion, encouraging hem o give furher argumens for p. This will push credences in p sill 16

18 furher in he posiive direcion, and so on. A he same ime, he growing sense of being confirmed by he ohers will lead o increased muual rus among he inquirers, adding furher momenum o heir polarizaion. Moreover, his also shows ha, in normal cases, polarizaion on he belief level is accompanied, in a sense, by polarizaion on he rus level: he iniially shared aiude of rus is reinforced as he effec of deliberaion. Our sudy raises he furher quesion wha happens in more unusual cases, e.g. when people do no rus each oher or hey lie in he slighly echnical sense of giving argumens for a conclusion hey do no believe in. There are hree cases o consider: people rus bu lie, people disrus bu ell he ruh, people disrus and lie. Using our simulaion program, we esed hese hree cases while keeping all he oher assumpions inac. The resuls are summarized in able 2. Trus Disrus Truh-elling Polarizaion Divergence Lying Divergence Polarizaion Table 2 As we see, here are wo siuaions ha lead o polarizaion, as always in he sense ha likeminded people are srenghened in heir iniial convicions as he effec of deliberaion. One is he normal siuaion which we sudied in he previous secion, i.e. when people rus oher people and do so for good reasons because he ohers are in fac rusworhy. The oher is when people disrus ohers, again for good reasons because he ohers are in fac unrusworhy. These wo cases exemplify wha we migh call siuaions of social calibraion: people s aiudes owards oher people adequaely reflec he acual rusworhiness of he laer. In he wo remaining cases, in which here is lack of social calibraion, we ypically ge a divided sociey: one camp believing he ruh and he oher camp believing he falsehood wih he members of one camp disrusing he members of he oher. 5 5 A he end of a bach simulaion, Lapua oupus he disribuion of average final degrees of belief for all inquirers in all socieies ha were considered. For an example see he upper diagram of Figure 4. Lapua, as i sands, does no oupu he disribuion of final degrees of belief for paricular socieies. Hence, we canno conclude ha socieies ha are no socially calibraed will divide from he daa ha we ge from Lapua while in bach mode. However, during a bach simulaion Lapua randomly selecs socieies for 17

19 Probabiliy We will close his secion by sudying an example of how a sociey consising of serious (ruh-elling) inquirers iniially inclined o believe he same hing can sill end up divided on he issue as he effec of a lack of social calibraion. We will sudy a simple sociey consising of only wo inquirers: Inquirer 1 (Inq 1) and Inquirer 2 (Inq 2). We se lisen chance for Inquirer 1 o 0.94 and for Inquirer 2 o 0.88, and he hreshold for boh inquirers o We choose a normally disribued rus funcion for boh inquirers wih expeced value Figure 6 shows how he inquirers degree of belief (DB) in p, and he expeced value of heir rus funcions, change wih ime. 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, Inq 1 DB Inq 1 rus Inq 2 DB Inq 2 rus Figure 6 We see ha afer some flucuaions he general rend is ha one inquirer will sar believing p while he oher will sar believing no-p. Lapua allows us o inspec he relevan parameers in a sep-wise fashion o see wha causes his resul. This reveals ha he following ranspires: 1. Boh inquirers iniially give argumens for p because heir DB in p is above he hreshold of asserion. 2. Since hey disrus each oher, hey will ake each oher s argumens as evidence for nop and lower heir DB in p (see cell (g) in able 1). 3. Inquirer 1 sill has a DB in p which is above he hreshold, and so she gives an argumen for p. 4. Given her disrus in Inquirer 1, Inquirer 2 becomes raher confiden ha no-p, so she gives an argumen for no-p (cell (g) in able 1). visual represenaion on he compuer screen. Tha visual informaion was used as addiional daa when concluding ha a divided sociey resuls under he condiions given in able 2. 18

20 5. Given her disrus in Inquirer 2, his is aken by Inquirer 1 o be evidence for he opposie, namely p (cell (i) in able 1). 6. By he same oken, Inquirer 1 will coninue o argue for p while Inquirer 2 will coninue o argue for no-p, and hey will become ever more confiden in he conclusions of heir argumens. 7. Evenually Inquirer 1 will become cerain of p and Inquirer 2 cerain of no-p. 8. Meanwhile hey will coninuously downgrade heir degree of rus sill furher because, as hey see i, hey repeaedly receive surprising messages from a disrused source (see cell (i) in able 1). We noe ha while divergence occurs wih respec o credence in p, polarizaion occurs wih respec o rus: he inquirers iniially disrused each oher and his iniial endency is reinforced as he effec of deliberaion. 6. Conclusion and discussion I have argued ha he original inerpreaion of Lapua as a model of opinion disclosure is somewha problemaic due o a problem of repeiion. Insead, I proposed an inerpreaion according o which inquirers are exchanging novel argumens for or agains a arge proposiion. I wen on o show ha he model exhibis polarizaion much like real argumenaive bodies. Inquirers in Lapua, if iniially disposed o believe in a given proposiion, will see heir credences in ha proposiion increase as a resul of group deliberaion. This lends addiional credibiliy o he model as a reasonably realisic represenaion of he phenomena in quesion. We also sudied condiions under which inquirers diverge in heir opinions. To he exen ha Bayesian reasoning is normaively correc, he perhaps mos surprising, and disurbing, resuls of his sudy is ha polarizaion and divergence are no necessarily he resul of mere irraional group hink bu ha even ideally raional inquirers will predicably polarize or diverge under realisic condiions. I remains o compare he presen heory wih he influenial Persuasive Argumen Theory (PAT) which also predics polarizaion. 6 According o PAT an individual s posiion on an issue is a funcion of he number and persuasiveness of pro and con argumens ha he person recalls from memory when formulaing his or her own posiion. Thus in assessing he guil or innocence of an accused in rial, jurors come o predeliberaion decisions on he basis of he relaive number and persuasiveness of argumens favoring guil or innocence. Group deliberaion will cause an individual o shif in a given direcion o he exen ha he discussion exposes ha individual o persuasive argumens favoring ha direcion raher han o argumens favoring he opposie 6 The following accoun is based on he overview in Isenberg (1986), pp

21 direcion. How persuasive an argumen is o a given individual is deermined by he validiy and novely of he argumen. One facor, among several, affecing perceived validiy is he exen o which he argumen fis ino he person s previous views. Novely has o do wih how new and unusual he argumen is o he person in quesion. Everyhing else equal, a novel argumen has a greaer persuasive force han a commonplace argumen. Lapua, as I have proposed o inerpre i, is clearly in he spiri of PAT. Thus, Lapua is also based on he assumpion ha he persuasive effec of an argumen depends essenially on wo facors: is perceived validiy (including he rusworhiness of he presener) and novely. There are also differences. For insance, Lapua is more specific han PAT in assuming ha individual inquirers updae heir degrees of belief in a paricular way, namely ha dicaed by Bayesianism. PAT as such does no posulae any more specific updaing mechanism, le alone a Bayesian one. Lapua assumes, in addiion, ha individuals degrees of rus are dynamically updaed in a Bayesian fashion. Furhermore, inquirers in Lapua engaging in group deliberaion updae heir credences in a piecemeal or sequenial fashion. The presenaion of a novel argumen, or collecion of argumens, will normally affec he receiving inquirer s credence in he conclusion. As PAT is normally formulaed, inquirers are supposed o collec in memory all he argumens hey are presened wih during group deliberaion, posponing heir own verdic on he maer unil deliberaion has come o an end. When he deliberaion has ended he inquirer akes a sand on he basis of a holisic assessmen of he number and meris of he pro and con argumens reained in memory. This holisic aspec of PAT is no unproblemaic in he ligh of experimens indicaing ha he order in which argumens are presened will affec he conclusion reached. Thus, Kaplan and Miller (1977) found ha subjecs end o recall persuasive argumens ha hey had been exposed o mos recenly raher han he ones hey had been exposed o firs. While here may be doubs abou some of he deails of PAT, here are many experimenal sudies poining o is broad empirical adequacy. I is reasonable o suppose ha a fair number of hese sudies will give (indirec) suppor for Lapua under he argumenaion inerpreaion considering he fac ha he laer is, by and large, subsumable under he former. A more careful assessmen of his claim which has he saus of a reasonable conjecure is, however, ouside he scope of he presen aricle. 7 Appendix 7 Acknowledgemen: I am graeful o Saffan Angere and Sephan Harmann for heir commens on previous versions of his paper. 20

22 To illusrae he role played by he condiion of source independence, we consider he case of one inquirer α receiving, a ime, posiives messages from wo sources, σ 1 and σ 2. By (Cond), C +1 α (p) = C α (p S + σ1 α S + σ2 α) = C α (p)c α (S + σ 1α S + σ 2α p) C α (S + σ S 1α + σ ) 2α (Bayes heorem) = C α (p)c α (S + σ 1α S + σ 2α p) C α (S + σ S 1α + σ p)c 2α α (p)+c α (S + σ S 1α + σ p)c 2α α ( p) (Toal probabiliy) = C α (S + σ 1α C α (p)c α (S + σ p)c 1α α S( + σ 2α p) p)c α (S + σ p)c 2α α (p)+c α (S + σ p)c 1α α (S + σ p)c 2α α ( p) (Source independence) = C α (p) τ σ 1α τ σ 2α C α (p) τ σ τ 1α σ +C 2α α ( p) τ σ1α τ σ2α (By (*) and (**)) This means ha we only need hree pieces of informaion in order o compue α s poserior credence in p: α s prior credence in p, he expeced value of α s rus funcion for σ 1 and for σ 2. Supposing hese values o be 0.8, 0.7 and 0.9, respecively, we ge a 0.99 poserior credence in p. The example can be generalized as follows: Theorem 1: Suppose ha α a receives messages from exacly n sources σ 1,, σ n, and ha all messages are posiive. Then C α +1 (p) = C n α (p) τ i=1 σi α C α (p) n τ i=1 σi α +C α ( p) n i=1 τ σi α Proof: Lef o he reader. We can generalize his sill furher. Theorem 2: Suppose ha α a receives messages from exacly n sources σ 1,, σ n, Le Pos α be he se of all indices of sources giving posiive messages, and Negs α be he se of all indices of sources giving negaive messages. Then C α +1 (p) = C n α (p) τ n σi α i Pos τ σi α α iεneg α C α (p) n τ σi α i Pos n τ σi α α iεneg +C α ( p) n τ σi α α i Pos n τ σi α α iεneg α 21

23 Proof: Lef o he reader. Corollary 1: Suppose ha α a receives messages from exacly n sources σ 1,, σ n, for an even n > 0, ha τ σi α Then C α +1 (p) = C α (p). = τ σj α, and ha here is an equal number of posiive and negaive messages. Proof: Follows from heorem 2. References Angere, S. (forhcoming), Knowledge in a Social Nework, submied paper. Goldman, A. I. (1999), Knowledge in a Social World, Clarendon Press, Oxford. Grice, P. (1975), Logic and Conversaion, in Cole, P. and Morgan, J. (eds.), Synax and Semanics, 3: Speech Acs, New York: Academic Press: pp Hegselmann, R., and Krause, U. (2006), Truh and Cogniive Division of Labour: Firs Seps Towards a Compuer-Aided Social Episemology, Journal of Arificial Socieies and Social Simulaion 9 (3). Isenberg, D. (1986), Group Polarizaion: A Criical Review and Mea-Analysis, Journal of Personaliy and Social Psychology 50 (6): Kaplan, M. F. (1977), Discussion Polarizaion Effecs in a Modified Jury Decision Paradigm: Informaional Influences, Sociomery 40: Kaplan, M. F., and Miller, C. E. (1977), Judgmens and Group Discussion: Effec of Presenaion and Memory Facors on Polarizaion, Sociomery 40: Myers, D. G. (1975), Discussion-induced Aiude Polarizaion, Human Relaions 28: Moscovici, S., and Zavalloni, M. (1969), The Group as a Polarizer of Aiudes, Journal of Personaliy and Social Psychology 12: I Olsson, E. J. (2005), Agains Coherence: Truh, Probabiliy, and Jusificaion, Oxford Universiy Press. Olsson, E. J. (2011), A Simulaion Approach o Veriisic Social Episemology, Episeme, in press. Olsson, E. J., and Vallinder, A, (in press), Norms of Asserion and Communicaion in Social Neworks, Synhese. Vallinder, A., and Olsson, E. J. (in press a)), Trus and he Value of Overconfidence: A Bayesian Perspecive on Social Nework Communicaion, Synhese. Valinder, A., and Olsson, E. J. (in press b)), Does Compuer Simulaion Suppor he Argumen from Disagreemen?, Synhese. 22

24 Soner, J. A. F. (1961), A Comparison of Individual and Group Decisions Involving Risk. Unpublished maser's hesis, Massachuses Insiue of Technology, Cambridge, MA. Sunsein, C. R. (2002), The Law of Group Polarizaion, The Journal of Poliical Philosophy 10 (2): Vinokur. A., and Burnsein, E. (I978), Novel Argumenaion and Aiude Change: The Case of Polarizaion Following Group Discussion, European Journal of Social Psychology 8: Zollman, K. J. (2007), The Communicaion Srucure of Episemic Communiies, Philosophy of Science 74 (5):