OPINION DYNAMICS AND BOUNDED CONFIDENCE MODELS, ANALYSIS, AND SIMULATION

Jouna of Atfca Socetes and Soca Smuaton (JASSS) vo.5, no. 3, 02 http://jasss.soc.suey.ac.uk/5/3/2.htm OPINION DYNAMICS AND BOUNDED CONFIDENCE MODELS, ANALYSIS, AND SIMULATION Rane Hegsemann Depatment of Phosophy, Unvesty Bayeuth Uch Kause Depatment of Mathematcs, Unvesty Bemen Abstact When does opnon fomaton wthn an nteactng goup ead to consensus, poazaton o fagmentaton? The atce nvestgates vaous modes fo the dynamcs of contnuous opnons by anaytca methods as we as by compute smuatons. Secton 2 deveops wthn a unfed famewok the cassca mode of consensus fomaton, the vaant of ths mode due to Fedkn and Johnsen, a tme-dependent veson and a nonnea veson wth bounded confdence of the agents. Secton 3 pesents fo a these modes majo anaytca esuts. Secton 4 gves an extensve expoaton of the nonnea mode wth bounded confdence by a sees of compute smuatons. An appendx suppes needed mathematca defntons, toos, and theoems. Keywods: opnon dynamcs, consensus/dssent, bounded confdence, nonnea dynamca systems.. INTRODUCTION Consde a goup of nteactng agents among whom some pocess of opnon fomaton takes pace. The goup may be a sma one, e.g., a goup of expets asked by the UN to mege the dffeent assessments on, say, the magntude of the wod popuaton n the yea, nto one snge judgement. On the othe exteme, the goup may be an ente socety n whch the ndvduas ued by vaous netwoks of soca nfuences deveop a wde spectum of opnons. In each case thee s a pocess of opnon dynamcs that may ead to a consensus among the agents o to a poazaton between the agents o, moe genea, that esuts n a cetan fagmentaton of the pattens of opnons. An undestandng f not an anayss of the nvoved pocess of opnon fomaton s hady possbe wthout an expct fomuaton of a mathematca mode. An eay fomuaton of such a mode was gven by J.R.P. Fench n 956 n ode to undestand compex phenomena found empcay about goups (Fench 956). See aso the eated mathematca anayss by F. Haay n 959 (Haay 959). Anothe souce of opnon dynamcs s the wok by M.H. De Goot n 974 (De Goot 974) and by K. Lehe n 975 (Lehe 975) and subsequent wok by C. G. Wagne, S. Chattejee, E. Seneta, J. Cohen among othes. (See (Wagne 978), (Chattejee 975), (Chattejee and Seneta 977), (Cohen et a. 986) and the efeences gven theen. Thee s an nteestng connecton to the Deph technque fo poong opnons of expets, (cf. (De Goot 974)). The focus of these contbutons s on consensus and how to each t. Ths s tue n patcua fo the encompassng study of Lehe and Wagne (Lehe and Wagne The majo pat of the smuaton esuts n ch. 4 wee poduced dung the academc yea 999-00 when Rane Hegsemann was a membe of the eseach goup "Makng Choces" at the Cente fo Intedscpnay Reseach (ZF) of the Unvesty of Beefed. He woud ke to thank ZF fo hosptaty and the stmuatng nteectua envonment. Many thanks as we to the patcpants of the thee ntenatona confeences (998, 999, and 00, a oganzed by ths eseach goup) whee R. Hegsemann coud pesent the basc appoach and fst esuts.

2 99) whch emphaszes atona consensus as a fundamenta featue fom justce to epstemoogy. The mpotance of modeng dsageement besde consensus was ponted out especay by R. P. Abeson (Abeson 964) and by N.E. Fedkn and E.C. Johnsen (Fedkn and Johnsen 999). Fo exampe, thee s the foowng emak n (Abeson 964, p. 53): Snce unvesa utmate ageement s an ubqutous outcome of a vey boad cass of mathematca modes, we ae natuay ed to nque what on eath one must assume n ode to geneate the bmoda outcome of communty ceavage studes. Offhand, thee ae at east thee potenta ways of geneatng such a bmoda outcome. Fom Fench to Fedkn/Johnsen, as dffcut as these modes mght be n the detas, they ae a compaatvey smpe n the sense that they ae a nea modes. Ths means n patcua that the anayss needed can be caed out by powefu nea technques such as matx theoy, Makov chans and gaph theoy. The fst nonnea mode wthn ths ne of thnkng was fomuated and anayzed n (Kause 997) and (Kause 00). Fo ths mode see aso (Beckmann 997), (Dttme 00), (Dttme 0) and (Hegsemann and Fache 998). A mode sma n spt has ecenty been nvestgated n (Wesbuch et a. 0) and (Deffuant et a. 00). Of couse, havng an expct mathematca mode does not mean at a that one has expct mathematca answes. Aeady R. Abeson obseved that a Moe dastc evson of the mode can be ntoduced by makng the system nonnea. (Abeson 964, p. 50). Though eementay, the mode s nonnea n that the stuctue of the mode changes wth the states of the mode gven by the opnons of the agents (see Secton 2). Not ony that hepfu mathematca toos ke Makov chans ae no onge appcabe, t tuns out, moeove, that goous anaytca esuts ae dffcut to obtan. Fo that eason we cay out the anayss of the above nonnea mode to a age extent by smuatons on the compute. Indeed, though we ae poud to pesent anaytca esuts, too, we want to emphasze the mpotance of caefu compute smuatons fo soca dynamcs n genea, and opnon dynamcs n patcua, wheneve nonneates ae nvoved. (See (Hegsemann and Fache 998). Fo compute smuatons on bnay opnon dynamcs see (Hoyst et a 0), (Latané and Nowak 997), (Stauffe 0), (Stocke et a. 0). In ths pape we addess the cassca queston of eachng a consensus as we as that of dsageement eadng to poazaton and, moe genea, the dynamcs of opnon fagmentaton of whch consensus and poazaton ae just two nteestng speca cases. Secton 2 of the pape deveops and evews wthn a unfed famewok fou modes of contnuous dynamcs: The cassca mode of consensus fomaton, a vaaton of ths mode due to Fedkn and Johnsen, a mode wth tmedependent (nhomogeneous) stuctue and a nonnea mode deang wth bounded confdence. Secton 3 then pesents fo a these modes majo anaytca esuts of whch some ae we-known and some ae new. Fo the convenence of the eade we state the esuts wth a mnmum of mathematcs. Pecse mathematca defntons and theoems togethe wth addtona hnts ae deegated to an Appendx. Secton 4 contans an extensve expoaton of the nonnea mode wth bounded confdence by a sees of compute smuatons. In the pesentaton we took cae to expan step by step the compute expements made. 2. MODELLING OPINION DYNAMICS Consde a goup of agents (o expets o ndvduas of some knd) among whom some pocess of opnon fomaton takes pace. In genea, an agent w nethe smpy shae no stcty dsegad the opnon of any othe agent, but w take nto account the opnons of othes to a cetan extent n fomng hs own opnon. Ths can be modeed by dffeent weghts whch any of the agents puts on the opnons of a the othe agents. Ths pocess of fomng the actua opnon by takng an aveage ove opnons can be epeated agan and eads, theefoe, to a dynamca pocess n dscete tme. Intutvey, one may expect that ths pocess of epeatedy aveagng opnons w bng newy fomed opnons of dffeent agents cose to each othe unt they fow nto a consensus among a agents. In the next two sectons we w show that the dynamcs of opnon fomaton can be much

3 moe compex than one woud ntutvey expect, even n the above smpe mode. The cuca pont hee s that the weghts put on the opnons of othes may change fo easons expaned beow. Ony fo the cassca case of constant weghts and enough confdence among agents the phenomenon of consensus s a typca one (cf., e.g., (De Goot 974), (Lehe 975), and the poneeng but ess foma (Fench 956)). Let n be the numbe of agents n the goup unde consdeaton. To mode the epeated pocess of opnon fomaton we thnk of tme as a numbe o ounds o of peods, that s a as dscete tme T = { 0,, 2, }. It w be assumed that the opnon of an agent can be expessed by a ea numbe as, e.g., n the case of an expet who has to assess a cetan magntude. Ths assumpton s made fo smpcty, because aeady n that case the opnon dynamcs consdeed can be qute ntcate. Ths case s sometmes efeed to as contnuous opnon dynamcs n contast to the, even moe estcted, case of bnay opnon dynamcs (cf. (Wesbuch et a. 0)). Late on we w aso consde hghe dmensona opnons (see the tme-vaant mode n the next secton). Fo a fxed agent, say whee n, we denote the agents opnon at tme t (n ound t) by x ( t ). Thus x ( t ) s a ea numbe and the vecto x () t = ( x (), t, xn ()) t n n- dmensona space epesents the opnon pofe at tme t. Fxng an agent, the weght gven to any othe agent, say j, we denote by a j. To keep thngs smpe we ntoduce a such that a + a2+ + an=. Futhemoe, et aj 0 fo a, j. Havng these notatons, opnon fomaton of agent can be descbed as aveagng n the foowng way x ( t+ ) = a x ( t) + a x ( t) + + a x ( t). (2.) 2 2 n n That s, agent adjusts hs opnon n peod t + by takng a weghted aveage wth weght a j fo the opnon of agent j at tme t. Of couse, weghts can be zeo. Fo exampe, f agent dsegads a othe opnons, ths means a = and a j = 0 fo j ; o, f foows the opnon of j then a j = and a k = 0 fo k j. It s mpotant to note that the weghts may change wth tme o wth the opnon, that s aj = aj (, t x()) t can be a functon of t and/o of the whoe pofe x () t. By coectng the weghts nto a matx, Atxt (, ()) = ( aj (, txt ())), wth n ows and n coumns, we obtan a stochastc matx,.e., a nonnegatve matx wth a ts ows summng up to. Thus, usng matx notaton, the genea fom of ou mode (GM) can be compacty wtten as xt ( + ) = Atxt (, ( )) xt ( ) fo t T. (GM) The man pobem we ae deang wth n ths pape s the foowng one: Gven an nta pofe (stat pofe) x (0) and the dynamcs specfed by the weghts what can be sad about the fna behavo of the opnon pofe,.e., about x () t fo t appoachng nfnty? In patcua, when does the goup of agents appoach a consensus c,.e., t hods m x ( t) = c fo a agents =,, n? In ths geneaty, howeve, one cannot hope to get an answe, nethe by mathematca anayss no by compute smuatons. Theefoe, we w teat vaous nteestng specazatons of the above genea mode (GM). Fst, we begn wth the cassca mode of fxed weghts,.e., xt ( + ) = Axt ( ) fo t T, (CM) whee A s a fxed stochastc matx and () x t the coumn vecto of opnons at tme t. Ths mode has been poposed and empoyed to assess opnon poong by a daogue among expets (De Goot 974), (Lehe 975). t

4 Thee s an nteestng vaaton of ths mode deveoped by (Fedkn and Johnsen 990, 999). Ths mode addesses opnon fomaton unde soca nfuence and assumes that agent adhees to hs nta opnon to a cetan degee g and by a susceptbty of g the agent s socay nfuenced by the othe agents accodng to a cassca mode. By ths vaaton the cassca mode becomes ( ) x ( t+ ) = g x (0) + ( g ) a x ( t) + a x ( t) (2.2) n n o, n matx notaton, xt ( + ) = Gx(0) + ( I G) Axt ( ) fo t T. (FJ) Hee G s the dagona matx wth the g, 0 g, n the dagona and I s the dentty matx. Obvousy (CM) s a speca case of (FJ), namey fo g = 0, n. Ths mode has been used to estmate fom expements the susceptbty of agents to ntepesona nfuence (Fench 956). A mode sma to (CM) but n contnuous tme, that s a system of dffeenta equatons nstead of dffeence equatons, has been expoed and apped athe eay by (Abeson 964). Though thee ae smates n the seach fo consensus among agents, n the pesent atce we w stck to the dscete tme famewok. The modes (CM) and (FJ), as we as Abeson s mode, ae both nea whch makes them systematcay tactabe by anaytca methods. The next type of mode s st nea but tme-vaant (o, n the theoy of Makov chans, nhomogeneous), that s xt ( + ) = At ( ) xt ( ) fo t T, (TV) whee the entes of matx At, ().e., the weghts, ae dependent on tme ony. The tme vaant mode (TV) potays, e.g., the so caed hadenng of postons whee agents put n the couse of tme moe and moe weght on the own opnon and ess weght on the opnon of othes. In the next secton t w be shown that anaytca esuts ae st avaabe, even fo hghe dmensona opnons, but the esuts ae ess shap than n the tme-nvaant cassca mode. The most dffcut type of mode occus f the weghts depend on opnons tsef because then the mode tuns fom a nea one to a nonnea one. Thus, the mode s of type (GM) whee A( x( t )) does not expcty depend on tme. It s st competey hopeess to anayse the mode n ths geneaty. Thee s, howeve, a patcua knd of nonneaty whch captues an mpotant aspect of eaty and seems at the same tme tactabe. But, compaed wth the othe modes, anaytca nsghts ae not so easy to obtan. Fo that eason we w nvestgate ths mode detaed and extensvey by compute smuatons n Secton 4. The mode we ae gong to exhbt potays bounded confdence among the agents n the foowng sense. An agent takes ony those agents j nto account whose opnons dffe fom hs own not moe than a cetan confdence eve ε. Fxng an agent and an opnon pofe x= ( x,, x n ) ths set of agents s gven by { } I(, x) = j n x xj ε (2.3) whee denotes the absoute vaue of a ea numbe. To make thngs not too compcated we assume that agent puts an equa weght on a j I(, x). That s to say, n the ght of the genea mode (GM) we et the weghts gven by aj ( x ) = 0 fo j I(, x) and aj ( x) = I(, x) fo j I(, x) ( fo a fnte set denotes the numbe of eements). Thus the mode wth bounded confdence s gven by

5 ( ) (, ()) x ( t+ ) = I, x( t) x ( t) fo t T. (BC) j I x t Ths mode has been deveoped by (Kause 997, 00); see aso (Beckmann 997) and (Dttme 00, 0). Fo anothe attempt to mode a ack of confdence see (Deffuant et a. 00), (Wesbuch et a. 0). Thee, n a pawse compason between agents, opnon adjustments ony poceed when opnon dffeence s beow a gven theshod ((Deffuant et a. 00), p. 2). 3. ANALYTICAL RESULTS FOR THE VARIOUS MODELS OF OPINION DYNAMICS In the foowng we pesent sevea majo esuts fo the vaous modes. Some ae we known, othes ae ess we known and some ae new esuts. A. The cassca mode The cassca mode (CM) was gven by + = fo t T { 0,, 2, } x ( t ) Ax( t) =, t whee A s a stochastc matx. Obvousy, xt ( ) = Ax(0) fo a t T and, hence, the anayss amounts to anayze the powes of a gven matx. Resut : On consensus If any two agents put jonty a postve weght on a thd one then fo evey nta opnon pofe a consensus w be appoached (whch consensus, of couse, depends on the nta pofe). Moe genea, a consensus w be appoached fo evey nta opnon pofe f and ony f fnay fo t bg enough any two agents put jonty a postve weght on a thd one. Fo a foma statement of ths esut togethe wth addtona hnts see Theoem n the Appendx. A speca case of Resut s gven f evey agent puts a postve weght on any othe agent. Ths coesponds to the ntuton that the goup shoud appoach a consensus f evey agent takes the opnon of any othe agent nto account. On the othe exteme s the case whee evey agent stcks to hs opnon wthout takng cae of any othe opnon. In ths case A s the dentty matx and, of couse, thee s no consensus, except the speca case of a consensus aeady n the nta opnon pofe. Resut descbes the confdence patten between these extemes that aows fo a consensus. An exampe fo thee agents s gven by the stochastc matx A 0 2 2 2 = 0. 3 3 3 0 4 4 2 Fo ths matx A the powe A s stcty postve (does not contan any zeo). A matx fo whch some powe s stcty postve s aso caed pmtve. By Resut a pmtve matx A s suffcent fo consensus. (The second pat of Resut povdes a condton that s not ony suffcent but necessay, too.) Moeove, by empoyng what s caed the noma fom of Gantmache, a moe efned esut s possbe (see Theoem 2 n the Appendx). Intutvey, the goup of agents can be spt up nto subgoups such that the fst g soated subgoups consst ony of essenta agents n the sense that an agent puts weght ony on agents n hs goup. Such a subgoup has pmtve stuctue f ts submatx of weghts s pmtve. j

6 Resut 2: On opnon fagmentaton Fo any gven nta pofe the opnon dynamcs appoaches a stabe opnon patten f and ony f the subgoups of essenta agents ae a pmtve. Fo the fna stabe opnon patten ony the nta opnons of essenta agents pay a oe. In patcua, the stabe opnon patten educes to a consensus f and ony f thee exsts just one subgoup of essenta agents ( g = ). (Fo moe nfomaton see theoem 2 n the Appendx.) An exampe of fou agents s gven by A 0 0 2 2 2 0 0 3 3 = 3. 0 0 4 4 0 0 2 2 Thee ae two subgoups of essenta agents fomed by agents and 2 and agents 3 and 4, espectvey. The fna stabe opnon patten conssts of a pata consensus between agents and 2 and a pata consensus between agents 3 and 4. In genea one obtans no consensus between these subgoups. B. The Fedkn-Johnsen mode Ths mode (FJ) was gven by x ( t+ ) = Gx(0) + ( I G) Ax( t) fot T. If G= 0 then (FJ) specazes to the cassca mode and t suffces to dscuss G 0. Resut 3: On opnon fagmentaton wth postve degees If thee s at east one agent wth postve degee and thee ae enough postve weghts then fo any gven nta pofe the opnon dynamcs appoaches a stabe opnon patten that can be computed fom the weghts, the degees, and the nta pofe. The above stabe patten epesents a consensus f and ony f thee pevas aeady a consensus among a agents wth postve degee. (Fo moe nfomaton see Theoem 3 n the Appendx.) Thus, n the case of postve degees one can expect a consensus ony fo vey speca nta pofes. Assumng a pmtve stuctue fo the whoe goup ths s vey dffeent fom what hods fo the cassca mode. An exampe fo fou agents s gven by.2..359.300.47.25.344.294 A =, G = dagonaof A 0 0 0.089.78.446.286 In ths case a agents have postve degee and a consensus s possbe ony f thee s a consensus at the begnnng. Fo x (0) = (25,25,75,85), e.g., one obtans as fna opnon patten (60, 60, 75, 75) (Fedkn and Johnsen 999, p. 6). Snce A has a stcty postve coumn, n the cassca mode one woud obtan fo x (0) as above a consensus accodng to Resut.

7 C. The tme-vaant mode The mode wth tme-vaance (TV) was gven by x ( t+ ) = A( t) x( t) fo t T, whee the weghts coected n matx At () depend on tme t. As one mght expect fom the cassca mode, the tme vaance w be no obstucton to a consensus as ong as the weghts eman suffcenty postve. If, howeve, the weghts tend to zeo vey qucky then consensus dsappeas. Ths nteestng phenomenon we ustate by a smpe exampe whch woks aeady fo a goup of two agents. Suppose agent does neve cae about the opnon of agent 2. In a fst scenao et agent 2 put at tme t, fo t 2, the weght t to agent and the emanng weght t to hmsef. Thus, the poston of agent 2 s hadenng by puttng ess and ess weght to agent. It s not dffcut to vefy that x2() t = ( t ) x(2) + t x2(2) and, because of x () t = x (0) fo a t, that x () t tends to x (0) 2. Theefoe, the two agents appoach aways a consensus n spte of hadenng the postons. In a second scenao et agent 2 hadenng hs poston faste, by puttng at tme t, fo 2 t 2, the weght t to agent and t 2 to hmsef. A tte computaton shows that x () t tends 2 to x ( x ) (2) + 2 2(2) and, of couse, x () t = x (0) fo a t. Thus, fo many nta pofes a consensus w not be appoached. Ths tte exampe shows that appoachng a consensus does depend on the speed by whch the weghts ae changng (cf. (Cohen 986)). Ths vague nsght s made moe pecse by the foowng esut. Resut 4: One tme-vaance consensus Consensus w be appoached fo evey nta opnon pofe, povded thee exsts a sequence of tme ponts wth the foowng popety: Fo the accumuated weghts b j between tme ponts t hods that fo any two agents and j thee exsts a thd one k such that b k and b jk ae postve and the mnma of both sum up to nfnty fo summng ove a tme ponts. Roughy speakng, the esut says that fo a consensus the weghts cannot tend too fast to zeo because the sum shoud be nfnte. Fo a pecse statement see Theoem 4 n the Appendx. The above esut can be fomuated and poved aso fo the case of mutdmensona opnons, that s, x ( t ) s not just a numbe but a vecto n hghe dmensons. The poof fo ths esut s teng because t shows that the set n hghe dmensons whch s spanned by the agents opnons must shnk to the pont of consensus f tme deveops. (See the Lemma befoe Theoem 4 n the Appendx.) D. Opnon dynamcs wth bounded confdence (BC) The (BC) mode was gven by whee I(, x) = { j n x xj ε} ( ) j I(, x()) t x ( t+ ) = I, x() t x ()fo t t T, and ε > 0 s the gven confdence eve of agent. Though some popetes hod st fo dffeent eves of confdence, n the foowng we w assume thoughout a unfom eve of confdence,.e., ε = ε fo a agents. Late on, n the next secton when deang wth smuatons, we w consde not ony symmetc confdence ntevas,.e., [ ε, ε] j +, but aso asymmetc confdence ntevas,

8.e., [ ε, ε ] + whee ε = ε, ε = ε ae postve but can be dffeent fom each othe. The set eft ght I (, x ) n the asymmetc case s then gven by { ε ε } I(, x) = j n xj x. In the asymmetc case ε ε we can have a one sded spt between two agents and j, namey ε< xj x ε f ε < ε and ε < x j x ε f ε < ε. In a one sded spt one agent ( f ε < ε and j f ε< ε) takes the othe agent (j and, espectvey) nto account, but not vce vesa. The confdence eves ε, ε seve as paametes of the mode and the set of possbe vaues of ( ε, ε ) s caed the paamete space. A patcua featue of ths mode, compaed wth the othe modes dscussed, s that a consensus w be eached n fnte tme, f thee s a consensus at a. Fo an opnon pofe x= ( x,, x n ) we say that thee s a spt (o cack) between agents and j f x xj > ε. An opnon pofe x= ( x,, x n ) we ca an ε -pofe f thee exsts an odeng x x x of the opnons such that two adjacent opnons ae wthn confdence,.e., x x fo a k ε k n + k 2 n Fst we coect some fundamenta popetes of ths mode fo the case ε= ε(see ( Kause 00)). Popetes I. The dynamcs does not change the ode of opnons,.e., x() t xj() t fo a j mpes that x ( t+ ) x ( t+ ) fo a j. j II. If a spt between two agents occus at some tme t w eman a spt foeve. III. If fo an nta pofe a consensus s appoached then the opnon pofe must be an ε - pofe fo a tmes. IV. Fo n= 2, 3, 4 a consensus s appoached f and ony f the nta pofe s an ε -pofe. The ast popety (IV) s no onge tue fo n= 5. That s, fo fve agents t can happen that they don t appoach a consensus though the nta opnons ae cose n the sense of an ε -pofe. If, howeve, the nta pofe s an equdstant ε -pofe,.e., the dstance between adjacent opnons s exacty ε, then fo a numbe of fve agents, too, a consensus w be appoached. A caefu consdeaton shows that fo a numbe of sx agents even n the case of an equdstant ε -pofe a consensus cannot be eached. These emaks ndcate that the numbe of agents nvoved can matte fo the dynamcs. Fo the smpest case of just two agents t s not dffcut to gve a compete anayss of the dynamcs. Fo an abtay n, howeve, the mathematca anayss of the dynamcs tuns out to be athe dffcut and up to now thee ae ony a few genea esuts avaabe. Fo that eason we w pesent n the next secton an extensve anayss of ths mode fo hghe vaues of n by compute smuaton. Of the few genea esuts we pesent two fundamenta ones. Resut 5: On consensus fo bounded confdence Consensus w be appoached fo a gven nta opnon pofe, povded fo an equdstant sequence of tme ponts the foowng popety hods: Fo any two agents and j thee exsts a thd one k such that a chan of confdence eads fom to k as we as fom j to k. In ths case, the consensus s eached n fnte tme.

9 (See Theoem 5 n the Appendx fo a pecse foma statement.) Resut 6: On opnon fagmentaton fo bounded confdence Fo any gven nta pofe thee exsts a fnte t * T and a dvson of a agents nto maxma * subgoups such that wthn each of the subgoups thee hods a consensus fo a t t. (Of couse, those pata consenses w be dffeent fom each othe n genea.) (See Theoem 6 n the Appendx fo a pecse fomuaton.) 4. SIMULATIONS In ths secton we w expoe the mode wth bounded confdence by means of smuatons. We want to do that, fsty, n a systematc way, and, secondy, foowng the KISS-pncpe: Keep t smpe, stupd! Unde that pncpe t s fay natua to stat n a fst step wth homogenous and symmetc -ntevas. Homogenety means that the sze and shape of the confdence nteva s the same fo a agents. Symmety means, that the nteva has the same sze to the eft and to the ght,.e.. In a second step we w anayse dffeent types of asymmety. 4. SYMMETRIC CONFIDENCE Wth contnuous, one dmensona, and nomazed opnons x taken fom the nteva x 0, ony confdence ntevas 0, make sense. Thus, the paamete space s gven by the unt squae. The dagona epesents symmetc confdence. A systematc appoach coud be 'wakng aong the dagona', statng the tp at the pont 0,0. eft ght 0 Fgue : The paamete space, wakng aong the dagona. We w andomy geneate a stat dstbuton of 625 opnons. Updatng s smutaneous. Fgue 2 shows thee stops on the tou aong the dagona. The stops ae snge uns. They a get gong wth the same stat dstbuton. The odnate ndcates the opnons. Snce t s usefu fo the foowng anayss the opnons ae addtonay encoded by coous angng fom ed ( x 0 ) to magenta ( x ). The abscssa epesents tme and shows the fst 5 peods. Obvousy t takes ess than 5 peods to get a stabe patten: Wth the qute sma confdence nteva of 0.0 exacty 38 dffeent opnons suvve n the end. Unde the much bgge confdence of 0.5the agents end up n two camps, and wth 0.25 the esut of the dynamcs s consensus. As t ooks, the sze of the confdence nteva eay mattes.

0 Fgue 2 shows ony snge uns. To get a bette feeng of what s gong on we un systematcay smuatons wakng aong the dagona. The smuatons stat wth 0.0, 0.02,, 0.4. (Fo easons that become obvous beow, thee s nothng new and nteestng n the paamete space fo, 0.4.) Fo each of these steps we epeat the smuaton 50 tmes, aways statng wth a dffeent andom stat dstbuton. Each un s contnued unt the dynamcs becomes stabe. Fgue 3 gves an ovevew. (a) 0.0 (b) 0.5 (c) 0.25 Fgue 2: Stops whe wakng aong the dagona

z 0.2 0.5 0. 0.05 30 y x 60 80 0 00 Fgue 3: Wakng aong the dagona smuaton esuts. The x-axs n Fgue 3 epesents the opnon space 0, dvded nto 00 ntevas. Ou step wak aong the dagona s epesented by the y-axs. (Note: the steps ae not tme steps!) The z-axs epesents the aveage (!) eatve fequences of opnons n the 00 opnon ntevas of the opnon space afte the dynamcs has stabsed. Fgue 3 deseves caefu nspecton: At the begnnng of ou wak aong the dagona of Fgue,.e. the y-axs of Fgue 3, thee s ony tte confdence. Fo exampe, step 2 o 3 means that 0.0o 0.02. In tems of snge uns we ae speakng about opnon dynamcs ke that n Fgue 2a. The z-vaues show that as an aveage we fnd unde that tte confdence a sma facton of the opnons n a ntevas of the opnon space. (Note: That does not mpy that n a snge un a ntevas ae occuped afte stabsaton; see Fgue 2a and Fgues 2a, 2b.) No pat of the opnon space seems to have especay hgh o especay ow fequences. That changes as we step futhe aong the dagona. Look, fo nstance, at step 5,.e. 0.5. A snge un exampe fo a dynamcs based on confdence ntevas of ths sze s gven by Fgue 2b, whee we end wth two camps hodng dffeent opnons. Fgue 3 makes t cea that ths s a fay typca esut: As we step fowad aong the dagona the aveage dstbuton of stabsed eatve fequences becomes ess and ess unfom. To the eft and to the ght of the cente mountans of nceasng heght emege. As we contnue ou wak on the dagona the andscape changes damatcay agan: At about step 25,.e. 0.25the 'Lefty' and the 'Rghty Mountans' come to a sudden and steep end. At the same tme a new and steep cente mountan emeges. An exampe fo a snge un n ths egon of the paamete space s gven by Fgue 2c. Fgue 3 shows that ths snge un wth a tota confdence nteva ncudng haf of the opnon space s a typca un, eadng to a consensus ncudng a o amost a agents. Except n the cente amost a opnon ntevas ae empty and the coespondng opnons ae emnated. Summng up and geneasng what we see n Fgue 3 one mght say: As the homogeneous and symmetc confdence nteva nceases we tanst fom phase to phase. Moe exacty, we step fom fagmentaton (puaty) ove poasaton (poaty) to consensus (confomty).

2 That a dynamcs, govened by aveagng among those opnons whch ae wthn a cetan fay sma confdence nteva, eads to an eveny dstbuted vaety of opnons (puaty) w pobaby not supse that much (see Secton 3, pat D, Resut 6). That fay age confdence ntevas dve ou dynamcs towads confomty s pobaby even ess supsng (see Secton 3, pat D, Resut 5). Bg confdence ntevas shoud dve a opnons decton cente and thee they convege. (Fo a vaues, 0.4 the esut s aways confomty, and that s the eason why we stop the wak on the dagona at the pont 0.4.) Thus, the ea task of undestandng s the phase n between: What mechansms dve ou dynamcs unde mdde szed confdence nto poasaton? Exteme opnons ae unde a one sded nfuence and move decton cente. The ange of the pofe shnks. At the extemes opnons condense. The -pofe spts n t 6. Fom now on the spt sub pofes beong to dffeent 'opnon wods' o communtes whch do no onge nteact. Condensed egons attact opnons fom ess popuated aeas wthn the each. In the cente opnons > 0.5 move upwads, opnons < 0.5 move downwads. Fgue 4: Regua stat pofe, 0.2. Fgue 4 gves decsve hnts to undestand poaty. The dynamcs stats wth a egua pofe,.e. a pofe fo whch the dstance between any two neghboung opnons s the same. A gey aea between two neghboung opnons ndcates that the dstance between the two s not geate than,. Caefu nspecton shows: Exteme opnons ae unde a one sded nfuence and move decton cente. As a consequence the ange of the pofe shnks. At the extemes opnons condense. Condensed egons attact opnons fom ess popuated aeas wthn the each. In the cente opnons > 0.5 move upwads, opnons < 0.5 move downwads. The pofe spts n peod t 6. Fom now on the spt sub pofes beong to dffeent 'opnon wods' o communtes whch do no onge nteact. Fgue 5 shows one aspect of the dynamcs n Fgue 4 moe n deta. The x axs gves the peods, the y axs of Fgue 5 shows the change fom one peod to the next,.e. ( t ) = x ( t) x ( t ) t=,2, (4.) fo a opnons fom Fgue 4. The opnons ae ndcated by the coou.

3 Fgue 5: Opnon changes fom peod to peod (50 opnons, egua stat pofe, 0.2 ). Fgue 5 makes cea that opnon changes stat at the extemes and ae thee (at the stat!) most exteme fo the most exteme opnons. Opnons next to (magenta) move downwads (negatve ), opnons next to 0 (ed) move upwads (postve ). Fom peod 0 to nothng changes moe n the mdde of the opnon space. But as tmes goes by changes wok though the opnon space decton cente. The opnons decty n the cente ae the ast to be affected. They make the bggest jumps. Fom peod 7 to 8 onwads nothng changes anymoe,.e. 0 fo a opnons x. The effects descbed above become n some espect moe obvous wth moe opnons. Fgue 6 shows fo 500 opnons what Fgue 5 showed fo 50. The opnons of the egua pofe ae agan ndcated by the coous. Snce cooung of the pofe s done sequentay, an eae cooued opnon may ate become hdden by othe opnons. Fgue 6, top shows a cooung of the pofe n an ascendng ode (0 to ) whe Fgue 6 foows a descendng ode ( to 0). By vsua nspecton t becomes mmedatey cea that changes stat at the extemes and each the cente ony wth some deay. The decsve key fo an undestandng of poaty ae obvousy the spts n the -pofe, nduced by a one sded nfuence at the extemes, a shnkng ange of opnons combned wth an nceasng fequency of opnons n cetan aeas of the opnon space whch attact opnons n ess popuated aeas wthn the each. Unde smutaneous updatng cacked pofes can neve get connected agan. Agents/opnons outsde ones own sub pofe ae out of ange/each n a qute sevee sense: They ae not ony outsde ones own confdence nteva, but aso outsde the confdence ntevas of a othes one takes seousy, outsde the confdence ntevas of a othes whch those othes take seousy etc.

4 Fgue 6: Opnon changes fom peod to peod (500 opnons, egua stat pofe, 0.2 ). Top: Cooung n an ascendng ode. Bottom: Cooung n an descendng ode. The spts do not ony expan poaty, they expan the stabsaton of ou dynamcs n genea. Fo fay sma confdence ntevas the stabsaton eads to a fay hgh numbe of suvvng opnons,.e. puaty. Fgue 7 shows a egua stat pofe wth 00 opnons and 0.05. The pofe spts 8 tmes. Agan a gey aea between two neghboung opnons ndcates that the dstance between the two s not geate than,. Fo mdde szed confdence ntevas we get ony a sma numbe of suvvng opnons,.e. poasaton; Fgue 4 gves an exampe. Unde age confdence ntevas the pofe spts neve o the spts eave aone exteme and qute sma mnotes whe an ovewhemng majoty conveges n the cente of the opnon space. Fgue 8 gves an

5 exampe of tota consensus. Note that even the fou opnons n the cente of the opnon space move fo a shot whe out of the cente. But they mege thee agan much eae than the uppe and owe pat of the pofe aves n the cente as we. Fgue 7: 00 opnons, 0.05, 8 spts. Fgue 8: 00 opnons 0.25, no spt, tota consensus. The smuaton esuts fom Fgue 3 ae based on andomy geneated stat pofes (unfom dstbuton), not on egua stat pofes. But the effects descbed so fa do not essentay depend on the eguaty of the stat pofe. What eguaty adds ae densty fuctuatons n the nta dstbuton of opnons. They ae addtona causes fo spts and nduce opnon changes deep nsde the stat pofe wthout any deay ght at the begnnng of the pocess.

6 4.2 ASYMMETRIC CONFIDENCE Up t now we consdeed waks aong the dagona of the paamete space (Fgue ),.e. symmety n the sense that. But confdence may be asymmetc. One can thnk of sevea types of asymmety. In the foowng we w anayse two cases. In the fst case (4.2.) the asymmety s ndependent of the opnon an agent hods. Whateve the opnon mght be, a agents have the same confdence ntevas, wth. In the second case (4.2.2) the asymmety s dependent upon the opnon the agent hods: An agent wth an opnon moe to the ght [eft] has moe confdence n the ght [eft] decton. 4.2. OPINION INDEPENDENT ASYMMETRY How to get an ovevew about what s gong on unde opnon ndependent asymmetes? Agan ou appoach s a systematc wak though the paamete space. But nstead of takng the oute aong the dagona we now wak on staght nes beow (o above) the dagona as ndcated by Fgue 9. Fo ths type of asymmety t does not matte whethe confdence s based to the ght o based to the eft. Theefoe t s ony one of the tanges, ethe the one beow o the one above the dagona, that has to be anaysed. We confne ouseves to the tange beow,.e. a bas to the ght. Fo a effects we fnd n that aea of the paamete space thee exst coespondng effects n the tange above (bas to the eft). eft ght 0 Fgue 9: Reseach stategy fo opnon ndependent asymmetes. To get a fst feeng we ook at the thee snge un exampes of asymmetc confdence n Fgue 0. We fnd phenomena we ae aeady fama wth, fo nstance fast stabsaton, puaty, poaty, and confomty. But obvousy the asymmetc confdence dves the dynamcs somehow nto the decton favoued by the asymmety,.e. to the ght.

7 (a) 0.02 0.04 (b) 0.03 0.5 (c) 0.0 0.25 Fgue 0: Snge uns, 625 opnons, andom stat pofe. To get a moe systematc ovevew we pesent the esuts of fou stepwse waks beow the dagona. In the fst wak we foow the staght ne 0.9. We stat wth 0.0and make

8 steps unt we get to the pont 0.36, 0.4. Fo each vaue of these steps we epeat the smuaton 50 tmes, aways statng wth a dffeent andom (unfom) stat dstbuton. Each un s contnued unt the dynamcs becomes stabe. The othe thee waks foow n the same way the nes 0.75, 0.5, and 0.. We aways stop when 0.4 s eached. Fgue gves an ovevew. 0.2 0.5 0. 0.05 30 (a) 0.9 60 80 0 00 0.2 0.5 0. 0.05 30 (b) 0.75 60 80 0 00

9 60 80 00 0 30 0.05 0. 0.5 0.2 60 80 00 (c) 0.5 60 80 00 0 30 0.05 0. 0.5 0.2 60 80 00 (d) 0.25

0.2 0.5 0. 0.05 30 (e) 0. 60 80 0 00 Fgue : Wakng beow the dagona smuaton esuts. Inspecton of Fgue a e suppots the foowng obsevatons: As (and theeby ) nceases a fou waks fnay ead agan nto a egon of the paamete space whee consensus pevas. But as compaed to becomes smae and smae the esutng consensus moves nto the favoued,.e. hee nto the ght decton. Fo a vey sma and the dynamcs stabses wth a ot of suvvng opnons. Thus agan we have a phase one mght con puaty. As and nceases poasaton emeges. As ong as s ony a tte bt smae than (cf. Fgue a, 0.9) t s the type of poasaton we know fom the symmetc case: n the eft and n the ght of the of the opnon space exteme opnon camps emege, gow, and get cose to each othe wth an nceasng (and theeby nceasng ). It s a somehow a 'symmetc' poasaton: The camps have about the same sze and the same dstance fom the cente (o the bodes, espectvey) of the opnon space. As becomes sgnfcanty geate than we obseve a new type of asymmetc poasaton. The most obvous effect s that a bg opnon camp emeges at the ght bode of the opnon space. Ths effect s exteme f s ony a sma facton of (cf. Fgue e). To the eft of ths man camp n a cetan dstance, but st to the ght of the cente of the opnon space we obseve smae but nevetheess outstandng fequences. Ths efects the fact, that asymmetc confdence tends to poduce n a cetan egon of the paamete space two o few opnon camps of dffeent sze: The bgge one nomay moe to the bode of the opnon space. Fgues a e (and smay Fgue 3) show how many opnons on an aveage ove 50 smuaton uns end up (afte stabsaton) n each of the 00 ntevas n whch the opnon space was dvded. Thus, one does not geneay see, how many opnons on the aveage suvve at a. Ths nfomaton s gven by Fgues 2a and 2b. Both fgues show the aveage numbe of dffeent

2 opnons that suvve afte the dynamcs has stabsed. The esuts ae based on 25 smuaton uns fo each pa e,,, 0, 0.02, 0.04, 0.06,..., 0.4. A smuatons stat wth 625 andomy geneated opnons. Fgue 2a shows the numbe n of suvvng opnons wth n 0, whe Fgue 2b s a deta of Fgue 2a wth n 2 as the uppe mt of the z axs. Both fgues cafy a bt moe ou speakng of puaty, poazaton, and consensus as thee dffeent phases: Fo vey sma confdence symmetc o asymmetc ntevas ots of opnons suvve (puaty). But as ethe o o both ncease we obseve a shap decne of suvvng opnons. Soon one gets to the geen yeow base whee fo the most pat ony 2 opnons suvve (poazaton). Fgue 2b n pncpe a magnfcaton of Fgue 2a shows that wth an futhe ncease n o ths poazaton tuns nto consensus. Fgue 2c shows the fna aveage opnon n the whoe popuaton fo a ponts, wth, 0.4, agan based on 25 smuaton uns wth 625 andom opnons at the begnnng. It s no supse that fo symmetc confdence ths fna aveage opnon s about 0.5. Wth asymmetc confdence the mean opnon moves nto the decton favoued by the asymmety. Ths effect s exteme f thee s ony tte confdence n the non favoued decton. Fgue 2d shows that the effect becomes mde as the confdence n the non favoued decton nceases as we. A decsve step fo an expanaton of the phenomena stated above s an undestandng of the new type of spts that occu unde asymmetc confdence. Snce t may be the case that x () t x() t, whe x () t x() t, whee x and x ae two neghboung opnons n ou pofe. In such a stuaton the opnon x () t affects the opnon x ( t) snce x () t s wthn the each of x ( t ). But at the same x () t s not affected (any onge) by x ( t) snce x ( t) s outsde the each of x () t. We thus have a one sded spt (see defnton n Secton 3, pat D). In the opnon dynamcs gven by Fgue 3 sevea one sded spts occu: Whee eve a gey aea between two neghboung opnons has ony whte nes n the decton top ght, thee we have a one sded spt. In contast to that a patten esutng fom both, whte nes decton bottom ght and whte nes decton top ght, ndcates that the two neghboung opnons ae mutuay wthn the eevant each: x ( t) n the each of x () t, and x () t n the each of x ( t ). 0 8 6 4 2 0 2.5 0.5 0 Fgue 2a: Numbe of emanng opnons afte stabsaton. Fgue 2b: Numbe of emanng opnons afte stabsaton, magnfcaton of Fgue 2a.

22 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 Fgue 2c: Fna aveage opnons u afte stabsaton Fgue 2d: Deta of Fgue 2c: fna aveage opnons 0.3 u 0.7 afte stabsaton. 0.4 ght 0.4 eft 0 Legend fo 2a 2d Fgue 2: Smuaton esuts: Numbe of suvvng opnons and fna aveage. Even wthout any spt n the pofe bae asymmety of confdence dves the opnon dynamcs n the favoed decton. But one sded spts ampfy ths effect damatcay. How the mechansm woks can be seen n Fgue 3: Opnons ght above a spt become the new exteme eft opnons of a emanng sub pofe that contnues to convege. At east fo a whe ths sub pofe s no onge nfuenced by moe eft opnons. The spt off sub pofe beow the one sded spt conveges and moves upwads. It s st unde the nfuence of opnons above the one sded spt. By that the whoe dynamcs s dven to the ght. Note that, contay to two sded spts one sded spts can cose agan.

23 One sded spt. One sded spts can cose. Two sded spts neve do that. Opnons ght above a spt become the new exteme eft opnons of a emanng -sub-pofe that contnues to convege. The sub pofe s at east fo whe not nfuenced by moe eft opnons. The spt off -sub pofe ght beow the one-sded spt conveges and moves upwads. It s st unde the nfuence of opnons above the one sded spt. x () t x () t x () t x () t Fgue 3: One sded spts (00 opnons, 0.8, 0.24 ). 4.2.2 OPINION DEPENDENT ASYMMETRY The confdence ntevas anaysed n 4.2. ae asymmetc. But the asymmety s ndependent of the opnon tsef. Howeve, t s a qute common phenomenon that that those hodng a moe eft (ght) opnon often sten moe to othe eft (ght) opnons whe beng sceptca upon moe ght (eft) vews. So t seems natua to mode an opnon dependent asymmety n the foowng way: The moe eft (ght) an opnon s, the moe the confdence nteva s based decton eft (ght). Fo the speca case of a 'cente' opnon,.e. x 0.5, the confdence nteva shoud be symmetc. 0 0.5 Fgue 4: Opnon dependent confdence ntevas. Fgue 4 ustates how foowng ths dea a confdence nteva of a gven tota (!) sze s shfted to the eft (ght) fo an opnon to the eft (ght) of the cente. Fo the cente opnon the eft and the ght confdence nteva ae of equa sze. How to mode opnon dependent asymmetes of confdence ntevas? One possbe stategy s to ntoduce an opnon dependent bas to the eft and a bas to the ght such that, 0 and. Havng done that we use and to dvde any gven confdence nteva nto a eft

24 and a ght pat. Foowng ou ntuton stated above, the vaues shoud be geneated by a monotoncay nceasng functon f of the opnon x wth x 0,. By f ( x) we get the monotoncay deceasng functon that we need to geneate the vaues agan foowng the ntuton stated above. Dffeent sopes fo the functon f woud then aow to mode the stength of the bas. bas 0.5 0 0.5 opnon x Fgue 5: Paamete space fo opnon dependent asymmety. Fgue 5 ustates an easy way to eaboate n deta ths type of asymmety. The x-axs ndcates the opnon. The y-axs o, espectvey, the bue nes ndcate the opnon dependent bas x to the ght and xto the eft. The bue gaphs ae geneated by otatons aound the bue pont m m 0.5,0.5,.e. accodng to the functon f( x) mx. It s xmx and 2 2 x x. These bases ae used to detemne how a confdence nteva of any gven sze s pattoned nto an and u u u u. Then t aways hods that x x. It s aso guaanteed that 0.5 0.5. In ths settng t s the m sope m n f( x) mx that contos the stength of the bas. Fo m 0 we do not have any. We defne and 2 bas. Both pats of the confdence nteva have, whateve the opnon mght be, the same sze. As m nceases (by otatng the bue gaph ant cockwse aound 0.5,0.5 ) the bas becomes stonge and stonge: Peope wth a moe eft (ght) opnon sten ess and ess to the ght (eft) sde of the opnon space. We w confne ouseves to sopes wthn the ange 0 m. To gve an exampe: Fo an opnon x 0.6, a tota confdence nteva of 0.4, and m 0.5 we get 0.8 and 0.22. Fo any postve m t hods that the moe one's opnon s ocated to the eft (ght), the moe one's confdence nteva s shfted to the eft (ght).

25 Ths appoach offes an easy way to anayse the effects of opnon dependent asymmetes of confdence ntevas (Fgue 6): Fo dffeent absoute szes of confdence ntevas we stat wth symmety. In each case we et m stepwse ncease and study the esutng dynamcs by means of smuaton. The anaysed aea of the paamete space w be 0 m. 0 m Fgue 6: Opnon dependent asymmetes: Anaysng the paamete space. Fgue 7 gves an ovevew. The gaphcs ae of the same type as n Fgues 3 and. The x-axs epesents the opnon space 0, dvded nto 00 ntevas. The z-axs epesents the aveage (!) eatve fequences of opnons n the 00 opnon ntevas of the opnon space afte the dynamcs has stabsed. In contast to the fome fgues the y-axs does not epesent changes n ; t now epesents changes of the paamete m whch contos the stength of the opnon dependent bas of. Aong the y-axs m s nceasng fom 0 to n 26 steps of sze 0.04 (whe n the fome gaphcs we saw steps of an nceasng o, espectvey). Thus, each gaphcs epesents a wak aong one of the bue hozonta nes n Fgue 6. Fgues 7a to c show the smuaton esuts based on 0.2, 0.4, 0.6. 0.2 0.5 0. 0.05 (a) 0.2 60 80 0 00

26 0.2 0.5 0. 0.05 (b) 0.4 60 0 80 00 0.2 0.5 0. 0.05 (c) 0.6 60 80 Fgue 7: Inceasng bas m (26 steps, m 0,..., ) fo thee dffeent confdence ntevas. 00 The smuatons n Fgue 7a ae based on 0.2. Aong the y-axs we stat (step ) wth m 0, what mpes 0.. Thus, we now get gong whee we got by step 0 when wakng aong the dagona n ou fst expements wth symmetc confdence ntevas (Fgue 2 and Fgue 3). Fo a symmetc confdence nteva of that sze a vey md poazaton stats to emege: The z- vaues show that as an aveage we fnd (on the aveage!) sma factons of emanng opnons n a ntevas of the sghty shunk opnon space. At the extemes the fequences ae a bt hghe, a 0

27 consequence of the one-sded nfuence whch dves the opnons decton cente. But we ae fa away fom a fu fedged poazaton as we w get fo 0.25 (step 25 n Fgue 3). Fgue 7a, fsty, shows that a suffcenty hgh opnon dependent bas w poduce a batant poazaton even unde the condton of a compaatvey sma confdence nteva. That poazaton s, secondy, moe sevee n the sense, that the opnon dstances of the two majo camps ae geate than the dstances we obseve n the symmetc cases. As m nceases the dstance between the two majo camps becomes geate and geate. Fo an m= the two camps occupy the most exteme postons 0 and. Fgue 7b shows the smuaton esuts fo an nceasng m based on 0.4. m 0 coesponds step of ou wak aong the dagona n the fst expements wth symmetc confdence ntevas (Fgue 2 and Fgue 3). Fo a symmetc confdence nteva of that sze we got a cea poazaton. Wth an nceasng opnon dependent bas the poazaton becomes even moe damatc unde both pespectves, sze of the camps and the dstance between them. The smuaton esuts fo a tota confdence nteva of 0.6 ae shown n Fgue 7c. m 0 coesponds step 30 of ou wak aong the dagona n the fst expements wth symmetc confdence ntevas (Fgue 2 and Fgue 3). An a ncudng consensus s the esut and that emans tue fo a md opnon dependent bas m. But a cetan pont (about step,.e. m 0.4 ) that consensus beaks down. A shape and shape poasaton s the fna esut. (a) m 0 (b) m 0.25 (c) m 0.5 (d) m 0.75

28 x () t x () t, x () t x () t, (e) m 0.99 Fgue 8: Fve dffeent bases fo 0.6 Fo a bette undestandng of the effects descbed so fa t s hepfu to ook at snge uns. Fgue 8 shows a sequence of snge uns n whch the opnon dependent bas gets stonge and stonge. A uns ae based on the confdence nteva 0.6 and an nceasng m. To keep thngs smpe we use a egua pofe of 50 opnons. The uns show: As m nceases t takes onge to get to a consensus. In Fgue 8c the bas s suffcenty stong to cause a beak down of the fome consensus. In peod 4 the pofe spts fnay and two poased opnon camps eman. As m nceases futhe, the dstance between the two camps becomes geate. The pncpe cause fo a these effects s that wth an nceasng m those at the extemes become ess and ess affected by opnons moe n the cente of the opnon space. The decsve pont becomes obvous by a compason of Fgues 8a and 8e: Unde symmetc confdence those at the extemes ae unde a one sded nfuence that dves them decton cente, theeby causng a shnkng of the whoe opnon space (Fgue 8a). Wth an nceasng opnon based bas the dve decton cente dsappeas o dependng upon the sze of the confdence nteva s sgnfcanty weakened at the extemes. In Fgue 8e based on an heavy bas of m 0.99 the opnons at the extemes stay whee they ae. At the same tme they attact step by step opnons n whose (at east) one sded each they ae. Thus, nstead of a dve decton cente the opnon based bas geneates a dft to the extemes. 4.3 MAIN RESULTS We can summase ou esuts, fsty, fo the case of symmetc and opnon ndependent asymmetes: Wth an nceasng symmetc o asymmetc confdence we step fom puaty to poasaton and then to consensus. Unde (a)symmetc confdence poasaton s (a)symmetc as we. In the symmetc case the majo causes fo poasaton ae spts n the opnon pofes. They ae caused by shnkng and condensng at the extemes, condensng nduced by condensng, and condensng nduced by densty fuctuatons n andomy geneated stat pofes. In the case of asymmetc confdence an asymmetc poasaton s caused and ampfed especay by one sded spts. At east tempoay one of the two esutng sub pofes nfuences the othe one, whe not onge beng nfuenced by the othe one tsef. Ths favous convegence nto the decton favoued by the asymmetc confdence.

29 Wth asymmetc confdence mean and medan move nto the favoued decton. Ths effect s exteme f thee s ony tte confdence n the non favoued decton. The effect becomes mde as the confdence n the non favoued decton nceases. As to the effects of opnon dependent asymmetes we can, secondy, concude: Wth an nceasng opnon dependng bas the dft decton cente s sgnfcanty weakened at the extemes and dependng upon the sze of both, the bas and the confdence nteva may even totay dsappea. Fo sma confdence ntevas whch poduce puaty n the symmetc case t hods: wth an nceasng bas at east a modeate poasaton stats eae. Wth an nceasng bas poazaton s ampfed: The two opnon camps at the extemes become bgge and the fna poston s moe to the extemes. Wth an nceasng bas eachng a consensus becomes moe and moe dffcut. Poasaton s the esut nstead. If consensus s st feasbe, t takes moe tme to get thee. One mght ask whethe the esuts depend upon smutaneous updatng. The answe s: no. Random sea updatng gves exteme opnons a sghty bette chance to suvve. But none of the esuts stated above depends cucay on smutaneous updatng. Futue dectons of eseach w, fsty, ncude the anayss of opnon spaces of hghe dmensons. (Fo a fst anaytca esut see theoem 4 n the Appendx). Secondy, we w anayse the effects of netwok stuctues n whch nteactons ae estcted to neghboung othes,.e. ndvduas vng, fo nstance, wthn ones v. Neumann o Mooe neghbouhood. Fst smuatons show that ths type of ocaty mattes damatcay: If the neghbouhoods n whch the agents nteact ae fay sma (though oveappng!), then the phase n between puaty and consensus,.e. poazaton, dsappeas. At east unde bounded confdence ocaty of nteactons may pevent socetes fom shap poasaton. APPENDIX: Theoems and Hnts A. The cassca mode (GM) The mode x ( t+ ) = Ax( t), t T has the consensus popety f fo evey x (0) n thee exsts a c such that m x ( t) = c fo a {,, n}, a t T. Theoem t If fo any two, j {,, n} thee exsts some k {,, n} then the consensus popety hods. such that a > 0 and a > 0 The consensus popety hods f and ony f thee exsts some t 0 T such that the matx 0 powe A t contans at east one stcty postve coumn. Fo the fst pat of Theoem see (De Goot 974), fo the second pat see (Bege 98). Theoem 2 Let A be n Gantmache noma fom wth dagona bocks A, s, g s. m x ( t) t exsts fo evey x (0) n f and ony f the A ae a pmtve fo g o, equvaenty, f s the ony oot of A of moduus. The consensus popety hods f an ony f g = and A s pmtve o, equvaenty, f s the ony oot of A of moduus and s a smpe oot. k jk

30 Fo a poof see (Gantmache 959). B. The Fedkn-Johnsen mode (FJ) Fom the mode (FJ),.e., x ( t+ ) = Gx(0) + ( I G) Ax( t) fo t T one obtans by nducton x () t = V () t x(0) fo t T, whee t t k V () t = M + M G wth M = ( I G) A. k= 0 t Fo G= 0, mode (FJ) specazes to (CM) and M = AV, ( t) = A. Theoem 3 Let G 0 and suppose A s an educbe matx. Fo evey x (0) n thee exsts x ( = ) m xt ( ) and one has the fomua x ( = ) ( I M) Gx(0). x Consensus x ( = ) ( c,, c) hods f and ony f x (0) = c fo a wth g > 0. See (Fedkn and Johnsen 990, Appendx). C. Tme-vaant mode (TV) Consde mutdmensona opnons,.e., the opnon of agent at tme t s gven by x ( t) m, whee m s the numbe of opnons consdeed. The (TV) mode then eads n j x ( t ) a ( t) x ( t) + = fo, j= j n t T. Ths shows that x ( t+ ) s a convex combnaton of n x (), t, x () t n m. Fo ponts p z,, z m the set of a convex combnatons of these ponts s denoted by conv{ p z,, z }. Fo a subset M m the damete of M s { } d( M) = sup a a' a, a' M, whee s the Eucdean nom on m (but t coud be any nom on m ). Lemma Fo n x,, x m and n k y = ak x, n the foowng estmate hods k= n n n d( conv { y,, y }) mn mn { ak, ajk } d( { x,, x })., j n k = Ths estmate shows that the damete of the set spanned by the agents opnons shnks by a cetan facto fom one peod to the next one. Ths Lemma s the cuca step n povng the foowng theoem. Let fo s, t T wth s t < the matx B (, ts) ( bj (, ts) ) = denote the matx poduct At ( ) At ( 2) As ( ) whch modes the accumuated weghts between peods s and t.

3 Theoem 4 Suppose thee exst a sequence 0= t0< t< t2< n T and a sequence δ, δ 2, n n k= [ 0, ] wth δm = such that m= { } mn b ( t, t ), b ( t, t ) δ k m m jk m m m fo a m, a, j n. Then fo any n x (0),, x (0) n m thee exsts a consensus n { x } * * x conv x (0),, (0),.e., m x ( t) = x fo a n. t Futhemoe, fo n y (0),, y (0) n m wth coespondng consensus popety that * * j x y x y, j n max (0) (0). * y one has the senstvty Fo Theoem 4 n the speca case of m = see (Chattejee 975), (Chattejee and Seneta 977). Fo Theoem 4 n case of moe genea aveage pocedues see (Kause 00). Theoem 4 as above fo mutdmensona opnons s new. D. Opnon dynamcs wth bounded confdence (BC) and s< t, s and t n T, a sequence ( 0,,, t s) of agents s caed a confdence chan fom to k fo (s, t) f t hods that 0 =, t s= k and Fo two agents k, {,, n} ( j )( ) I, x( t j) x(0) gven fo j=, 2,, t s. j Theoem 5 Consensus w be appoached (n fnte tme) fo a gven nta pofe, povded thee exst h such that fo a m the foowng popety hods: Fo any two agents and j thee exsts a thd one k such that a confdence chan exsts fom to k and fom j to k fo (( m ) h, mh) The poof of Theoem 5 empoys Theoem 4.. Theoem 6 Fo any gven nta pofe thee exst t * T, natua numbes < n < < nk < n and nonnegatve numbes c j fo 0 j k such that fo evey j x() t = cj fo a (, ) * n < n n = n = n fo a t t. j j+ 0 k+ Theoem 6 can be deved by appyng Theoem 5 to cetan subgoups. In a dffeent manne, Theoem 6 was fst poved by J.C. Dttme (Dttme 00), (Dttme 0, p. 468). Thee, nstead of Theoem 5, the foowng esut s used (Dttme 0, p. 467): If the opnon pofe s an ε -chan fo evey tme pont then a consensus w be eached n fnte tme. (Ths can be obtaned aso as a speca consequence of Theoem 5.) REFERENCES ABELSON, R P (964), Mathematca modes of the dstbuton of atttudes unde contovesy. In Fedeksen N and Guksen H (Eds.), Contbutons to Mathematca Psychoogy, New Yok, NY: Hot, Rnehat, and Wnston.

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