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Probability of inconsistencies in theory revision

Probability of inconsistencies in theory revision
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  Eur. Phys. J. B (2012) 85: 44DOI: 10.1140/epjb/e2011-20617-8 Regular Article T HE  E UROPEAN P HYSICAL  J OURNAL  B Probability of inconsistencies in theory revision  A multi-agent model for updating logically interconnected beliefs under bounded confidence S. Wenmackers 1 , a , D.E.P. Vanpoucke 2 , and I. Douven 1 1 University of Groningen, Faculty of Philosophy, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands 2 Ghent University, Department of Inorganic and Physical Chemistry, Krijgslaan 281 – S3, 9000 Gent, BelgiumReceived 26 July 2011 / Received in final form 26 November 2011Published online 25 January 2012 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2012 Abstract.  We present a model for studying communities of epistemically interacting agents who updatetheir belief states by averaging (in a specified way) the belief states of other agents in the community. Theagents in our model have a rich belief state, involving multiple independent issues which are interrelated insuch a way that they form a theory of the world. Our main goal is to calculate the probability for an agentto end up in an inconsistent belief state due to updating (in the given way). To that end, an analyticalexpression is given and evaluated numerically, both exactly and using statistical sampling. It is shownthat, under the assumptions of our model, an agent always has a probability of less than 2% of endingup in an inconsistent belief state. Moreover, this probability can be made arbitrarily small by increasingthe number of independent issues the agents have to judge or by increasing the group size. A real-worldsituation to which this model applies is a group of experts participating in a Delphi-study. 1 Introduction Sociophysics studies social phenomena using existingmodels from statistical physics, such as the Ising spinmodel [1] or the concept of active Brownian particles [2], or tailor-made models that are physical in spirit, includ-ing various multi-agent models [3]. The use of agent-based models has become increasingly popular in social dynam-ics research [4,5]. The branch of sociophysics we are in- terested in here is opinion dynamics, which investigatesgroups of epistemically interacting agents. There is evi-dence from psychological experiments that agents adjusttheir opinion when they are informed about the opinion of another agent [6]. Partly inspired by such accounts, opin-ion dynamics studies how the opinions or belief states of agents evolve over time as a result of interactions withother agents, which may lead to cluster formation, polar-ization, or consensus on the macro-level [3]. Opinion dy- namics is also of interest for social epistemology – a branchof philosophy, that focuses on social aspects of knowledgeof beliefs [7]. It studies, for example, how to deal with peerdisagreement, the best response to which is found to becontext-sensitive; among other factors, it depends on thegoal of the investigation [8]. In general, the processes involved in opinion dynam-ics are very complex. Apart from analytical results, alsocomputer simulations of agent-based models are used tostudy these large, complex systems. Simulations allow re-searchers to perform pseudo-experiments in situations in  Supplementary Online Material is available in electronicform at a e-mail: which real-life experiments are impossible, impractical, orunethical to perform. For some seminal contributions tothe field of computational opinion dynamics, see [1,9,10]. In most approaches to opinion dynamics, an agent’sbelief state is modelled as an opinion on either a singleissue or multiple unrelated issues. We propose a model foropinion dynamics in which an agent’s belief state consistsof multiple interrelated beliefs. Because of this intercon-nectedness of the agent’s beliefs, his belief state may be inconsistent  . Consider the statement “It is raining and it isnot raining”.Even if one has no information on the currentweather, one can see that this statement is false: its logicalform is a contradiction, which is always false. Likewise, if one considers several aspects of the world simultaneously,one of the resulting theories about the world – to be madeprecise below – can be rejected out of hand as being logi-cally inconsistent.Our goal is to study the probability that an agent endsup with an inconsistent belief state. In order to explainhow the beliefs are interrelated, how the agents revise orupdate their belief state, and how this leads to the possi-bility of inconsistency, we briefly review six aspects of ourmodel: the content of an agent’s opinion, the update ruleaccording to which agents adjust their own opinion uponinteraction with others, aspects related to the opinion pro-file, the time parameter, the group size, and the main re-search question. The notions introduced in the currentsection will receive a more formal treatment further on.We compare our model with earlier approaches, in par-ticular with the arguably best-known model for studyingopinion dynamics, to wit, the Hegselmann-Krause (HK)  Page 2 of  15 Eur. Phys. J. B (2012) 85: 44 model [3]. In this model, the agents are trying to deter-mine the value of an unspecified parameter and only holdone belief about this issue at any point in time. In themost basic version of the HK model, an agent updates hisbelief over time by averaging the beliefs of all those agentswho are are within his ‘bound of confidence’; that is, theagents whose beliefs are not too distant from his own. Themodel we present can be regarded as an extension of theHK model. 1.1 Content of an agent’s opinion Agent-based models of opinion dynamics come in two fla-vors: there are discrete and continuous models. In discretemodels [1], an agent’s belief is expressed as a bit, 0 or 1 (cf. Ising spin model in physics). The opinion may rep-resent whether or not an agent believes a certain rumor,whether or not he is in favor of a specific proposal, whetheror not he intends to buy a particular product, or whichparty the agent intends to vote for in a two-party system.This binary system can be generalized into discrete modelsthat allow for more than two belief states (cf. multi-spinor Potts spin), which makes it possible to model multi-ple attitudes towards a single alternative or to representpreferences among multiple options [11,12]. In continuous models [3,13], the agents each hold a belief expressed as a real number between 0 and 1. This may be used as amore fine-grained version of the discrete models: to rep-resent the agent’s attitude towards a proposal, a politicalparty, or the like. In such models, values below 0.5 repre-sent negative attitudes and values above 0.5 are positiveattitudes. Alternatively, the continuous parameter may beemployed to represent an agent’s estimation of a currentvalue or a future trend.These models can be made more realisticby taking intoaccount sociological and psychological considerations. Forinstance, they may be extended in a straightforward man-ner to describe agents who hold beliefs on multiple, inde-pendent topics (such as economic and personalissues [14]).As such, the models can account for the observation thatagents who have similar views on one issue (for instance,taste in music) are more likely to talk about other mattersas well (for instance, politics) and thus to influence eachother’s opinion on these unrelated matters [15] 1 . Neverthe-less, it has been pointed out in the literature that thesemodels are limited in a number of important respects, atleast if they are to inform us about how real groups of agents interact with one another [20,21]. One unrealistic feature of the current models is that the agents only holdindependent beliefs, whereas real agents normally havemuch richer belief states, containing not only numerousbeliefs about possibly very different matters, but also be-liefs that are logically interconnected.In the discrete model that we propose, the belief statesof the agents no longer consist of independent beliefs; theyconsist of   theories   formulated in a propositional language 1 This has been implemented for continuous opinionsin [16–19], and for discrete opinions in [14]. (as will be explained in Sect. 2.1). We will show that thisextension comes at a cost. Given that the agents in earliermodels hold only a single belief, or multiple, unrelatedbeliefs, their belief states are automatically self-consistent.This is not true for our model: some belief states consistingof interrelated beliefs are inconsistent. 1.2 Update rule for opinions The update rule specifies how an agent revises his opinionfrom one point in time to the next. A popular approachis to introduce a  bound of confidence  . This notion – whichis also called ‘limited persuasion’ – was developed first forcontinuous models, in particular the HK model [3] 2 , andwas later applied to discrete models as well [12]. Moreover, the idea of bounded confidence can be extended to updaterules for belief states which are theories: such an HK-likeupdate rule will be incorporated into our current model.There is some empirical evidence for models involv-ing bounded confidence. In a psychological experiment,Byrne [29] found that when an agent interacts with an- other agent, the experience has a higher chance of be-ing rewarding and thus oft leading to a positive relation-ship between the two when their attitudes are similar, ascompared to when their attitudes differ. According to this‘similarity attraction paradigm’, in future contacts, peo-ple tend to interact more with people who hold opinionssimilar to their own. Despite this evidence, some read-ers may not regard updating under bounded confidenceas a natural way for individuals to adjust their opinionsin spontaneous, face-to-face meetings. Those readers mayregard the agents as experts who act as consultants in aDelphi study 3 . In such a setting, the agents do not inter-act directly, but get feedback on each other’s opinions onlyvia a facilitator. When the facilitator informs each expertonly of the opinion of those other experts that are withintheir bound of confidence, an HK-like update rule seemsto apply naturally. 1.3 Opinion profile An opinion profile is a way to keep track of how manyagents hold which opinion. This can be done by keeping a 2 See also [22–24]. For related approaches, see [13,25,26]. This approach was partly inspired by [27,28]. 3 A Delphi study is a survey method typically applied toforecast future trends, to estimate the value of a parameter, orto come to a consensus regarding a decision. Multiple expertsanswer a questionnaire separately from each other. There areadditional rounds in which the experts receive additional in-formation, including feedback on the answers gathered in theprevious rounds. It has been observed that the experts tendto adjust their initial opinion in the direction of a consensus.The Delphi method was initiated by the RAND corporationfor a study commissioned by the US Air Force [30]: the goal of  this first published Delphi study was to estimate the number of bombs required to realize a specified amount of damage. Cur-rently, Delphi studies are particularly popular in health careand nursing [31].  Eur. Phys. J. B (2012) 85: 44 Page 3 of  15 list of names of the agents and writing each agent’s currentopinion behind his name. An anonymous opinion profilecan be obtained by keeping a list of possible opinions andtallying how many agents currently hold a opinion; we willemploy the latter type of profile. Opinion dynamics can bedefined as the study of the temporal evolution of opinionprofiles. 1.4 Time Many studies in opinion dynamics investigate the evolu-tion of opinion profiles in the long run. Usually, a fixedpoint or equilibrium state is reached. Hegselmann andKrause, for instance, investigate whether iterated updat-ing will ultimately lead groups of agents to full or par-tial consensus [3]. Mas also investigates consensus- versus cluster-formation, as a function of the sociological make-up of the group under consideration [32].For sociologists, the behavior of opinion profiles atintermediate time steps may be more relevant than itsasymptotic behavior. Research on voting behavior, for ex-ample, should focus on intermediate time steps [33]; after all, elections take place at a set date, whether or not theopinion profile of the population has stabilized at thatpoint in time.In our study, we calculate the probability that an agentcomes to hold an inconsistent opinion by updating. Wedo not investigate the mid- or long-term evolution of theopinion profile, but focus on the opinion profiles resultingfrom the very first update. In other words, we considerthe opinion profile at only two points in time: the initialprofile and the profile resulting from one round of updates. 1.5 Group size Another interesting parameter to investigate in opiniondynamics is the group size. We are interested in updateswhich lead to inconsistent opinions, which may occur al-ready for groups as small as three agents (see Sect. 2.4below). The social brain hypothesis [34] states that 150 relations is the maximum people can entertain on aver-age: Lorenz presents this as an argument to model groupsof agents of about this size [19]. Whereas this figure seems moderate from the sociological point of view, this is notnecessarily the case from a mathematical viewpoint. Asobserved by Lorenz [19] (p. 323), “[c]omplexity arises with finite but huge numbers of agents”. Therefore, opiniondynamics is often studied in the limit of infinitely manyagents, which makes it possible to express the equationsin terms of ‘density of agents’. We will not do this inour current study: because of the previous observations,we should at least investigate the interval of 3 up to150 agents. 1.6 Research question As we have remarked, the agents in our model may end upin an inconsistent belief state, even when all agents startout holding a consistent theory. The main question to beanswered in this paper is: how  likely   is it that this pos-sibility will materialize? More exactly, we want to knowwhat the probability is that an agent will update to aninconsistent belief state and how this probability dependson the number of atomic sentences in the agents’ languageand on the size of their community. To this end, an ana-lytical expression is given and evaluated numerically, bothexactly and using statistical sampling. It is shown that, inour model, an agent always has a probability of less than2% of ending up in an inconsistent belief state. Moreover,this probability can be made arbitrarily small by increas-ing the number of atomic sentences or by increasing thesize of the community. 2 Preliminaries In this section, we first present the logical framework weassume throughout the paper. Then we specify the repre-sentation of the opinion profile and the employed updaterule. Finally, we relate our work to previous research on judgment aggregation and the discursive dilemma. 2.1 Logical framework 2.1.1 Language and consequence relation The agents in our model will have to judge a number of independent issues; we use the variable  M   for this num-ber (where  M   ∈  N + ). Throughout this section, we willillustrate our definitions for the case in which  M   = 2, theeasiest non-trivial example. Each issue is represented byan atomic sentence. If the agents are bankers, the issuesmay be investment proposals; if they are scientists, theissues may be research hypotheses. As an example, oneatomic sentence could be ‘magnetic monopoles exist’, andanother ‘it will rain tomorrow’. Atomic sentences can becombined using three logical connectives: ‘and’, ‘or’, and‘not’. The collection of sentences that can be composed inthis way is called the language  L .We assume a classical consequence relation for the lan-guage, which, following standard practice, we denote bythe symbol   . If   A  is a subset of the language (a set of sentences) and  a  is an element of the language (a partic-ular sentence), then  A    a  expresses that  a  is a logicalconsequence of   A . That the consequence relation is classi-cal means that it obeys the following three conditions: (1)if   a  ∈  A  then  A    a ; (2) if   A    a  and  A  ⊆  B  then  B    a ;and (3) if   A    a  and for all  b  ∈  A  it holds that  B    b ,then  B   a . Semantically speaking, that  a  is a logical con-sequence of   A  means that, necessarily, if all the sentencesin  A  are true, then so is  a . 2.1.2 Possible worlds If we were to know which of the atomic sentences are truein the world and which are false, we would know exactly  Page 4 of  15 Eur. Phys. J. B (2012) 85: 44 Table 1.  With  M   = 2, there are  w max  = 2 M  = 4 possibleworlds,  w  = 0 ,...,w  = 3. m  = 1  m  = 0 w  = 0 0 0 w  = 1 0 1 w  = 2 1 0 w  = 3 1 1 what the world is like (at least as far as is expressiblein our language, which is restricted to a finite number of aspects of the world). The point is that our agents do notknow what the world is like. Any possible combination of true-false assignments to all of the atomic sentences is away the world may be, called a  possible world  .Formally, a possible world is an assignment of truthvalues to the atomic sentences. Hence, a language with  M  atomic sentences allows us to distinguish between  w max  =2 M  possible worlds: there is exactly one possible world inwhich all atomic sentences are true; there are  M   possibleworlds in which all but one of the atomic sentences aretrue; there are  M  2   possible worlds in which all but twoof the atomic sentences are true; and so on.We may represent a possible world as a sequence of bits (bit-string). First we have to decide on an (arbitrary)order of the atomic sentences. In the bit-string, 1 indi-cates that the corresponding atomic sentence is true inthat world, 0 that it is false. Let us illustrate this for thecase in which there are M   = 2 atomic sentences: call ‘Mag-netic monopoles exist’ atomic sentence  m  = 0 and ‘It willrain tomorrow’ atomic sentence  m  = 1. Then there are w max  = 4 possible worlds,  w  ∈ { 0 ,..., 3 } , which are listedin Table 1. Also the numbering of the possible worlds is arbitrary, but for convenience we read the sequence of 0’sand 1’s as a binary number. The interpretation of possibleworld  w  = 2, for example, is that sentence  m  = 0 is falseand sentence  m  = 1 is true: in this possible world, it holdsthat magnetic monopoles do not exist and that it will raintomorrow. 2.1.3 Theories A theory is a subset of possible worlds 4 . Let us explainthis: an agent believes the actual world to be among thepossible worlds that are in his theory; he has excluded theother possible worlds as live possibilities. To see that atheory may contain more than one specific possible world,consider an agent who is sure that ‘Magnetic monopolesexist’ is false, but has no idea whether ‘It will rain to-morrow’ is true or false. If these are the only atomicsentences in his language, the agent holds a theory withtwo possible worlds. Given that we can order the possibleworlds, we can represent theories as sequences of 0’s and 4 By defining a theory in terms of possible worlds, we havechosen for a semantic approach. The equivalent syntactical ap-proach would define a theory as a subset of sentences in theagents’ language closed under the consequence relation for thatlanguage. 1’s, which in turn can be read as binary numbers. (Thisprocedure is similar to the one used above for represent-ing possible worlds by binary numbers). Note that thereare  t max  = 2 w max theories that can be formulated in alanguage with  M   atomic sentences.Table 2 below illustrates this set-up for the case where M   = 2. In that table, theory  t  = 0 is the inconsistenttheory, according to which all worlds are impossible; syn-tactically, it corresponds to a contradiction. We know be-forehand that this theory is false: by ruling out all pos-sible worlds, it also rules out the actual world. Theory t  = 15 regards all worlds as possible; syntactically, it cor-responds to a tautology. We know beforehand that thistheory is true – the actual world must be among the onesthat are possible according to this theory – but preciselyfor that reason the theory is entirely uninformative. Theother theories are all consistent and of varying degreesof informational strength. The most informative ones arethose according to which exactly one world is possible; alittle less informative are those according to which twoworlds are possible; and still less informative are the the-ories according to which three worlds are possible.In Table 2, we have numbered the theories by inter- preting their bit-string notation as a binary number. Thereverse order of the worlds in the top line is so as to makeworld  w  correspond with the  w th bit of the binary repre-sentation of the theory. 2.2 Opinion profile So far, we have focused on the belief state of a singleagent, which is expressed as a theory. Now, we consider acommunity of   N   agents. The agents start out with (pos-sibly different) information or preferences, and thereforemay vote for different theories initially. The only assump-tion we make about the agents’ initial belief states is thatthey are consistent. Subsequently, the agents are allowedto communicate and adjust there preference for a theoryaccordingly. In particular, we model what happens whenthe agents communicate with all other agents whose belief states are ‘close enough’ to their own – that are withintheir bound of confidence, in Hegselmann and Krause’sterminology – and update their belief state by ‘averaging’over the close enough belief states, where the relevant no-tions of closeness and averaging are to receive formallyprecise definitions. The totality of belief states of a com-munity at a given time can be represented by a stringof   t max  numbers,  n 0 , ...,  n t max − 1 , where the number  n t indicates how many agents hold theory  t  at that time. Wemay also represent these numbers as a vector, −→ n . We referto this string or vector as the (anonymous) opinion profileof the community at a specified time. Because each agenthas exactly one belief state, the sum of the numbers in anopinion profile is equal to the total number of agents,  N  .Also, given that initially no agent has the inconsistenttheory as his belief state,  n 0  is always zero before anyupdating has taken place. Later this may change. By up-dating, an agent may arrive at the inconsistent theory;  Eur. Phys. J. B (2012) 85: 44 Page 5 of  15 Table 2.  With  M   = 2, there are  w max  = 2 M  = 4 possible worlds,  w  = 0 ,...,w  = 3, and  t max  = 2 w max = 16 different theories, t  = 0 ,...,t  = 15. The penultimate column gives the sum of bits (bit-sum),  s t , of each theory. The last column represents theopinion profile of the community. w  = 3  w  = 2  w  = 1  w  = 0  s t  Opinion profile t  = 0 0 0 0 0 0  n 0 t  = 1 0 0 0 1 1  n 1 t  = 2 0 0 1 0 1  n 2 t  = 3 0 0 1 1 2  n 3 t  = 4 0 1 0 0 1  n 4 t  = 5 0 1 0 1 2  n 5 t  = 6 0 1 1 0 2  n 6 t  = 7 0 1 1 1 3  n 7 t  = 8 1 0 0 0 1  n 8 t  = 9 1 0 0 1 2  n 9 t  = 10 1 0 1 0 2  n 10 t  = 11 1 0 1 1 3  n 11 t  = 12 1 1 0 0 2  n 12 t  = 13 1 1 0 1 3  n 13 t  = 14 1 1 1 0 3  n 14 t  = 15 1 1 1 1 4  n 15 we shall call such an update a  zero-update   (because theinconsistent theory is represented by a string of only 0’s).In most opinion dynamics studies, a random opinionprofile is used as a starting point. Because our questiondeals with a probability in function of the initial opinionprofile, we explicitly take into account  all possible   initialopinion profiles, or – where this is not possible – takea large enough statistical sample out of all possible ini-tial opinion profiles. The different opinion profiles can bethought of as resulting from the individual choices theagents make regarding which world or worlds they deempossible. Here, we assume that the adoption of a theoryas an initial belief state can be modeled as a sequenceof 2 M  independent tosses of a fair coin, where the agentis to repeat the series of tosses if the result is a sequenceof only 0’s. As a consequence, all consistent theories havethe same probability – namely, 1 / ( t max  −  1) – of beingadopted as an initial belief state by an agent. That is tosay, we are studying what in the literature are sometimesreferred to as ‘impartial cultures’ (cf. Sect. 2.4). Further-more, the agents are assumed to choose independently of each other. 2.3 Update rule Theorists have studied a variety of update rules, depend-ing on the application the authors have in mind. For in-stance, to model gossip communication, Deffuant et al. usea rule in which updates are triggered by pairwise interac-tions [13]. To model group meetings, the updates should rather be simultaneous within the entire group of agents.During a conference, the agents meet each other face-to-face; in that case, additional effects should be takeninto account, such as the ‘primacy effect’, which demon-strates that the order in which the agents’ opinions arepublicly announced may influence how the others revisetheir opinion.As mentioned before, we may think of our group of agents as a group of scientists, bankers, or other expertswho act as consultants in a Delphi-study. The choices inthe selection of the update rule follow from that. Delphi-studies are typically conducted in a way such that theexperts do not have any direct interaction [35]. Thus, we need a model with simultaneous updating but withoutprimacy effects: in this respect, the update rule of the HKmodel [3] applies to this situation in a natural way.Another relevant aspect of the HK model is that anagent may not take into account the opinions of   all   theagents in the group. This may occur when the agent knowsall the opinions but does not want to take into account theopinions of agents who hold a view that is too differentfrom the agent’s own, or because the facilitator of theDelphi-study only informs the agent about the opinionsof experts who hold an opinion similar to the agent’s.In order to quantify what counts as a similar opinion,we introduce the ‘maximal distance’ or ‘bound of confi-dence’,  D . This parameter expresses the number of bitsthat another agent’s opinion may maximally differ fromone’s own if that agent’s opinion is to be taken into ac-count in the updating process. To quantify the differencebetween two theories, we use the so-called Hamming dis-tance of the corresponding bit-strings, defined as the num-ber of digits in which these strings differ [36].It is possible to consider heterogeneous populations,where agents may have different bounds of confidence [22]. Because Hegselmann and Krause report no qualitative dif-ference between the homogeneous and the heterogeneouscase [22], we choose the simpler, homogeneous approach: D  has the same value for all agents in any population weconsider. We investigate the influence of the value of  D  onthe probability of updating to the inconsistent theory. Byan agent’s ‘neighbors’ we refer to the agents whose opin-ions fall within the bound of confidence of the given agent.Note that, however  D  is specified, an agent always countsas his or her own neighbor.
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