AxiomaticTheoryandSimulation: A Philosophy of Science Perspective on Schelling’sSegregationModel KlausG.Troitzsch1 1Institut für Wirtscha�s- und Verwaltungsinformatik, Universität Koblenz-Landau, Universitätsstraße 1, D- 56070 Koblenz, Germany Correspondence should be addressed to kgt@uni-koblenz.de Journal of Artificial Societies and Social Simulation 20(1) 10, 2017 Doi: 10.18564/jasss.3372 Url: http://jasss.soc.surrey.ac.uk/20/1/10.html Received: 17-12-2016 Accepted: 08-01-2017 Published: 31-01-2017 Abstract: The paper uses Schelling’s famous segregation model and a number of extensions to show how a reconstruction of the theory behind these models along the lines of the ‘non-statement view’ on empirical sci- ence can contribute to a better understanding of these models and a more straightforward implementation. A short introduction to the procedure of reconstructing a theory is given, using an extremely simple theory from mechanics. The same procedure is then applied to Schelling’s segregation theory. A number of extensions to Schelling’s model are analysed that relax the original idealisations, such as adding di�erent tolerance levels between the two subpopulations, assuming inhomogeneous subpopulations and heterogeneous experiences of neighbourhoods, among others. Finally, it is argued that a ‘non-statement view’ reconstruction of a mental model or a verbally expressed theory are relevant for a useful specification for a simulation model. Keywords: Theory Reconstruction, Non-Statement View, Schelling Model, Segregation, Axiom Introduction 1.1 This paper is intended to show the similarities between simulation modelling in general and a method of for- malisingtheories,whichwasdevelopedsomethirtyyearsago(Sneed1979;Balzeretal.1987)andhasbeenused to reconstruct theories in sciences such as physics, but only rarely in sciences such as psychology (Westmeyer 1992), economics (Stegmüller et al. 1981; Alparslan & Zelewski 2004), sociology and political science (Druwe 1985; Troitzsch 1987, 2012b). Only in a few cases has the analogy between the ‘non-statement view’ of recon- structing and formalising theories and the simulation of theory-derived models been shown: In sciences such as physics, this is not necessary as many dynamic phenomena can be described with classical mathematics, suchassystemsofordinary,partialorstochasticdi�erentialequations,which lendthemselvestoareformula- tion in terms of this philosophy of science approach (to be described in paragraph 2.7). This also holds for the neoclassicmethodology ineconomics. However, in manycases whereemergent phenomenaon amacro level resultingfrominteractionsbetweenelementsofamicrolevelneedtobedescribed,evenstochasticdi�erential equations might not be su�icient to explain the emergent phenomena. This is particularly the case when the elements of the micro level are inhomogeneous, which is typical in systems which economics, sociology and political science are interested in. Where the elements of social systems can be simplified as consisting of ho- mogeneous elements, an approach with stochastic di�erential equations is sometimes su�icient, as has been shown by Weidlich & Haag (1983), Helbing (1991/92) and, more recently, by Johansson et al. (2008) (for a more detailed discussion see Troitzsch (2009, 2012a)). In the case of social science research that looks at systems of inhomogeneous, interacting and interpreting human actors, only few papers have discussed the analogy between simulation and ‘non-statement view’ reconstruction (Troitzsch 1992, 1994, 2012b; Balzer & Moulines 2015). 1.2 The paper is structured as follows. In the next section, the use of the terms ‘axiom’ and ‘axiomatisation’ will be discussed, and a short description of theory reconstruction according to the ‘non-statement view’ will be given. Section3willexemplifythisreconstructionprocesswiththefamoussegregationmodelofSchelling(1971) JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 kgt@uni-koblenz.de whileSections4and5willapplythis formalisationmethodtoanextensionofSchelling’smodelwithstructural inhomogeneityandbehaviourrulesthatchangeovertime. It isworthnotingthat inthesecaseschangesonthe macro level in turn change due to the individual changes, as described with the ‘boat’ or ‘bathtub’ metaphor firstcoinedbyColeman(1990,p.8). Finally,Section6triestoassesstheadvantagesofthe‘non-statementview’ for computational social science at large against a slightly less formal agent-based simulation approach. AxiomsandAxiomatisationofaTheory Axiomsinthesocialsciences 2.1 Theword‘axiom’seemstohavebeenusedforthefirsttimeinthecontextofEuclid’sgeometrywhereit isunder- stoodasastatementwhichneednotandcannotbeprovenas“anestablishedprincipleoraself-evidenttruth” (Merriam-Webster, “axiom”) or a “maxim, that has found general acceptance or is thought worthy of common acceptance whether by virtue of a claim to intrinsic merit or on the basis of an appeal to self-evidence” (Ency- clopædia Britannica 2014, “axiom”). 2.2 In a talk given in June 1971, Suppes (1974, p. 472), a�er having talked mainly about axiomatisation approaches in physics, stated: Many problems of interest in the behavioral and social sciences have also been treated from an axiomaticstandpoint. Muchofthecontemporarywork inmathematicaleconomicssatisfiesahigh standardofaxiomatization,andwhennotexplicitlysostated, itcaneasilybeputwithinastandard set-theoretical framework without di�iculty. On the other hand, with the exception of some of the problemsofmeasurement Imentionedearlier, the impactof thetheoryofmodelsasdevelopedin logicandthekindmetamathematicalquestionscharacteristicof that theoryhavenotbeenwidely applied in the social sciences, and the relation of these sciences to fundamental questions of logic has not had the history of examination characteristic of problems of long standing in physics. 2.3 In what followed in his talk, he gave an example of an axiomatisation of stimulus-response theory inspired by previouswork(Suppes1969)andmentionedanumberofsimilarattemptsmainly inequilibriumeconomics— whichisfavouredthankstoitshighlevelofmathematisation. This isalsowhyoneoftheearlytopRussianread- ers in mathematics named only “political economy” as a social science subdiscipline apt for mathematisation and axiomatisation: Of course, in the study of such complicated phenomena as occur in biology and sociology, the mathematical method cannot play the same role as, let us say, in physics. In all cases, but espe- ciallywherethephenomena aremostcomplicated, wemustbear in mind, ifwearenot to loseour way in meaningless play with formulas, that the application of mathematics is significant only if the concrete phenomena have already been made the subject of a profound theory. In one way or another, mathematics is applied in almost every science, from mechanics to political economy. (Aleksandrov 1999, p. 4) 2.4 Onemust,however,admitthat(nearly)all thoseaxiomatisationsoftheories inthesocialsciencesat largewere appliedtocaseswhereeitheronlythemacrolevelwasconsidered(ineconomics)orwhereonlythemicrolevel (psychology and sociology of small groups) was considered. Indeed, the problem of the interaction between these two (and potentially even more) levels was only very rarely the object of axiomatisation attempts — at least before the era of agent-based modelling and its predecessors in multilevel modelling. Economics used as “one of the best examples ... the systems of equations of the mathematical theory of prices ... to describe thegeneralcharacterof theorder thatwill formitself” (Hayek1967,p.261)whereassociologyo�enusedgame theoryas inColeman’s reconstructionofanexperimentconductedbyMintz (1951) (Coleman1990,p.203–215), to name just two extreme examples. 2.5 In thecontextof thispaper, weuse theword ‘axiom’ inthesenseofa“condition ... thatha[s] tobesatisfied by thebasicnotionsof thetheory inquestion” (Balzeretal. 1987, p.24), suchthat it isnotastatementthat isheld to be true but the predicate that the theory makes about its intended applications. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 The‘non-statementview’andsimulation 2.6 Tointroducetheproceduresofreconstructingatheoryalongthelinesofthe‘non-statement’view(“reconstruc- tion procedures”, (Balzer et al. 1987, p. 23)1), we use a very simple theory from classical mechanics, namely Hooke’s early theory of elasticity — Petroski (1996, p. 9–11) tells the story of Hooke’s discovery — which says “that up to a limit, each object stretches in proportion to the force applied to it”. Hooke’s experiment consists ofaspringwhoseupperendis fixedandwhoselowerendcanbeloadedwithoneormoresmallweightswhich will extend the spring by a measurable amount. The law says that (within a certain range) the extension is pro- portionaltothenumberofthesmallweightshangingfromthespring. Thus, this lawcanbedescribedwithjust two terms which are measurable without any knowledge of springs: the number N of identical weights and the lengthLof theextension. Hookefoundoutthat foranyspringthetwonumericalvalueswereproportional with di�erent proportionality factors ks for di�erent springs s: N = ksL such that this law could be used for weighing things with unknown weights. 2.7 A certain Hooke-like experiment can be understood as a model2 of classical particle mechanics (CPM). In this simplificationofthediscussionin(Balzeretal.1987,p.29–34), it isunderstoodthatN andLarenon-theoretical termswithrespecttoCPM (orrather: withrespecttoHooke’sspringlawHSL,aswewillcall thisextremelysim- plified version from now on), given that counting identical weights and measuring the length of the extension havenothingtodowithsprings. Ontheotherhand,the‘deviceconstant’ks forsprings isnotevenconceivable and hence unmeasurable without using HSL. Thus it has to be considered as a “theoretical term with respect toHSL”asbeforestatingHooke’s lawit is totallyunclearwhetherks alsodependsonthenumberofweightsN appended to the spring. In terms of (Balzer et al. 1987) we can now formulate: Mpp(HSL): x isapartialpotentialmodelofHooke’sspringlaw(x ∈ Mpp(HSL)) i�thereexistS,W,N,L,k∗such that 1. x = 〈S,W,N,L,k〉 2. S is a finite set [of springs] 3. W is a finite set [of identical weights] 4. N : P(W) ×S →N [N(w̄,s) yielding the number of identical weights loaded at the lower end of the spring where w̄ ⊂ W denotes the collection of weights hanging from spring s ∈ S] 5. L : P(W)×S →R+ [L(w̄,s) yieldingthelengthoftheextensionproducedbyasubsetof identical weights at the lower end of the spring] 6. k∗ : P(W) ×S →R+ [k∗(w̄,s) yielding the quotient N(w̄,s)/L(w̄,s) ] 2.8 In this definition, k∗ is not yet a device constant as it does not only depend on the spring but also on the col- lectionofweightshanging fromthespring. OnlywhenHookedetectedthatat least forsmallextensionsk only depended on the spring, he could formulate: Mp(HSL): x is a potential model of Hooke’s spring law (x ∈Mp(HSL)) i� there exist S,W,N,L,k such that 1. x = 〈S,W,N,L,k〉 2. S is a finite set [of springs] 3. W is a finite set [of identical weights] 4. N : P(W) ×S →N [N(w̄,s) yielding the number of identical weights loaded at the lower end of the spring] 5. L : P(W)×S →R+ [L(w̄,s)]yieldingthelengthoftheextensionproducedbyasubsetofidentical weights at the lower end of the spring] 6. k : S → R+ [k(s) yielding the quotient N(w̄,s)/L(w̄,s) where w̄ ⊂ W denotes the collection of weights hanging from spring s ∈ S] and finally M(HSL): x is a model of Hooke’s spring law (x ∈M(HSL)) i� there exist S,W,N,L,k such that 1. x = 〈S,W,N,L,k〉 2. x ∈Mp(HSL) 3. ∀w̄ ⊂ W ∀L < L0 and∀s ∈ S: k(s) = N(w̄,s)/L(w̄) isaconstantwhichdoesnotdependon w̄. where 3 can be called the axiom of Hooke’s spring law as it postulates that the extension of the spring is pro- portional to the weight at its lower end. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 Schelling’sSegregationModelRevisited Schelling’smodelreconstructed 3.1 The famous Schelling model (Schelling 1971) has been programmed very o�en but rarely has it been used to analysethedependencyof thesegregation indexoninputparameterssuchasdensity,groupsizesandthresh- old(exceptperhapswhenSquazzoni(2012,p.90–92)comparedthesimilarityindexdynamicsforthreedi�erent thresholdvaluesdenotingthe“preferenceoflikeneighborsat25,33and50%”—seealso(Epstein&Axtell1996, p. 165–171)). Density, for instance, was typically set to 85 per cent (Bruch & Mare 2006, p. 674, footnote 13), and group sizes were typically equal, also in the models using, for instance, three subpopulations (Muldoon et al. 2012,p.42),butoccasionallyan“empiricalrace-ethniccomposition”wasused(Bruch&Mare2006,p.674, foot- note 12) . Furthermore, it has never been used to analyse the behaviour of the model systematically when the individualsdonothaveidenticalthresholdsbutthresholdsfollowingacertaindistributionwhichmightalsobe di�erent between groups (except (Gilbert 2002)). A mathematical analysis of the Schelling model and some of itspossibleextensionwasgivenbyZhang(2004)whoshowedthatsegregationis“stochasticallystable”(p.148). 3.2 Inthissection,anattemptismadetoreconstructSchelling’smodel intermsofthe“non-statementview”intro- duced above. The “reconstruction procedure” is quite similar to the one on in paragraph 2.7. 3.3 A run of a Schelling simulation model written in NetLogo (Wilensky 1997) can be understood as a model of Schelling’s segregation theory (SST), where it is understood that in any real-world context: • the individuals occupying houses or apartments or, more generally, city blocks in their world, • their density, • their individual ‘colours’ and • thesegregationindexwhichcanbeeasilycalculatedfromthedatadefiningwhichcityblocksareoccupied by which individual agent(s) aremeasurablewithoutanytheoryofsegregationwhereasthe individual tolerance levelsareunobservableas human beings are rarely in a position to give their individual tolerance levels (or, more generally speaking, any kind of propensity or action probability) a numerical value. 3.4 Hence, a potential model of SSTcan be defined as Mp(SST): x is a potential model of Hooke’s spring law (x ∈Mpp(SST)) i� there existW,W,P,`,T,θ,b,c,φ,δ,ς such that 1. x = 〈W,W,P,`,T,θ,b,c,φ,δ,ς〉; 2. W is a set of pairs 〈W,P〉 [each consisting of a city and its inhabitants or, in the simulation model, the ‘world’ of a Netlogo model interface together with the turtles on it]; 3. W is a finite set [of city blocks or, in the simulation model, of patches, collecting all city blocks of the target system or, in the simulation model, the ‘world’ of a NetLogo model interface]; 4. P is a finite set [of persons or households moving between city blocks or, in the simulation model, of turtles moving between patches]; 5. ` : P →{`1,`2} [`(p) yielding the feature of a person in question, for instance their language or, in the simulation model, the colour of a turtle]; 6. T isafiniteset[ofpointsintimewhencensusrecordsaretakenor, inthesimulationmodel,ofticks]; 7. θ : P ×T → [0, 1] [θ(p,t) yieldingathresholdvaluehelpingpersonorturtlep todecidewhetherto stay or to move at time t]; 8. b : P ×T → W [b(p,t) yieldingthecityblockbwherehouseholdp livesatacertaincensustimetor, inthesimulationmodelthepatchb theturtlepoccupiesatacertaintick tofthesimulationmodel]; 9. c : W → {cxmin, ...,cxmax}× {cymin, ...,cymax} [c(b) yielding the integer coordinates of a city block or, in the simulation model, of a patch]; 10. φ : P × T → [0, 1] [φ(p,t) yielding the proportion of persons of the same colour in the Moore neighboorhood of a certain person at a certain census time or, in the simulation model, the same]; JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 11. δ : P×T → W [δ(p,t) yieldingthecityblockor, inthesimulationmodel, thepatchtowhichperson (or turtle) p will move at time t; φ(p,t) > θ(p,t) → δ(p,t) = b(p,t),φ(p,t) ≥ θ(p,t) → δ(p,t) 6= b(p,t), i.e. i�theproportionofpersonsofthesamecolourintheMooreneighbourhoodofp isbelow thethresholdθ(p,t) thispersonorturtlewillmovetothenearest freecityblockorpatchortoacity block or patch where the neighbourhood seems to be more convenient]; 12. ς : W → [0, 1] [ς(〈W,P〉) yielding the segregation index for the whole collection of city blocks and their inhabitants or, in the NetLogo model, of the simulated world and its turtles]. 3.5 The segregation index D mentioned in item 12 is defined (Duncan & Duncan 1955) as D = 1 2 n∑ i=1 ∣∣∣xi X − yi Y ∣∣∣ (1) where xi and yi are the local numbers of persons belonging to each of the two subpopulations in n subareas and X and Y are the overall sizes of the two subpopulations. In the current context, we have to consider that the n subareas are overlappingas each patch counts the turtles in a squareneighbourhood of 49 patches. Fur- thermore, we use the segregation index in the range of 0 to 100 instead of 0 to 1 such that ς = 100D/49. 3.6 Some more derived terms used later on need to be mentioned here: • the minority size ν defined as |{p∈P|`(p)=`1}||P| • the density d defined as |P||W| Intendedapplicationsof STT 3.7 Some of these terms might not be measurable in intended real-world applications of SST: • θ is quite di�icult to measure when asking people for a real number in the interval [0, 1] to describe be- yond which percentage of similar neighbours in their vicinity they are happy or below which threshold they would take a certain action. Other sources of information about such propensities — census data or data from registration o�ices, from which removal frequencies can be obtained — do not yield more reliable information about actual individual propensities. Approaches to overcome this di�iculty have been made for instance by da Fonseca Feitosa et al. (2011), Wong (2013) and Benenson et al. (2003)). In most simulation models published so far based on SST implementations, θ has been a constant for all members of both subpopulations in each simulation run, much like the device constant in HSL, but, see below, this is, of course, not the only possible interpretation of θ. • δ isalsoquitedi�iculttomeasure—onewouldhavetoaskinterviewees“wherewouldyouwanttomove in case you find that in your neighbourhood there are too many people speaking another language?”, as wasdonebyXie&Zhou(2012)andinamoresophisticatedmannerbyBruch&Mare(2006)andLewisetal. (2011). Suchaquestion,however,containstwohypotheticalconditions—whichisusuallydiscouragedby textbooksonsurveymethodology(cf. e.g.,Converse&Presser1986,p.23). Thisiswhyinmost“Schelling” simulationmodelsδ justpointstoanarbitraryfreepatchinthevicinityofthecurrentplacealthoughthere is a lot of empirical evidence that people choose deliberately where to move, and there exist simulation models like the ones cited above which take this into account. 3.8 Thefunctionpcaninprinciplebereconstructedfromindividualdataofsubsequentcensuses(whenindividual dataarekeptbetweencensusdates)or fromrecordsofresidentregistrationo�ices(if theseexist inthecontext in question). 3.9 Intended applications are usually partial potential models of a theory that do not include terms which are the- oretical with respect to the theory in question, and here is where intended applications of SST have serious problems for several reasons: • If, as is usually the case although not in Schelling’s original paper, the world is understood as a torus, there is no real-world correspondence possible at all, but this restriction can easily be solved. The fact thatSchelling’soriginalandnearlyallsimulationmodelsdescribetheworldstructuredasacheckerboard is not so much of a problem as Flache & Hegselmann (2001, 5.1) showed that social dynamics “may be widely robust against changes of the underlying standard assumption of rectangular grids”. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 • Having only two more or less homogeneous subpopulations which di�er in exactly one binary feature is a simplification — Gilbert (2002) has pointed this out and showed a number of relaxations and its con- sequences — and it will be di�icult to find a social system which can be described in so simple terms. However, there are modelling attempts which try to overcome this and other simplifications, too, for in- stance Muldoon et al. (2012) and Durrett & Zhang (2014) with larger neighbourhoods, Lewis et al. (2011) and Wong (2013) with more than two subpopulations. • Describingneighbourhoodsonlyone-dimensionallywiththeproportionsofinhabitantsbelongingtodis- tinguishablesubpopulationsisobviously inadequate,astherearemanyothermotivestomovefromone city district to another which were for instance taken into account by da Fonseca Feitosa et al. (2011); for the inclusion of the housing market see (Zhang 2004). Thepartialpotentialmodel of SSTanditssimulation implementation 3.10 Leaving the problems in the two previous paragraphs aside for a while, one can now easily map this descrip- tion of the potential model of SST on Wilensky’s NetLogo simulation model (Wilensky 1997) and the extension described inthispaper—seeTables1and4. Theextensiondescribedherecanberunwithexactly thefeatures of Wilensky’s original. SST term NetLogo model component W the set of all possible runs of the model W the patches in a certain run of the model P the turtles in a certain run of the model ` the built-in turtle variable color T NetLogo’s ticks θ the global variable %-similar-wanted — which shows that in original SST the tolerance level is not an individual variable but a global constant, a re- striction first relaxed in Gilbert (2002) b the NetLogo built-in function patch-here c NetLogo’s built-in turtle variables xcor and ycor φ the value of this function is calculated in the procedure update-turtles in Wilensky’s code (similar-nearby) δ the function move-unhappy-turtles in Wilensky’s code ς the value of this function is calculated in a few lines in the procedure update-globals added to Wilensky’s code but can also be calculated as a single function Table 1: Correspondence betweenSST terms and NetLogo components 3.11 NeitherSchelling’soriginalpapernoranyofthefollowingworkyieldsaclosedformulaconnectingthesegrega- tionindexς tothetolerancethresholdθ —whichsofarwasmostlyassumedtobeconstantforallagentsandat all times, with the exception of Gilbert (2002) — or to the density d = |P |/|W | < 1 (which must be strictly < 1 asotherwiseunhappyagentshavenochancetoswerve)ortothefractionsofthetwogroups(usuallyassumed equal, but it is also — and perhaps even more — interesting to find out how segregation works with respect to a minority; the fractions of the groups can easily be expressed in the terms of Mp(SST)). But, multiple runs of the simulation model give an opportunity to derive at least a linear or nonlinear regression equation between the segregation index ς (certainly a macro variable) and one or more of the other macro or micro variables. θ, although a constant in the original version of SST, is a feature of the individuals and hence a micro variable. In extended versions, however, building on Gilbert (2002), θ will become a function of the macro variables µθ andσθ,andtheindividualθp,t willevenchangetheir individualvaluesovertimedependingonlocalneighbour- hoods. Firstresults 3.12 Figure1givesafirst impressionofthedependenceofthesegregationindexonthetolerancethreshold: Itseems that the dependence is nonlinear — as already observed by Squazzoni (2012, p. 92)) — but obviously entirely di�erentfortolerancethresholdsbelowandabove80percent. Indeed,abovealevelof80percent,segregation cannotbeachievedas itbecomesextremelydi�icult for theagentstobecomehappywithsostrongademand. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 http://ccl.northwestern.edu/netlogo/models/community/Segregation http://ccl.northwestern.edu/netlogo/models/community/SchellingAnalysed http://ccl.northwestern.edu/netlogo/models/community/SchellingAnalysed 3.13 Here, it is importanttonotethat inWilensky’s implementationunhappyagents justmovetosomeotherempty patcheswithouttakingintoaccountwhetherthesepatchesmeettheirneedsbetterthanthepatchestheycome from ( “keep going until we find an unoccupied patch”; the extended version stops a run when over the last 20 ticks the standard deviation of percent-unhappy was below 1). We will first analyse the results for tolerance levels below 80 per cent in more detail to return to the problem of agents’ unintelligent search for alternative patches. Sheet2 Page 1 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 % similar wanted s e g re g a ti o n i n d e x Figure 1: Segregation index as dependent on tolerance threshold for 45 per cent of one group and a density of 0.8; three runs per combination 3.14 Tothisend,aMonteCarlosimulationwithpartlyrandomparametercombinationsisruntosearchthecomplete parameter space (reasonably leaving out tolerance thresholds above 80 per cent) and to find out how much of the variance of the segregation index can be explained by tolerance threshold (uniformly distributed between 5 and 75 per cent), density (.75,. 85 and .95) and size of the minority group (10, 30, 50 per cent) with 300 runs foreachcombinationof thetwolatter factors, resulting in2,700 individual runs. A�erwards,wewillextendthe model along the lines of the ideas presented by Gilbert (2002). 3.15 Thisyields thescatterplots presented in Figure2. Notethat in these plotsonly those runswere usedwhere the tolerance threshold did not exceed 65 per cent. Figure 1 shows that the emergent behaviour of the system is di�erent for these high tolerance levels. Most of these graphs show the cubic dependence between tolerance threshold and segregation index. 3.16 A first attempt at analysing the outcome of this model is a Monte Carlo simulation with 3,000 runs varying the tolerance threshold, the size of the minority and the density. Here, we want to find out how strong the depen- dence of the segregation index on these three input parameters is. This analysis shows a variance reduction of nearly 90 per cent (R2 = 0.872). The tolerance threshold is the most important input parameter with a stan- dardisedβ = 0.901, the influenceoftheminoritysize isweakerwithβ = −0.281 (thesmaller theminority, the higher the segregation index), whereas the influence of the density is not even significant (in spite of the high number of runs, for the relevance or irrelevance of significance in simulation analysis see Ziliak & McCloskey (2007)) with a standardised β = −0.028. 3.17 Thisfindingcanbegeneralisedtoacubicregressionofthesegregationindexontolerancethresholdθ,minority sizeν anddensityd inthisMonteCarlosimulationwith3,000runs. Thevariancereductionisslightlyhigherthan inthelinearcase(R2 = 0.934)andthesegregationindexcanbe‘predicted’withastandarderrorofabout3.66 percentage points. The le�-hand diagram of Figure 3 shows how perfect this regression is. However, it is even more interesting for our current concern that the segregation index, the density and the minority size can be usedtomeasurethetolerancethreshold3 —herethevariancereductionisalsoabove90percent(R2 = 0.915) and the standard error is about five percentage points (see the right-hand diagram of Figure 3). 3.18 ThismeansthatSSTyieldsaproceduretomeasurethevalueofatermthatcouldnototherwisebemeasuredin real-worldscenarios,hencethetolerancethresholdisatheoreticalvariablewithrespecttoSST—muchlikethe case of the device constant of Hooke’s springs which can be measured with HSL. The regression equation can bedefinedastheaxiomof SSTstatingthattheexpectedvalueof thetolerancethresholdof twohomogeneous subpopulations is a cubic function of the three terms specified above and that the parameters of this function are just the 16 regression coe�icients (not given here, as it is entirely unclear what the coe�icients β111 for the product θνd or β201 for the product θd mean). So, one could conclude that a “black white segregation index” in New York, Northern New Jersey and Long Island of 81.5, as reported by Frey (2016) and Frey & Myers (2005), and the same index for Tucson AZ of 36.9 can be interpreted as a tolerance level (of both subpopulations the same!) of more than 70 and less than 12, respectively. 3.19 This said, one must also ask whether this is of any use if we know that Schelling’s model is an idealisation of whatcanbeobservedintherealworld. Wecan,ofcourse,extendthismodeltobeat leasta littlemorerealistic JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.1, number: 1875 Page 1 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.1, number: 2125 Page 2 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.1, number: 2375 Page 3 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.3, number: 1875 Page 4 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.3, number: 2125 Page 5 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.3, number: 2375 Page 6 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.5, number: 1875 Page 7 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.5, number: 2125 Page 8 %-similar-wanted 806040200 s e g re g a ti o n -i n d e x 80 60 40 20 0 percentagered: 0.5, number: 2375 Page 9 Figure 2: Segregation index as dependent on tolerance threshold for three levels of minority size and three density levels; 300 runs per combination andmakethetolerancethresholdavariablethatcanvaryamongindividuals,betweenthetwosubpopulations and, lastly, over time. This is what we will analyse in the next section. AddingMoreComplexitytoSchelling’sModel 4.1 Asalreadydiscussedinearliersections,theversionofthemodeldescribedinthefollowingsubsectionsextends Wilensky’s implementation inseveral respects. While theextensionsaboveweremerely technical (e.g.,adding a formula for calculating the segregation index, adding a stopping mechanism when the model run seemed to have stabilised), the extensions dealt with in this section are more substantial and are as follows: • tolerancerelatedsearchofanewneighbourhood, i.e. agentsdonotonlysearchforanunoccupiedpatch but they look for an unoccupied patch which fits their needs better than the current patch4; • tolerance levels can be di�erent for the two subpopulations5 and , i.e. population red might like to live togetherwithpopulationgreenwhichinturnpreferstoliveapartfromred—examplesare: arichminor- ity preferring to live in gated communities and a middle class majority taking no o�ence at rich people livingintheirneighbourhoodoraminorityofhooliganswhodonotcarefortheirneighbourhoodbutwho influence majority people to move away; this leads to θ(p,t) = θi i� `(p) = i; • tolerance thresholds may di�er within each subpopulation (i.e. they have distributions with di�erent statisticalparameters,here: meansandvariances)6; thismeansthatθ isnolongeraglobalconstantas in thedefinitionofthepotentialmodelofSST, item7,butinsteadarandomvariableapproximatelynormally distributedwithineachsubpopulationiwithmeanµθ,i andstandarddeviationσθ,i censoredtotherange of [0.05, 0.95]; • tolerancethresholdschangeover timeasaconsequenceofcommunicationbetweenagents; thismeans that θ is no longer an individual constant but a variable changing over time (see Section 5). JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 segregation-index 6040200 p re d s e g re g a ti o n -i n d e x 80 60 40 20 0 Page 1 %-similar-wanted 6040200 p re d % -s im il a r- w a n te d 80 60 40 20 0 Page 2 Figure 3: Le�: Segregation index as dependent on tolerance threshold for minority size and density. Right: Tolerancethresholdasdependentonsegregation index, tolerancethreshold,both inaMonteCarlosimulation with 3000 runs Sophisticatedsearch, inhomogeneousanddi�erentsubpopulations 4.2 InanextbigMonteCarlosimulationwith6,000runs,weexperimentwiththefirstthreeextensionslistedabove. We make a twofold di�erence: • one between the simple search of a new place (as coded by Wilensky 1997) and the tolerance related searchwheretheagentslookforanunoccupiedpatchwhichisatleastpopulatedwithslightlylessagents of the other colour or language — if none is found the agent does not move — and • betweenhomogeneoussubpopulations(all individualsofasubpopulationhavethesamethreshold)and inhomogeneous subpopulations (within each subpopulation the tolerance threshold follows a censored normal distribution with a mean — usually di�erent for the two subpopulations — and a variance of 15 percentage points; censoring makes sure that the individual tolerance threshold remains between five and 95 per cent). 4.3 This leadsto1,500simulationrunsforeachofthefoursubexperimentsdefinedbysearchstrategyandsubpop- ulation homogeneity, and in each of the four subexperiments density, minority size and the two means of the tolerance threshold are randomly varied. 4.4 The outcome of this experiment is analysed with a linear regression which yields the variance reductions and standardised βs collected in Table 2. input parameters simple search tolerance related search homogeneous inhomogeneous homogeneous inhomogeneous all R2 0.627 0.720 0.689 0.802 minority size β -0.168 –0.340 -0.207 -0.404 density β -0.127 -0.198 -0.133 -0.182 tolerance mean minority β 0.388 0.560 0.389 0.556 tolerance mean majority β 0.670 0.533 0.696 0.513 tolerance means cubic R2 0.757 0.649 0.748 0.647 Table 2: Variance reduction and standardised regression coe�icients for the linear dependence of the segrega- tion index ondensity, minoritysize, searchstrategies andthreshold distributionsand fora cubic regression on the two tolerance means (all coe�icients are significantly di�erent from 0, α < 0.0005) 4.5 Table 2 shows that the strength of the dependence of the segregation index on the four more or less contin- uously varied input parameters decreased considerably due to the fact that the two subpopulation now have di�erent tolerance levels—afindingthatneeds furtheranalysis. Ontheotherhand, it is interestingtoseethat bothinternalinhomogeneityandamoresophisticatedsearchstrategyincreasethestrengthofthedependence. Here, it isworthnotingthatat least the former (internal inhomogeneityallowsforamoreprecisepredictionor JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 explanation of the segregation index) comes as a surprise and also calls for further analysis. Unlike the case with thresholds identical between the two subpopulations, density now makes a di�erence, although not a remarkable di�erence, when thresholds di�er between and within subpopulations. Finally, the standardised regression coe�icients of the tolerance means are now clearly below the level marked in the previous analysis with subpopulation-independent tolerance thresholds. This is particularly true for the simple search strategy which additionally leads to relatively low regression coe�icients of the minority’s tolerance level. %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 10 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 homogeneous, simple search Page 1 %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 10 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 inhomogeneous, simple search Page 3 %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 10 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 homogeneous, tolerance related search Page 2 %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 inhomogeneous, tolerance related search Page 4 Figure 4: Segregation index as dependent on tolerance threshold for homogeneous and inhomogeneous sub- populations and two search strategies; 1500 runs per combination; the vertical axis is the unstandardized pre- dictedvalueofthesegregationindexfromacubicregressioninthetwotolerancethresholdmeanswhereasthe coloured dots represent the approximate values of the dependent variable 4.6 Figure4showshowthetwotolerancethresholds(or, respectively, theirdistributions) influencethesegregation index. Thesediagramsshowthesegregationindexvaluesaspredictedbyacubicpolynomial(itsR2 isalsogiven in Table 2) in the two tolerance means (the coloured dots, however, show the approximate segregation index value as they were yielded by the simulation). 4.7 Obviously, it does not matter whether the tolerance threshold distributions of the two subpopulations are dif- ferent or similar — otherwise the colour shades of the dots in the four diagrams of Figure 4 would have been separated by borders running top down. On the contrary, the colour shades are quite distinctly separated by borders which run parallel to the plane spanned by the two input parameters. Hence, the fiercest segregation occurs when the overall mean tolerance threshold is high: if both subpopulation thresholds are above 50 per cent, a segregation index above 46 can be expected (red and dark red dots in the top far corners of the dia- grams) whereaswhen bothare below 30per cent the expected segregation index willbe below30 (violet, blue and dark green dots in the bottom foregrounds of the diagrams). The overall impression given by the four di- agrams does not point to big di�erences caused by the choice of the two binary input parameters (tolerance levelstandarddeviation0vs. 15,simpleortolerancerelatedsearchstrategy). However,perhapstheboundaries betweenthedi�erentlycolouredregionsofthediagramarelesssharpforthediagramsshowinghomogeneous subpopulationsandforthediagramsshowingsubpopulationsapplyingthesimplesearchstrategy(at leastthis is what one would expect from Table 2). The only remarkable di�erences between the four diagrams are per- JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 haps the clearer symmetry of the surfaces with respect to the input parameter %-similar-red-mean in the diagrams for homogeneous subpopulations (best visible at the right-hand edge of the surfaces) compared to anasymmetricparabolaat theright-handedgesof thediagramsfor inhomogeneoussubpopulations. Further- more, it is interestingtoseethatboththelowestandthehighestsegregationindicesarereachedforthesimple search in homogeneous subpopulations — as if inhomogeneity and a more sophisticated search strategy lead to a smaller range of the segregation index. A Further Extension: Individually Di�erent Tolerance Thresholds Chang- ingoverTimeDuetoCommunication 5.1 ThefinalextensionofSchelling’soriginalmodel introducesane�ectof theexperienceofagents intheirneigh- bourhoods on their tolerance threshold. The idea behind this extension is that an agent surrounded by a high proportionofagentsofthesamesubpopulationwill increaseitstolerancethreshold, i.e. willwanttohaveanin- creasingproportionofsimilaragentsarounditself,whereasanagentsurroundedbyahighproportionofagents oftheothersubpopulationwilldecreaseitstolerancethreshold, i.e. willacceptanincreasingproportionofdis- similar agents around itself. Hence, θ is now a function which yields an agent’s individual tolerance threshold as a function of φ defined above in the definition of the potential model of SST, item 10, as follows: θ(p,t + 1) = θ(p,t)(1 −ε) + εφ(p,t) (2) where ε can be viewed as a parameter describing how fast tolerance or intolerance are learned. 5.2 The typical outcome of this extended version is also segregation in most cases. This is even more pronounced thaninthetime-constantversions. However,ascanbeseeninthetwoplotsatthetoprightofFigure4,usually the mean of the distribution of tolerance threshold in the majority subpopulation increases while its variance decreases; for the minority, the opposite holds — this e�ect is the more graphic the smaller the minority is —, and the most tolerant individuals of each subpopulation can be found at the borders of the clusters. 5.3 Another interestingobservationis that forthesimplesearchthisversionofthemodelproducesaneverending wandering of members of one or both subpopulations: Whenever a large proportion of both groups is ‘happy’, the more tolerant population moves to places where they are not welcome from the point of view of the other population. This leads to an oscillation of the segregation index, of the percentage of similar agents in the neighbourhood and of the percentage of ‘unhappy’ agents. This is in line with observations made by Weidlich & Haag (1983, p. 86–112) who analysed “the migration of two interacting populations between two parts of a city”, which is certainly an object of analysis that is quite similar to Schelling’s problem, and observed that under certain circumstances, namely one population wanting to live together with the other population and the other population trying to avoid this, the expected or most likely trajectory of the system would become a stablespiralorevenalimitcycle. InthecurrentversionoftheSchellingmodel, it isusuallyaspiral—giventhat the simulation runs are partly stochastic, it is undecidable whether limit cycles really evolve. 5.4 Oscillations do not evolve in the case of the tolerance related search discussed above (see Section 4). Further- more,theyarethemorefrequentthehigherthetolerancemeansofthetwosubpopulationsare(withbothlow, no oscillation at all evolves). On the contrary, the segregation stabilises preferably when both initial tolerance levels are low at the same time. 5.5 Figure5showsthesituationofsuchanoscillatingsimulationrunandtheoscillationswhichcouldbeobserved duringtherun. Thesimulationstartedwithmeantolerancelevelsof63percent(inthe11percentminority)and 52 per cent (in the 89 per cent majority). From the very beginning, the minority agents were mostly unhappy whereasthemajorityagentsweremostlyhappy. Inthefirstround,theminorityconcentratedwhichmademost ofthemhappy. Thedistributionoftheirtolerancelevelsmoveddown,whereasthedistributionofthetolerance levels of the majority moved up and became very narrow. This implies that they tried to move away from the minorityagentswho, inthemeantime,hadbecomemoreandmorefriendlytowardsthemajorityandfollowed themwhichmadethemajorityagentsmoreandmoreunhappy(andtheminorityagentsaswell). Finally,when nearly all minority agents had become unhappy the process repeated. 5.6 Realworldscenariosofthekinddiscussedinparagraphs5.2–5.4aredi�iculttofindaslongitudinaldataforseg- regationindicesarerarelyavailableandusuallytooshorttocovermorethanonecycle. However,gentrification ofadisadvantagedquarteranditslaterneglectbeforeanewgentrificationphasestarts isanobservationwhich is more o�en than not, although unsystematically, made. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 Figure5: Screenshotoftheextendedmodelwithoscillations;minority: di�erentshadesofred,majority: di�er- ent shades of green, the less tolerant the darker; the plots at the right-hand side show the history of the run in terms of segregation index, percentage of similars in the neighbourhoods and percentage of unhappy agents, the latter two separately for minority, majority and whole population, whereas the plots at the far right show the tolerance distributions of minority and majority as well as the history of their means and their divergences (µθ ± 1.0σθ) input parameters tolerance related search homogeneous inhomogeneous all R2 0.767 0.798 minority size β -0.294 –0.390 density β 0.141 0.121 tolerance mean minority β 0.791 0.758 tolerance mean majority β 0.247 0.225 tolerance means cubic R2 0.716 0.683 Table 3: Variance reduction and standardised regression coe�icients for the linear dependence of the segrega- tion index ondensity, minoritysize, searchstrategies andthreshold distributionsand fora cubic regression on the two tolerance means (all coe�icients are significantly di�erent from 0, α < 0.0005) 5.7 In the remainder of this section, we will only deal with the version where the search for an alternative patch is tolerancerelated. The linear regressionof thesegregation indexonthesameinputparametersasaboveyields the variance reductions and standardised βs collected in Table 3. 5.8 Table 3 shows higher variance reduction than in Table 2. Here, the e�ect of the tolerance of the majority is considerably reduced, and it seems that the segregation index depends mainly on the initial tolerance level of the minority (which, as in all experiments, ranges between five and 75 per cent). 5.9 Finally, the two diagrams in Figure 6 show considerable di�erences as compared to the two diagrams in the bottomofFigure4: highinitial thresholdlevelsmainly intheminoritybutalso inthemajoritycanleadtomuch highersegregationindicesthaninthenon-adaptiveversion. Unlikethenon-adaptiveversion,itisnowsu�icient for a high segregation index that one of the two subpopulation has a tolerance level distribution with a high mean, and the tolerance level mean of the minority is even more important than the one of the majority. JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 10 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 adaptive, homogeneous, tolerance related search Page 584 %-sim ilar- red-m ean 806040200 U n s ta n d a rd iz e d P re d ic te d V a lu e 70 60 50 40 30 20 %-similar-green-mean 80 60 40 20 0 54.12..81.39 48.49..54.12 43.42..48.49 39.39..43.42 35.81..39.39 32.42..35.81 28.87..32.42 25.32..28.87 21.28..25.32 12.25..21.28 adaptive, inhomogeneous, tolerance related search Page 585 Figure 6: Segregation index as dependent on adaptive tolerance threshold for homogeneous and inhomoge- neoussubpopulationsandtolerancerelatedsearchstrategy; 1500runspercombination; theverticalaxis is the unstandardisedpredictedvalueofthesegregationindexfromacubicregressioninthetwotolerancethreshold means whereas the coloured dots represent the approximate values of the dependent variable Conclusions 6.1 The paper has shown that the formalism introduced by the ‘non-statement view’ is quite similar to the for- malism introduced in simulation models. If one starts with the definition of a potential model of a theory instead with a simulation model (as in the case of a ‘non-statement view’ reconstruction above), the former can be used as a specification of the simulation model before it is written. This can lead to a more straight- forward and perhaps to a more transparent simulation program. To show this we refer to another version of the extended Schelling model7 which makes the similarity between specification and program much clearer than in the original version of Wilensky (1997). For instance, by comparing Table 1 and Table 4, it is evident thatthemodelversioninspiredbythe‘non-statementview’reconstructionismuchmorestraightforwardthan the usual attempts (Wilensky 1997). Only the program code for θ looks unnecessarily complicated. This is, however, mainly due to the fact that the extended version contains additional features, which were not fore- seen in Schelling’s original publication: in Schelling’s version and many other implementations, θ is just the global variable %-similar-wanted which in the extended version is replaced with the three global variables %-similar-red-mean, %-similar-green-meanand%-similar-wanted-std-devallowingfortwodi�erent inhomogeneous subpopulations. SST term NetLogo model component W the set of all possible runs of the model W the patches in a certain run of the model P the turtles in a certain run of the model ` the built-in turtle variable color T NetLogo’s ticks θ the turtle variable my-%-similar-wanted which is initialised as a random normally distributed variable with mean either %-similar-red-mean or %-similar-green-mean and standard devia- tion %-similar-wanted-std-dev and — in the version with adaptive tolerance — updated every tick according to Equation 2 b the NetLogo built-in function patch-here c NetLogo’s built-in turtle variables xcor and ycor φ the function phi δ the function delta ς the function duncan Table 4: Correspondence betweenSST terms and NetLogo components in the rewritten extended version 6.2 Finally, two issues need to be discussed: • Did the ‘non-statement view’ reconstruction lead to new insights into real-world segregation processes JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 as intended applications of Schelling’s original model? • Did the various extensions systematically analysed in this paper lead to any explanations of observable macro behaviour in real-world populations? 6.3 Thefirstquestionhasapositiveanswer: Underthe(perhapsunrealistic)assumptionthatthetolerancethresh- oldisthesameforallpersonsofbothsubpopulations,thistolerancethresholdcanbeestimatedinmoreorless the same way as the device constant of Hooke’s springs. This is perhaps not very helpful as this assumption is indeed unrealistic — both with respect to the equality of this threshold in the two subpopulations and to the homogeneity within each subpopulation. However, with di�erent thresholds for the two subpopulations both Figures4and6indicatethatthecurvewhichisdefinedbythesurfacedefinedbythecoloureddotrepresenting the individual simulation runs and a horizontal plane defined by the observed segregation index of a popula- tion (for instance in a metropolitan area) represents a multitude of combinations of the two tolerance thresh- olds: for instance,allyellowdotsrepresentallcombinationsofthetwoθsof thetwosubpopulationswhichare compatiblewithsegregation indicesofapproximately40. Hence, ifweknewthedistributionsof individual tol- erance thresholds in both subpopulations, we could both predict and explain the resulting segregation index. Predicting and explaining the threshold, however, is only possible under the unrealistic assumption that the distributions in the two subpopulations are identical (θ1 = θ2 or µθ1 = µθ2). In this case, the best estimate of θ1 = θ2 or µθ1 = µθ2 is the coordinate on the θ axes of a point in the coloured curved surface in Figure 4 and 6 whosevertical (ς)coordinate istheempiricalsegregationindexusedforestimatingthe(meanof) thetolerance threshold (for the case of identical means between the subpopulations). 6.4 Beside this result, the ‘non-statement view’ reconstruction of Schelling’s model led to a slightly more straight- forwardimplementation,which—bytheway—resemblesalittlemoreadeclarativeprogramsuchthatHLogo (Bezirgiannis et al. 2016) could be an alternative tool for modelling such a reconstructed theory. 6.5 The second question may be answered in a way that all of these extensions were developed in order to over- come the empirical simplifications of Schelling’s original model. For instance, one of the phenomena that is currently observed in di�erent parts of Germany — intolerance of an overwhelming majority faced with a very smallminority, toleranceofamodestmajorityfacedwithalargeminority—canbeexplainedwithasimulation run showing growing intolerance of an initially moderate majority (level 30 growing to 56) confronted with a small (10 percent), less intolerant minority (level growing from 30 to 38). However, the problem remains: the morecomplex(andrealistic) themodel isdesigned,themoreits falsifiabilitydecreases,asmostoftheparame- tersaddedtotheoriginalselectionareverydi�icult tomeasure. Thiscalls foradditional theories linkedtoSST (Balzeretal. 1987,pp.57�.) defininghow,for instance, individualtolerancelevelscanbemeasured. Thiswould leaveonlyε—theparameterwhichdefinesthe learningof toleranceandintolerance intheadaptiveversionof Section 5 — as a newSST-theoretical term and newSST would turn into a theory explaining how populations learn to be tolerant or intolerant. Notes 1Balzer, Moulines and Sneed used Je�rey’s decision theory (Je�rey 1965) as an example to “make the re- constructionprocedureseasytograsp”(Balzeretal. 1987,p.23). Thistheoryhadalreadybeen“reconstructed” by Sneed (1982) and seems to have been one of the first theories from the social sciences at large ever having been dealt with in terms of the ‘non-statement view’. 2When the word model is used in the sense of the ‘non-statement view’ it is italicised. 3A similar experiment was done by Forsé & Parodi (2010) but only for an 8x8 checkerboard and with a dif- ferentmetric forsegregation,arrivingata linear relationshipbetweentolerance levelandsegregation(Forsé& Parodi 2010, p. 459). 4AsimilarapproachwasusedbyBruch&Mare(2006),seealsothediscussionbetweenthemandvandeRijt et al. (2009). 5This has also been studied by Stoica & Flache (2014). 6Empirical evidence for this can be found in Xie & Zhou (2012). 7This version is available at http://ccl.northwestern.edu/netlogo/models/community/ SegregationExtended JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 http://ccl.northwestern.edu/netlogo/models/community/SegregationExtended http://ccl.northwestern.edu/netlogo/models/community/SegregationExtended References Aleksandrov, A. D. (1999). A general view of mathematics. In A. D. Aleksandrov, A. N. Kolmogorov & M. A. Lavrent’ev (Eds.), Mathematics. Its Content, Methods and Meaning. Three Volumes Bound as One, (pp. 1–64). Dover. Translation edited by S.H. Gould Alparslan, A. & Zelewski, S. (2004). Moral hazard in JIT production settings: A reconstruction from the struc- turalist point of view. 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Ann Arbor: The University of Michigan Press JASSS, 20(1) 10, 2017 http://jasss.soc.surrey.ac.uk/20/1/10.html Doi: 10.18564/jasss.3372 http://ccl.northwestern.edu/netlogo/models/Segregation http://ccl.northwestern.edu/netlogo/models/Segregation Introduction Axioms and Axiomatisation of a Theory Axioms in the social sciences The `non-statement view' and simulation Schelling's Segregation Model Revisited Schelling's model reconstructed Intended applications of STT The partial potential model of SST and its simulation implementation First results Adding More Complexity to Schelling's Model Sophisticated search, inhomogeneous and different subpopulations A Further Extension: Individually Different Tolerance Thresholds Changing over Time Due to Communication Conclusions