Rapport de stage du M2 de physique de l’ENS LyonSoutenance le 3 Septembre 2008

Effect of Local Coordination on aSchelling-Type Segregation Model

Sebastian GRAUWIN

AbstractaaIn his 1971 article dealing with racial dynamics called Dynamic Models of Segregation,the economist Thomas C. Schelling showed that a small preference for one’s neighborsto be of the same color could lead to total segregation, even if total segregation doesnot correspond to a residential configuration which maximizes the collective happiness,or utility. Fruitful analogies between the physicical and sociological concepts have led usto create an enhanced version of the sociologists’ Schelling model by introducing severalforms of local coordination between the agents in a Schelling-type model. We find outthat, under a certain range of parameters, coordination is sufficient to reduce and evenbreak Schelling segregation patterns.

aaKeywords : Schelling, segregation, coordination, utility, socio-physics.

Institut des Systemes Complexes (IXXI),Laboratoire de Physique de l’ENS Lyon

Directeur de stage : P. JENSENRapporteurs : E. BERTIN, P. BORGNAT

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AcknowledgementsThis work would not have been possible without the help and support of several

persons that I would like to thank :Mark Fossett and Andre Ourednik who kindly answered my questions and helped

me better understand their work,Florence Gofette-Nagot and John McBreen for fructuous discussions,The Laboratoire d’Economie des Transports of Lyon which provided paper copies of

many articles, Charles Raux and Fabrice Marchal, for their kind interest and advices,And, of course, a special thanks to Pablo Jensen, who led me through the front door

of the socio-physical world and offered me a work subject full of lessons and opportuni-ties.

A Few RemarksThis paper presents a study of the possible impact of different kinds of urban policies.

Physicists having no scolar authority to take sides in socio-political issues, we tried hereto be as objective as we could be. Nonetheless, beneath the physicist lies an attentivewitness of the times. The opinions that may still be implicitely expressed in this paperare those of the sole author, and should therefore not be attributed to any of the citedreferences.

This report is an extended version of an article that we are planning to submit tothe Journal of Artificial Society and Social Simulation. This is the reason why it waswritten in english. Another article dealing more specifically with the differences andanalogies between the economists’ Schelling model and the physicists’ Ising model, tobe submitted to the American Journal of Physics as a response to [37], is in preparation.

During the internship, we sent a paper (presenting some parts of the results we hadat that time) as a proposal for a contribution to a colloquium entitled ‘La nouvelle ques-tion spatiale’ that will take place in September in Marne-La-Vallee. That contributionwas accepted. This would give us the opportunity to present our work to practitionersof social sciences.

We used the Java-based multi-agent programmable modeling environment Netlogo toimplement our model and run simulations. The graphical outputs presented in this paperhave been generated by this software. Netlogo can be freely download at http ://ccl.northwestern.edu/netlogo/. Our model had been added to the Netlogo User Community Mo-dels share page and can also be freely download at http ://ccl.northwestern.edu/netlogo/models/community/segreg-vs-coord.

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Table des matieres

1 Introduction 11.1 Residential Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Schelling-Type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Analogies with Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Local Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The model 32.1 Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The City and the Agents . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Neighborhoods and Co-properties . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Agents’ Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Dynamic Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Without Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 With Local Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 With Global Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 A Measure of Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Simulation Results 93.1 Dynamic Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Without Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Introduction of Coordination . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Stationary Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 How to make these measures . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Influence of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Study of the Robustness of our Model . . . . . . . . . . . . . . . . . . . . . . . 15

4 Analytical Study 164.1 A Markov Chain Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Resolution for T →∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Resolution of the Global Coordination Case . . . . . . . . . . . . . . . . . . . . 184.4 Stability of the Highly Segregated States for T → 0 . . . . . . . . . . . . . . . . 19

5 Limits of the Model 22

6 Conclusion 23

A Empirical Evidence of Residential Segregation 27

B Additionnal Simulations results 28B.1 Influence of T on the Patterns of the Stationary Configurations . . . . . . . . . 28B.2 Caracteristic Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28B.3 Robustness of our Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30B.4 Simulation Results with Other Utility Functions . . . . . . . . . . . . . . . . . 30

C Ising vs Schelling 32

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1 Introduction

1.1 Residential Segregation

If we examine metropolitan areas, we find that many dissimilarities exist betweendifferent types of block district. Some of them contain large homes with prosperous po-pulations, others are neighborhoods of modest or even run-down homes. This kind ofdissimilariy reflects the ordering of the city population according to the socio-economicstatus of its inhabitants. Another important way in which neighborhoods differ is intheir racial composition. For example, numerous studies [3, 4, 5, 11, 12, 18] have beenconducted in the United States, where the older metropolises of the Midwest and Nor-theast are characterized by older and poorer centers where most residents are AfricanAmericans and newer and wealthier suburban rings where most residents are whites (seeannex A for empirical evidences of the New-York metropolitan area). In european cities,the old city centers are on the contrary more valued and tend to attract the wealthierpart of the population, inducing also a segregation based on the socio-economic status.Ethnic-related segregation also exist in european cities.

The problems generated by this residential segreagation are more than actual. Theethnic segregation is strongly linked to immigration/integration issues. The economic-related residential segregation is a reflect of an elarging gap in the society that we dailyhear about. The resolution of these two kinds of issues certainly constitutes two of themost important human and politic challenges of the next decades. An essential stepbefore applying any urban policy is to determine whether or not there exists underlyinglaws governing the evolution of the population repartition in the cities, and if so, to wellunderstand them.

Sociologists have long identified three kinds of mechanisms which are at stake in thesegregation phenomena [17] :discriminative processes, that can be either negative or positive, promoted by the lawor being a reflect of the day-to-day behaviour of the main population. For examplethe contemporary racial segregation seen in America in residential neighborhoods hasmainly been shaped by public policies and mortgage discrimination [4, 5, 12, 11].socio-economic reasons,individual preferences, which are called upon to explain the existence of segregation bet-ween two (economically similar) minority groups (see [15] and the case of the blacks andlatinos in New-York in annex A.).These three proposed mechanisms seem simple enough, however one has to go a littlefurther to understand the persistence of segregated patterns once the (negative) dis-criminative laws and behaviours have disappeared and when people express a generaltolerance for others which do not belong to their ethnic or socio-economic group.

1.2 Schelling-Type Models

At the end of the 60’s, Thomas Schelling introduced a model of segregation [31, 32, 33]where individuals, living on a lattice, chose the place where they lived according to thecolors of their neighbors (his model is thus only based on individual preferences). He

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showed that if the individuals had even a mild preference for living near people of theirown color, and if they moved to satisfy their preferences, complete segregation at thecity scale occured. This result is considered as surprising and has generated a large lite-rature. Many variants of the Schelling model have been proposed since very recently 1,with sociologist [3, 4, 5, 11, 12, 39], economist [13, 14, 15, 19, 23, 35, 40, 41, 42], gametheoritician [9, 10, 16, 22, 30], or physicist [36, 37, 38] approaches. All these models,adding some levels of complexity to the original Schelling model, have reinforced Schel-ling main finding : that segregation may occur at the city level even if all the agents arerather tolerant. Hence the aura of fatalism that sometimes seems to surround this kindof model [8, 45].

The structure of the segregated areas is related to the ‘demographic’ parameters cho-sen, to the exact specification of the preferences of the agents and to the rules governingtheir moves. The original model was very simple. Suppose the population is divided intwo groups of agents 2, red and green. Place a certain number of agents on a large chessboard, leaving some free places. An agent prefers to be on a square where half or moreof the agents on the 8 surrounding squares are of his own color (where he has a utilityof 1) to the opposite situation (where he has a utility of 0). At each iteration, one ofthe agents with a zero utility is chosen at random and is moved to a preferred loca-tion. This model, when simulated, yields to complete segregation, even though people’spreferences for being with their own color are not so strong (people do not make anydifference between neighborhoods composed of 50% or 100% of members of their owngroup ; the utility function is still strongly asymmetric).

Even so, with the original Schelling choice of utility function, the agents are all fullysatisfied in the final segregated configurations, and the collective utility increases whenwe get from a initial random configuration to a final segregated configuration. Moreparadoxical is the case where the agents strictly prefer 50-50% neighborhoods, but re-latively prefer an all-similar neighborhood to an all-dissimilar one [23, 42]. The agents’selfish moves yields once again to highly segregated configuration, even at the cost ofloss of collective utility. This kind of phenomenum, well known in the economic context,takes its roots in the lack of coordination between the agents. We will come back to itin section 3.1.

1.3 Analogies with Physics

Some physicists have tried to show that the economists’ Schelling model was basicallyjust a more complicated version of the physicists’ ferromagnet Ising model [21, 24, 27,34, 36, 37]. It is true that powerful analogies can be made between the two domains[38]. An agent, his color and his utility computed with respect to the composition of hisneighborhood is comparable to a particle, his spin and its energy computed with respectto the interaction with the neighboring particles. The two models may generate highlysegregated patterns. According to us however, the parallels between the two models

1. Schelling’s ideas have reached the scientific world outside the economists’ community since hewas awarded the 2005 Nobel Memorial Prize in Economics Sciences.

2. in the context of social science simulations, individuals or any decision making entity is referedto as an agent

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are limited and, even if they seem to produce the same results, they are fundamentallydifferent 3.

Our main point is that in the Schelling model, agents move selfishly to increase theirown utility, even if their moves make the utility of other agents decrease. The decisionto move an agent is thus a local decision. In the Ising model, the spin of a particle mayflip if it improves the interaction between this particle and its neighbors. The flippingoccurs only to benefit the global energy. Ising models are thus, contrary to Schellingmodels, characterized by the presence of a form of coordination between the particles.

1.4 Local Coordination

These physical analogies therefore allow us to identify the concept of coordinationbetween the agents which absence has to be noticed in the Schelling models. Anootherincentive to focus on this concept comes from the real-world issues related to socialhousing. A current trend of thoughts (see for example a recent article published in Li-beration [43]) proposes to develop social housing no more on the city scale (which yieldsto the creation of segregated areas) but on the residential building scale. The idea isto stimulate the coordination between neighbors to make them locally take care of onesocial housing. Coordination would thus induce better socially integrated cities.

The central concept of the work presented here is thus coordination. We present in thenext section the Schelling-Type model we developped in which we have introduced localor global coordination. We then present in section 3 some simulations results that showthat the introduction of coordination yields to city configuration where the collectiveutility is quantitatively higher than in the case without coordination and where Schellingsegregation patterns are shaken. Section 4 provides some analitycal confirmations of theresults of our simulations and finally, section 5 gives a critical analysis of the realism ofour model.

2 The model

2.1 Basic Setup

2.1.1 The City and the Agents

The model uses a two-dimensional NxN square lattice with periodic boundary condi-tions, ie, a torus containing N2 cells. Each cell corresponds to a dwelling unit, each ofthem being of equal quality. We suppose that a certain characteristic divides the po-pulation in two groups of agents that we will refer to as red and green agents. Eachhousehold may thus be occupied by a red agent, a green agent, or may be vacant. Wedenote by V the (constant over time) number of vacant cells, and by NR and NG the(constant over time) number of respectively red and green agents. Demographically, the

3. We do not develop too much our refutation of the parallel between Schelling and Ising models inthe following, a response paper to [36, 37] is in preparation. See annex C.

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parameter N controls the size of the city, the parameter v = V/N2 its vacancy rate,and the parameter nG = NG/(NR + NG) its composition. In the simulations presentedbelow, we took N = 30 - a good compromise between the necessity to take a large valueto simulate a realistic city and the convenience to take small N to have relatively shortcomputation times and nice graphical displays -, nG = 0.5 and v = 10%.

2.1.2 Neighborhoods and Co-properties

We define the neighbors of an agent as the agents living on the H nearest cellssurrounding him and his co-proprietors as those on the h nearest cells surroundinghim. Here, H and h are two fixed integers which verify h ≤ H (the idea is that theco-proprietors represent in a stylised way next-door neighbors or the people living inthe same residential building whereas the neighbors represent the people living in thesame street or in the same district). Examples of possible forms of neighborhoods andco-properties are shown below in Fig 1. The main results of our model are qualitativelyindependent of any specific definition of neighborhood, provided its size is relativelysmall compared to the size of the city - to keep the “local” property of neighborhood.We thus mainly work here with a H = 8 so-called “Moore” neighborhood, but we alsopresent in annex some results with larger neighborhoods to point out the robustness ofour conclusions.

Figure 1 – Different forms of neighborhood (or co-property) used in our simulations. An agent(in red) is placed at the center of a configuration composed respectively by the 4, 8, 24 and 44 cellssurrounding him (in yellow).

2.1.3 Agents’ Utility

Each agent computes his own level of satisfaction via an utility function which de-pends on his neighborhood’s composition, which is supposed to be the main thing theagents care about. Following Schelling [33], we characterize the composition of agent k’sneighborhood with the sole parameter sk, which is the fraction of agent k’s neighborswho are similar to him. We assume here that the members of each group share the sameutility function, ie we only have to specify two utility functions (uR and uG). In thefollowing, we will also denote by U the global (or collective) utility which is the sum ofall agents’ utility computed for each configuration of the city and by U∗ = U/[(1−v)N2]its normalized value, which is also the mean utility of the agents in the city.

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The range of conceivable individual utility functions is very wide, going from realis-tic data-based functions to stylized imaginary ones. Some possible choices are displayedfigure 2. In his original 1969 model, Schelling used stair-like utility functions whichdescribed agents that could be either totally happy (with a unitary utility) or totallyunhappy (with a zero utility), the state of the agents depending on the position of theirfraction of similar neighbors with respect to a certain ‘happiness threshold’ here fixed at50%. We could imagine however that the level of satisfaction of the agents can presentmore degrees of graduation that just the two states happy/unhappy. The ‘linear’ utilityfunction displayed figure 2 is an example corresponding to agents who are progressivelyhappier as their fraction of similar neighbors increases. Various reasons could explainthe inclination towards one’s own race, including cultural or socio-economic concerns,fear of potential hostility from the other group, or fear of isolation (this utility functioncould corresponds to the white in the United Sates, see [3, 4], but also to people witha good socioeconomic status who wish to live in an area with high quality houses andwith the ‘best’ schools for their children).

Figure 2 – Exemples of agents’ utility fonctions. In the simulations presented in this paper, wealways took uR = uG = uap, where uap is the asymetrically peaked function defined by uap(s) = 2s fors < 0.5 and uap(s) = 3/2− s for s ≥ 0.5.

Finally, the two peaked functions on the right side of figure 2 can represent in a sty-lised way agents whose main preference goes to a 50/50 mixed neighborhood, whereasin the case of the asymetrically peaked function, the agents still prefer to live strictlybetween themselves rather than being strictly isolated. Survey data suggest that this isa plausible assumption. For example, recent multi-city studies shows that individualsfrom all racial groups prefer highly integrated neighborhoods, but at the same timethey do have a bias in favor of own-group members [3, 11, 18]. We could also imaginethat this utility function could correspond in the case of socio-economic segregation topoor families with social ambition for their children who would seek the ‘best’ possibleenvironment even if it comes with a high financial cost.

We choose in the following to work only with the sole asymmetrical peaked utilityfunction uap. It has already been shown in a variant of Schelling original model [42] thatthe asymmetry in favor of the all-similar neighborhood in this utility function leads tosegregation patterns at the city scale, even at the cost of a low collective utility. It is thusthe perfect candidate for testing if the introduction of coordination allow to increase thecollective utility.

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2.2 Dynamic Rules

At each iteration, we pick one agent and one vacant cell at random and we givethat agent the opportunity to move in that vacant cell. His deciding between moving ornot moving affects not only him, but also all the agents living in his present neighbo-rhood and those living in his potentially new neighborhood. We introduce here differentrules which allow the potential mover to choose between the two options he is facing,classifying them according to the amount of involved agents they take into account.

2.2.1 Without Coordination

This is the case which corresponds to usual socio-economic models, where the actionsof each agent are solely based on their own profit. In the context of every Schelling-typemodels we know about, this comes back to only taking into account the utility of thepotential mover. Following Zhang [40, 41, 42], we assume that the probability that thispotential mover chooses to move is determined by :

P (M) =eu(.|M)/T

eu(.|M)/T + eu(.|M)/T=

1

1 + e−∆u/T, T > 0

where we have noted ∆u the gain in utility the potential mover would realize if he wasto move. This choice of P (M) is commonly assumed in the literature and is known asa log-linear behavioral rule. Its justification lies in the logit model of econometrics [1] :a random noise following an i.i.d. extreme value distribution is added to agent’s utilies.This noise can be seen here either as a way to take into account the preferences of theagents over any other thing than their neighborhood’s composition (and not correlatedto it : the quality of the house, the presence of markets and school, the level of soundnoise...) or as a way to modelize the agents’ bounded rationality : they sometimes makemistakes and take utility-decreasing moves. Clearly, the probability for an agent to takean utility-decreasing move drops down as T → 0.

In a more physical sense, we can understand the introduction of ‘temperature’ in thisdynamic rule as a way to ‘fluidify’ the system (the city). This analogy was done in [38]and it was shown that he introduction of fluidity always creates completely segregatedpatterns compared to the original Schelling model which sometimes jam itself in partiallysegregated configurations, which does not fit to the reality.

2.2.2 With Local Coordination

We introduce local coordination by taking into account the potential change of uti-lity of the co-proprietors of the vacant cell considered by the potential mover. In thefollowing, we will denote by C this set of agents. The idea here is that it is more logicalto introduce local coordination through the potentially new co-proprietors (who haveyou take an ‘admission exam’) than through the current co-proprietors (that you canquit on your free will). We propose three different ways to counterbalance the wish ofthe potential mover by the opinion of his potentially new co-proprietors :

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‘Qualified vote’ rule

Each co-proprietor i of C computes apart from the other the gain ∆ui he would realizewould the move happen. He then vote ‘for’ the move with a probability 1/

(1 + e−∆ui/T

).

The move then takes place with a probability :

P (M) =1

1 + e−∆u/Tf({ 1

1 + e−∆ui/T

}i∈C

)where f = 1, 1/2 or 0 if respectively more than half, exactly half or less than half of theco-proprietors vote ‘for’ the move.

‘Pondered vote’ rule

Here, the co-proprietors do not take their decision apart from each other but by bearingin mind the utility of the co-property as a whole. All that matter is thus the change∑

i∈C∆ui of the utility of the co-property and the probability that the move effectivelytakes place is then determined by 4 :

P (M) =1

1 + e−∆u/T

1

1 + e−∑

i∈C ∆ui/T

This rule can be interpreted as the existence of a compensation mechanism between theco-proprietors : if one of them can gain more in the move than the loss of another one,they will negotiate over a mutually beneficial deal (e.g. I accept this new guy but you’llhave to take care of my house plants/my dog during the holidays or I agree not to accepthim if you allow me to paint the stairs in fuschia).

‘Sum’ rule

In this third example of local coordination, we place the potential mover and his poten-tial new co-proprietors on equal footing by taking :

P (M) =1

1 + e−(∆u+∑

i∈C ∆ui)/T

We could interprete this rule as the involvement of a real estate agent, whom the po-tential mover has adressed to and who tries not to offend his future customers.

4. We implicitely admit in the two ‘vote’ rules that the co-proprietors deciding whether to acceptor not the potential mover are subject to the same noise T that the potential mover deciding whetherhe want to move or not. We could argue that the reasons (apart from their neighborhood composition)the agents take into account in their decision-making being totally different in the two cases, we haveto introduce a level of noise T2 to replace the parameter T in the co-proprietors acceptance probability.However, it would also be arguable that realistic T and T2 should be correlated, ie we would have tochoose and justify a function T2(T ). As we only try here to build a simple stylised model, we assumedthat this correlation function was equal to the identity function.

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2.2.3 With Global Coordination

Finally, we could imagine that a central authority is managing the move of the agentsand allow them according to the change in the collective utility ∆U they induce. Theprobability that the move happens is then :

P (M) =1

1 + e−∆U/T

Note that this rule is much closer than the other one to an Ising-like model (seeannex C).

2.3 A Measure of Segregation

Let’s introduce the function ρ defined on the space of all possible configurations ofthe city as the number of pairs of red-green neighbors. It is pretty obvious that the valueof ρ serves as a natural index of segregation as it measures the degree of exposure (orpotential contact) between the members of the two groups. A lower value of ρ meanshigher degree of segregation. Note that not only the values of ρ will vary with the sizeN of the city, but for a given configuration of the city, the value of ρ will also stronglydepend on the size H of the neighborhood considered by the agents. We thus need tonormalize ρ in order to have an index that will allow us to compare different situations.

We define our normalized index of segregation as :

ρ∗ =ρ− ρρ

where ρ = HN2nG(1− nG)(1− v)2. We will justify in section 4.2 that ρ corresponds tothe expectancy of ρ for a randomly generated city of size N , with a vacancy v, a frac-tion nG of green agents whose neighborhood is of size H. We verified this by generatingrandomly 10000 configurations and computing their segregation indices ρ∗. The figure3 presents an histogram displaying the frequency with which different values of ρ∗ wereobserved. The distribution is effectively centered on ρ∗ = 0, with a variance of 10−4.

The value of reference (0) of ρ∗ corresponds to an average random configuration,positive values of ρ∗ means that red and green agents are less exposed to each otherthan in the average random case, positive values of ρ∗ bigger than say two times thewidth of the frequency distribution of figure 3 (ρ∗ > 0.1) corresponds to segregatedconfigurations which are very unlikely generated by mere coincidence : their occurenceimplies the will of some agents of not being too exposed to agents not similar to them.On the opposite, negative values of ρ∗ means that the red and green agents are moreexposed to each other than in the average random case. Note finally that the ρ∗ indexcannot grasp every aspects of the city configurations. It can’t for example make thedifference between the ‘checkerboard’ ordered configuration of figure 3 and a randomconfiguration.

The index of segregation we introduced here is an adaptation of the ‘potential func-tion’ ρ of Zhang [42]. In empirical analysis, researchers (sociologists and geographers)

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Figure 3 – Up : four possible configurations of the city. Bottom : in red, frequency distribution ofthe ρ∗-values of 10000 randomly generated configurations with N = 30, v = 10%, H = 8, nG = 0.5.The width of the histogram bars δρ∗ = 0.015 was chosen in order to fit to the range of the obtaineddata. Note that distribution curve is centered on ρ∗ = 0 and that it is very narrow : its width isapproximatively 0.05. In blue : values of the ρ∗-values of the four configurations up.

have developed many indices to measure residential segregation, based on the composi-tion of the people living in the census tracts (areas containing typically a few thousandpeople), whereas here, we based our index on the dwelling units (cells). See [20, 28, 29]for a more detailed discussion on segregation indices.

3 Simulation Results

3.1 Dynamic Evolutions

Since at a given iteration of the model it is unsure whether a move will or won’t takeplace, the number of iterations is not a convenient chronological reference to follow thedynamic evolution of the city. In order to compensate for this weakness, we characterizea given moment in time with the parameter τ which is the number of periods whichhave already taken place, where a period is defined as (1− v)N2 performed moves. Anincrease of one unity of τ thus means that, on the average, the agents have all movedone time.

Before starting any simulation, we have to choose a value for the parameter T ,which controls the influence of the preferences of the agents concerning anything buttheir neighborhood composition (we will in the following refer to those other preferencesas noise with respect to the main preferences on neighborhood compositions). If wechoose too high a value, the noise effects would wipe out the influence of the agents’decision-making process. If we choose too low a value, the ‘fluidity’ brought by the noise

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Figure 4 – Evolution towards a highly segregated configuration starting from a random configurationin the case of the ‘without coordination’ rule. Up : some snapshots of the evolution of the city. Bottom :evolution with τ of the index of segregation and of the global utility, the marked points correspondingto the snapshots. T = 0.1, N = 30, v = 10%, H = 8.

would be less important and the system would have very long reaction times. In thesimulations presented in this section, we fixed the noise level to T = 0.1. This valuecan be compared to the absolute value of a typical gain |∆u| an agent realizes whenhe moves. Qualitatively, with this value of T , an agent would thus ‘mistakingly’ (withrespect to what is supposed to be his main preference) choose to realize or endorse anutility-decreasing move with a probability of 1/(1 + e) ' 27%.

3.1.1 Without Coordination

We present fig 4 a typical evolution of the city in the case where the agents movewithout coordination. Starting from a random configuration, although the agents’ pre-ferences go to mixed configurations, we observe the rapid formation of segregated areaswhich then slowly melt into one another leading to the emergence of a ‘2 blocks’ confi-guration. In parallel, we observe that after a transitory phase corresponding to theformation of the ‘2 blocks’ configuration, the segregation index ρ∗ and the collectiveutility U∗ reach a stationary phase where their values fluctuate with a low amplitudearound a certain mean value.

We verify here a seemingly paradoxical result that has already been observed el-sewhere [33, 41, 42] : whereas the agents decide to move in order to improve their ownutility, their moves lead to a highly segregated configuration at the city level in whichmost of the agents are far from being fully satisfied.

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This apparent paradox can be understood if we think in terms of stability of thecity’s configurations. Let’s imagine an isolated and perfectly mixed area within the citywhere one half of the agents are green and the other half red, the agents thus maximi-zing their individual utility. Suppose that one green agent mistakingly leaves that area.The half-half equilibrium being perturbed, the agents utilities thus all decrease, but thegreen’s will decrease a bit more than the red’s because of the asymmetry in the utilityfunction. Assuming that the range of available destinations for the green and the redagents of our area is the same, the remaining green agents will thus be more likely toleave than the remaining red agents. Thus the perturbation of the mixed equilibrium ismore likely to grow than to resorb itself. This kind of argument was used by Schelling[33] to explain why integrated patterns are unstable.

One the other hand, it is easy to see that the segregated patterns are very stable.Let’s take for example a red agent of the all-red area of the last snapshots of figure 4.That agent may only face three options : move elsewhere in the red area (which wouldchange nothing to his utility and to the collective situation), move somewhere in thered-green border area (which would improve his situation but not the collectice one)or move in the green area (which would benefit to the collective situation but largelydisavantage him). The agent would thus very unlikely choose the third option and evenif he did so, he would very likely return to the red area as soon as the occasion presentsitself since he would prefer to be among his own with a utility of 0.5 than isolated witha utility of 0.

In Zhang’s words [42] : although nobody likes complete segregation, the residentialpattern is very stable. Only moving across the color line by a considerable number ofagents could disturb the segregation equilibrium, but nobody has incentive to do so be-cause it causes a loss of utility. [...] Segregation is stable not because people like it, butbecause any individual who wants to change the situation unilaterally will have to goacross the color line, which may not be the desirable thing to do from the individual’sperspective. [...] The failure of the system to escape complete segregation is similar tothe phenomenon of “coordination failure” studied by economists in many other contexts.It is the agents’ inability to move simultaneously that make them stuck in a situationnobody likes...

3.1.2 Introduction of Coordination

...hence the idea to introduce a form of coordination between the agents. In figure5, we present typical simulations where the chosen dynamical rule includes either a lo-cal coordination between the agents (for which the potential mover requires a kind ofauthorization from his potential new coproprietors to move) or a global one (where herequires a kind of authorization from his potential new and former neighbors). To en-hance the effects of the introduction of coordination, we started the simulations with ahighly segregated configuration similar to the ones the city converged to in the ‘withoutcoordination’ case. The level of noise is still fixed to T = 0.1.

The first thing to notice is that the highly segregated state is not stable anymore : ina relatively few number of periods, the agents get away from the one-colored areas either

11

Figure 5 – The introduction of coordination destabilizes the highly segregated configurations andleads to configurations which present different kinds of local mixity. Up : some snapshots of the evo-lutions of the city with the different rules. Bottom : evolution with τ of the index of segregation andof the global utility, the marked points corresponding to the snapshots. T = 0.1, N = 30, v = 10%,H = 8.

12

by creating some kind of nucleus in the area of the other color or by progressing towardsit in some kind of pseudopod which base is on the red-green areas frontier zone. Then,as the system evolves, locally mixed patterns appear and remain stable. Once again thesegregation index and the collective utility after a transitory phase reach a stationaryphase in which their values fluctuate around certain mean values. The produced mixedpatterns allow the agent to improve their utilities relatively to what their utility were inthe highly segregated configuration and of course they produce a segregation decrease.

The introduction of coordination, even at the local level, thus leads to thedesagregation of the segregated pattern. This first result is in itself a truenovelty in the context of Schelling-type models 5

The second thing to notice is that the different dynamical rules create different mixedpatterns. Qualitatively speaking, the local coordination rules create locally ordered areaswhere the agent’s utility is maximalized and the global coordination rule creates a singleordered area that overflow the entire city. In the local coordination cases, some agentsremain stuck at the border of two locally ordered areas where their utility is not maxi-malized and the collective utilities reach only values between 0.8 and 0.9. In the globalcase, there is no such stucked agents, and the collective utility reaches really maximumvalues (around 0.99). In the physicist language, the introduction of coordination impliesa correlation length between the agents which remains finite in the local coordinationcases, but which becomes infinite in the ‘global coordination’ case. One can also unders-tand the difference between the effects of the local coordination rules :Differences between the qualified vote and the pondered vote rules : In the case of thequalified vote rule, the transitory phase is much longer, locally segregated areas remainsin the stationary phase - inducing higher segregation indices and lower collective uti-lities - and the fluctuations have a higher amplitude. In a sociological point of view,those differences can be explained by the fact that the qualified vote rule emphasizesthe individual opinion of each coproprietor, whereas the pondered vote rule smooth theirdifference of opinion. In a more physical point of view, one could say that the qualifiedvote rule implies only an interaction between the potential movers and each of theirpotential coproprietors, whereas the pondered vote rule implies also an interaction bet-ween these coproprietors, inducing a higher correlation length.Differences between the pondered vote and the sum rules : First, the transitory phase isshorter with the sum rule, which can be understood with the same kind of argumentsthat we just used. Second, however the collective utilities are higher in the ‘ponderedvote’ case, the segregation indices are not lower as one could have thought (intuitively,higher collective utility means that more agents have a 50-50% neighborhood and thattheir is more red-green pairs). The explanation for this counter-intuive result must liein the form of the distributions of the sk in the respective stationary configurations (ahigher proportion of agents having 3/7 similar neighbors, an equal proportion of agentshaving 4/8 similar neighbors and a lower proportion of agents having 4/7 similar neigh-bors in the ‘sum’ case are sufficient to account for a lower collective utility and a lowersegregation index).

5. Until now, every time it was tried to add a level of complexity in the original Schelling model,the robustness of Schelling’s conclusion was reinforced.

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3.2 Stationary Measures

3.2.1 How to make these measures

Previously, we established that for a noise level T = 0.1, the introduction of coordi-nation has an impact on the dynamics of the city and creates locally ordered patterns.We want here to study the influence of the level of noise on the stationary values of ρ∗

and U∗.One preliminary condition to this study is to make sure that we can make the dif-

ference between a real stationary phase and a very slowly evolving transitory phase. Itwill be theoretically established section 4.1 that a stationary phase of the system alwaysexists and is always independent of the initial configuration. The figure 6 shows howwe used that property to make sure to recognize this stationary phase : in most cases,if we start with a ‘2 blocks’ configuration such as the one of figure 3, the segregationindex will decrease toward its final value ; on the other hand, if we start from a randomconfiguration, the segregation index will increase toward its final value. The recipe torecognize a stationary phase is thus simple : start two simulations, one with a randominitial configuration, the other with a ‘2 blocks’ initial configuration ; wait for them toproduce the same measures and then you can make your measures.

Figure 6 – Exemple of temporal evolution of ρ∗ and U∗ starting from different initial configuration.After a transitory phase, the system evolves towards a stationary set configurations which doesn’tdepend on the initial configuration. N = 30, v = 10%, H = 8

3.2.2 Influence of T

We present in figure 7 the stationary values of ρ∗ and U∗ for different values oftemperature 6. The first thing to notice is that, as we had predicted it, for high values ofT the chosen dynamic rule doesn’t matter : the stationary phase is dominated by noiseand the curves corresponding to the different rules converge towards a certain limit. Onthe other hand, the more T is lowered, the more the stationary phases are determinedby the dynamic rule. The case ‘without coordination’ presents what we would call in aphysical context a phase transition that can be interpreted in terms of the stability of

6. see annex B1 for typical snapshots of the stationary phases at different T -values

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the highly segregated configuration : whereas the fluctuations generated by the strongnoises allow the agents stucked in segregated areas to escape these areas, under a certainthreshold of noise (between T = 0.1 and T = 0.2), these fluctuations become lessimportant and the segregated domains remain stable. The ‘global cordination’, ‘sum’and ‘pondered vote’ cases also present a smoother phase transition : the lower the noiseis, the more the ordered local patterns remain stable.

Figure 7 – Stationary mean values of the segration index and of the collective utility with respectto the level of noise T , for the different dynamic rules. The error bars give the standard deviation ofthe fluctuations of ρ∗ and U∗ once the system has reached its stationary phase. The mean value andstandard deviation were taken on the data of 10 periods of the stationary phases (when the fluctuationwere important, we used larger temporal windows). N = 30, v = 10%, H = 8.

Unlike the other rules including coordination, in the ‘qualified vote’ case, the intro-duction of coordination has an effect only for a short range of T -values, here betweenT = 0.1 and T = 0.2. We observe an abrupt phase transition between T = 0.05 andT = 0.1, a transition value under which the segregation and collective utility curves ofthe ‘qualified vote’ case merge with those of the ‘without coordination’ case. The factthat the coordination through a qualified vote of the potential new coproprietors hasnot any effect for low values of T can be understood by a greater stability of the highlysegregated configuration (as it will be explained section 4.4), but also by a relativelyweak stability of integrated states (the ‘mistakes’ of the coproprietors are amplified withthis rule by the stair-like form of the function f({1/(1+e−∆ui/T )i∈C}) introduced section2.2.2).

Note that the effect of the introduction is stronger when the size of the copropertiesh is higher. Note also once again by comparing the ‘sum’ and the ‘pondered vote’ curvesthat higher U∗ does not necessarily imply lower ρ∗.

3.3 Study of the Robustness of our Model

Our model includes many other parameters apart from the level of noise T that haveto be tested in order to check the robustness of our results. Does the introduction ofcoordination destroy segregated patterns only for a restrained range of those parametersor is it rather regular ? We present in annex B3 some stationary measures correspondingto different values of the vacancy rate v and large values of the size of the neighborhood

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H. The results remain qualitatively unchanged : the introduction of coordination alwaysimprove the collective utility level compared to the case without coordination. Morequantitatively, the vacancy rate has an influence on some temperatures of transition.

We also present in annex B4 results corresponding to other choices of utility functionsfor the red and the green group. The simulations shows that whenever one group isstrictly in favor of integration and the other not, the stationary patterns results from acompetition between the coordination tendancy of the tolerant group and the segregativetendancy of the other one.

4 Analytical Study

4.1 A Markov Chain Process

We define a state x as a N2-vector, each element labeling a cell of the NxN lattice.Each state x thus represents a specific configuration of the city. Obviously, each of thedynamic rules implies that the probability that the state at the tth iteration xt is equalto a given state x only depends on the state at the previous iteration xt−1 :

Pr(xt = x|xt−1, . . . , x1, x0) = Pr(xt = x|xt−1)

Each dynamic rule thus corresponds to a finite Markov process.Let’s denote by X the set of all possible states of the city once the demographic

parameters N , v, nG have been chosen. It is easy to figure out that the Markov chaindescribing our system is irreductible (it is possible to get to any state from any state 7),aperiodic (given any state x and any integer k, there is a non-zero probability that wereturn to state x in a multiple of k iterations) and recurrent (given that we start in statex, the probability that we will never return to x is 0). These three properties ensure 8

that the probability to observe any state x after t iterations starting from a state yconverges towards a fixed limit independent of the starting state y as t→∞.

In other words, for each set of parameters and dynamic rule, there exists a stationarydistribution

Π : x ∈ X → Π(x) ∈ [0, 1] ,∑x∈X

Π(x) = 1

which gives the probability with which each state x will be observed in the long run.This confirms what we observed in the simulations : in the long run, our dynamicalsystem (the city) is evolving towards an attractor composed of a relatively small subsetA of X. It follows that any measure M performed on the states space X - such as thesegregation ρ∗ or the global utility U - will in the long run fluctuates around a meanvalue M∞ =

∑x∈A Π(x)M(x).

For the following, it will be useful to define two states as immediately communicatingstates if we can switch one in the other by moving one agent.

7. Since T > 0, each imaginable move has a non-zero probability to happen.8. references

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4.2 Resolution for T →∞The resolution in the limit T → ∞ may seem unimportant, but its interest lies in

the fact that it provides a solid reference common to all the dynamical rules.When T →∞, the probability to go from one state to another one which communicatesdirectly with it does not depend on the configuration of these two states (the P (M) ofthe different dynamic rules all converge towards a constant independent of the changesof the agents’ utilities as T → ∞). Therefore, in the long run each state of X will beobserved with the same probability. In the physical point of view, T →∞ correspondsto the case when there is no interaction between the agents. This absence of interactionmake it very easy to predict the different outcomes of our model for high values of T .

For mathematical convenience, we place ourselves in the grand canonical ensembledescribing cities with fixed ‘volume’N2 and ‘temperature’ T →∞ and where the numberof each kind of ‘particles’ (cells occupied by a red agent, cells occupied by a green agent,vacant cells) is free. We introduce the fixed ‘chemical potentials’ µR = (1−nG)(1−v)N2,µG = nG(1−v)N2 and µV = vN2 as the probabilities to observe respectively a red agent,a green agent or a vacancy on a given cell of the city at T →∞.

Expectation value of ρ.On the toric lattice representing the city, there is HN2/2 pairs of neighboring cells. Theprobability to get one red agent and one green agent on a given pair is, for T →∞ andin the grand canonical ensemble, 2µRµG. Hence the expression of the expectation valueof ρ that we used section 2.3 :

ρ = HN2/2 · 2µRµG = HN2nG(1− nG)(1− v)2

By definition, ρ∗ = (ρ − ρ)/ρ must then necessarily converges to 0 as T → ∞, thisholding for each dynamical rule.

Expectation value of U∗

To determine the expectation value of the normalized collective utility U∗, we will pro-ceed by determining the expected utility of the individual agents. Let’s first focus on ared agent. His neighborhood composed of the H neighboring cells, can be characterizedby the total number of agents t ≤ H that live in his neighborhood, and among themby the number s ≤ t of agents similar to him. It is straightforward to see that thereexists

(Ht

)(ts

)= H!

s!(t−s)!(H−t)! such configurations and that the probability to get such a

configuration is, for T → ∞ and in the grand canonical ensemble, µsRµt−sG µH−tV . The

utility of this agent in such a configuration would be uR(s/t).Putting all these pieces together and doing the same work for a green agent, the

expection value of U∗ in the grand canonical ensemble can be written as :

U∗ −→T→∞

H∑t=0

t∑s=0

H!

s!(t− s)!(H − t)!µH−tV

[nGµ

sGµ

t−sR uG

(st

)+ (1− nG)µt−sG µsRuR

(st

)]

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This formula holds for any choice of utility function for the green an red agents. Itcan also be easily adapted to fit to cases where the members of a same group don’tshare the same utility function. Figure 8 validates the grand canonical approch sincethe measures of collective utility for T = 10 we made in our simulations fit with thepredicted values for t→∞.

Figure 8 – In pink : grand canonical theoretical values of the collective utility for T → ∞ withrespect to the vacancy rate v in the case of uR = uG = uap. In blue : values of U∗ for T → ∞ takenfrom all the simulations’ results presented in this report. As v → 1, the remaining agents have more andmore probability to have no neighbors and therefore an utility of zero. That’s why U∗ → 0 as v → 1.

4.3 Resolution of the Global Coordination Case

In the ‘global coordination’ case, one can find an analytical expression for the sta-tionary distribution function :

Theorem 1. In the case of global coordination, the stationary distribution takes theanalytical form :

Π(x) =eU(x)/T∑z e

U(z)/T

It follows that for sufficiently small values of T , the configurations observed in the longrun are those which maximize the collective utility.

The first part of this theorem theoretically allows us to predict anything about thesystem in the long run. In practice, however this analytical form of the stationary distri-bution is not very useful : one would have to generate the N2!

V !NG!NV !possible configurations

and compute their collective utility to have an idea of the collective utility distributionin the configuration space phase. And in our nG = 0.5 reference case, these amounts to(using Stirling formula, for N = 30, v = 10%) :

2(1/v)vN2(2/(1− v))(1−v)N2√

v(1− v)2πN2∼ 10368 !!

a number that is for example many times larger than 1080, the estimated number ofatoms in the universe. And the number of possible configuration become even larger

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when we take realistic values of N !The second part of the theorem is more interesting, since it is a more practical pre-

diction which is furthermore consistent with what we observed in the simulations.

Proof of Theorem 1.This result is a special case of theorem 6.1 in [39], (pp. 95-97). The proof here follows

the same arguments presented in [39]. Let Pxy be the transition probability from statex to y and π the function defined as π : X → [0, 1]; x→ π(x) = eU(x)/T/

∑z e

U(z)/T . Itis straightforward to check that π(x) satisfies the detailed balance condition π(x)Pxy =Pyxπ(y). If x and y are two different and not communicating states, the relation istrivially satisfied since in this case Pxy = Pyx = 0. If x = y, the detailed balance conditionis also trivially verified. In the case where x 6= y and x and y are two communicatingstates, to change x in y and vice-versa, one must only switch the position of one agentand one vacant cell. Because there is a total of (1 − v)N2 agents and vN2 vacant cell,this agent and this vacant cell will be picked with a probability γ = 1/[v(1− v)N4]. Itfollows that

π(x)Pxy =eU(x)/T∑z e

U(z)/Tγ

1

1 + e−[U(y)−U(x)]/T=

eU(x)/T∑z e

U(z)/Tγ

eU(y)/T

eU(x)/T + eU(y)/T

=eU(y)/T∑z e

U(z)/Tγ

eU(x)/T

eU(x)/T + eU(y)/T=

eU(y)/T∑z e

U(z)/Tγ

1

1 + e−[U(x)−U(y)]/T= π(y)Pyx

Hence the detailed balance condition is always verified and∑x∈X

π(x)Pxy =∑x∈X

π(y)Pyx = π(y)∑x∈X

Pyx = π(y) · 1 = π(y) ,

which means that π is a stationary distribution of the process. Because the Markovchain is finite and irreductible, it has a unique stationary distribution. Hence, for eachstate x, Π(x) = π(x) = eU(x)/T/

∑z e

U(z)/T .

Define then S has the subset of X of the states that strictly maximize the collectiveutility :

S = {y/∀x ∈ X U(y) ≥ U(x)}.The second part of Theorem 1 can now be prooved as follow : for two states x and yof S, we will have Π(x)/Π(y) = e[U(x)−U(y)]/T = 1, which means that the states thatmaximize the collective utility are observed in the long run with the same probability ;for two states x ∈ X \ S and y ∈ S, we will have Π(x)/Π(y) = e[U(x)−U(y)]/T → 0 asT → 0, which means that for T → 0, the probability to observe a state that doesn’tmaximize the collective utility becomes in the long run infinitesimaly small.

4.4 Stability of the Highly Segregated States for T → 0

Let’s imagine that the city is an highly segregated state such as the one representedone the left of fig 9. From the agents’ point of view, the city is divided into three areas :

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Figure 9 – In order to investigate the stability of a highly segregated state, we are trying to estimatethe probability to form a red ‘nucleus’ (composed of 4 red agents in a square formation) in the green-dominated area in four successive moves but also the probability that this nucleus resorbs itself. Wehave here enlighted in yellow the moving red agents and in white the vacant cells they leave or moveinto.

a strictly red-dominated area, a stricly green-dominated area and a mixed area, wherean agent has one or several neighbors of the other group. With the geometry of the cityintrinsic to our model, this mixed area is separated into two non connected sub-areawhose width depends on the size of the agents’ neighborhood H.

We saw section 3.1 that one way to disrupt such a segregated state was to createred (resp green) nucleus in the green (resp red) area. We want here to give a qualitativeestimation of the probability to create such nucleus, but also the probability that sucha nucleus resorbs itself. The balance we will obtain this way will give us some insight ofthe stability of the highly segregated states depending on the dynamical rule.

Lemma 1. The probability to get from a typical highly segregated state to a state witha nucleus composed of 4 red agents in a square formation in the green area in foursuccessive iterations (the red agents moving from the strictly red-dominated area) can bewritten as

p = Ap1p2p3p4 ,

where A > 0 provided v is high enough and where the pi’s depends on the dynamic ruleand the values of T , H, and eventually h.

Proof of Lemma 1.We implicitely made the assumption that the initial configuration was such that therewere four vacant cells in a square formation within the strictly green-dominated area.This is not so absurd since, starting from a random highly segregated state, such aconfiguration can be obtained by utility-neutral moves of green agents. And v has to behigh enough to ensure that at least 4 vacant cells are present in the green-dominatedarea.It is then straightforward to see that A is the probability to pick one of the 4 previousvacant cells and a red agent of the red-dominated area at each of the four iterations,and that each pi’s corresponds to the probability that the ith move happens.

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Lemma 2. The pi can be written in the form

pi =1

1 + e−αi/T,

where αi > 0 in the ‘sum’ and global coordination cases and α1 < 0, α2 < 0, α3 > 0 andα4 > 0 in the ‘without coordination’ and pondered vote cases.

Proof of Lemma 2.The form of the pi is given by the probability functions P (M) corresponding to the dif-ferent rules presented section 2.2. The αi correspond to the gain in utility of the agentsconcerned by the moving rule. For example, in the ‘without coordination’ rule case, thereis only one agent concerned : the red agent coming from the red-dominated area. Theutility of the four red agent goes from 0.5 to respectively uap(0) = 0, uap(1/6) = 0.33,uap(2/7) = 0.57, uap(3/8) = 0.75, hence the negative gain for the first two agents andpositive gain for the last two. The proof of Lemma 2 for the other moving rules goessimilarly.

We can now look at the limit T → 0, which affects strongly the probability of themove according the the sign of the utility gain : while pi becomes infinitely small forαi > 0, for αi < 0, pi remains finite. On figure 10, we indicate the order magnitude ofeach pi by ε or 1 whether it converges towards 0 or remains finite when T → 0.

At this point, we have found an estimation of the probability to create a nucleusin a particular path of the Markov chain, ie we give a minimization of the probabilityto create a nucleus. To estimate the stability of the nucleus, we should find an maxi-mization of the probability to resorb it. The difficulty lies in the fact that a red agentleaving the nucleus can either go back to the red-dominated area or go elsewhere in thegreen-dominated area.

Lemme 3. The probability pi that a red agent leave a nucleus can be written in theform

pi =1

1 + e−βi/T,

where |βi| depends on where this leaving agent move.

Checking every possibilities, one can find as before the order of magnitude of theprobability that a red agent leaves the nucleus for T → 0 after a finite number of itera-tions (this probability being equal to 1 for an infinite number of iterations).

Looking at the obtained balance on fig 10, one can say that with the ‘sum’ and‘global coordination’ rules, the probability to create nuclei starting from a completelysegregated states is non-zero, even when T → 0. On the other hand, the probability toresorb the nucleus tends to 0 when T → 0. Thus the highly segregated states are uns-table and are not frequently observed in the long run. This confirmed what we observedin the simulations.

With the ‘qualified vote’, the ‘pondered vote’ and the ‘without coordination’ rules,

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Figure 10 – Order of magnitude of the probabilities corresponding to successive moves leading to orleaving a nucleus of 4 red agents in the green-dominated area. For the ‘sum’ and ‘global coordination’rules, the probability to create a nucleus remains finite as T → 0 whereas once it exist, the probabilityfor one agent to leave it become infinitely small. A ‘2 blocks’ configuration is thus less stable than one ofits nucleus-disturbed version. For the ‘vote’ and ‘without coordination’ rule, a nucleus is very unlikelyto begin to form itself and

the probability to create nuclei tends to 0 as T → 0 whereas the probability to resorba nucleus (of less than 2 agents) remain finite. At this point, we can only say that thehighly segregated states are at least very stable under these dynamic rules when T → 0.This also confirmed what we observed in the simulation : the stationary configurationsin the ‘without’ coordination’ and ‘qualified vote’ cases correspond are highly segrega-ted one and the time needed to leave such a configuration increases exponentially with1/T in the ‘pondered vote’ case (see annex B2). This study yet do not allow us to pre-dict whether or not the ‘pondered vote’ rule always finally breaks the highly-segregatedstates.

5 Limits of the Model

Comments on the time scale :Our chronological reference, the period, was defined to correspond to the amount of timeneeded in order that each agent move on the average one time. This definition allowed usto avoid to transcribe each temporal units explicitely or implicitely present in the model(time of an iteration divided in one duration for the visit, one for the decision process ofthe coproprietors, one for the potential move ; in reality, several agents may try to movesimultaneously and not succesively as we supposed in the model). In the real world,people are moving typically once each 5 (in Hong Kong) to 10 (european cities) years.One can thus convert the periods in real time, at least approximatively. According tothe simulations we presented earlier, the stationary phases are thus sometimes reachedonly after several centuries or millenia. Apart from the fact that the agents should notbe immortal (we can suppose that they are replaced by their children), it is very un-

22

reealistic to suppose that the size and composition of the city but also the preferencesof its inhabitants do not change over such a long time.

Realistic values of T :Considering the transition at T ∼ 0.05 between a phase where the coordination has aneffect and a phase where the coordination has no effect anymore we just describe in the‘qualified vote’ case, one could raise the question whether such a transition may alsoexist in the case of the other local coordination rules, with a temperature of transitionlower than those for which we run the simulations presented at the previous section. Wepresent in annex B2 some simulations results which show that typical times needed toreach the stationary phases increase exponentially with 1/T . The exploration of the sta-tionary phase for T -values lower than say T = 0.01 would therefore be rather unrealisticsince it would sometimes imply to wait time scale comparable to hundreds of milleniabut it would also be unpracticable since it would require larger and larger computationtimes.

Our model also has another flaw that questions its relevance for low values of T .One can check during the simulations the successful move rate which is related to theaverage number of cells an agent visits before he effectively moves. For medium andhigh values of noise (T ≥ 0.1), it takes realistic values (between 10 and 20%, whichcorrespond to 5 to 10 visits). But for low T , it takes values that do not reflect the rea-lity : the magnitude of the successful move rate is typically 1 over 10000 for T ∼ 0.02 !This unrealistic aspect could be improved for example by picking specifically (and notrandomly) the vacant cells that are more susceptible to greet the picked potential mover.

6 Conclusion

The introduction of the concept of coordination in a Schelling-type model representsthe main achievement of the work presented in this report. The coordination enables tobreak the segregation patterns that were until now obtained in conventional Schelling-Type models at the expense of the collective utility. Our hope is that this breakthroughwill contribute to wipe out the aura of fatalism that usually go along with these modelsand will enhance our understanding of the human society and our capacity to regulateit throught new kind of policies. This is of course a huge task, considering the amount ofparameters to take into account, and the relative susceptibility of the system. We showhow closely related rules could lead to different results (the pondered vote rule breakingthe segregation while the qualified vote rule being without impact), hence the interestto have tested in our models different local coordination rules.Beyond this, two things have to be kept in mind : first, that to reach the highest levelof collective utility, it is not necessary to invoke a global authority who will regulate themove of all the agents, the local coordination of the neighbors may well be sufficient toreach already high level of collective utility. Second, that the introduction of coordina-tion can not break the segregation without the agents having a certain preference formixed patterns. Coordination is only a way to reinforce on the large scale the wishes of

23

the agents on the local scale : if they are intolerant, segregation will occur, if they aretolerant, integration may occur.

24

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Annexes

A Empirical Evidence of Residential Segregation

Numerous studies concerning residential ethnic segregation have been conducted inthe United States. Figure 11 shows the 1990 repartition of the inhabitants of New-Yorkaccording to their ethnic origin. One can only observe the existence of highly segregatedareas.

Figure 11 – Empirical evidence of residential ethnic segregation in New-York, 1990. Each area’s colorindicates which group (if any) constitutes a numerical majority (i.e., at least 50% ) of the populationin the area. If the predomominate group is a ”super majority” (i.e., 80% or more), the area is shown ina darker shade of the group’s color. Note that the color red indicates ”mixed areas” – areas where noneof considered groups is a numerical majority. Image taken from M. Fossett SegMaps program [44].

Residential segregation according to the socio-economic status is yet another kindof segregation. Figure 12 shows the 1990 repartition of the inhabitants of New-Yorkaccording to their wealth level. Once again, one can but only observe the existence ofsegregated areas. This SES segration is highly correlated to the ethnic segregation (it ismore than obvious on the Manhattan peninsula for example), but yet exists indepen-dantly from it (see for example the segregation between richer and poorer white familiesin the suburban areas).

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Figure 12 – Empirical evidence of residential segregation with the socio-economic status. Each area’scolor indicates its percentile rank compared with other areas of the city based on the median familyincome of families in the neighborhood. Five shades of blue are used to indicate SES quintiles. If thearea ranks high on income, it is shown in a darker shade. If it ranks low on income, it is shown in alighter shade. Image taken from M. Fossett SegMaps program (see [44])

B Additionnal Simulations results

B.1 Influence of T on the Patterns of the Stationary Configu-rations

We present on figure 13 snapshots of the city corresponding to the different dynamicrules of our model, at different values of T . These snapshots correspond to typicalconfigurations of the stationary phases on which the measures presented figure 7 weremade.

B.2 Caracteristic Times

We present fig 14 temporal evolution of the segregation index for six different si-mulations run with the ‘pondered vote’ rule with different values of T . The times nee-ded to reach the stationary configurations starting from a ‘2 block’ one fit roughly aln τstat ' 1/T law. The same kind of law also exists in the ‘qualified’ vote case, butalso in the ‘without coordination’ case (starting from a random initial configuration andevolving towards a ‘2 block’ one.)

These laws are closely related to the form of the probability P (M) of the dynamics

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Figure 13 – Typical stationary patterns for the different rule at different temperatures. The noisedominates at high temperature and structured patterns appear at low temperatures. Note for the ‘sum’and ‘pondered vote’ cases that whereas at T = 0.2 locally segregated areas still exist thanks to thenoise fluctuations, at T = 0.1 these small areas have disappeared. N = 30, v = 10%, H = 8.

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Figure 14 – Left : temporal evolutions of the segregation index for six simulations starting from a‘2 block’ configuration, the agents moving according to the ‘pondered vote’ rule with a different valueof T for each of the simulation. Right : corresponding times needed to reach the stationary phases. Thecoefficient of correlation of the ln τ vs 1/T regression is R2 ' 0.93.

rules, especially the 1/(1+e−∆u/T ) part 9 which concerns only the potential mover. Suchlaws do not exist in the ‘sum’ and ‘global coordination’ case where the potential moveris not singularized anymore in the dynamic rule.

B.3 Robustness of our Model

We present on figure 15 stationary measures similars to the ones presented figure7 for different values of the vacation rate v. Note that the transition between a phasewhere the coordination with ‘qualified vote’ has an effect and a phase where it has nonedepends on the values of v. Note also that the effect of the coordination is always moreimportant when the opinions of a greater number of coproprietors are taken into account.

Finally, we present on figure 16 some stationary measures corresponding to the largerneighborhood size H = 44, taking into account only the opinions of the coproprietorsliving on the h = 8 surrounding cells in the local coordination cases. The results are verycomparable to those obtained with H = 8, which shows the robustness of our modelwith this parameter.

B.4 Simulation Results with Other Utility Functions

In the main body of this report, we stick to the case when the utilities of the agentswere all computed with the ‘asymetrical peaked’ function uap. However, our model allowsto try other utility functions. The range of possible outcomes is thus much larger. Wepresent on figure 17 typical stationary configurations in two new hypothesis.

9. the crucial move in the evolution in the transitory phase are those for which the potential movermake a ‘mistake’. Such move are proposed with a rather constant probability to an agent at eachiteration, and the probability that such a move happen is 1/(1 + e−∆u/T ). The time needed to reachthe stationary configurations is thus qualitatively τstat =< 1 + e−∆u/T >' eδ/T for T → 0 δ > 0.)

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Figure 15 – Stationary mean values of the segration index and of the collective utility with respectto the level of noise T , for the different dynamic rules.

The first hypothesis corresponds to the case where all the agents are intolerantand preferentially live only with members of their own group, computing their level ofsatisfaction with the linear utility function. With or without coordination, the outcomeis the same : highly segregated configurations which maximalize the collective utility.

The second hypothesis corresponds to the case where red agents are intolerant, butwhere green agents have a strict preference for mixed neighborhoods, while still preferingan all-similar neighborhood to an all-dissimilar one. This situation implies a competitionbetween highly segregated global configuration and globally mixed configuration becausein the two cases only one group could be satisfied. The simulations show that withoutcoordination, the segregation wins, but that the introduction of coordination move theequilibrium towards more mixed states.

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Figure 16 – Stationary mean values of the segration index and of the collective utility with respectto the level of noise T , for the different dynamic rules.

C Ising vs Schelling

Note : we only aim here at making a few remarks. A more developped argumentationon the questions we deal with in this section is in preparation for an article to be submitto the American Journal of Physics.

In a recent series of article [36, 37], Stauffer claimed that “the Schelling model ofsegregation” was “a modification of the Ising magnet at zero temperature”. We alreadysaid that it was tempting to make the parallel between the Schelling and the Ising mo-dels : an agent, his color and his utility computed with respect to the composition of hisneighborhood is comparable to a particle, his spin and its energy computed with respectto the interaction with the neighboring particles. In economic model such as Schelling’s,we have a maximization principle (each agent try to maximizes is own utility) and inphysical model such as Ising a minimization one (the equilibrium state minimizes theglobal energy). And the two models may generate highly segregated patterns.

These analogies can nevertheless not be pushed too far. Within the physical Isingmodel, the force exerced on a particle by its neighbors are additive and reciprocal. Inthe economic Schelling model (which correspond to our ‘without coordination’ case),the utility of an agent can not be written as a sum of contributions with each of hisneighbors and there is no reciprocity in the interaction between two agents.

In fact, in order to have at least an additivity of the interaction between neighbors,one must necessarily suppose that the agents utility vary linearly with their fractionof similar neighbors sk. This is the only case when the comparison between Ising andSchelling can be envisaged (and that’s roughly what is done by Stauffer). The utilityuap that we mostly used in this paper is in that sense non physical.

Let’s put aside these additivity/reciprocity of the interaction issues. We saw that onemostly important concept in Ising models is the minimization of the global energy, whichcounterpart in a classical (without coordination) Schelling model should be the maxi-mization of the global utility. We saw that this is not what is found in the simulationswhen the agents utility is compute with uap. On the contrary, the global coordinationrule was introduce to do exactly this.

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Figure 17 – Up : typical stationary configurations for the case when the two groups compute theirlevel of satisfaction with the linear utility function : uR = uG = ulinear. Since the agents all wantto live preferentially only with agents similar to them, the collective utility is already maximalized inthe case without coordination. The introduction of coordination thus doesn’t change anything. Noticethat since nobody want to live on the red-green frontier, the frontier is composed by the vacant cells.Bottom : typical stationary configuration for the case when the red group is intolerant (uR = ulinear)and the green group is tolerant (uG = uap). Notice than the dissymmetry between the two group leadsto configuration where the vacant cells are all in the green areas. N = 30, v = 10%, H = h = 8.

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