_ournal of Mathematical 'Sociology �� Gordon and Breach Science Publishers Printed in Birkenheadj England !_971, Vol. 1, pp 143-186 SOCIOLoG DYNAMIC MODELS OF SEGREGATIONt of Pittsburgh THOMAS C. SCHELLING Harvard University 'Jniversity Somesegregation resultsfrom the practicesof organizations,somefrom specialized communication ffstems, some from correlation with a variable that is non-random and some results from the interplayof individual choices.This is an abstract study of the interactivedynamicsof discriminatory :ty individual choices. One model is a simulation in which individual members of two recognizable groups distribute themselves in neighborhoods defined by reference to their own locations. A second University model is analytic and deals with compartmented space. A final section applies the analytics to oundation 'neighborhood tipping.' The systemic effects are found to be overwhelming: there is no simple correspondence of individual incentive to collective results. Exaggerated separation and patterning result from the dynamics of movement. Inferences about individual motives can usually not be drawn from aggregate patterns. Some unexpected phenomena, like density and vacancy, are generated. gan A general theory of 'tipping' begins to emerge. sity :hicago People get separated along many lines and in many ways. There is segregation by olorado Sex, age, incom,, language, religion, color, taste, comparative advantage and the accidents of hi rt,,'_ical location. Some segregation results from the practices of 9fade organizations s,me is deliberately organized and some results from the interplay at'onto of individual cho ces that discriminate. Some of it results from specialized com- munication systc_ls, like different languages. And some segregation is a corollary York of other modes of segregation: residence is correlated with job location and transport. _f California, Berkeley If blacks exclude whites from their church, or whites exclude blacks, the segregation is organized, and t may be reciprocal or one-sided. If blacks just happen to be Baptists _r and whites Methodists, the two colors will be segregated Sunday morning whether ,calf they intend to be tA not. If blacks join a black church because they are more comfortable among their ow'_ color, and whites a white church for the same reason, undirected individual choic_ can lead to segregation. And if the church bulletin board is where reas of mathematical socioJogy. !q people advertise :ooms for rent, blacks will rent rooms from blacks and whites from ding with the logic of measurement, whites because of a communication system that is correlated with churches that are :ussions of the relationship of measurement correlated with color. ,sychological phenomena. Articles dealing ._of mathematical modelsin socialscience Some of the same mechanisms segregate college professors. The college may own Tnificantmethodological or substantive some housing, from which all but college staff are excluded. Professors choose housing g with computersor computer programs n to a substantive sociological or social commensurate with their incomes, and houses are clustered by price while professors 611 also considerspeculativearticlesthatare ently precise, general,and abstractso "f This study was sponsored by The RAND Corporation with funds set aside for research in tical sociologyby stimulating mathematical areasof special interest,and was issuedas RM-6014-RC in May 1969. The views expressed are not t have usuallybeen dealt with only verbally, necessarilythoseof RAND or its sponsors. Publishers Ltd. 143 r Brothers Limited, Birkenhead.

144 T.C. SCHELLING DYNAMIC MO] are clustered by income. Some professors prefer an academic neighborhood any this paper might be put in third pl differential in professorial density will cause them to converge and increase the local segregation in the United States, even density. And house-hunting professors learn about available housing from other It is not easy, though, to draw the li professors and their wives, and the houses they learn about are the ones in neighbor- tion, the more organized kind, and the hoods where professors already live. are substitutes for organization. Fear( The similarity ends there, and nobody is about to propose a commission to the fearisjustified, andwhether the san desegregate academics. Professors are not much missed by those they escape from in Common expectations can lead to col their residential choices. They are not much noticed by those they live among, and, sanctioned by ostracism, is often self-, though proportionately concentrated, are usually a minority in their neighborhood. The economically induced separal While indeed they escape classes of people they would not care to live among, they are To choose a neighborhood is to cho more conscious of where they do live than of where they don't, and the active choice good schools is to pick a neighborh( is more like congregation than segregation, though the result may not be so different, people who want to be with the kind c This article is about the kinds of segregation--or separation, or sorting--that can furthermore rely, even in making eco result from discriminatory individual behavior. By 'discriminatory' I mean reflecting color-discriminating believing that da_ an awareness, conscious or unconscious, of sex or age or religion or color or whatever than lighter-skinned, one may consciot the basis of segregation is, an awareness that influences decisions on where to live, of poverty (or, believing that others rel whom to sit by, what occupation to join or to avoid, whom to play with or whom to indices in order to coincide with them). talk to. The paper examines some of the individual incentives, and perceptions of strangeness, fear, hostility, and sheer difference, that can lead collectively to segregation. The paper also examines the avoidance. If the sentiment is recip_ extent to which inferences can be drawn, from the phenomenon of collective segrega- segregating tendencies of both groups tion, about the preferences of individuals, the strengths of those preferences, and the Economic segregation might statisticl facilities for exercising them. if that degree were enough to cause The ultimate concern is segregation by 'color' in the United States. The analysis, discrimination could complete the job. though, is so abstract that any twofold distinction could constitute an interpretation might not cause the collapse of the segr --whites and blacks, boys and girls, officers and enlisted men, students and faculty, For all these reasons the lines dividi teenagers and grownups. The only requirement of the analysis is that the distinction enforced, and the economically induce be twofold, exhaustive, and recognizable, are furthermore not the only mechan At least two main processes of segregation are omitted. One is organized action communication systems---especially di --legal or illegal, coercive or merely exclusionary, subtle or flagrant, open or covert, influence that, though interacting with kindly or malicious, moralistic or pragmatic. The other is the process, largely but a different mechanism.) Still, they ar not entirely economic, by which the poor get separated from the rich, the less educat_l separately understood. from the more educated, the unskilled from the skilled, the poorly dressed from the This paper, then, is about those me well dressed, in where they work and live and eat and play, in whom they know and behavior into collective results. whom they date and whom they go to school with. Evidently color is correlated with income, and income with residence so even if residential choices were color-blind Individual Incentives and Collective R and unconstrained by organized discrimination, whites and blacks would not be Economists are familiar with systems randomly distributed among residences._" neither intends nor needs to. be aware This is not to claim that the organized discrimination or the economically induced counterpart at the level of the indivi_ segregation is less powerful, or less important, or less a matter of social concern, bankingsystemisone thewaysavingsd( than the segregation that results from individual action. Indeed, aside from the question Similarly, biological evolution is re of which mechanism may account for the greater part of observed separation by but the little creatures that mate and color, the organized segregation involves civil rights and the economically determined to know that they were bringing abo segregation raises questions of social equity. On those grounds alone the subject of the extinction of species. Among so second languages is a phenomenon that is a massive 'free market' activity with t A treatment socioeconomic differentials between whites and nonwhites,ia, choice. relationcomprehensive to residentialpatterns, isinof Pascal(1967). , _

G DYNAMIC MODELS OF SEGREGATION 145 an academic neighborhood any paper might be put in third place. Still, in a matter as important as racial to converge and increase the local on in the United States, even third place deserves attention. )ut available housing from Other _ It is not easy, though, to draw the lines separating 'individually motivated' segrega- rn about are the ones in neighbor, lion, the more organized kind, and the economically induced kind. Habit and tradition are substitutes for organization. Fear of sanctions can coerce behavior whether or not )ut to propose a commission to the fear is justified, and whether the sanctions are consensual, conspiratorial or dictated. issed by those they escape from in Common expectations can lead to concerted behavior. ('Guilt by association,' when _d by those they live among, and, sanctioned by ostracism, is often self-enforcing.) a minority in their neighborhood. The economically induced separation is also intermixed with discrimination. dd not care to live among, they are To choose a neighborhood is to choose neighbors. To pick a neighborhood with e they don't, and the active choice good schools is to pick a neighborhood of people who appreciate schools (or of the result may not be so different, people who want to be with the kind of people who appreciate schools). People may r separation, or sorting--that can furthermore rely, even in making economic choices, on information that is itself 'discriminatory' I mean reflecting color-discriminating believing that darker-skinned people are on the average poorer ge or religion or color or whatever than lighter-skinned, one may consciously or unconsciously rely on color as an index lences decisions on where to live, of poverty (or, believing that others rely on color as an index, adopt their signals and d, whom to play with or whom to indices in order to coincide with them). And if the process goes far enough, alienation, ml incentives, and perceptions of strangeness, fear, hostility, and sheer habit can accentuate the tendency toward m. The paper also examines the i avoidance. If the sentiment is reciprocated, positive feedback will amplify the _henomenon of collective segrega. . segregating tendencies of both groups. gths of those preferences, and the Economic segregation might statistically explain some initial degree of segregation if that degree were enough to cause color-consciousness, a superstructure of pure t the United States. The analysis, discrimination could complete the job. Eliminating the economic differentials entirely could constitute an interpretation might not cause the collapse of the segregated system that it had already generated. nlisted men, students and faculty, For all these reasons the lines dividing the individually motivated, the collectively :he analysis is that the distinction enforced, and the economically induced segregation are not clear lines at all. They are furthermore not the only mechanisms of segregation. (Separate or specialized omitted. One is organized action communication systems---especially distinct languages---can have a strong segregating .ubtle or flagrant, open or covert, influence that, though interacting with the three processes mentioned, is nevertheless other is the process, largely but a different mechanism.) Still, they are very different mechanisms and have to be ed from the rich, the less educated separately understood. lied, the poorly dressed from the This paper, then, is about those mechanisms that translate unorganized individual nd play, in whom they know and behavior into collective results. Evidently color is correlated with idential choices were color-blind Individual Incentives and Collective Results hites and blacks would not be Economists are familiar with systems that lead to aggregate results that the individual neither intends nor needs to be aware of, results that sometimes have no recognizable tion or the economically induced counterpart at the level of the individual. The creation of money by a commercial less a matter of social concern, banking system is one the way savings decisions cause depressions orinflations is another. n. Indeed, aside from the question Similarly, biological evolution is responsible for a lot of sorting and separating, part of observed separation by but the little creatures that mate and reproduce and forage for food would be amazed and the economically determined to know that they were bringing about separation of species, territorial sorting, or ase grounds alone the subject of the extinction of species. Among social examples, the coexistence or extinction of second languages is a phenomenon that, though affected by decrees and schoolcurricula, ials between whites and nonwhites, in is a massive 'free market' activity with results that correspond to no conscious collective choice.

146 T. C. SCHELLING DYNAMIC Romance and marriage are exceedingly individual and private activities, at least Some Quantitative Constraints in this country, but their genetic consequences are altogether aggregate. The law and ( Counting blacks and whites in a the church may constrain us in our choices, and some traditions of segregation are how they get along. But it tells s enormously coercive but outside of royal families there are few marriages that are to the people who are moving in ( part of a genetic plan. When a short boy marries a tall girl, or a blonde a brunette, it And with quantitative analysis th_ is no part of the individual's purpose to increase genetic randomization or to change analogous to the balance-sheet ide some frequency distribution within the population, contain no news unless one just In some cases small incentives, almost imperceptible differentials, can lead to The simplest constraint on dic strikingly polarized results. Gresham's Law is a good example. Some traditions, boundaries, not both groups (colo furthermore, are sternly self-enforcing" passing to the right of an oncoming car. the population as a whole, the nul Some collective actions have almost the appearance of being organized--fads in locally, in a city or a neighborhooc clothing, dancing and car styles, be a majority. But if each insists ol Some of the phenomena of segregation may be similarly complex in relation to that will satisfy them---complete se the dynamics of individual choice. One might even be tempted to suppose that some Relaxing the condition, if white: 'unseen hand' separates people in a manner that, though foreseen and intended by least one-third, it won't work. If no one, corresponds to some consensus or collective preference or popular will. But no fewer than one-fifth, there is a in economics we know a great many macro-phenomena, like depression and inflation, And not everybody can be in the that do not reflect any universal desire for lower incomes or higher prices. Similarly Other constraints have to do wit with bank failures and market crashes. What goes on in the 'hearts and minds' of the teacher is one color mixed mal small savers has little to do with whether or not they cause a depression. The hearts a three-man team cannot represent and minds and motives and habits of millions of people who participate in a segregated i each member has company exclusi_ I society may or may not bear close correspondence with the massive results that _ In spatial arrangements, like a collectively they can generate, next to somebody. A neighborhooc We also know that people who would not support the government with private neighbor on either side, the minim donations may vote a system of mandatory taxes to finance public goods. The worth If people draw their boundaries d of a new turnpike may depend on keeping traffic below the level of congestion that at dinner, with men and women would just equalize its attractiveness with that of alternative routes. 'Freedom of to one locally by the opposite sex choice' is sometimes nothing more than the lack of enforceable contract. Thus his horizon to the next person on unregulated behavior does not necessarily reflect preferences about its results. People in a hospital and there are four bed acting individually are often unable to affect the results they can only affect their all-white rooms. own positions within the overall results. _ Transitions involve the usual re Evolutionary processes may lead to typewriter keyboards, weights and measures college that wants suddenly to have and the pitches of screws, systems of coinage, and lefthand or righthand automobile i black freshmen, only to discover th drive that are self-perpetuating in spite of inefficiency until a heroic effort can bring ' more. Professions, occupations an about concerted change. Yet, also, some massive concerted changes can occasionally relations, whether it is color, sex, n be brought about by some simple manipulation, as when daylight saving goes into effect / or when a round table replaces a rectangular one in a highly stratified conference room. t A special reason for doubting any social efficiency in aggregate segregation is that f Separating Mechanisms the range of choice is often so meager. The demographic map of almost any American ! The simple mathematics of ratios metropolitan area suggests that it is easy to find residential areas that are all white comes are logically possible, but t or nearly so and areas that are all black or nearly so but hard to find localities in that leads away from, particular ou which neither whites nor nonwhites are more than, say, three-quarters of the total, i or integration may result from in_ And, comparing decennial maps, it is nearly impossible to find an area that, if integrated by which various mixtures and se within that range, will remain integrated long enough for a man to get his house paid the incentives and the behavior t for or his children through school. The distribution is so U-shaped that it is virtually way that different individuals coml a choice of two extremes. _ and react to each other's presence. i

DYNAMIC MODELS OF SEGREGATION 147 _ ._. ,1 and private activities, at least Quantitative Constraints together aggregate. The law and l _ounting blacks and whites in a residential block or on a baseball team will not tell me traflitions of segregation are . Ja0w they get along. But it tells something, especially if mtmbers and ratios matter :here are few marriages that are to the people who are moving in or out of the block or being recruited for the team. dl girl, or a blonde a brunette, it And with quantitative analysis there are usually a few logical constraints, somewhat etic randomization or to change analogous to the balance-sheet identities in economics. Being logical constraints, they ' contain no news unless one just never thought of them before. ptible differentials, can lead to The simplest constraint on dichotomous mixing is that, within a given set of :ood example. Some traditions, boundaries, not both groups (colors, sexes) can enjoy numerical superiority. Within the right of an oncoming car. the population as a whole, the numerical ratio is determined at any given time but ce of being organized--fads in locally, in a city or a neighborhood, a church or a school, either blacks or whites can be a majority. But if each insists on being a local majority, there is only one mixture similarly complex in relation to that will satisfy them--complete segregation. tempted to suppose that some Relaxing the condition, if whites want to be at least three-fourths and blacks at aough foreseen and intended by [ least one-third, it won't work. If whites want to be at least two-thirds and blacks ,' preference or popular will. But I no fewer than one-fifth, there is a small range of mixtures that meet the conditions. na, like depression and inflation, And not everybody can be in the mixtures if the aggregate ratio is outside the range. :omes or higher prices. Similarly Other constraints have to do with small numbers. A classroom can be mixed but on in the 'hearts and minds' of the teacher is one color mixed marriages can occur only in the ratio of one to one, y cause a depression. The hearts a three-man team cannot represent both colors equally, and even in a two-man team le who participate in a segregated each member has company exclusively of one color. e with the massive results that In spatial arrangements, like a neighborhood or a hospital ward, everybody is next to somebody. A neighborhood may be 10Yo black or white but if you have a )rt the government with private neighbor on either side, the minimum nonzero percentage of opposite color is 50. finance public goods. The worth If people draw their boundaries differently we can have everybody in a minority: _low the level of congestion that at dinner, with men and women seated alternately, everyone is outnumbered two alternative routes. 'Freedom of to one locally by the opposite sex but can join a three-fifths majority if he extends of enforceable contract. Thus his horizon to the next person on either side. If blacks occupy one-sixth of the beds ferences about its results. People in a hospital and there are four beds to a room, at least 40 _o of the whites will be in sults they can only affect their all-white rooms. Transitions involve the usual relations among numbers and their derivatives. A _yboards, weights and measures college that wants suddenly to have 10 _o of its students black will have to admit 40 _o :fthand or righthand automobile black freshmen, only to discover that it must then pass three classes before accepting :y until a heroic effort can bring more. Professions, occupations and residences are constrained by these numerical lcerted changes can occasionally relations, whether it is color, sex, nationality, age or degree status that is involved. m daylight saving goes into effect aighly stratified conference room. ' in aggregate segregation is that Separating Mechanisms .hic map of almost any American The simple mathematics of ratios and mixtures tells us something about what out- ddential areas that are all white comes are logically possible, but tells us little about the behavior that leads to, or so but hard to find localities in that leads away from, particular outcomes. To understand what kinds of segregation say, three-quarters of the total, or integration may result from individual choice, we have to look at the processes to find an area that, if integrated by which various mixtures and separations are brought about. We have to look at t for a man to get his house paid the incentives and the behavior that the incentives motivate, and particularly the s so U-shaped that it is virtually way that different individuals comprising the society impinge on each other's choices and react to each other's presence.