The chequerboard model suggests that residential segregation may be an un- intended consequence of the dispersed actions of the individuals. The focus of the analysis is on the ‘segregation that can result from discriminatory behavior’ (Schelling 1978: 138).
That is, in the chequerboard model, segregation is caused by discriminatory behaviour. In the context of the model, ‘discriminatory’ implies that individu- als are aware (consciously or unconsciously) of the differences between two (or more) groups of individuals. This awareness may or may not affect individual behaviour. If it does, it may cause a range of discriminatory behaviour from strong discrimination (i.e. very intolerant) to weak discrimination (i.e. very tolerant). Let us suppose that there are two groups of individuals, As and Bs. In the context of the chequerboard model, strong discriminatory preferences imply that As (or Bs) want to be the majority in the neighbourhood, mild discriminatory preferences imply that As (or Bs) try to avoid a certain minority status.
As a first step to get a grip of the chequerboard model, let us start thinking about strong discriminatory preferences. It is quite simple to understand why strong discriminatory preferences (e.g. racism) cause residential segregation. As Schelling nicely argues,
The simplest constraint on dichotomous mixing is that, within a given set of boundaries, not both groups can enjoy numerical superiority. For the whole population the numerical ratio is determined at any given time; but locally, in a city or a neighbourhood, a church or a school or a restaurant, either [As] or [Bs] can be majority. But if each insists on being local majority, there is only one mixture that will satisfy them – complete segregation.3
(Schelling 1978: 141)
If As and Bs are not content with any minority status, a process of moving to other places where they can be content will start. This process will not stop until every individual is content. This logical constraint implies that the only one mixture that can satisfy every individual is complete segregation – where every neighbourhood is composed of either As or Bs. Let us consider this as our first model (Model I) of segregation.
The specified mechanism is very simple. For descriptive purposes, we can think of every individual as conforming to a simple rule: IF the ratio of the other group to the neighbourhood population is larger than or equal to 1/2 THEN move, IF NOT stay. More precisely, given that the sum of the ratios of As (α) and Bs (β) to the neighbourhood population is equal to one (α + β = 1); the following rules for As and Bs will bring about complete segregation. As: IF β ≥ 1/2 THEN move, IF NOT stay; Bs: IF aα ≥ 1/2 THEN move, IF NOT stay.
These rules can be interpreted as the mechanisms that may bring about com- plete segregation within this simple model. Although the model is simple, it makes us aware of the fact that strong discriminatory preferences bring about complete residential segregation. If we knew that every individual in ethnically segregated neighbourhoods hold strongly discriminatory preferences, we could easily use this simple model to explain all instances of segregation. This is not the case, however. There are people who have milder discriminatory preferences. Moreo- ver, as far as we can observe, there may be other causes of residential segregation, such as welfare differences among different ethnic groups. It is not possible to explain segregation by simply arguing that racism causes segregation. We need to inquire other possibilities.
Our simple model does not tell us whether different groups of individuals may be segregated even if they are happy with mixed neighbourhoods. Now, let us ex- amine whether such preferences may cause segregation. Assume that agents (in- dividual As and Bs) are not concerned – and not informed – about the mixture of their neighbourhood, but they care about their immediate neighbours. Moreover, assume that both As and Bs are happy living in a mixed neighbourhood, they can also accept a minority status, but they do not want to live as an extreme minority. These assumptions bring us to our second model of segregation (Model II), which is the widely discussed chequerboard model of segregation.
In the chequerboard world, As and Bs live in a chequerboard city. This city, like every other city, has boundaries for a given time period and it has a shape. There are sixty-four places to live (e.g. sixty-four houses) and some of these places are not occupied, thus agents can move to these unoccupied places if they wish. The chequerboard city is shown in Figure 4.1, in where every ‘_’ represents a possible place to live in.
The idea is to place two types of agents in this ‘city’ and see what happens if individuals have mildly discriminatory preferences (i.e. if they could tolerate a minority status, but would not want to live as an extreme minority). Let us sup- pose that (a) there are twenty-two As and twenty-three Bs: forty-five people in total; (b) agents are randomly distributed; and (c) ‘each [agent] wants that more than one-third of his neighbours are like himself’ (Schelling 1978: 148).
Figure 4.1 The chequerboard city.
In Figure 4.2 As and Bs respectively show the places occupied by As and Bs. Every agent will have at most eight neighbours (the cells surrounding them). Now consider what happens if every agent wants that more than one-third of his neigh- bours are like himself (Schelling 1978: 148). Given this specification, circled letters (As and Bs) in Figure 4.2 show the nine agents who are not satisfied with their place. Since there are some unoccupied places, residents who are discontent can move to places where they could be content. However, as they move they will change the state of their old neighbours. When an A moves to another place he reduces the number of As for his previous neighbours, so the As in those places are more likely to become discontent after he moves. If they are, they will try to move as well. As dissatisfied residents start moving, the residential distribution of As and Bs changes.
Figure 4.3 shows one of the possible results. It could be seen in Figure 4.3 that the process that was triggered by dissatisfied residents leads to residential segre- gation. Note that it is possible for each individual to move to a couple of places, however, though different moves will lead to different outcomes (i.e. different distributions of residents), the overall result – segregation – does not change. In Figure 4.3 all the individuals are content and they are more segregated. ‘This is more than just visual impression’, says Schelling. If you compare the number of neighbours that are the same type for each group and the average number of unlike neighbours – compare Figure 4.2 with Figure 4.3 – you will see that a lot has changed. Also, the number of individuals who have no neighbours of the other type has increased.
The surprising thing about this demonstration is the following: in Figure 4.2
Figure 4.2 Randomly distributed residents.
Figure 4.3 Segregated city.
there were only nine out of forty-five people who were dissatisfied. Their decision to move to another place triggered a process that brought about residential segre- gation. The model shows that even if people in this representative city have mild preferences about the type of people living around them (e.g. every agent wants more than one-third of his neighbours to be like himself) and even if the number of dissatisfied agents is not more than the 20 per cent of the whole population, they might end up living in separate places. If the readers try the model on an actual chequerboard with different preferences of agents and with different initial distributions they can see that various combinations of different individual prefer- ences and initial distributions will bring about the same result: more segregation.4 But if they coincidentally start with a mixed neighbourhood where everybody is content, they will see that segregation will not occur. However, this is not a stable distribution. If you move a couple of individuals and make some of the residents unhappy, the process will take you to segregation instead of restoring the mixed neighbourhood.
Schelling argues that the ‘instructive’ thing ‘about the experiment is the unre- vealing process’ (1978: 150, emphasis added). We can get a better understanding of this process by examining the constituent mechanisms of the model. As for the first simple model, we can define the mechanisms at the individual level as simple behavioural rules for each agent. Movements of the agents might be defined in terms of use of simple IF . . . THEN rules. We only have to assume that agents are capable of computing the fraction of the neighbours who are their own type (see Epstein and Axtell 1996: 165). If we call this fraction x, agents move according to the following rule: IF 1/3 ≥ x THEN move. IF 1/3 < x THEN stay.
There are no other specifications about where to move. Agents may randomly move until they find a suitable place, or they may know where to move in ad- vance, but this does not change the results of the model. Interactions of the agents are also defined according to these basic IF . . . THEN rules. These rules implicitly define the responses of the individuals to the existence of others. As some agents execute these rules, the states of some other agents (e.g. their previous and sub- sequent neighbours) change from ‘content’ to ‘discontent’. This, in turn, causes further execution of IF . . . THEN rules – until every agent is content.
If we interpret the IF . . . THEN rules as the mechanisms at the individual level, the connection of these individual mechanisms can be interpreted as an aggregate mechanism (or a process). To define the mechanisms at the individual level, we have to define the states of the agents. Every individual could be content or discontent.5 If the agent is content the mechanism is not triggered and the state of the individual does not change. But if the individual agent is discontent, the mechanism is triggered and the individual moves to another place – thus, the state of the agent changes to ‘content’. The inputs are, then, the mixture of immediate neighbours and the preferences of the agent.6 Because the mixture of the immedi- ate neighbours is one of the inputs, every individual (and, thus, every mechanism) is connected to his immediate neighbours (to its neighbour mechanisms). If the agent moves, not only his own state changes, but also the states of his previous and subsequent neighbours may change. This network of mechanisms can be con-sidered as an aggregate mechanism that is responsible for the transition between the different states of the city. The states of the city can be defined by the ‘number of neighbours that are the same type for each group’, by the ‘average number of unlike neighbours’ or by the ‘number of people who have no neighbours of the other type’. Note that under this interpretation reinforcement is not a mechanism, rather, it is a property of the aggregate mechanism:7 as individuals execute the rules and move, execution of the rules by other individuals is reinforced (also see Holland 1995). It should be noted that in this model there is no feedback from the state of the city, for the state of the city is not one of the inputs of the individual mechanisms.
Table 4.1 summarises our depiction of the mechanisms in the chequerboard model. The model is mainly based on the individual mechanisms that define the actions of the dispersed individuals and on the interactions between them. The actions of the agents are defined in terms of IF . . . THEN rules. IF . . . THEN rules also implicitly define the response of the individuals to the existence of others. Since there are as many mechanisms as the number of agents and they all react to the distribution of the other agents near them, we can think of the chequerboard city as a network with nodes connecting one agent to at most eight agents. Finally, we can think about the relation between the lower level mechanisms (rules for agents) and higher-level mechanisms, that is, the relation between the changes in individual states and changes in the state of the system as a whole. Under this in- terpretation basic individual mechanisms can be considered as the building blocks for the higher-level mechanisms (see Holland 1995). The interaction of the indi- vidual mechanisms (behavioural rules of the agents) constitutes a social mecha- nism that may as well be defined as a process, which transforms an integrated city into a segregated one. Thus, what we have here is an ‘aggregate mechanism that which takes as “input” the dispersed actions of the participating individuals and produces as “output” the overall social pattern’ (Ullmann-Margalit 1978: 270).
We may continue conjecturing by way of changing some of the assumptions of the chequerboard model. For example, we may ask whether the results change if individuals are concerned about the composition of their neighbourhood. Or, we may use heterogeneous agents who have different tolerance levels. These conjectures may help us analyse the conditions under which mild discriminatory preferences bring about segregation. In Chapter 7, we will consider some of the recent follow-ups to the chequerboard model and discuss the strength of Schell- ing’s conjecture. Yet, at this step it is important to understand the nature of this conjecture.
Basically, Schelling conjectures about some of the mechanisms that may work at the individual level, investigates how they may interact (the aggregate mecha- nism), and whether or not they produce segregation. There is one chief reason for the selection of this methodology: while it is possible to observe the effects of an aggregate mechanism for a particular real city, it is not possible to find out what individual mechanisms might be at work from these observations, because the states of the city and effects of the aggregate mechanism give very little informa- tion about the underlying individual mechanisms. That is, the observation of the fact that a particular city is residentially segregated does not give us any infor- mation about the motivations and preferences of the agents. Moreover, several different types of micromotives may be responsible for residential segregation. For these reasons, Schelling uses the chequerboard model to get an idea about the types of individual mechanisms that may produce segregation. The next section examines these issues in detail.
Source: Aydinonat N. Emrah (2008), The Invisible Hand in Economics: How Economists Explain Unintended Social Consequences, Routledge; 1st edition.