The Schelling's model of segrégation (1971), (1978).

version française

Schelling's model of spatial segregation (Schelling, 1969, 1971, 1978) is a precursory example of a strong emerging phenomenon, clearly based upon social interaction. Schelling's aim is to explain how segregationist residential structures (like ghettos) may occur spontaneously, even if people are not so very segregationist. The absence of a global notion of segregationist structures (like the notion of ghettos) in the agent's attributes (preferences) is a very important feature of this model. Agents have only local preferences over their neighbourhood. Moreover, people have only very weak segregationist behaviour, but the play of interactions \index{interactions!local interactions} generates global string segregationist results. In the original Schelling model, agents are localised within a 8-by-8 checkerboard  Figure 1.

Taking the ``colour`` (on the checkerboard) as the criteria of discrimination, the problem of each agent is to choose a location given an individual threshold of acceptation for the proportion of other colours in their neighbourhood. That is, agents interact only locally, with their 8 direct neighbours (a so-called "Moore'' Neighbourhood). There are not any global representations at all about the global residential structure. Agents have only weak segregationist local behaviour, in the following sense: each agent agrees to stay in a neighbourhood with people that are mainly of another colour, on condition that there are at least 37,5% with the same colour in the neighbourhood. More specifically, Schelling uses the following rule: an agent with one or two neighbours will try to move if there is not at least one neighbour of the same colour (with a tolerance of 50% in the neighbourhood); an agent with three to five neighbours needs at least two like him (33\%, 50% and 60% tolerance), and one with six to eight wants at least three agents of the same colour (50%, 57,1%, 62,5% tolerance). Schelling  denotes by a fully integrated structure of the population a structural pattern where there is alternately one agent of each colour in all directions; in other words, each agent (except at the edges) has four neighbours of one colour and four of the other. There is no agent in the corners. At the edges, there are two (or three) similar agents alternately among five neighbours, and two of each colour at the corners. Under Schelling's behavioural assumption, a fully integrated structure is an equilibrium (an order) because no agent wants to move. But, from this stable configuration, a slight perturbation is sufficient to induce a chain reaction and the emergence\index{emergence} of local segregationist patterns. Specifically, Schelling extracted twenty agents at random, and added five at random in the free spaces. By moving discontented agents, local segregationist patterns appear, like in the java applet below..

Figure 1 : equilibrium with
integrated population

Local interactions are sufficient for the occurrence of spatial homogeneous patterns; spatial segregation is an emerging property of the system's dynamics, while spatial segregation is not an attribute of the individual agents. Sometimes, integrated (non-homogeneous) patterns may survive. Integrated structures are easily perturbed by random perturbations, while homogeneous structures are more stable (frozen zones).

Graphique 2 : 14 mécontents

In Figure 2 the discontented agents are shown by changing color from dark to grey and from red to magenta. These agents move at random towards a new location in agreement with their preferences. This move generates new discontented agents by a chain reaction  until a new equilibrium is reached. This may be a state of perfect segregation, with clearly delimited ghettos, like in Figure 3, or locally integrated patterns may survive in some niches within homogeneous patterns of populations.

Graphique 3 :
convergence après 4 itérations

Bibliographie.

• Schelling, T.S. (1971) ; "Dynamic Models of Segregation" ; Journal of Mathematical Sociology ;1 p. 143-186.
• Schelling, T.S. (1978) ; Micromotives and Macrobehavior ; W.W. Norton and Co, N.Y.

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