Explorations of the chequerboard world

It may be argued at the outset that if a model’s conjecture is interesting and prom- ising enough it will be subjected to many tests and explored in different ways. The chequerboard model is a good example in this respect. First of all, the cheq- uerboard model is widely cited in sociological and geographical studies concern- ing residential segregation (e.g. Aaronson 2001; Aberg 2000; Bayer, McMillan and Rueben 2001; Clark 1991; Denton and Massey 1991; Downs 1981; Farley, Fielding and Krysian 1997; Fielding 1997; Friedrichs 1998; Huttman et al. 1991; Iceland 2002; Ihlanfeldt and Scafidi 2002; Massey and Denton 1993; Torrens and Benenson 2005; Zhang 2000. There are also many studies on residential segrega- tion that do not mention Schelling’s chequerboard model. For example, Gürke (2006) gives an extensive list of the causes of segregation, but does not mention mild discriminatory preferences as a possible cause of segregation).

Second, the chequerboard model has been theoretically and empirically tested several times. This gives us a chance to evaluate Schelling’s conjecture further. To see the strength of Schelling’s initial conjecture, let us have a look at the different ways in which the chequerboard model has been explored.

Epstein and Axtell (1996) demonstrate that Schelling’s initial hypothesis holds under a wide variety of conditions and for a variety of initial starting points. (For an overview of Schelling-type models and related discrete choice models see Meen and Meen (2003). For an overview of interaction-based approach to social science see Blume and Durlauf (2001).)

Pancs and Vriend (2003) test the chequerboard model under the conditions of strict preference for perfect integration. Remember that in Schelling’s model in- dividuals who have mild discriminatory preferences do not strictly prefer a mixed neighbourhood. They are content as long as they do not have an extreme minority status. That is, As (or Bs) do not care whether the neighbourhood is segregated or integrated as long as As (or Bs) do not have an extreme minority status. The as- sumption of strict preference for perfect integration implies that individuals prefer a mixed (integrated) neighbourhood to a segregated one. Pancs and Vriend (2003) show that Schelling’s results hold even if individuals have a strict preference for perfect integration. Similarly, Zhang (2004a) tests the results of the chequerboard model in an evolutionary game-theoretical framework and shows that its results hold even if all individuals strictly prefer to live in mixed neighbourhoods (also see Benito and Hernandez 2004; Young 1998; Zhang 2004b).

Portugali, Benenson and Omer (1994) introduce a couple of new elements to Schelling’s model, such as stochastic household behaviour and heterogeneous agents, and find that Schelling’s results hold under these conditions (also see Por- tugali and Benenson 1997; Portugali, Benenson and Omer 1997).

Another line of exploration focuses on the structure of the neighbourhoods. Fagiolo, Valente and Vriend (2005) and Flache and Hegselmann (2001) test the robustness of the chequerboard model by changing the topological characteristics of the neighbourhoods and find that Schelling’s results hold under a wide variety of neighbourhood forms. These studies basically explore and change certain prop- erties of the chequerboard model and lend support to Schelling’s conjecture.

However, not all types of explorations support Schelling’s results under all conditions. For example, Sethi and Somanathan (2004) assume that individuals are also affected by the affluence of their communities and argue that both high- and low-income disparities lead to residential segregation when combined with mildly discriminatory preferences. Yet, they also show that intermediate levels of income disparity produce multiple equilibria and both integration and segregation becomes possible (also see Somanathan and Sethi 2004). In a similar manner, Benenson (1998) integrates new elements to the chequerboard model. In Ben- enson’s evolutionary model agents may adapt their behaviour to local or global environments and vacant places are scarce. He finds that the tendencies to adapt to the local environment and to the global environment may be in conflict. If agents adapt to their local environment, then they become more neutral to differences and that residential distribution is somewhat random. However, when they adapt to the global environment residential segregation is observed in the long-run. This implies that residential segregation is more likely when individuals care about the ethnic composition of their neighbourhood rather that their immediate neigh- bours.

Two other studies consider the role of vision in neighbourhood formation. Lauri and Jaggi (2003) argue that when individuals are able to observe the neighbour- hood structure of a wider area, integrated neighbourhoods may become stable and Schelling’s results do not hold. (Also see Ellen (2000), who emphasises the role of expectations in the process of segregation.) Fosset and Waren (2005), on the other hand, argue that Lauri and Jaggi’s (2003) results are caused by the specifica- tions of their model. They argue that residential distribution freezes in their model because individuals move only when they can improve their satisfaction, and be- cause individuals occupy their place forever if they are satisfied. They suggest that these are implausible assumptions and show that increased vision does not lead to a stable integrated neighbourhood when these assumptions are relaxed. Yet, Ed- monds and Hales (2005) also show that if the chequerboard is more crowded than that of Schelling’s and if intolerance levels are higher, the chequerboard model would suggest that segregation is decreased. They interpret this result by arguing that Schelling’s model does not provide a general theory and that it should not be interpreted as such. These studies confirm the incompleteness of the chequerboard model, yet they generally support the main hypothesis that mild discriminatory preferences may bring about segregation under some conditions.

Aforementioned studies that explore the chequerboard model do not test it against real-world data. There are a couple of studies that undertake this task. Some survey data suggest that Schelling’s insights may be true and that they need further examination. For example, Bobo and Zubrinsky (1996) and Farley (1997) argue that many individuals are highly tolerant of mixed neighbourhoods. This suggests that strong discriminatory preferences alone cannot explain residential segregation. This gives us a reason to believe that the individual mechanisms (i.e. tendencies) depicted in Schelling’s model may actually exist.

Another interesting way to explore and test the chequerboard model is to in- tegrate some real-world data to the model. Sander et al. (2000a,b) and Zhang (2000) use survey data to determine the preferences of different types of agents in the chequerboard city. That is, they employ assumptions that are consistent with real individuals’ preferences. Their survey data suggest that individuals are tolerant to mixed neighbourhoods, but that whites are less tolerant than blacks. By way of integrating this information, Sander et al. (2000a,b) and Zhang (2000) demonstrate that Schelling’s insights hold.

Sander et al. (2000a,b) assume larger neighbourhoods and integrate housing costs and costs of moving into the model. Moreover, they use survey data to give shape to the preference functions of the individuals. The actual discriminatory preferences of blacks and whites are represented in the model by defining, con- sistently with survey data, three types of agents of each group. Sander et al. then simulates the model under some ‘what-if’ scenarios to see how different factors (e.g. housing costs, moving costs, discriminatory preferences) may be related to each other. What we have in this case is confrontation with data, as well as new conjectural scenarios.

Another way to explore Schelling’s model is to confront certain aspects of the model with statistical data. This is exemplified by Clark (1991). By study- ing statistics for certain particular segregated cities, he confirms that integrated equilibria (i.e. mixed neighbourhoods) are not stable. In another study, Benenson (2004) simulates the residential dynamics of Yaffo (near Tel Aviv) between 1955 and 1995 with a model that is similar to the chequerboard model, and demonstrate that Schelling’s insights may be explanatory.

However, not all of the empirical tests support the chequerboard model. Bruch and Mare (2003) find that the shape of the utility functions of individuals influ- ence Schelling’s results. In the chequerboard model individuals have a threshold utility function. That is, they consider moving only when a certain threshold of neighbourhood ethnic mixture is exceeded. Bruch and Mare (2003) show that if individuals have a continuous utility function, that is, if they continuously con- sider moving whenever they find a neighbourhood where they can be more satis- fied, then the levels of residential segregation may decrease. They also show that utility functions of real-world individuals are continuous and argue that levels of segregation are lower than those suggested by the chequerboard model (also see Bruch and Mare 2004).

Finally, we should also note that Easterly (2004) tests the macro implications of Schelling’s segregation models and finds that data does not support macro- consequences of Schelling’s segregation models. For this reason, his study is at odds with the sprit of Schelling’s analysis and does not really test Schelling’s insights.

These studies give us enough evidence that the chequerboard model has re- ceived considerable attention and its results and implications have been explored and tested in different ways. These explorations give us reason to believe that Schelling’s insights may be relevant for the real world. On the other hand, they also imply that mildly discriminatory preferences may fail to bring about resi- dential segregation under certain circumstances. More properly, the individual tendencies to avoid minority status may not cause segregation when other tenden- cies (e.g. the preference to live in a wealthy environment) interfere.

If the argument that the chequerboard model contributes to a meta-model of residential segregation is accepted, explorations of the chequerboard model may be considered as future refinements and expansions of this meta-model. Hence, Schelling’s initial hypothesis and his chequerboard model helped researchers to expand their conceptual toolbox to include more explanatory factors. Our view of residential segregation is more refined after the chequerboard model and its explorations, because we now have a better idea of possible interactions among different explanatory factors.

Source: Aydinonat N. Emrah (2008), The Invisible Hand in Economics: How Economists Explain Unintended Social Consequences, Routledge; 1st edition.

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