1At first sight this article seems to show something quite remarkable. Despite a huge volume of literature it suggests that the famous Schelling segregation result is not surprising and is « totally explained » by the structural constraints of the model. It suggests further that the sort of preferences that Schelling proposed lead to segregation independent of the dynamic involved and that segregation should not be regarded as an emergent property of the system. The paper argues that the result of the self-organisation of the system is not « additional undesired segregation » compared to that which would satisfy the individuals involved. This, it is claimed is because, segregation is identified with order and the sort of system in question should not tend to self organise to produce more order than that generated by the intentions of the actors in the system. Thus the received view that the system self organises to generate an unintended aggregate outcome is, they claim, incorrect.
2Before proceeding, let us be absolutely clear about what Schelling claimed. He argued that even though people might have no intention to achieve segregation, their efforts to satisfy some threshold criterion for the number of unlike neighbours they would like to have, might lead to almost total segregation. This, even if the threshold was not very high. He never suggested that tolerance for racial mixing was the explanation for the existence of ghettos. There are many explanations for their existence and Schelling has always been well aware of this. All that he observed was that despite people being relatively racially tolerant their individual actions could lead to a situation in which individuals of different races were separated.
3The authors are right to point out some of the difficulties with Schelling’s model such as the fact that the threshold of 1/3 for the proportion of acceptable neighbours does not correspond to an integer number. But this, within a model, which is no more than a conceptual exercise, is not really important. This is a problem with taking the Moore neighbourhoods with 8 neighbours. However, taking other neighbourhoods and other thresholds does not change much as several authors have pointed out. The bias is there but can be minimised by more judicious choices. An alternative is to make a continuous approximation of the discrete lattice as is done in Vinkovic and Kirman (2006).
4The essential argument of the authors is that systems such as Schelling’s tend to return to their most probable states and that one has only to perform a statistical analysis of the possible configurations of the system to see what the natural result of this will be. Yet the essence of Schelling’s argument is that there is a dynamic process involving individual choices, which selects certain outcomes. Whilst the authors are right to say that what happens is the result of the intentional behaviour of the individuals they are not right to say that the global result is what they intended. The individuals have no preferences over global states. Thus the terminal states of the global system should be attributed to the dynamics of the system engendered by the purposeful movements of the individuals but should not be regarded as what those individuals intended.
5What are the terminal states of the dynamics? Schelling’s dynamic process stops when no dissatisfied individual can find a free place in which he would be satisfied or when there are no dissatisfied indivi-duals. Situations of this sort are « attractors » for his dynamics. Counting the number of possible states, which satisfy one of these criteria, does not seem to be relevant here. To say that certain types of configuration are « most probable » is misleading. This would be the case if one had a uniform distribution over the possible configurations. However, the whole point here is that we are looking at a dynamic process and analysing its limit states. The authors say that it is difficult to do this analytically and the fact that most of the research on the subject has been done by simulations bears witness to this. An exception is given by the analytical results for the deterministic process provided by Pollicot and Weiss (2005).They however, examine the limit of a Laplacian process in which individuals’ preferences are strictly increasing in the number of like neighbours. In this situation, it is intuitively clear that there is a strong tendency to segregation. Yet, Schelling’s result has become famous precisely because the preferences of individuals for segregation were not particularly strong whereas in the paper by Pollicot andWeiss there is a strong preference for neighbours of one’s own colour.
6In much of the literature attention has been focused on a 50% tolerance level. That is people are happy as long as there is not a strict majority of neighbours of the other colour. Yet here we see clearly how important the measure of segregation is. The authors have adopted an index which, they claim, has many favourable properties but these are not spelled out. To see how the measure of integration matters, consider the following example due to Stauffer and Solomon (2007).
7In this situation nobody is unhappy but the result is difficult to interpret as segregation since 8 out of 12 Bs have half A neighbours and half B neighbours. The tolerance level here is less than the 1/3 described by Forsé and Parodi but, as they point out, when all the squares on the board are not filled the 1/3 level becomes effectively close to 1/2.
8A number of authors have suggested adding a random element to the model, for instance, in the above example having those who have exactly half their neighbours of each colour, move with probability ½. Here, one has to be careful in interpreting the results, since, each simulation is just one realisation of such a process. However, if one proceeds like this, what one is interested in is the distribution over all the realisations of the final states of the process. If these are essentially concentrated on those which are segregated, then it is reasonable to describe this as an « emergent » property. Incidentally, contrary to the assertion of the authors the relationship between the degree of segregation and the aversion to people of different races is not linear. In this respect their figure is somewhat misleading.
9Here, it is worth observing that it is perfectly possible to satisfy everyone in the Schelling model even if they all want to be in a strict majority in their neighbourhood, provided that there are enough empty spaces. If there are 56 individuals, 28 black and 28 white all of whom cannot tolerate more than 3 unlike neighbours then it suffices to place the blacks on one side of the board and the whites on the other with a strip of 8 empty squares in the middle.
10This illustrates the importance of the number of empty spaces in the Schelling model. The more empty spaces there are the more the system will tend to form small isolated clusters protected by a band of empty spaces. Note here that the intentions of the individuals are the same as when there are fewer spaces but the aggregate result is different. It is in this sense that the aggregate results are an emergent property of the system and not a direct result of the intentions of the individuals.
11The authors spend some time on the analogy with physics and point out the family relation with the Ising model but observe that in that model the particles change their « spins » as a result of their interaction with their neighbours whereas in the Schelling model the individuals do not change their race. Stauffer and Solomon (2007) discuss a generalisation of the Ising model to take this into account. In that article the temperature of the system is taken to be tolerance, the more tolerant of other races the higher the temperature.
12A final remark concerning the satisfaction of the individuals is in order. Schelling has often been criticised for assuming the discontinuous level of satisfaction of the individuals in his model as Forsé and Parodi observe. Many people have suggested that there is a built in bias towards segregation since people are always happy as long as the number of their like neighbours exceeds their threshold. Thus increasing segregation neither improves nor worsens peoples’ utility.
13But suppose that people have a strict preference for mixity as suggested by Pancs and Vriend (2006). In this case the utility of individuals is maximal when they are in neighbourhoods with exactly half of their neighbours of each colour and as the inequality in the numbers of the neighbours of the two colours increases so the utility of the individual decreases. Surely, this is the most favourable situation for complete integration. However, if one simulates the process with this condition, what happens is very odd. Individuals are very rarely happy since they are unlikely to be in the ideal situation with four neighbours of each colour. As a result they are constantly moving to try to improve their utility. This means that people will always be on the move, the placement of the individuals will change but the overall pattern at any point in time will be very similar to that found in the original Schelling model. The real difference is that now the picture at the micro level is totally different with most people trying to move to increase their satisfaction. This is in contrast to the original Schelling model where people cease to move after a certain time.
14Thus, the authors are right to point out that the choice of utility function is important but not right to suggest that the relation between tolerance and segregation is simple and linear.
15Many other features of the Schelling model could be evoked here and a number of people such as Sethi and Somanathan (2004) have evoked the importance of other factors such as income level. In Kirman and Vinkovic we showed that if income is a criterion in addition to race the rich of one colour will tend to be segregated within a region of their own colour. Thus there is a sort of double segregation.
16The possibilities are almost limitless but it is worth concluding by observing that Schelling made a simple observation about the possibility of segregation even though people were not particularly racist. It was an abstract example far removed from reality. Despite uninformed criticisms suggesting that Schelling tried to show that segregation was an innocent by-product of a tolerant society, Schelling did not make any such claim. The authors are right to reject such claims but wrong to attribute them to Schelling. Furthermore, the segregation that occurs in his model, is not a simple consequence of the direct intentions of the individuals, it is an emergent property. The intentions of the individuals are the source of the force that drives the emergence of aggregate segregation but the latter is not what the individuals intended. Hence, in Schelling’s view, the discrepancy between « Micro motives and Macro behavior ».
