A Review on Schelling Model
POON Ming Kei 26 Mar 2020 Instructor name: Shingyu Leung Abstract
In this paper, we would investigate how different parameters of the Schelling’s model on social segregation affects the end result such as the overall satisfaction level and the number of iterations taken to reach convergence.
1. Introduction
In any countries, we can easily discover a place where a culturally similar group is living together. The phenomenon of such separation among homogenous groups of population is called social segregation. Shelling (1969) studied this phenomenon and suggested that such a result can be modelled by some simple rules over many iterations. He discussed the movements of two types of agents on a one- dimensional array (p.491). In this paper, we would extend such study into a two-dimensional matrix and observe the result of the movement of the agents by varying different values of the parameters.
2. Numerical methods
A programme using MATLAB is used to stimulate the result of the movement of agents. Here we would confine the size of the matrix to 50-by-50 and the number of agents to two. The neighbourhood of a cell is defined as the collection of the cells one unit vertically, horizontally or diagonally from it. An agent is said to be unsatisfied (satisfied) if the proportion of the same type of agents in its neighbourhood is less than (more than or equal to) the set tolerance level.
For each set of parameters, the agents will be randomly assigned with a position on a 50-by-50 grid. For each iteration, the unsatisfied agents will be removed temporarily and then randomly assigned into an empty cell. The process will continue until all agents are satisfied or the number of iterations reaches the pre-determined maximum number of iterations. The matrix is said to be reaches convergence if all the agents in it are being satisfied.
3. Results
(i) Variation on the Tolerance Level
Here’s the results when the tolerance level is set to be 0.5 and 0.8 respectively. We can see that the convergence of the case with tolerance level 0.5 occurs with only 32 iterations. However, for the case with tolerance level 0.8, convergence cannot be reached. Furthermore, for tolerance level 0.8, the number
of unsatisfied agents oscillates without convergence. The convergence of matrices is seen to be dependent on the values of the tolerance level.
(a) (b) (c)
(d)(i) (ii)
Figure 1: Location of agents (a) at the beginning (b) at the final iteration (32nd) with tolerance level at 0.5, and at the final iteration (100th) with tolerance level 0.8. (d) Plots of the number of unsatisfied agents at the nth iterations with tolerance (i) 0.5 and (ii) 0.8.
Figure 2: Average total number of iterations required for (i) all agents from both group are satisfied (blue plot), (ii) all agents from one of the group are satisfied (green plot), and (iii) average number of remaining unsatisfied agents (red plot) against the level of tolerance across 10 test runs.
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From Figure 2, a pattern could be seen interacting with tolerance and the degree of convergence. We suggested that Since the neighborhood of most of the agents consists of 8 grids, the pattern of convergence behavior changes at tolerance level equal to the integral factors of 1/8, as summarized in Table 1.
TABLE 1
Tolerance Level Against its General Properties
Tolerance
General Properties
Tolerance
General Properties
0 – 0.125
Convergence occurs in around 4 iterations.
0.5 – 0.625
No convergence can be reached. Number of remaining unsatisfied agents is around 120 (5% of the total number of agents).
0.125 – 0.25
Convergence occurs in around 10 iterations.
0.625 – 0.75
No convergence can be reached. Number of remaining unsatisfied agents is around 770 (32% of the total number of agents).
0.25 – 0.375
Convergence occurs in around 18 iterations.
0.75 – 0.875
No convergence can be reached. Number of remaining unsatisfied agents is around 2300 (95% of the total number of agents).
0.375 – 0.5
Number of iterations required to reach convergence increase to 40 – 60 with a great variance. Also, one of the groups is more likely to be all satisfied 25 iterations before both groups are satisfied.
0.875 – 1
No convergence can be reached. Nearly all the agents are unsatisfied.
Such pattern is persistant for different initial proportions of agents and board sizes. This is also verified by the emergence of the more complex patterns when we enlarge the neighbourhood to a 25-by-25 grid around the concerned agents.
(ii) Variation on the initial proportion of agents
By setting the tolerance to 0.5, we varied the initial proportion of the agents to observe the criteria for convergence. Here, we denoted the initial proportion of agent A and agent B by PA and PB respectively, and assume PA ≥ PB . Hence, in social science context, agent B could represent a social minority group.
It can be seen from Figure 3(a) that convergence is reached only with PA being 0.4 – 0.5 and PB being 0.35 – 0.5. However, observing from (b), convergence is likely to be reached by the majority agent A for PA > 0.5. For extreme majority of agent A, i.e. 0.8 < PA < 0.9, from (c), nearly all of the agents B are unsatisfied at the end.
This may have real-life implications as immigrants or ethnic groups may be in extreme minority in some countries. Thus, over generations, according to the Schelling’s model, the minority might be scattered among the territory.
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(a) (b)
(c) (d)
Figure 3: (a) Total number of iterations, (b) total number of iterations for any one group of agents being satisfied, (c) difference between final number of agent A and that of agent B, (d) total number of final unsatisfied agents, against different PA and PB.
5. Conclusions
Schelling’s model, as a simplistic formulation to a social situation, is seen to be able to generate complex result by adjusting various parameters. The tolerance level and the initial proportion of the agents not only affects the average time needed for convergence, it also affects the variability of the time and the degree of final satisfaction of the agents. Further studies can be investigated into applying such model into real- life situations with more humanistic factors.
Acknowledgments
The author would like to thank Professor Shingyu Leung for his dedicate instruction on how to stimulate the problem into computer implementation and how to organize findings into an academic report.
References
Schelling, T. (1969). Models of Segregation. The American Economic Review, 59(2), 488-493. Retrieved March 29, 2020, from www.jstor.org/stable/1823701
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