Schelling's Dynamic Models of Segregation: A cellular automata ...

Schelling’s Dynamic Models of Segregation: A cellular automata approach Juan Miguel Benito∗and Pen´elope Hern´andez† June 10, 2004

Abstract Schelling presented one-dimensional landscapes populated with agents of two distinct ‘types’ in which micro-level agent preferences involve macro-level effects. Schelling’s model exhibits the sectoral dynamics of individual preferences. A crucial feature in Schelling’s model is the dynamic given the asynchronous movements of each agent. Given an order, each player chooses to move or to stay depending on his neighborhood configuration. Each agent looks for a position where he becomes happy because in it his neighborhood configuration is admissible. A happy society verifies that all agents are happy therefore we obtain a stable society. We study a variant of the one-dimensional Schelling’s model, capturing the myopic behavior of players in a geometric environment. We propose a cellular automata approach acquiring the dynamics microlevel preferences. A cellular automata consists of a lattice of cells, or sites over a finite alphabet, and a rule. At each time, the type of agent at site i is updated according to the fixed rule or (transition function) that depends on both the present value and the corresponding of its immediate neighbors. Under our approach, we characterize the set of stable society and the set of environment in which converges to a stable situation. Keywords: C63, C79, D89, R23.

∗ Departamento Econom´ıa. Universidad P´ ublica [email protected] † University of Alicante. e-mail: [email protected]

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1

Introduction

The analysis of neighborhoods effects is neither new in economics, or in Game Theory that often have been called local interactions1 . Individuals take choices, and sometimes their alternatives are influenced by another agents that are in their environments. The interest of neighborhood effects was first recognized by Thomas C. Schelling who stressed the relationship between individual decisions and aggregate outcomes. Schelling presented a model in which an agent is sensitive to the colors of its neighbors. Depending on his neighborhood each agent is defines as contented or discontented. Discontented agents are moved to places on which they are content until everybody is satisfied, or else no places remain where discontented agents would be content2 . This approach captured the idea that short-range, (or, like Schelling [14] called they later, micromotives), interactions produce large-scale structures, (macrobehaviors). Agents care only about the type of their immediate neighbors, and the result is a line divided into large segregated neighborhoods. The reason, is based on the effect of movement to its immediate neighbors, and the consequence to other neighbors, and so on. Hence, local interactions generating global structure emerge in a natural way. We study a variant of the linear model of Schelling, capturing the myopic behavior of players in a geometric environment. We propose a cellular automata approach acquiring the dynamics micro-level preferences. A cellular automata consists of a lattice of cells, or sites over a finite alphabet, and a rule. At each time, the type of agent at site i is updated according to the fixed rule or (transition function) that depends on both the present value and the corresponding of its immediate neighbors. Under our approach, we characterize the set of stable society and the set of environment in which converges to a stable situation. Section 2 gives a concise introduction to Schelling’s linear model of segregation. Notation, definitions, and ideas of relevance to the subsequent discussion are introduced in section 3. Section 4 gives some cases of Schelling model, that will be use to implement our approach in section 5. Section 5, also, focuses on a particular family of automata, the called Cellular Automata. In section 6, we present the topic on complexity for this problem. Likewise, section 6 provides a concise introduction to information theory, which will be useful to the boundary mark that we show in this section. In section 7 we conclude. 1

See Durlauf [6] to an extensive review of neighborhood effects. This model has been treated as example of spontaneous order. For an extensive explanation see Binmore [3]. 2

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2

The Schelling Linear Model

Schelling, ([10], [11], [12], [13], [14]), presented both one and two-dimensional landscapes populated with agents of two distinct ‘colors’ and studied how micro-level agent preferences for like-colored neighbors manifested themselves al the macro-level. We study the one-dimensional segregation model. In this model, people is distributed uniformly along a line. Everybody defines his neighborhood by reference to his own location. An individual moves if he is not content with the color mixture of his neighborhood, moving to the nearest place where the color mixture does meet his demands. For simplicity, Schelling supposed that everyone of a given color has the same preferences regarding the color mixture of his own neighbors. In the next subsections, we show the ingredients which guide Schelling’s linear model.

2.1

Schelling’s segregation linear model

As we have said before, Schelling presented two different models of segregation. In this subsection we show the ingredients of Schelling’s segregation linear model3 . Suppose that we have a population distributed randomly along a segment. This population is composed of two well differentiated types of individuals; we can think of them, like Schelling do it, as black and white. Each agent of the population cares about the types of his neighbors, which it is defined as the occupants of the four adjacent neighbors4 . Each agent requires that some fraction of these neighbors be the same type as himself, then this agent is happy, otherwise he is unhappy. If this fraction is not satisfied, the agent moves to the nearest site that meets his fraction. As Schelling, (1971, p. 150), said: Now we need a rule about they move. Let me specify that a dissatisfied member moves to the nearest point that meets his minimum demand -the nearest point at which half his neighbors will be like himself at the time he arrives there. ‘Nearest’ means the point reached by passing the smallest number of neighbors on the way; and he merely intrudes himself between two others when he gets there. In this model, unhappy agents move to another site sequentially starting from left to right of the segment. Note that agents near to the start and the end of the segment have a smaller neighborhood than the rest of them. It is obvious that if each agent demands a majority of his neighborhood be the same type as himself, then this fraction will lead to segregation. 3

There are some extensive explanations of Schelling’s two-dimensional model of segregation such as Krugman [9], Epstein & Axtell [5], or Binmore [3]. 4 Note that the neighborhood could be composed of two, six, or another even number of neighbors.

3

What Schelling presented was that preferences which are compatible with integrated structures, normally lead to segregated environments. There are two important features in this model. First is that local interactions produce global structures. Agents care about the type of their adjacent neighbors, and the outcome is a segment divided into segregated neighborhoods. This is because in this model exist chain reactions: the movement of an agent influence his neighbors, but it may induce them to move, which then affects their neighbors, and so on. The second important feature is that while segregated neighborhoods always emerge, the details of the final composition of them depends on the initial conditions of the distribution of population along the segment. This kind of process are also called path-dependent

3

The model

Let S t be the society at stage t generated by the position of n agents, and let A = {0, 1} be their types. We denote S t = (sti )i={1,...,n},t∈N as a vector which represent the mixture of agents at stage t. For instance, for t = 5 and n = 7, we have S 5 = (s51 , s52 , s53 , s54 , s55 , s56 , s57 ) = (0, 0, 1, 0, 0, 1, 0). A neighborhood V (i, d) of an agent i is a vector in M(2d+1)×1 centered at the position i and radio d. We define it by: V (i, d) = (si−d , . . . , si , . . . , si+d ), where d means the number of neighbors for each side at si . Let V (i, d)∗ be the corresponding neighborhood without si defined by: V (i, d)∗ = (si−d , . . . , sbi , . . . , si+d ).

For instance, considering the last example, for t = 5, n = 7, we have = (s51 , s52 , s53 , s54 , s55 , s56 , s57 ) = (0, 0, 1, 0, 0, 0, 1), and for the third element, i = 3, and d = 1 we have V (3, 1) = (s52 , s53 , s54 ) = (0, 1, 0), and V (3, 1)∗ = (s52 , s54 ) = (0, 0). In this case, the neighborhood is composed of the first adjacent agent in each side of the agent of reference, s53 . Every agent of the society, S t , has a tolerance about the mixture of his neighborhood. Everyone cares about the types of his neighbors. Let d, m ∈ N. Denote by P = {(d, m)} the set of tolerances. The agent i prefers (d, m) to (d, m0 ), and we write it (d, m)  (d, m0 ), if: S 5 (0, 1)

(d, m)  (d, m0 ) ⇔ m ≥ m0 where m and m0 are the number of neighbors of the same type. Utility of agent i of the society S t has a preference about the mixture of his neighborhood. Everyone cares about mixture among whom he lives, he 4

cares about the type of others close by. The composition of his neighborhood determines if he is happy or not. Schelling’s model has a binary utility function for every different element of S t . Each agent has a preference about his neighborhood, and this neighborhood’s structure determines if they are happy or non-happy. Schelling relates individual’s tolerance relation with an utility function. An utility function Ui assigns a real value to each element in S t , ranking the elements of S t in accordance with the individual’s tolerance. Utility for the agent i ∈ {1, . . . , n} and tolerance in m is:  1 if |{sj ∈ V (i, d)∗ |sj = si }| ≥ m d t Ui (S ) = 0 if |{sj ∈ V (i, d)∗ |sj = si }| < m

3.1

Movement

The movement is one of the most important point in Schelling’s linear model, since given an original society S t and tolerances (d, m), the movement determines the final society’s landscape. Therefore, different ways of movement yield different final outcomes S t+1 . Consider S t+1 be the society at stage t + 1 in which agent i moves to position j of S t . Suppose w.l.o.g that j > i, therefore: S t+1 = {1, . . . , i − 1, i + 1, . . . , j − 1, j, i, j + 1, . . . , n} Define the set Jt of the position of happy agents in a society S t : Jt = {j ∈ {1, . . . , n}|Uid (S t ) = 1} The movement is an application which transforms the initial society vector, S t to S t+1 , with the same elements but changing the position of some agents. To determine who changes and where, Schelling use his neighborhood demand. Then, a non-happy agent looks for the nearest position, j, in which he is happy. Suppose that there exists such a j and it is unique. He supposes that a non-happy agent moves to “nearest” point j, that satisfies his neighborhood demand or tolerance level. By definition j is the nearest position at S t such that the agent i is happy. t be a n × n matrix, defined by: Let Ci,j t Ci,j = {Fn×n ∈ Mn×n |S t × Fn×n = S t+1 |Ujd (S t+1 ) = 1} t the movement matrix of agent in position i to position j. We call Ci,j Design of movement Suppose that i ∈ {1, . . . , n} is not happy in S t . We proceed to detail the movement of i to the nearest position in which he becomes happy. Let j be the lowest positive integer i ≤ j ≤ n such that

|{sj ∈ V (j, d)∗ |sj = si }| ≥ m. 5

For any non-happy element sti , the movement is an application which goes from S t to S t+1 , moving the unhappy individual to a new position with a new neighborhood that satisfies his neighborhood’s demand. We can understand the movement as a n × n matrix composed of ones and zeros. Now, we explain how to build up the movement matrix for an element, sti . The movement matrix is a permutation matrix 5 , which is obtained from the identity matrix, I, by exchanging columns. For the case that j > i, [i > j], the i-th column is exchanged for j-th column, and all the columns between the i-th and the j-th, included j-th, are displaced one preceding t by: [following] position. For j > i, we define Ci,j



i Ct = j

                     

i

j



1 0 ... 0 0 1 ... 0 .. . . . . . .. .. . 0 ... ... 1 0 ... 0 1 1 0 ... 0 .. . . . . . .. .. . 0 ... 1 0 1 0 .. .

0 1 .. .

... 0 ... 0 . . .. . . 0 ... ... 1

                     

t , is repreThe characteristic polynomial of any movement matrix, Ci,j n−j+i−1 j−i+1 sented by (1 − x) (−x) .

4

Some cases of Schelling model

In following subsections, we prove that the model of Schelling arrives to an equilibrium. Definition 1 We say that a society, S t is an equilibrium, if for all sti ∈ S t , Uid (S t ) = 1. Let B t (d, m) = |{si ∈ S t |Uid (S t ) = 1}| be the number of happy elements of S t given d, and m. Next subsections prove some particular cases of the linear model of Schelling. 5

Permutation matrix is a square matrix which has in each row and in each column exactly one entry 1 whereas all other entries are zero

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4.1

Case m = d = 1

Next proposition asserts that the number of happy agents is an increasing function on t for the particular case when d = m = 1. t such Proposition 2 Let i a non-happy agent in S t . Then, there exist a Ci,j t+1 t that B (1, 1) > B (1, 1).

Proof. We suppose that i is the agent in S t with the smallest position such that Ui1 (S t ) = 0. Then, sti−1 = sti+1 6= sti . Let j the nearest position where the agent i is happy. Hence after the movement of i to j, agents i − 1 and i + 1 are together and therefore, they satisfy the happiness condition if one of them was not happy before the movement. t the matrix associated to the movement of the agent i Consider now Ci,j t the society at stage t + 1. Then st 6= st and to the position j. Fix S t × Ci,j i j will be happy even if he was stj+1 = sti 6 . Hence, i becomes happy and st+1 j+1 t+1 t+1 t+1 7 t not before the movement . This is so because si = sj = sj+1 = sj . Then B t+1 (1, 1) = |{si ∈ S t+1 |Ui1 (S t+1 ) = 1}| t ) − U 1 (S t )+ ≥ B t (1, 1) + Ui1 (S t × Ci,j i+1 1 t t t ) +Ui+1 (S × Ci,j ) − Uj1 (S t ) + Uj1 (S t × Ci,j 1 (S t )+ = B t (1, 1) + Uj1 (S t+1 ) − Ui+1 1 t+1 1 t 1 (S t+1 ) +Ui (S ) − Uj (S ) + Uj+1 ≥ B t (1, 1) + 1

4.2

Case d = 2 and m = 1

Next proposition affirms that the number of happy agents is an increasing function on t for the particular case when d = 2, and m = 1. t such Proposition 3 Let i a non-happy agent in S t . Then, there exist a Ci,j that B t+1 (2, 1) > B t (2, 1).

Proof. We suppose that i is the agent in S t with the smallest position such that Ui1 (S t ) = 0. Then, sti−2 = sti−1 = sti+1 = sti+2 6= sti . Let j the nearest position where the agent i is happy. Hence after the movement of i to j, agents i − 1 and i + 1 are together. t the matrix associated to the movement of the agent i Consider now Ci,j t the society at stage t + 1. Then we have two to the position j. Fix S t × Ci,j possible cases: 6 7

We are supposing that j > i, for i > j then stj−1 = sti For i > j it would be st+1 j−1

7

• j > i stj−1 = stj = stj+1 6= sti and stj+2 = sti . Hence, i becomes happy and st+1 j+2 will be happy even if he was not before the movement. This is so because sti = st+1 = st+1 j j+2 . Then B t+1 (2, 1) = ≥ = ≥

|{si ∈ S t+1 |Ui2 (S t+1 ) = 1}| t ) − U 2 (S t ) + U 2 (S t × C t ) B t (2, 1) + Ui2 (S t × Ci,j j+2 j+2 i,j t 2 t+1 2 (S t ) + U 2 (S t+1 ) B (2, 1) + Uj (S ) − Uj+2 j+2 B t (2, 1) + 1

• i > j stj−1 = stj = stj+1 6= sti and stj−2 = sti . Hence, i becomes happy and st+1 j−2 will be happy even if he was not before the movement. This is so because sti = st+1 = st+1 j j−2 . Then B t+1 (2, 1) = ≥ = ≥

5

|{si ∈ S t+1 |Ui2 (S t+1 ) = 1}| t ) − U 2 (S t ) + U 2 (S t × C t ) B t (2, 1) + Ui2 (S t × Ci,j j−2 j−2 i,j 2 (S t ) + U 2 (S t+1 ) B t (2, 1) + Uj2 (S t+1 ) − Uj−2 j−2 B t (2, 1) + 1

Cellular Automata Approach

We study a variant of the one-dimensional model of Schelling, capturing the myopic behavior of players in a geometric environment. We propose a cellular automata approach. Cellular automata are simple mathematical model which captures natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set. The values of the sites evolve in discrete time steps according to deterministic rules that specify the value of each site in terms of the values of neighboring sites.

5.1

Elements of Cellular Automaton

A Cellular Automata, (henceforth CA), starts with a unidimensional linear array of cells of length n over a finite alphabet K, K = {0, 1}, with a neighborhood ratio r, r ∈ {1, . . . , n}. In each cell we have a type of agent 0 or 1. We call K the alphabet of our CA. Then, S t = Kn . We define the global transformation function, φ¯ by φ¯ : Kn −→ Kn where,

φ¯ = (φr (i))i∈{1,...,n} 8

5.1.1

Rules

In the CA model, it is assumed that each individual interacts with its 2r nearest neighbors on a lineal array. This determines which individuals will be allowed to move, and which location will be its new place. The CA rule is the function φr (i) from a neighborhood configuration to a new cell-value. The CA’s equation of motion is given by applying the rule to each point, i, separately: φr : K2r+1 −→ K where the value of cell in position i of S t is given by: st+1 = φr (sti−r , sti−(r−1) , . . . , sti−1 , sti , sti+1 , . . . , sti+(r−1) , sti+r ) i The numbers of different rules for a CA, depend on k and r, where k is 2r+1 the cardinality of K. For a given CA, the possible rules are equal to k k . 2r+1 Let rule number be each of the k k rules of the CA, and we denote each rule number by a decimal number8 . Suppose that k = 2, r = 1, and φ1 = 94, the value of cells obtained of the eight possible neighborhoods are combined to form a binary number that is quoted as a decimal integer. 94 in binary numbers is represented by 01011110, and the rule is given by (Table 1). Neighborhood Rule

111 0

110 1

101 0

100 1

011 1

010 1

001 1

000 0

Table 1: Example of rule With a convenient assignation of rule, there exists a CA which implements the movement of the linear model of Schelling.

6

Complexity of CA

The sets of CA configurations may be viewed as formal languages, consisting of sequences of symbols (cell values) forming words according to definite grammatical rules. Then, cellular automaton configurations may be viewed as formal languages, consisting of sequences of symbols, (site values), forming words according to definite grammatical rules. In particular, the language generated is isomorphic to the same generated by Turing machines9 . CA are capable to emulate a Turing machine which is able to carry out universal computation, that it was proved by Berlekamp et al. [2], Lindgren et al. [8], and Smith [15]. Hence, any algorithm can be implemented by CA, namely the Schelling movement. 8

Following the notation of Wolfram (1983). Turing machine is a theoretical computing machine to serve as an idealized model for mathematical calculation. A Turing machine is able to implement any algorithm. 9

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The complexity of a CA, and therefore, the complexity of his behavior, depends on the number of rules that guide it. The number of rules depend on the neighborhood ratio, r, and the cardinality of the alphabet, k. Then, 2r+1 as it has been mentioned, the possible rules for a CA are equal to k k . 2r+1 The number of different rules with given k, and r grows as k k , and therefore becomes immense even for rather small k and r. Wolfram [17] proposed a classification to identify the behavior of any CA as a member of one of the following categories: • Class 1 ⇒ Evolution leads to a homogeneous state in which, for example, all sites have value 0; • Class 2 ⇒ Evolution leads to a set of stable or periodic structures that are separated and simple; • Class 3 ⇒ Evolution leads to a chaotic pattern. • Class 4 ⇒ Evolution leads to complex structures, sometimes long-lived. In an experimental way, Wolfram [18] gives the fractions for various sets of CA in each of the four classes. Class 1 2 3 4

k = 2, d = 1 0.50 0.25 0.25 0

k = 2, d = 2 0.25 0.16 0.53 0.06

k = 2, d = 3 0.09 0.11 0.73 0.06

k = 3, r = 1 0.12 0.19 0.60 0.07

Table 2: Fractions of Wolfram classification. In Table 2 we can see that, in particular most CA for the family of d > 2 tend to be in the third class, in which evolution leads to a chaotic pattern, then they tend to be chaotic. To reduce complexity we propose a CA approach of a version of Schelling model.

6.1

Some elements of information theory

In this section we present briefly, some of the elements of information theory. Let us to start with the concept of entropy, which is a measure of a random variable. Let X be a discrete random variable with alphabet θ and probability mass function p(x) = P r{X = x}, x ∈ θ. The entropy H(X) of a discrete random variable X is defined by X H(X) = − p(x)log[p(x)] (1) x∈θ

where the log is to the base 2. By convention 0log[0] = 0, which is easily justified by continuity since xlog[x] → 0 as x → 0. 10

Example 4 Let X= Then



1 with probability 0 with probability

1 5 4 5

1 1 4 4 H(X) = − log − log = 0.721928 5 5 5 5

Let X1 , X2 , . . . , Xn be a sequence of n symbols from an alphabet θ = {a1 , a2 , . . . , a|θ| }. We will use the notation xn and x interchangeably to denote a sequence x1 , x2 , . . . , xn . The type Px (or empirical probability distribution) of a sequence x1 , x2 , . . . , xn is the relative proportion of occurrences of each symbol of θ. Let Pn denote the set of types with denominator n. If P ∈ Pn , then the set of sequences of length n and type P is called the type class of P , denoted Tn (P ). For example, if θ = {0, 1}, then the set of possible types with denominator n is       1 n−1 n 0 0 n , ,..., , , , Pn = n n n n n n Example 5 Let θ = {0, 1}, an alphabet. Let x = 11001. Then the type Px is 3 2 , Px (1) = · Px (0) = 5 5 The type class of Px is the set of all sequences of length 5 with two zeros, and three ones. T5 (Px ) = {00111, 01011, . . . , 11100}

·

And the cardinality of T5 (P ) is   5 | T5 (P ) |= = 10 2

6.2

·

Cellular Automata: One-Dimensional Model of Schelling Revisited

We focus on a variant of Schelling’s model. Every agent is a member of one of two well differentiated groups, and has a fixed tolerance for the composition of his neighborhood. We suppose as Schelling approach, that everyone of a given type has the same tolerances regarding the type mixture of his own neighbors. Also, for simplicity, we suppose that both types of agents have the same tolerance. Therefore, each agent computes the fraction of the adjacent neighbors, 2d, who are its own type, if this number is greater than, or equal to, its tolerance level, m, the agent is considered happy, otherwise he is nonhappy. In the model of Schelling since it is a asynchronous model a happy agent remains in the same site meanwhile a non-happy agent looks for the nearest site that satisfies its tolerance threshold and moves there. 11

We represent a society and a movement rule by a CA. By definition of a rule of a CA each agent is able to see the neighbors up to a given distance, r. Therefore, the movement for each agent is defined in the following way: • In our model, an agent who decides to move, always the agent moves one site towards the right. • For a given cell, i, movement is showed by change the type of element in i, and the type of some adjacent neighbor. Then, the rule of our CA, φr (i), over a given site i, for d = m = 1 is defined by: 1. Any element sti = a, a ∈ K, which is happy, does not change its type from a to b, b ∈ K. Except when sti−1 is non-happy and sti−2 is happy, because sti−1 will move before sti , as we explain after. 2. Any non-happy element sti = a, changes its type from a to b, except when sti−2 is non-happy, in this case, it remains without change. Because it is waiting for the movement of sti−2 , that affects it directly. Having in mind this two steps, we have to know if sti−2 is happy or not, for this reason, the length of our automaton neighborhood configuration has to be as far as sti−3 , then r = 3. Now, we need an order of moving. The elements which have decided to move, they do in turn, counting from left to right. Whereas for Schelling is the same order but with non-happy elements. We need an order to take the choice of to move or not, and this order is sequential starting from left to right of the society. A neighborhood in our CA model, V (i, d), consists of a cell and the r cells to its left and to its right. However, some cells at the left of the society, S t don’t have r neighbors to them left, and some cells at the right of the society lacks r neighbors on them right. This problem is solved by pretending that the two ends of row are joined. The cell at the left end is considered to neighbor on the cell on the right end. In effect, the world is really a circle of cells rather than a line. This assumption is common in the literature of CA10 . In the model of Schelling, all agents are able to observe the number of two types of agents that occupy the society. This is because each agent needs to know where are the places at which neighbors demand is met, to select the nearest one. However, this assumption is relaxed, so information available fro each agents is only the neighbors up to a given distance r. This suppose that the information that each agent has in our model is a lesser quantity than in the model of Schelling. 10

Albin [1] shows CA models with others assumptions

12

For a given society, there are many rules that obey the two steps defined before. But the set of rules that we are looking for, has to obey that any society, S t in a typical set of length n, Tn (P ), verifies that r(S t ) = S t+1 ∈ Tn (P ), i.e.: the proportion of 0 and 1 remains constant. Having in mind last norm of our rule, given the type PS t ,that we call it P by simplicity, of a society S t of dimension n. The number of rules that obey this particular feature is much lesser than the number of all rules that they are for a CA composition. It leads us to Lemma 6. Lemma 6 If n > 1. Then Y

n)

| Tn (P ) ||Tn (P )| < (2n )(2

(2)

P ∈Pn

Proof. Induction proof over n. • n=2

⇒ First case, it is verified. 11 22 11