Exploring Simulated Residential Spending Dynamics in Relation to Income Equality with the Entropy Trace of the Schelling Model

Abstract

The Schelling model of segregation has provided researchers with a simple model to explore residential dynamics and their implications upon the spatial distribution of resident identities. Due to the simplicity of the model, many modifications and extensions have been produced to capture different aspects of the decision process taken when residents change locations. Research has also involved examining different metrics for track segregation along the trace of the simulation states. Recent work has investigated monitoring the simulation by estimating the entropy of the states along the simulation, which offers a macroscopic perspective. Drawing inspiration from empirical studies which indicate that financial status can affect segregation, a dual dynamic for movements based on identity and financial capital has been produced so that the expenditure of a monetary value occurs during residential movements. Previous work has only considered a single approach for this dynamic and the results for different approaches are explored. The results show that the definition of the expenditure dynamic has a large effect on the entropy traces and financial homogeneity. The design choice provides insight for how the housing market can drive inequality or equality.

1. Introduction

This paper looks at how an extended Schelling model of residential segregation [1,2] which incorporates a monetary variable [3] can be used to explore different expenditure dynamics of residential mobility, which can bring about greater social equality. The Schelling model (to be described later on) provides a flexible framework that can be extended with more dynamics to capture an array of societal features such as the financial status of residents [4]. This extention will be used to study housing spending patters which drive social equality [5], and provide insight into how this can be adjusted in practice to sense the state of the financial homogeneity of an urban environment. The unique approach offered here is that the model relies upon an entropic definition of the simulation traces so that a balance can be sought between the residential entropy changes and that of the financial aspect of residents to attempt to find an plausible equal exchange.

There are various reasons for why this study focuses on the residential expenditure dynamics for equality. The housing market is a big driver of the economy in many nations which impacts other markets directly and indirectly [6]. This does not appear to be a recent phenomenon as even from early housing ownership records from the 17th century in London, housing quality has been shown to be linked to the standard of living [7]. Although there are many interesting aspects of how the housing market and trends affect the economy and the flow of capital it is also important to look at the greater social aspect which is the value of equality. How equality arises is often misunderstood as being a simple objective, but studies show how it is intertwined with housing that affects equality in a type of mutual dependency [8]. There is research into taxation measures which explores the ideas of re-distribution focusing on the largest holders of capital [9].

The methodological approach proposed here aims to help researchers explore how different plausible expenditure mechanisms in the real-estate market could potentially assist in addressing societal equality. The Schelling model, although rudimentary in the simplicity of the dynamics, provides a model which to build a more complex model from and this extended model by incorporating a financial expenditure dynamic upon agent movements offers a novel manner in which income inequality can be explored and understood. Different functional forms for the pricing system are proposed with results visualized for the examiner to assess quantitatively and qualitatively how they may shape society on a major financial variable for individuals over the course of time. Given the work on income inequality measures in [10], a study of regions with examples in [11], and the empirical evidence in [12] there is need for a researcher to have a manner to visualize possible impacts of policy on house purchase costs which this work strives to offer.

Shelling Model

The Schelling model proposes an ideal grid (lattice) where the cells represent blocks for possible residency by a set of agents that can occupy a cell. Each cell can be occupied by a single agent or be left as unoccupied (empty). The cells are designated a label which identifies them as belonging to one group or another and is commonly used to designate race, although another identity differentiation can be used such as wealth/class [13] or even an identity label derived from soccer team support [14]. Over a set of discrete time points, the agents are allowed to change positions where they can leave a current cell in order to occupy another cell that is empty (swap), or they are allowed to remain in their current position. The criteria as to whether these cells remain in the same position, and if not to which cell they move to, depends upon their local homogeneity. If, from the immediate neighbors of an agent, a certain number do not share the same identity label (treating cells occupied by different identities and empty cells the same), the agent re-locates to another position from the set of empty cells which provide a sufficient number of homogeneously labeled agents adjacent to it. If an agent lacks the sufficient homogeneity in its surroundings and an empty cell which can provide a sufficient number of homogeneous adjacent agents does not exist, then the agent remains in place until such a position arises or not. The initial state has a uniform distribution of the agents of the different identities and are allowed to then make moves at each time step, and it is possible for this simulation of residential movements to enter a state where no agents move and some are left without homogeneity satisfaction.

There are N cells in the grid and a cell can produce a membership in 3 groups $m_n\in \{m_{group1},m_{group2},m_{empty}\}$. With the use of linear indices, we can map the cell number to row and column coordinates as $m_n⟺m_{(ni,nj)}$. The homogeneity of a cell position can be calculated via:

$$l\left(m_n\right)=\sum _{i=-1}^1\sum _{j=-1}^1\left({\delta }_{m_{(ni,nj)},m_{(ni+i,nj+j)}}:i,j\ne 0\right).$$

(1)

where the $\delta$ is the Kronecker delta. The local homogeneity threshold value h is used to evaluate the criteria for the agent homogeneity satisfaction via:   

$$r\left(m_n\right)=\left\{\begin{array}{cc}(l\left(m_n\right)\ge h)\hfill & \mathrm{if}{m}_n\notin \left\{m_{empty}\right\}\hfill \\ 0\hfill & \mathrm{if}{m}_n\in \left\{m_{empty}\right\}\hfill \end{array}\right..$$

(2)

The cells not on an edge require 4 similar cells adjacent to it ($h=4$), cells on an edge but not a corner need 3 ($h=3$), and the cells in a corner require 2 equally labelled cells ($h=2$) or more to be satisfied on this constraint. In [15], more detail is provided for the reader to understand the equation based perspective of the Schelling model. Since the satisfaction can be calculated for each agent occupied cell, these can be placed into a vector, ${\mathbf{r}}_t=[r(m_1,t),\dots ,r(m_N,t)]$, and from this an overall satisfaction of the agents which will ‘remain’ in the same position in the next iteration t can be found via:

$$R_t=\sum _{n=1}^{N}r\left(m_{n,t}\right).$$

(3)

This value over the simulation is used to determine the amount of homogeneity that has been achieved for the groups at each time point in the simulation.

The work of [4,16,17] provides an important insight into the practical applications of the Schelling model and insight into the effect of financial dynamics in residential movements in an urban environment. The studies look at residential regions in cities of Israel and examines the segregation patterns in relation to the wealth status. The findings are used to augment the canonical Schelling model to allow for the financial variable to be incorporated into the movement decision making of the agents in the model. In [3], the Schelling model incorporates a monetary variable for each agent along with its identity variable. The agents are then allocated an initial amount which is then ‘spent’ as a utility cost upon each movement. This ’dynamic’ of an expenditure acts independently of the residential dynamics and does not influence the original Schelling model definitions. The purpose of its introduction is to study plausible parallel effects to the effects of the residential segregation. Algorithm 1 shows the algorithm used as a financial distribution dynamics in [3]. Each agent is allocated an ‘income’ i, which it then spreads upon cell movement for other agents to accumulate (those adjacent to the destination cell of the moving agent). It forces an agent to distribute 5% of its income (or wealth) across its new neighbors, and this value is distributed according to a uniform sampling of the amount. This dynamic was introduced having drawn inspiration from the anthropological studies of [18], which discusses how residential migrations distribute money into the communities they arrive in due to various costs and even through the increasing of property values.

Algorithm 1: Income dynamic

incomeDynamic$(grid,n,i,j,income$) 1: $spread=0.05$ 2: $neighbors\leftarrow getAdjacentAgents\left(grid\right[i,j\left]\right)$ 3: $distribute\leftarrow (spread\times income)\times \parallel neighbors\parallel$ 4: $grid{[i,j]}_{income}=grid{[i,j]}_{income}-distribute$ 5: $du\leftarrow unif(\parallel neighbors\parallel )$ 6: $du\leftarrow du/sum\left(du\right)$ 7: $tmp\leftarrow 1$ 8: $\mathbf{for\; all} (\forall (i,j)\in neighbors) \mathbf{do}$ 9: $grid{[i,j]}_{income}\leftarrow du\left[tmp\right]+grid{[i,j]}_{income}$ 10: $tmp\leftarrow 1+tmp$ 11: end for 12: return$grid$

Instead of directly using the numerical value of wealth, an ‘income bracket’ [19] is used. This allows for a more objective similarity distance to be produced from the income differences and is produced by the approach described in the Data section. As one agent can have an identity homogeneity with another agent, a satisfaction based upon the financial similarity (homogeneity with another agent) is defined as:

$$\begin{array}{c}\hfill l_b\left(m_{n_b}\right)=\sum _{i=-1}^1\sum _{j=-1}^1\left(\right(|m_{(ni,nj,b)}-m_{(ni+i,nj+j,b)}|<h_{b_1}):i,j\ne 0).\end{array}$$

(4)

The term $m_{n_b}$ is the income bracket of an agent, and $h_{b_1}$ the income bracket max distance threshold for homogeneity. This can be used to assess whether an agent is financially homogeneous in a cell position or not via:

$$b_h\left(m_{n_b}\right)=\left\{\begin{array}{cc}\left(l_b\left(m_{n_b}\right)\ge h_{b_2}\right)\hfill & \mathrm{if}{m}_n\notin \left\{m_{empty}\right\}\hfill \\ 0\hfill & \mathrm{if}{m}_n\in \left\{m_{empty}\right\}\hfill \end{array}\right..$$

(5)

This produces an income satisfaction across every agent in the grid ${\mathbf{b}}_t=[b_h\left(m_{1_b,t}\right),\dots ,b_h\left(m_{N_b,t}\right)]$ and an overall income homogeneity score for the grid state:

$$I=\sum _{n=1}^N{b}_h\left(m_{n_b}\right).$$

(6)

Figure 1 shows the income homogeneity across an example grid. In this example trace the income homogeneity $I=17$ at initialization.

Financial segregation [20,21] in relation to the Schelling model has received attention mainly from research in [3,4,17] but not from the perspective of exploring dynamics which can produce more equality. As will be shown in the results section, there are alternatives to Algorithm 1 for a spending dynamic which can produce a faster rate of income homogeneity with movements on the grid by the agents. The Methodology will present these different approaches for modeling the spending and the Results will show how they compare.

The work of [3,15] examines the Schelling model in a non-conventional manner by taking an assessment of the entropy trace of the simulation. The sampling of the entropy states allows the examiner to produce a macroscopic metric upon a social system to monitor the stability by providing a well supported metric for observing the ‘out of the ordinary’ in respect to some variables such as the identity label distribution among the agents. This can be considered as another possible tool available for sensing the state of a society.

2. Data

Ref. [22] provides data for the distribution of incomes from the tax payers of the USA. This data is used to produce a CDF (cumulative distribution function from empirical data) which can then be sampled and allocated to the agents on the grid as their monetary variable values. This is shown in Figure 2 where it can be intuitively seen how there is a strong skew to the right. The distribution mirrors the studies that show how in general this phenomenon exists ubiquitously and the work of [23] provides an explanation of how this arises.

3. Methodology

In this section, the methodology behind sampling the entropy of the dual dynamic Schelling model (identity and monetary) is presented as well as the proposed dynamics for agent spending. The purpose of exploring different spending dynamics is to assess how they can affect the overall income homogeneity I of the agents.

The same de-labelling factor $\frac{N!}{({\prod }_g^{group}{N}_g!)}$ is used in conjunction with the n-tuples for the income components to find the total permutation space. The sample probability for the grids is $p(R,I)=\frac{\parallel \mathbf{r},\mathbf{b}\in \mathbf{R},\mathbf{I}\parallel }{K}$ for K samples used,

$$\widetilde{\Omega (R,I)}=p(R,I)\times \left(\frac{N!}{({\prod }_g^{group}{N}_g!)}\times {\left(N^{\frac{\sum {\mathbf{m}}_{{\mathbf{n}}_{\mathbf{i}}}}{1000}}\right)}^2\right).$$

(7)

Here, R represents the residential macrostate value that is produced from the aggregate of how many agents in the grid have achieved homogeneity through their local neighbor comparisons based upon their Schelling identity (same group). I represents the homogeneity the income homogeneity (proposed in [3]) which agents achieve homogeneity via similar income bracket values that is aggregated across the agents to produce this macrostate value. $\widetilde{\Omega (R,I)}$ is the density of the joint macrostate space, $p(R,I)$ the sampled probability for that point in R and I, $m_{n_i}$ are the agent group membership in the grid using linear indices, N is the number of grid cells and $N_g$ the number of agents in a group. The density at the macrostate of the grid can then be used to find the entropy for each macrostate value the microstate configuration is in:

$$S_{R,I}=k_B\ln \widetilde{\Omega (R,I)},$$

(8)

which permits the estimation of the entropy for a trace of a simulation; $S_{R,I,t}=k_B\ln {\widetilde{\Omega \left((R,I)\right)}}_t$. The density for the macrostates is sampled via Monte Carlo. Since this exists on a 2D space, a contour map of the densities for $\Omega (R,I)$ can be produced.

An important part of this research is to showcase that different correlations between the income of the agents and the spending that they do has significant effects on the homogeneities of the system and especially the income homogeneity. To differentiate between the various behaviors of the agents, one has to mathematically describe their spending habits in terms of their income. Proposed here are functional forms based on income brackets to influence house purchases. This is integrated into the previous model where the new effects can be explored. Firstly, one logical thought is that people of lesser incomes would have to spend a greater percentage of their wealth in order to move, and that is what people with greater incomes do. How intense this decline of the percentage of expenditure is, is something that can be thought of as arbitrary. For this reason two different functions are provided, one with a steeper and one with a smoother decline. So the percentage of the income spent per agent is calculated for these two instances as:

$$P_{steep-decline}\left(x\right)={\displaystyle \frac{1}{x+log\left(x\right)}}$$

(9)

$$P_{smooth-decline}\left(x\right)={\displaystyle \frac{1}{1+log\left(x\right)}}$$

(10)

with x being the income of the agent.

This approach is logical, but if one wants to understand the behavior of the system then other distributions for the percentage of the income per agent spent according to their income need to evaluated and studied. Firstly, one can think of a societal behavior where the richer individuals spread their wealth more, meaning for this model that they spend a greater percentage of their income while moving. For this, another distribution function is formed, as seen below:

$$P_{increase}\left(x\right)={\displaystyle \frac{x^{\frac{1}{4}}}{180}}$$

(11)

The form of this function, as with the previous ones, has no physical meaning, and it is just a mathematical representations of a function describing the aimed behaviors.

Lastly, one other possibility that one ought to look into is how the residential and income homogeneities behave overall and towards one another when the spending is focused around a certain income bracket of the agents of the system. This case will provide a better insight into what one might expect if not all the agents spend their money but only some agents do so, and how this impacts the system.

$$P_{centered-high-class}\left(x\right)={\displaystyle \frac{{(x+100)}^{3.5}{e}^{-{(x+100)}^{\frac{1}{5.5}}}}{2.32·{10}^{16}}}$$

(12)

The Equation (12) is written in a mathematical formed loosely based on the Maxwell-Boltzmann distribution function where it was developed according to the percentage of wealth spent to be high around a specific area of the upper class, as this is expected to be the people that have a higher effect when they move. The logic behind these last 2 expenditure functional forms is to put larger income expenditures on higher income brackets. Their values grow as they reach the upper end of the income brackets.

Figure 3 demonstrates the different functions for the distribution percentages dependent upon the income of the resident. The main motivation to deviate from the previous approach is that the percentage was a fixed number across the income brackets. There can be situations where residents of certain income brackets pay higher or lower percentages of their income when moving. This would encapsulate a disproportional challenge for certain groups. It can be seen how the orange Equation (9) shows how the lowest income brackets would pay the largest percentages by far. In blue, the line follows Equation (10), where the line has a less steep decrease so that the purchasing burden extends to higher income brackets more uniformly. In green, the simulations explore a situation where the highest earners pay the largest percentages for residential movements from Equation (11). Finally, in red, a parabolic equation allows the largest financial burden to focus upon a specific range of income earners, as defined by Equation (12).

4. Results

Through the described mechanics it can be seen that the outcomes of the system are heavily dependent on the distribution function of the spending that the agents do according to their income. Below are presented the different figures for the residential and income homogeneities and their correlation for each of the different wealth spending distribution function. In the figures to follow, each will have two plots so that: Sub-figure (a) presents the behavior and evolution of the residential homogeneity of the system (R) on the left and the income homogeneity (I) on the right. Sub-figure (b) presents in the relation of the two types of homogeneity in the blue line initiating from the point and the phase space of the initial points in the ellipses presented at the top left.

Figure 4 is based on the evolution of the system given the Equation (9) for the percentage of income spending of the agents. Through Figure 4a, it is seen that the system reaches residential homogeneity in a very small number of iterations, and the income homogeneity is seen to be increasing converging to a final higher value. In Figure 4b, it is seen that the increasingly lower dependence of the wealth expenditure due to Equation (9) of the agents when moving according to their wealth leads to a low dependence of the income homogeneity to the residential homogeneity until the very last iterations when the maximum residential homogeneity is observed, where the income homogeneity is seen to be increasing but not to very high values. It will be shown that, when compared to the other distributions, income homogeneity is increasing in the system but it remains relatively low and reaches lesser overall values. The gradient of the income homogeneity as well as the final values it reaches against the residential homogeneity as seen in Figure 4b are key aspects because they showcase the final position of the system, how quickly it is reached, and how homogeneous the overall position is. The final values of the income homogeneity will be seen to showcase the difference between the various income spending distributions. In the same graph, the co-centric ellipses presented actually represent the space of the possible starting points of the system. Due to the extremely high number of possible arrangement of the agents, it was not possible to sample and present data for lower income homogeneities given the starting residential homogeneity. However, one can imagine that this phase space of initial points is the same as the one presented for lower I values just by moving the same figure further down in the graph.

In Figure 5, the behavior of the system given Figure 5 for the percentage of the income spending of the agents is shown. In Figure 4a, it can be observed that the system again reaches residential homogeneity quickly and that the income homogeneity also increases to a steady maximum value. Through Figure 5b, it can be observed that the income homogeneity again increases with the residential homogeneity, but this time it is seen that there exists a much higher correlation between the residential and the income homogeneities than that which was observed in the previous example in Figure 4b. Moreover, the income homogeneity reached has much greater values. This spending dynamic is based upon Equation (10).

Equation (11) is used for the following results. Again, similar behaviors as before can be observed in Figure 6a. In Figure 6b, an even higher dependence of the income homogeneity to the residential than before is shown.

Lastly, Equation (12) for the spending focused on one income is presented. Here it can be seen that the residential homogeneity is reached quickly but the income homogeneity takes a considerably higher number of iterations until it converges to its final maximum value. In Figure 7, it is seen that the income homogeneity initially barely increases in terms of the residential homogeneity and then very abruptly reaches its highest values.

Overall it can be seen that the equations that offer a higher percentage of spending to the higher incomes lead to systems that are overall more homogeneous in their incomes, meaning that the agents of higher incomes are actually holders of greater entropy, and getting these agents to further spend their wealth greatly increases the final income homogeneity of the system.

5. Concluding Remarks

This work investigates the effect of spending dynamics for residential movements alongside a canonical Schelling model. The dual dynamics of the Schelling models allows for the investigation of income inequalities when different percentages are incurred for moving agents from one cell in the grid to another. Previous work investigated this approach using a uniform percentage deduction dynamic and this work looks at how different expenditure models affect the income homogeneity of the agent collection. The question driving this study is whether different expenditure models can direct the model towards increased or decreased equality. Not all of the proposed functional forms may be feasible in practice, but they do offer insight into manifestation on the macroscopic effects from measures applied to the individual based upon income brackets.

The results show that certain dynamics, which require different income earners to pay different percentages, can change the amount of income homogeneity to a greater or lesser extent. Income homogeneity remains at the lowest numbers when the lower end of income earners pay the highest percentages and the highest income homogeneity is produced with increasing and parabolic functional forms. This is shown on a set of simulation traces and within a contour plot of the dual dynamic entropy space for the identity and income homogeneity values. Such studies can provide insight into policies aimed at improving the issues of equality and how these can be measured in society. Further work could entail the inclusion of constraints and thresholds for the purchasing costs that can be represented as barriers for agent movements.