This study addresses a gap in the existing literature on the Schelling segregation model by conducting a comprehensive qualitative assessment of various relocation policies. We introduce novel Schelling models driven by diferent relocation policies and analyse their impact on the convergence time and final segregation levels. Our findings demonstrate that all policies result in segregation levels within bounds established by policies where agents relocate to maximize their happiness. Notably, a policy ensuring the minimum improvement in agent segregation significantly reduces the model's convergence time. These results underscore the potential influence of relocation policies, such as those employed by online recommenders in real estate platforms, on societal segregation dynamics. The study provides valuable insights into potential strategies for mitigating and decelerating segregation through tailored recommendations.
In 1971, Thomas Schelling proposed the very first agentbased model to explain how individual actions could reIn Schelling’s simple spatial proximity model, a division between the two groups of the population emerged as a result of a homophily tendency of the agents that, he claimed, in real life can happen along many dimensions such as ethnicity, language, income, and class afiliation dimensional grid (city), with each agent having a preference for living next to people of his type. When an agent is surrounded by too many agents of a diferent kind it ifes its preferences. Schelling observed that even when agents are tolerant (low homophily threshold), the city gets segregated in a few simulation steps.
Several variants and enhancements of the Schelling model have been proposed so far. Some of them modify Published in the Proceedings of the Workshops of the EDBT/ICDT 2024 ∗Corresponding author. †Conceptualized the research, conducted the experiments, made the plots, wrote the code and the paper. ‡Conceptualized the research, supervised the experiments and wrote nEvelop-O LGOBE
0000-0001-8067-984X (G. Mauro); 0000-0002-1547-6007 (L. Pappalardo) the agents’ behaviour, for example associating to each agent an income status [5] or treating the problem with a reinforcement learning approach [6]. Other works analyse what happens to the model if the environmental conor if the dynamics take place on a network-like structure [12, 13]. In two of these works [12, 9], the agent picks the cell that maximizes its happiness. Other works included real-world segregation data along with strategies to validate simulated behaviour with observations [14, 15, 16] and sociological theories [17, 18, 19, 20]. A recent empirical study suggests a link between experienced income segregation and an individual’s tendency to explore new Gambetta et al. [22] show that imposing mobility constraints to agents in the Schelling model strongly afects convergence time and the final segregation level.
While previous research has explored various aspects of urban segregation using models like the Schelling model, there is still a gap in understanding how diferent strategies or guidelines, known as ”relocation policies,” directly influence the dynamics of urban segregation.
These policies could include government initiatives, algorithms employed by real estate platforms like Idealista, Booking, or Airbnb 1, or other mechanisms that shape the distribution of people across neighbourhoods.
These online real estate platforms are more and more actively suggesting housing options to users, playing a pivotal role in influencing urban development [ 23]. The choices individuals make, guided by these platforms or CEUR
ceur-ws.org other relocation policies, can contribute to scenarios of either increased or decreased segregation within the city, or the emergence of other phenomena like gentrification [24, 25]. Furthermore, these platforms have been proven to have a crucial impact on the urban scenario.
For example, in areas with a high AirBnB presence, rents and transactions substantially rise [26] and racial biases appear to be reinforced [27].
This work aims to fill the literature gap, underscor- Figure 1: Example of a happy agent (left) and an unhappy ing the need to systematically measure and understand agent (right) with a homophily threshold ℎ = 3. The dashed the numerical impact of diferent relocation policies on square represents the Moore neighbourhood Γ of cell . On urban dynamics. It does so by ofering relocation sug- the left, three yellow agents are in the neighbourhood of a gestions to a portion of Schelling model-like agents and yellow agent, so the agent is happy. On the right side, only scrutinizing how these recommendations afect both con- two agents share the type of the agent in cell , making the vergence time and observed levels of segregation. Our agent unhappy. ifndings reveal that policies focused on income (dis)similarity notably increase segregation times, while strategies encouraging agents to relocate where they would experi- use income data from the 2022 USA Social Security Adence minimal or maximal happiness expedite segregation ministration report,2 which delineates the US worker times. Notably, these latter policies establish both lower population percentages within specific income intervals. and upper bounds for the observed segregation levels of Every agent is assigned an income interval with a proball the analysed policies. ability proportional to the US population within , and the assigned income is picked uniformly at random 2. Policy-driven segregation model within . The majority agents are the richest 40% ones; the minority agents are the poorest 60% ones. Note that Schelling’s classical model illustrates how urban segre- the income assignment changes at each simulation, engation may emerge due to individual preferences for hancing the robustness of our results. Figure 2 shows similar neighbours. The city is represented as a grid the income distribution: as expected, a few agents have where agents of two types (initially placed randomly) a high income, while a heavy tail of agents have a low inhabit cells or leave them unoccupied (approximately income. 20% remain empty). The parameter ℎ controls agents’ homophily tendencies. At each simulation step, an agent 1.75M in position evaluates its Moore neighbourhood [28] Γ ) 1.5M – the surrounding eight adjacent cells in a square forma- SD1.25M tion. If an agent has fewer than ℎ neighbours of its type, (eU1.0M it becomes unhappy and relocates to a random, empty om0.75M cell. Figure 1 schematizes the Moore neighbourhood of a Icn0.5M happy cell (left) and unhappy cell (right). The simulation 0.25M terSmcihnealtliensgw’shaennaalyllsaisgernevtseaalrseshtraipkpiny.g outcomes: even 0.0M 0 500 Agents10(0ra0nked) 1500 2000 with a low ℎ value (e.g., ℎ = 3, indicating agents are Figure 2: Income distribution of the agents in the model. On happy with only 3/8 of their neighbours sharing their the x-axis, the agents are ranked by associated income. The type), the city segregates rapidly, maintaining an aver- y-axis represents the income. A few agents have a high income age segregation level higher than the agents’ minimum (around 1 million dollars), and the majority of the agents have requirement. a low income.
This paper aims to evaluate the impact of diverse relocation policies within the classical Schelling model in terms of convergence time and final segregation level. Simulation starts with agents randomly spread on the In our model, each simulation takes place on a 50 × 50 grid (see Figure 3, left). Each cell can either be occupied grid where 75% of its cells are randomly populated with by only one majority agent (yellow), occupied by a mi agents. The agents are categorised into two groups: nority agent (red) or be empty (white). At the end of majority agents (60%) and minority agents (40%). At the the simulation, the grid appears spatially clustered as in beginning of the simulation, each agent is associated Figure 3. Even if agents are tolerant (e.g., they are happy with a fixed income . To this purpose, as in [5], we
where ( , cell and cell ′: Γ = {(, ) ∶ | −
| ≤ 1, | − | ≤ 1}
We compute the number of agents in the Moore neighbourhood of cell that are of the same type of agent in cell as: Σ( , ) = ∑ ( , ′∈Γ ′ ) ′) denotes the equality of agents between In contrast with the original Schelling model, and fol- . For each agent, , its segregation score indicates the ( , ′) = { 1 if ( )
= ( 0 otherwise ′ ) where ( )
returns the type of the agent in cell number of agents of the same type of in its Moore neighbourhood divided by 8 (the maximum number of Moore neighbours): (3) (4) (5) (6) (7) (8) (9) Start occupancy
End occupancy
√( − )2 + ( − )2 when just 3/8 of neighbours are similar to them), the city The Moore [28] neighbourhood centered at a cell = (random policy). In our model, we introduce more sophisticated relocation policies.
When an agent leaves its cell because unhappy, our model assigns to an empty cell a score proportional to a policy , sorts the cells in decreasing order, and selects the top cells. The unhappy agent uniformly randomly picks one of these cells. We set = 30 to emulate realworld practices in online real estate platforms, typically suggesting 30 results per page. 3
We investigate six main policies: • Similar neighbourhood: the score of a cell is calculated as: () ∝ (Γ , Γ )
(10) The more the neighbourhood of a cell is similar to the neighbourhood of the original cell , in terms of average income of the agents, the higher the score of cell . • Diferent neighbourhood : the score of a cell is computed as • Distance-relevance: the score of a cell is directly proportional to the cell’s relevance and inversely proportional to the distance between the starting and arriving cell [22]: () ∝
() 2 (, ) 2
(15)
of human mobility, as postulated by the
Gravity model, wherein individuals seek to minimize
travel time while being drawn toward significant
locations [
(right) configurations resulting from the execution of the model, where all agents follow the baseline random policy.
Starting from the same initial configuration, Figure A1 reports the final configurations of simulations in which all agents follow the other six policies 2.2. Experimental settings () ∝ (Γ 1 , Γ ) (11) I[n0, 1o0u0r],exapperairma menettse,r wreepvreasreynttihneg atdhoeptpioerncernattaege ∈of agents following the suggested policy. At the beginning The score of the cell is inversely proportional of the simulation, each agent has a probability to follow to the economic similarity between the starting the policy during all steps of the simulation, and thus and ending neighbourhoods. a probability 1 − to follow the random policy. Each • Minimum improvement: the agents may move agent will be categorised as policy-follower or not at only to cells it would be happy. Among these cells, the beginning of the simulation based on probability . the score of each cell is inversely proportional The baseline of our experiments is the classical Schelling to the number of agents of the same class of the model, where all agents follow the random policy (this, agent in the starting cell : = 0 ).
() ∝ 1 , if Σ(, ) ≥ ℎ (12) of Wthee pmeorfdoerlm.E1a0c0h sciomnufigluartaiotinosnfocormeabcinhecsotnhfigeuvraatluioens Σ(, ) of two parameters: the policy and the adoption rate • Maximum improvement: the agents may move . Each simulation uses a diferent random spatial disonly to cells it would be happy. Among these cells, tribution of agents on the grid and a diferent random the score of each cell is directly proportional income assignment taken from the income distribution. to the number of agents of the same class of the Each simulation terminates when all agents are happy or agent in the starting cell after a maximum of 300 simulation steps. For each simulation, we calculate the convergence time, , the number () ∝ Σ(, ) , if Σ(, ) ≥ ℎ (13) of steps needed to reach an equilibrium state, and the • Recently emptied: we assign a higher score to final segregation level, ⟨⟩ , at the end of the simulation. the empty cells that have been emptied for the lower amount of time in the last steps:
() ∝ 1 () (14)
a reasonable choice for an RS, is to suggest users occupy locations that were already in conditions of being inhabited and that were recently free.
The analysis of convergence time as varies uncovers intriguing patterns (see Figure 5).
The baseline random model typically converges in around 27 steps. In accordance with the suggestion of Gambetta et al. [22] the more the users follow a distance relevance policy, the more the segregation process is slowed down (hence, the higher ). The two policies
a diferent economic composition significantly amplifies segregation, especially when influencing the majority of the population.
Conversely, four policies lead to a reduction in ⟨⟩ : recently emptied, distance-relevance, similar neighbourhood and especially minimum improvement. The recently emptied policy shows a negligible reduction until only 60% of the population adopts it but becomes increasingly efective with an increased adoption. The distance-relevance policy substantially decreases ⟨⟩ . However, the policy that most efectively reduces final observed segregation levels is minimum improvement: even with a small percentage of agents following this policy, the average ⟨⟩ reduction is substantial. 4. Discussion rooted in the neighbourhood income similarity, similar neighbourhood and diferent neighbourhood substantially increase convergence time. In particular, the model is not able to reach a stable equilibrium if 10% (or more) agents relocate to a similar neighbourhood. A similar result holds for the recently emptied policy.
Remarkably, the only two policies that efectively expedite segregation, reducing the value of as their adoption rate increase, are the policies that suggest agents to relocate in places where they would be happy: minimum improvement and maximum improvement.
Our study explores the intricate relationship between relocation policies and the dynamics of urban segregation.
Through a series of simulations, we unveil the impact of these policies on both convergence time and the final level of segregation.
The implications of policies grounded in neighbourhood composition, such as the similar neighbourhood and diferent neighbourhood , reveal intriguing trends. On the one hand, as one can expct, suggesting agents to relocate to a neighbourhood with a similar income, thereby 0.74 maintaining a comparable average income distribution 0.72 among neighbours, increases convergence times. In fact, 00..7608 if agents adhere strictly to this policy, the model fails to 0.66 converge. On the other hand, it is noteworthy that even S0.64 suggesting agents to relocate to a socioeconomically dif000...566802 rdmmeisiacnt.x...erimmieml.ppprtroieovdv.. rsdaiimfnf.d.nonemeiigghh.. fdeerceenltenraetigiohnboisurmhooostdpsrloonwosudnocwedn wsehgernegbaettiowneetinm4e0s%.Tahnids 0.56 60% of users relocate according to this policy. Interest0 10 20 30 40 50 60 70 80 90 100 ingly, having 100% of agents follow this policy produces p (%) a similar efect, in terms of convergence time, as only Figure 6: Average final segregation levels ⟨⟩ across 100 sim- 10% of agents following it. This dichotomy can also be ulations of models with a growing percentage of users ac- appreciated in the segregation levels ⟨⟩ analysis. cepting the suggestion of the RS. Counterintuitively, a policy that suggests agents relocate to a socioeconomically diferent neighbourhood , thus suggesting a mixing, increases the average observed segregation levels as its adoption increases. Surprisingly, suggesting agents relocate to neighbourhoods with a similar average income distribution reduces the final observed segregation levels.
The analysis of the observed final segregation level seems bounded by the outcomes of two extreme policies: the maximum improvement policy drives the final segregation level to its maximum, and the minimum improvement policy minimizes it. This distinction becomes particularly pronounced when the relocation policies are adopted by many agents (high adoption rate ). Indeed, minimum improvement for = 100% reduces the
Even more intriguing insights emerge from analysing the final segregation level varying the adoption rate (Figure 6). The final segregation level ⟨⟩ for the baseline random model stabilises around 0.66. Notably, for all policies, the more users adhere to a policy, the greater the change in ⟨⟩ , indicating that suggesting relocation policies other than the random one significantly impacts urban segregation.
Only two policies, diferent neighbourhood and maximum improvement amplify final segregation levels as adoption rate increases. In particular, the former leads to the most substantial rise, while even the policy suggesting an unhappy user move to a neighbourhood with distance relevance, 133 steps
minimum improvement, 6 steps recently emptied, 300 steps similar neighborhood, 300 steps ifnal segregation level by 16.67% compared to the baseline model. Conversely, maximum neighbourhood significantly increases the final segregation level by 13.64%.
From a sociological perspective, this observation emphasizes how policy choices significantly mould societal structures. The extremes represented by the accentuated segregation of the similar neighbourhood or maximum improvement policies and the minimized segregation of the minimum improvement policy delineate the wide spectrum of potential societal outcomes based on policy implementations. Recognizing these boundaries provides crucial insights into the intricate connection between policy decisions and the resultant societal dynamics. It clarifies how diferent policies can influence segregation levels, thereby guiding more informed and balanced interventions.
different neighborhood, 17 steps maximum improvement, 4 steps This paper investigates the efects of diferent relocation policies within the Schelling model on convergence time and final segregation levels. It sheds light on how these policies influence urban segregation dynamics, paving the way for future research and the development of more equitable urban strategies, particularly in understanding the impact of online real estate platforms on neighbourhood demographics.
This work can be improved and extended in several Figure A1: Examples of final configurations produced by each
directions. Inspired by Moro et al. [
Authors thank Dino Pedreschi for its precious intuitions as well as Emanuele Ferragina, Giuliano Cornacchia and Daniele Gambetta for their valuable suggestions.
In Figure A1, we present examples of final configurations produced by the execution of the model in which the 100% of agents follow one of the six policies. All the simulations starts from the same initial configuration reported in Figure 3 (left).
It is noticeable that the diferent neighbourhood and maximum improvement policies, depicted in the last row, result in a visually less mixed scenario compared to the
The code for replicating and reproducing our model and
experiments is available at
https://github.com/mauruscz/RS-chelling. The
simulation has been performed using the Python module MESA
[