# Polarization and Segregation through Conformity Pressure and Voluntary Migration: Simulation Analysis of Co-Evolutionary Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. The Base Model

**preference threshold**between $c=0$ and $c=p$.

**switching possibility frequency**. Upon receipt of a switching opportunity, the agent observes the average preference of each community at the end of the last period and his own current preference ${c}_{t}$, and calculates the payoff to be obtained from his current community and other communities. Then, he switches to the best among them. For simplicity of analysis, we prohibit agents in ethnicity ⊕ from choosing community − and agents in ethnicity ⊖ from choosing community +; community + is meant to be a hometown of ethnicity ⊕ and similarly for −, while community 0 represents a city that could integrate people of different ethnicities. Whether an agent received a switching opportunity or not, the agent’s preference ${\pi}_{t}\left(\omega \right)$ is assimilated toward the current average in the belonging community at the end of the period; if agent $\omega $ belongs to community c at the end of period t and ${\overline{\pi}}_{t}^{c}$, his preference is adjusted to

**speed of preference evolution**, representing the strength of conformity pressure. Apart from the simplifying restriction on feasible switches of community to city and their hometown, an agent’s ethnicity only determines its own initial preferences and initial affiliation to a community, ${\pi}_{0}(\xb7)$ and ${c}_{0}(\xb7)$. That is, if agent $\omega $ belongs to ethnicity $p\left(\omega \right)=\oplus $, its initial preference ${\pi}_{0}\left(\omega \right)$ is chosen from interval $[0,1]$ and the initial community is set to either ${c}_{0}\left(\omega \right)=0$ or ${c}_{0}\left(\omega \right)=+$; similarly, if agent $\omega $ is from ethnicity ⊖, the initial preference ${\pi}_{0}\left(\omega \right)$ and the initial community ${c}_{0}\left(\omega \right)$ are chosen from $[-1,0]$ and $\{-,0\}$, respectively.

**polarization**to mean a long-run outcome in which the average preference of each hometown’s residents is not neutralized, i.e., it converges to somewhere strictly positive or negative, not 0 like city residents. By looking at the long-run average preference of hometown residents, we can make quantitative statements such as whether and how much polarization is reinforced (the hometown preference goes farther from zero) or weakened (it goes closer to zero) by each factor in this model.

## 3. Results

#### 3.1. One-Shot Simulation

#### 3.2. Monte Carlo Comparative Statics

#### 3.2.1. Frequency of Switching Opportunity

#### 3.2.2. Asymmetry in Population Distributions

#### 3.2.3. Extension: Exogenous Utility of City

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1 | For example, see the discussion by moral philosophers and social advocates for global citizenship, such as Appiah [1]. |

2 | For example, the U.S. Department of Homeland Security [2] issued a concern on homegrown terrorists and states that the nation is facing “one of the most serious terror threat environments since the 9/11 attacks.” The survey by [3] compares domestic and transnational terrorism in empirical data. Hafez and Mullins [4] summarize socio-psychological causes of homegrown radicalism. The research note by international political scientists Klausen et al. [5] provides a behavioral framework to assess the risk of homegrown terrorists and analyzes 68 cases. |

3 | See Bisin and Verdier [14] for survey and summary of their papers. |

4 | McElreath et al. [16] also consider the evolution of social norms in a stag hunt game, while they introduce a kind of asymmetric assortative matching between different types of agents into the model. Our model also exhibits such an asymmetric matching structure, since city residents are matched with those of different ethnicities while hometown residents are matched only with those of the same ethnicity. |

5 | Notice that in their choice of parameters, migration even occurs from a payoff-disadvantageous community to a payoff-advantageous community. This assumption of migration may lie somewhere between ours and Kuran and Sandholm [17], as we will see. |

6 | |

7 | Precisely, this is a reduced form of the KS model. Originally, each agent $\omega $ in the KS model chooses action $a\left(\omega \right)$ from $[-1,1]$. The agents’ preference $\pi \left(\omega \right)$ represents an individually ideal action, though there is a conformity pressure to not differ from the average action $\overline{a}$ of the society. Thus, the agent’s utility from taking action $a\left(\omega \right)=a$ is formulated as $u\left(a\right)=-(1-w){(a-\pi \left(\omega \right))}^{2}-w{(a-\overline{a})}^{2}$, where w denotes the strength of conformity pressure; the optimal action is just the weighted average of the agent’s individualistic ideal and the average action: ${a}^{\ast}\left(\omega \right)=(1-w)\pi \left(\omega \right)+w\overline{a}$. Then, unconscious psychological assimilation drives the preference to fill the gap from the current action: $\dot{\pi}\left(\omega \right)=\mu ({a}^{\ast}\left(\omega \right)-\pi \left(\omega \right))$. This reduces to a continuous-time version of our reduced-form preference dynamic (3). |

8 | |

9 | As we all know, Schelling [24] is the seminal work on theoretical framework of segregation. See Clark [25], Schuman et al. [26], Bruch and Mare [27] for classical and recent empirical studies of racial segregation in U.S. From simulations and theoretical analysis of best response dynamics, Pancs and Vriend [28] strengthen Schelling’s results on the checkerboard model by allowing agents to have a discontinuous strict preference for perfect integration (exact equal proportion of each race in the agent’s neighborhood) and finding that this still leads to segregation. Dokumacı and Sandholm [29] provide a rigorous foundation of evolutionary dynamics in the Schelling’s tipping model and remove its underlying assumption of out-of-equilibrium sorting. Zhang [30] integrates the idea of tipping in the checkerboard model. |

10 | To show off its identity, there may even be hostile attitudes toward the social norm of different groups, especially if the opposite group is considered as an establishment: [31] for classical socioeconomic ethnology. |

11 | Although we add a circle to + and − to distinguish ethnic groups from their hometowns, we ignore this notational difference when relating them: for example, “hometown $c=p$ of ethnicity p” means hometown + of ethnicity ⊕ and hometown − of ethnicity ⊖. |

12 | The theoretical result of Kuran and Sandholm [17] may be parallel to the simulation result of De et al. [35]. They modified the local interaction model of a repeated Prisoner’s Dilemma as in [36] to add an observable group identity to each agent and also force agents to move their positions occasionally. From a simulation of replicator dynamics, they found that high mobility results in less discrimination against agents from a different group. |

13 | Springer [39] argues that many lone wolf terrorists once sought acceptance for themselves in the majority of society, but refusals and failures turned them to extremism. |

14 | Sageman [38] argues that online social networks contribute to the current emergence of domestic terrorism. |

**Figure 1.**Evolution of individual preferences by communities in each period: (

**a**) city residents; (

**b**) hometown residents; (

**c**) average preference of residents in each community. Each dot represents the preference of an agent at each period. Green is associated with the city, blue with town + and red with town −. In (

**c**), black lines indicate preference thresholds, as defined in Equation (2).

**Figure 2.**Evolution of individual preferences by the initial community and the ending community: (

**a**) staying in city, (

**b**) city to towns; (

**c**) towns to city; (

**d**) staying in towns. Here, blue is associated with ethnicity ⊕ and red with ethnicity ⊖.

**Figure 3.**The Monte Carlo simulation with switching opportunity frequency $\mu \in (0,1)$. (

**a**) average preferences of ethnicity ⊕; (

**b**) sizes of flows of ethnicity ⊕; (

**c**) initial average preferences of ethnicity ⊕ by the initial community and the ending community. In (

**a**), × represents the town, · overall average and ∘ the city, while blue markers show average preferences at the ending period and red markers show those at the beginning period. In (

**b**,

**c**), red markers show agents who start initially from the town and blue markers those initially from the city, while × represents ending at the town and ∘ ending at the city.

**Figure 4.**The Monte Carlo simulation with varying initial share of hometown among ethnicity ⊕. Notation follows that of Figure 3a. (

**a**) average preferences of ethnicity ⊕; (

**b**) population shares of city and town +; (

**c**) average preferences of ethnicity ⊖; (

**d**) population shares of city and town −.

**Figure 5.**The Monte Carlo simulation with varying exogenous utility of city ${v}_{0}$. Notation follows that of Figure 3a. (

**a**) average preferences of ethnicity ⊕; (

**b**) population shares of city and town +.

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Zusai, D.; Lu, F.
Polarization and Segregation through Conformity Pressure and Voluntary Migration: Simulation Analysis of Co-Evolutionary Dynamics. *Games* **2017**, *8*, 51.
https://doi.org/10.3390/g8040051

**AMA Style**

Zusai D, Lu F.
Polarization and Segregation through Conformity Pressure and Voluntary Migration: Simulation Analysis of Co-Evolutionary Dynamics. *Games*. 2017; 8(4):51.
https://doi.org/10.3390/g8040051

**Chicago/Turabian Style**

Zusai, Dai, and Futao Lu.
2017. "Polarization and Segregation through Conformity Pressure and Voluntary Migration: Simulation Analysis of Co-Evolutionary Dynamics" *Games* 8, no. 4: 51.
https://doi.org/10.3390/g8040051