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经济学 1971

隔离的动态模型

托马斯·谢林

哪怕只是温和的个人偏好,经由人群相乘,也能把一座混居的城市,推向泾渭分明的隔离。

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In depth · the introduction

模型里没有一个人是偏执狂——城市却仍旧干净利落地一分为二。这道鸿沟,就是那个发现。

核心想法

设想一群两种人,散落在一张棋盘上。每个人都很随和:在混居的地方住得很舒服,只有当自己家门口附近成了小小的少数派时才会不安——比方说,紧邻的邻居里,同类不到三分之一。一旦谁感到不安,就搬到最近的、不再不安的位置。规则,全部如此。

跑起来,混居的棋盘便慢慢把自己分拣成大片的一种、与大片的另一种,中间隔着清晰的边界。最令人惊讶的,是这效应之大:最终的分离,远比任何一个人曾经想要的更为极端。

一盘用硬币玩的棋

哈佛的经济学家托马斯·谢林,在 1969 至 1971 年前后想通了这件事——而且,是出了名地用手算的。他把一分与一角的硬币摆在棋盘上,给每枚硬币一条简单规则,规定它需要多少同类邻居,再把不满意的硬币一枚枚挪到附近的空格,反反复复。他眼看着一片「胡椒盐」般的混合,坍缩成泾渭分明的领地。最初的实验里并没有计算机;这份惊奇,是在一张桌面棋盘上涌现的,此后又在每一台计算机上被重现。

它为何重要

它斩断了一个人们以为牢不可破的联系——以为一个隔离的社会,必由怀有偏见的个体构成。谢林表明,哪怕是一群宽容的人,每个人都甘愿做少数派,也能仅仅因为「谁在何时搬家」的数学,而跌入尖锐的分离。这一课是双向的:隔离可以在无人有意的情况下延续,而温和的个人愿望,能复合成严酷的群体事实。而同样的形状——微小的个人选择、出人意料的群体形态——远不止出现在居住一事上。

像一场没人在奔跑的缓慢踩踏

想象一个半满的剧场,每个人只想身边有几张熟悉的面孔。一个人挪了挪座位,去靠近几位与自己相似的人;这便空出一个座、挤了另一个座,推着下一个人也挪动,再下一个。没有谁在冲向出口,可一座接一座,观众把自己分拣成了一团团。单看任何一次挪动,都不起眼——但整个剧场,最终被安排成了一种没有人选择的样子。

一张方格棋盘,方格有靛蓝、琥珀两色,再加上少量空格,起初彻底混居。一个滑块设定每个方格要求多少同色邻居;按下运行,不满意的方格便一轮一轮地跳到最近的、能让自己满意的空格,直到整盘重组为带清晰边界的大片单色区域。读数显示满意方格的比例,以及一个最终远高于各方格所提要求的隔离指数。

它身在何处

谢林的棋盘,是如今所谓「涌现」的一个奠基范例——简单的局部规则,如何造出复杂的全局图案,这正是贯穿康威「生命游戏」与统计物理的同一主题。它帮助开启了「基于主体的建模」与复杂性经济学这门领域,也为他赢得了 2005 年诺贝尔经济学奖的一份。下一次你听到「临界点」这个词,它便可追溯到这项工作。

The original document
Original source text
Thomas C. Schelling · Journal of Mathematical Sociology, Vol. 1 (1971), pp. 143–186
The question
People get separated along many lines and in many ways.
Schelling opens by cataloguing the axes of separation — sex, age, income, language, religion, colour — and by distinguishing its causes. Some segregation is imposed by organizations; some is deliberately organized; and some, he notes, “results from the interplay of individual choices that discriminate.” It is this last, decentralized kind — no authority, no conspiracy, only many small personal moves — that the paper sets out to model.
Two models
The first is the spatial proximity model: agents of two kinds sit on a line or a checkerboard, each content only if a high-enough fraction of its near neighbours share its kind, and each discontented agent moving to the nearest spot that satisfies it. The second is the bounded neighbourhood model — the “tipping” model — in which agents share one common area and differ in how large a minority of the other kind they will tolerate; departures shift the ratio against those who stay, so a neighbourhood can flip wholesale from mixed to uniform.
The surprise
The systemic effects are overwhelming: there is no simple correspondence of individual incentive to collective results. Exaggerated separation and patterning result from the dynamics of movement.
Worked by hand with coins on a board, the mixed start collapses into broad single-colour regions even though every agent was willing to be a local minority. The aggregate segregation is far more extreme than any individual demanded — the macro-pattern is an amplification of the micro-motive, not a mirror of it.
[ … ]
Journal of Mathematical Sociology · 1971