Thomas Schelling, methodological subversive

Rajiv Sethi

22 January 2017

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At a time when economic theory was becoming virtually synonymous with applied mathematics, Thomas Schelling managed to generate deep insights into a broad range of phenomena using only close observation, precise reasoning, and simple models that were easily described but had complex and surprising properties.

This much, I think, is widely appreciated. But what also characterised his work was a lack of concern with professional methodological norms. This allowed him to generate new knowledge with great freedom, and to make innovations in method that may end up being even more significant than his specific insights into economic and social life. 

Consider, for instance, his famous ‘checkerboard’ model of self-forming neighbourhoods, first introduced in a memorandum (Schelling 1969), with versions published in a 1971 paper (Schelling 1971) and in his 1978 book, Micromotives and Macrobehavior (Schelling 1978).
 
The model is simple enough to be described verbally in a couple of paragraphs, but has properties that are extremely difficult to deduce analytically. It is also among the very earliest agent-based computational models, it reveals some limitations of the equilibrium approach in economic theory, and it continues to guide empirical research on residential segregation (Card et al. 2008, Bayer et al. 2014).

Here's the model. There is a set of individuals partitioned into two groups, let's call them pennies and dimes. Each individual occupies a square on a checkerboard, and has preferences over the group composition of its neighbourhood. The neighbourhood here is composed of the (at most) eight adjacent squares. Each person is content to be in a minority in their neighbourhood, as long as minority status is not too extreme. Specifically, each wants strictly more than one-third of their neighbours to belong to their own group. 

Initially suppose that there are 60 individuals, arrayed in a perfectly integrated pattern on the board, with the four corners unoccupied. Then each individual in a central location has exactly half their neighbours belonging to their own group, and is therefore satisfied. Those on the edges are in a slightly different situation, but even here each individual has a neighbourhood in which at least two-fifths of residents are of their own type. So they too are satisfied.

Now suppose that we remove 20 individuals at random, and replace five of these, placing them in unoccupied locations, also at random. This perturbation will leave some individuals dissatisfied. Now choose any one of these unhappy folks, and move them to a location at which they would be content. Notice that this affects two types of other individuals: those who were previously neighbours of the party that moved, and those who now become neighbours. Some will be unaffected by the move, others may become happy as a result, and still others may become unhappy. 

As long as there are any unhappy people on the board, repeat the process just described: pick one at random, and move them to a spot where they are content. What does the board look like when nobody wants to move?

Schelling found that no matter how often this experiment was repeated, the result was a highly segregated residential pattern. Even though perfect integration is clearly a potential terminal state of the dynamic process just described, it appeared to be unreachable once the system had been perturbed. The assumed preferences are tolerant enough to be consistent with integration, but decentralised, uncoordinated choices by individuals appear to make integration fragile, and segregation extremely stable. Here's how Schelling (1978) summarised the insight:

“People who have to choose between polarized extremes... will often choose in a way that reinforces the polarization. Doing so is no evidence that they prefer segregation, only that, if segregation exists and they have to choose between exclusive association, people elect like rather than unlike environments.”

One can tune the parameters of the model – the population size and density, or the preferences over neighbourhood composition – and see that this key insight is robust. And for reasons discussed by Kirman and Sethi (2016), equilibrium reasoning alone cannot be used to uncover it. 

A very different kind of contribution, but also one with important methodological implications, may be found in Schelling's 1960 classic, The Strategy of Conflict. Here he considers the adaptive value of pretending to be irrational, in order to make threats or promises credible (emphasis added):

“How can one commit himself in advance to an act that he would in fact prefer not to carry out in the event, in order that his commitment may deter the other party? One can of course bluff, to persuade the other falsely that the costs or damages to the threatener would be minor or negative. More interesting, the one making the threat may pretend that he himself erroneously believes his own costs to be small, and therefore would mistakenly go ahead and fulfill the threat. Or perhaps he can pretend a revenge motivation so strong as to overcome the prospect of self-damage; but this option is probably most readily available to the truly revengeful.’”

Similarly, in bargaining situations, ‘the sophisticated negotiator may find it difficult to seem as obstinate as a truly obstinate man.’ And when faced with a threat, it may be profitable to be known to possess ‘genuine ignorance, obstinacy or simple disbelief, since it may be more convincing to the prospective threatener.’

Starting with three classic papers (Kreps et al. 1982, Kreps and Wilson 1982, Milgrom and Roberts 1982) in the same 1982 issue of the Journal of Economic Theory, a large literature in economics has dealt with the implications for rational behaviour of interacting with parties who, with small likelihood, may not be rational. While this work has focused on characterising rational responses to irrationality, Schelling's point speaks also to payoffs, and raises the possibility that departures from rationality may have adaptive value

The methodological implications of this are profound, because the idea calls into question the normal justification for assuming that economic agents are in fact fully rational. Jack Hirshleifer explored the implications of this in a wonderful paper on the adaptive value of emotions (Hirshleifer 1984), and Robert Frank wrote an entire book about the topic (Frank 1988). But the idea is right there, hidden in plain sight, in Schelling's parenthetical comments.

Finally, consider Schelling's burglar paradox, also described in The Strategy of Conflict:

“If I go downstairs to investigate a noise at night, with a gun in my hand, and find myself face to face with a burglar who has a gun in his hand, there is a danger of an outcome that neither of us desires. Even if he prefers to just leave quietly, and I wish him to, there is danger that he may think I want to shoot, and shoot first. Worse, there is danger that he may think that I think he wants to shoot. Or he may think that I think he thinks I want to shoot. And so on. ‘Self-Defense’ is ambiguous, when one is only trying to preclude being shot in self-defense.”

Baliga and Sjöström (2004) have shown exactly how such reciprocal fear can lead to a fatal unravelling, and explored the enormous consequences of allowing for pre-play communication in the form of cheap talk. I have discussed the importance of this reasoning in accounting for variations in homicide rates across time and space, as well as the effects of Stand-your-Ground laws.

There are a handful of social scientists whose impact on my own work is so profound that I can't imagine what I'd be writing if I hadn't come across their work. Among them are Glenn Loury, Elinor Ostrom, and Thomas Schelling. I can think of at least of my five papers – on segregation (Sethi and Somanathan 2004), on variations in homicide across regions and communities (O’Flaherty and Sethi 2010a, 2010b), on reputation in bargaining (Abreu and Sethi 2003), and on social norms (Sethi 1996) – that flow directly from Schelling's thought.

It may surprise some to know that Glenn Loury's Du Bois lectures (Loury 2009) are dedicated to Schelling, but it makes perfect sense to me. Here's how Glenn explains his choice in the preface:

“Shortly after arriving at Harvard in 1982 as a newly appointed Professor of Economics and of Afro-American Studies, I begin to despair of the possibility that I could successfully integrate my love of economic science with my passion for thinking broadly and writing usefully about the issue of race in contemporary America. How, I wondered, could one do rigorous theoretical work in economics while remaining relevant to an issue that seems so fraught with political, cultural and psychological dimensions? Tom Schelling not only convinced me that this was possible; he took me by the hand and showed the way. The intellectual style reflected in this book developed under his tutelage. My first insights into the problem of ‘racial classification’ emerged in lecture halls at Harvard's Kennedy School of Government, where, for several years in the 1980s, Tom and I co-taught a course we called ‘Public Policies in Divided Societies’. Tom Schelling's creative and playful mind, his incredible breadth of interests, and his unparalleled mastery of strategic analysis opened up a new world of intellectual possibilities for me. I will always be grateful to him.”

As, indeed, will I.

References

Abreu, D, and R Sethi (2003), ‘Evolutionary Stability in a Reputational Model of Bargaining’, Games and Economic  Behavior 44(2): 195-216.

Baliga, S, and T Sjöström (2004), ‘Arms Races and Negotiations’, Review of Economic Studies 71(2): 351-69.

Bayer, P, H Fang and R McMillan (2014), ‘Separate When Equal? Racial Inequality and Residential Segregation’, Journal of Urban Economics 82: 32-48.

Card, D, A Mas and J Rothstein (2008), ‘Tipping and the Dynamics of Segregation’, Quarterly Journal of Economics 123 (1): 177-218.

Frank, R (1988), Passions Within Reason: The Strategic Role of the Emotions, Norton

Hirshleifer, J (1984), ‘On the Emotions as Guarantors of Threats and Promises’, UCLA Economics Working Papers 337.

Kirman, A, and R Sethi (2016), ‘Disequilibrium Adjustment and Economic Outcomes’, in Complexity and Evolution: Toward a New Synthesis for Economics edited by David S. Wilson and Alan Kirman, Strüngmann Forum Reports.

Kreps, D, P Milgrom, J Roberts and R Wilson (1982), ‘Rational Cooperation in the Finitely Repeated Prisoners' Dilemma’, Journal of Economic Theory 27(2): 245-52.

Kreps, D, and R Wilson (1982), ‘Reputation and Imperfect Information’, Journal of Economic Theory 27(2): 253-79.

Loury, G (2002), The Anatomy of Racial Inequality, Harvard University Press.

Milgrom, P, and J Roberts (1982), ‘Predation, Reputation, and Entry Deterrence’, Journal of Economic Theory 27(2): 280-312.

O'Flaherty, B, and R Sethi (2010a), ‘Peaceable Kingdoms and War Zones: Preemption, Ballistics and Murder in Newark’, in The Economics of Crime: Lessons for and from Latin America, NBER.

O'Flaherty, B, and R Sethi (2010b),’Homicide in Black and White’, Journal of Urban Economics 68(3): 215-30.

Schelling, T (1960), The Strategy Of Conflict, Harvard University Press.

Schelling, T (1969), ‘Models of Segregation’, Rand Corporation Memorandum RM-6014-RC.

Schelling, T (1971), ‘Dynamic Models of Segregation’, Journal of Mathematical Sociology 1(2): 143-86.

Schelling, T (1978), Micromotives and Macrobehavior, Norton.
 
Sethi, R (1996) ‘Evolutionary Stability and Social Norms’, Journal of Economic  Behavior & Organization 29(1): 113-40.

Sethi, R, and R Somanathan (2004), ’Inequality and Segregation’, Journal of Political Economy 112(6): 1296-1321.

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Topics:  Frontiers of economic research

Tags:  game theory, residential segregation

Rajiv Sethi

Professor of Economics, Barnard College, Columbia University, and External Professor, Santa Fe Institute.

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