Week 8: Non-Schelling Models

It turns out that one of the reasons why developing precise mathematics for these models was crucial is that many of the models resembling Schelling’s do not hold up under closer examination.

First, there is the issue of “reducibility,” meaning that the simulation gets “stuck.” Classical Schelling models iterate over each agent until all are satisfied and then lock themselves into that configuration indefinitely. However, which configuration the model settles on is purely a function of which agents got their turn to act first. Thus, running the same simulation twice with a random permutation would yield vastly different levels of segregation.

Second, there is “periodicity.” In some simulations, the model can find itself locked into an infinite loop. Imagine that we have two dissatisfied agents, one of type A and one of type B, sitting adjacent to each other. Then they trade spots to maximize their satisfaction. As a result, they are still adjacent to each other and therefore remain dissatisfied, swapping again the next round.

And finally, there are poorly-defined move rules. Consider the statement: “Agents choose to go to the closest vacant cell that suits them.” How do agents determine such a cell? Do agents look at each vacancy in some specified order? This vagueness leads to the implementation making certain decisions that are not specified by the model but significantly impact its outcome.

The key benefit of our approach is that it eliminates this issue. Since we use the right Markov chain with precisely defined transition probabilities and acceptance probabilities, we can be sure that regardless of the initial state or update order, we reach some well-behaved stationary regime.