Xeno’s paradox

There is probably some very deep psychology behind the age-old tradition of blaming problems on foreigners. These days I’m a foreigner, in Switzerland, and so I get to see how things are and how I affect them. I’ve found that I can trigger a change in norms even by going out of my way to have no effect on them. It’s a puzzle, but I think I’ve got it modeled.

In my apartment there is a norm (with a reminder sign) around locking the door to the basement. It’s a strange custom, because the whole building is safe and secure, but the Swiss are particular and I don’t question it. Though the rule was occasionally broken in the past (hence the sign), residents in my apartment used to be better about locking the door to the basement. The norm is decaying. Over the same time period, the number of foreigners (like me) has increased. From the naïve perspective, the mechanism is obvious: Outsiders are breaking the rules. The mechanism I have in mind shows some of the subtlety that is possible when people influence each other under uncertainty. I’m more interested in the possibility that this can exist than in showing it does. Generally, I don’t think of logic as the most appropriate tool for fighting bigotry.

When I moved in to this apartment I observed that the basement door was occasionally unlocked, despite the sign. I like to align with how people are instead of how the signs say they should be, and so I chose to just remain a neutral observer for as long as possible while I learned the how things run. I adopted a heuristic of leaving things how I found them. If the door was locked, I locked it behind me on my way out, and if the door wasn’t I left it that way.

That’s well and good, but you can’t just be an observer. Even my policy of neutrality has side effects. Say that the apartment was once full of Swiss people, including one resident who occasionally left the door unlocked but was otherwise perfectly Swiss. The rest of the residents are evenly split between orthodox door lockers and others who could go either way and so go with the flow. Under this arrangement, the door stays locked most of the time, and the people on the cusp of culture change stay consistent with what they are seeing.

Now, let’s introduce immigration and slowly add foreigners, but a particular kind that never does anything. These entrants want only to stay neutral and they always leave the door how they found it. If the norm of the apartment was already a bit fragile, then a small change in the demographic can tip the system in favor of regular norm violations.

If the probability of adopting the new norm depends on the frequency of seeing it adopted, then a spike in norm adoptions can cause a cascade that makes a new norm out of violating the old one. This is all standard threshold model: Granovetter, Schelling, Axelrod. Outsiders change the model by creating a third type that makes it look like there are more early adopters than there really are.

Technically, outsiders translate the threshold curve up and don’t otherwise change its shape. In equations, (1) is a cumulative function representing the threshold model. It sums over some positive function f() as far as percentile X to return value Y in “X% of people (adopters early adopters (E) plus non-adopters (N)) need to see that at least Y% of others have adopted before they do.” Equation (2) shifts equation (1) up by the percentage of outsiders times their probability of encountering an adopter rather than a non-adopter.

If you take each variable and replace it with a big number you should start to see that the system needs either a lot of adopters or a lot of outsiders for these hypothetical neutral outsiders to be able to shift the contour very far up. That says to me that I’m probably wrong, since I’m probably the only one following my rule. My benign policy probably isn’t the explanation for the trend of failures to lock the basement door.

This exercise was valuable mostly for introducing a theoretical mechanism that shows how it could be possible for outsiders to not be responsible for a social change, even if it seems like it came with them. Change can come with disinterested outsiders if the system is already leaning toward a change, because outsiders can be mistaken for true adopters and magnify the visibility of a minority of adopters.

Update a few months later

I found another application. I’ve always wondered how it is that extreme views — like extreme political views — take up so much space in our heads even though the people who actually believe those things are so rare. I’d guess that we have a bias towards over estimating how many people are active in loud minorities, anything from the Tea Party to goth teenagers. With a small tweak, this model can explain how being memorable can make your social group seem to have more converts than it has, and thereby encourage more converts. Just filter people’s estimates of different group’s representations through a memory of every person that has been seen in the past few months, with a bias toward remembering memorable things. I’ve always thought that extreme groups are small because they are extreme, but this raises the possibility that it’s the other way around, that when you’re small, being extreme is a pretty smart growth strategy.

One Response to “Xeno’s paradox”

[…] subject has some relationship to some extensions to Schelling’s opinion models and to my dissertation work (on surprising group-scale effects of “what you […]

enfascination » Blog » PR as common knowledge arbitrage added these pithy words on Nov 22 14 at 16:54