schelling points

[hitchhiker]

You have to meet a stranger in New York, on a given day. You cannot coordinate place and time between you in advance. Where do you meet?

The answer involves something called a "focal point" or Schelling point (after the person who developed the theory), and looking it up leads to some very interesting ideas on coordination without cooperation.

Comments

[minoanmiss]

Intriguing concept! Also reminds me of past adventures. :)

[brooksmoses]

That reminds me of the clearly-labeled "Meeting Point" in SFO's terminal 3.

It also reminds me of learning about statistical distributions and predictions from one of my colleagues in grad school. In particular: A Gaussian bell curve is a special case of a minimal-Shannon-entropy distribution, for the case where you know that a variable has an infinite possible range and the only other things you know about it are are the mean and variance of the distribution. You can generalize this to other cases (e.g., IIRC a Poisson distribution is the case with a infinite-in-one-direction range and a known mean), and things in that direction were relevant to he was doing, but the thing that this puts me in mind of is just the basic idea of distributions that minimize Shannon entropy.

So, it seems to me that what's happening is that to find a Schelling point, people's thought process can be modeled as computing the distribution of possibilities that minimizes the Shannon entropy given the limited shared information, and then picking the mode of that distribution.

As an illustration, for picking the meeting time I could make a very rough model by claiming the mental representation starts with a triad: am, noon, or pm. Within "am" and "pm" you've got hours, and then for a given hour you have a dyad of "o'clock" or not, and so on. All those various "bits" of the time representation have expected distributions based on experience, but I expect for most people the dominant factor in picking the mode of the distribution is simply that "noon" is a complete answer, whereas "am" and "pm" get further subdivided.

(Yes, I ignored "midnight", which should be in the set with "am", "noon", and "pm". I ignored it mostly because I don't know what the 4-element equivalent word is for "triad", and I justify that because it's not during daytime -- and also it comes with the question of whether you mean the midnight at the beginning of the day or the one at the end of the day, which makes it something to avoid.)

The Wikipedia article doesn't mention this in so many words, but I assume it's not anywhere near an original idea.

Also, the Wikipedia article doesn't mention the obvious-to-me extension of the "level-n theory", which is to look at the case where the strategy for a level-n player converges to an limit value as n goes to infinity. This is often readily computed by looking for fixed points of the recursion expression, and for games such as "pick the same answer as the other people" the limit computation is essentially trivial since all levels have the same strategy.

The cognitive-hierarchy theory is also extensible that way, and the limit should converge for nearly all games, I imagine, since in the limit each additional level is not changing the expected distribution of strategies. It would probably be interesting to consider what games where it doesn't converge would look like, and whether there are any examples that are not just unplayable things such as "pick a larger number than your opponent".

[hitchhiker]

i like the identification of "salience" as a key factor, independent of probability distributions - i.e. if some options clearly stand out to you, you could reasonably assume that they would stand out to other people as well, and therefore be attractive parts of the total space of options. i.e. as a foreigner i would not have expected grand central terminal to be as salient as the empire state building, and indeed by the "probability distribution" framing i would not know whether to look at the choices as "there are several train stations and only one empire state building" or "there are several train stations and several notable skyscrapers, and now how rare are each category", but clearly grand central terminal has a cultural significance as a meeting point, which raises its salience independently of how many bits of information you need to pinpoint it.

i did expect the wikipedia article to talk about fixed points too, but maybe they are not as applicable a concept when considering a largely open set?