Hmm. This is the first I’ve seen it claimed that common knowledge in the technical sense (X knows that Y knows that X knows etc up to infinity) is actually sometimes achieved in real life. I don’t think I buy it. Say “knows” means “assigns >50% probability to”. Then the probability that common knowledge is achieved is the limit of the probability that nth level common knowledge is achieved as n goes to infinity. This is a decreasing sequence. Why should we expect it to stabilize at a positive number? The obvious thing to do would be to prove a lower bound on the probability of nth level common knowledge that is independent of n. But what would such a lower bound consist of? Maybe something like “the probability that X and Y are normally functioning human beings that have just made eye contact”? But this does not logically imply that X’s impression of Y was one of normal functionality, though it strongly suggests it. Repeatedly patching the attempt only leads to another infinite decreasing sequence of probabilities, yielding the same problem. Yes the differences between the successive probabilities in this sequence are small, but to show stabilization you need to show more than that the individual differences are small.
Of course, the question of whether common knowledge in the technical sense is ever achieved in real-life situations is irrelevant to such situations, because normal people’s decision algorithms don’t work purely by expected utility. The normal person’s analysis of the bus situation is: start by assuming that everyone will perform the default socially acceptable action, in this case getting off if and only if you have agreed to do so. Then perform tweaks based on the fact that people will have beliefs about each others’ actions and may change their plans accordingly. Repeat until your model of the world stabilizes. Then the point is just that the eye contact agreement changes what the default socially acceptable action is (in both people’s model of the world).
Notably, in the bus situation it would ordinarily be eye contact _agreement_ that would mean the people get off of the bus, not (necessarily) the eye contact by itself. This is because the main thing that you need common knowledge of is “Joining this acquaintance would be nice [though we care mainly about each other’s company]”—not just the fact that both people have heard the acquaintance. The concept of socially acceptable actions based on agreement encodes this important information, which might be forgotten in far mode (as it seems to have been in Chwe’s description of the situation).
Hmm. This is the first I’ve seen it claimed that common knowledge in the technical sense (X knows that Y knows that X knows etc up to infinity) is actually sometimes achieved in real life. I don’t think I buy it. Say “knows” means “assigns >50% probability to”. Then the probability that common knowledge is achieved is the limit of the probability that nth level common knowledge is achieved as n goes to infinity. This is a decreasing sequence. Why should we expect it to stabilize at a positive number? The obvious thing to do would be to prove a lower bound on the probability of nth level common knowledge that is independent of n. But what would such a lower bound consist of? Maybe something like “the probability that X and Y are normally functioning human beings that have just made eye contact”? But this does not logically imply that X’s impression of Y was one of normal functionality, though it strongly suggests it. Repeatedly patching the attempt only leads to another infinite decreasing sequence of probabilities, yielding the same problem. Yes the differences between the successive probabilities in this sequence are small, but to show stabilization you need to show more than that the individual differences are small.
Of course, the question of whether common knowledge in the technical sense is ever achieved in real-life situations is irrelevant to such situations, because normal people’s decision algorithms don’t work purely by expected utility. The normal person’s analysis of the bus situation is: start by assuming that everyone will perform the default socially acceptable action, in this case getting off if and only if you have agreed to do so. Then perform tweaks based on the fact that people will have beliefs about each others’ actions and may change their plans accordingly. Repeat until your model of the world stabilizes. Then the point is just that the eye contact agreement changes what the default socially acceptable action is (in both people’s model of the world).
Notably, in the bus situation it would ordinarily be eye contact _agreement_ that would mean the people get off of the bus, not (necessarily) the eye contact by itself. This is because the main thing that you need common knowledge of is “Joining this acquaintance would be nice [though we care mainly about each other’s company]”—not just the fact that both people have heard the acquaintance. The concept of socially acceptable actions based on agreement encodes this important information, which might be forgotten in far mode (as it seems to have been in Chwe’s description of the situation).