Quick clarification question that i think is already addressed in the post but I'm being lazy. How much of this depends on resource--in particular, memory--limitations? Seems to me like even with perfect memory, causal uncertainty and limited data is enough to get the general shape of your results. Is the idea that with better memory you converge on steady (when it's true) more quickly, and so have less time to exhibit the gambler's fallacy?
Yeah, that's right. Most of the graphs are with just limited data about the process, so really there are no "bounded-rationality" style constraints there—perfect Bayesians will do it. The main issue is that if they have perfect memory, they won't do it for *that long*: as the graph right above "second upshot" shows, once they've seen a sequence of 150 tosses, they already are only (on average) < 5% confident of Switchy, so will only commit the gambler's fallacy to a very small degree.
Limited memory explains how we can have the same effects for people who've seen tons of trials (eg people playing slots in a casino, or weirdos like me who've tossed lots of coins).
I wonder if a lot of the same work that can be done by limited memory could also be done by still more causal uncertainty. You let agents be uncertain about whether they're drawing from a steady, sticky or switchy distribution, but they're always sure they're drawing from the same distribution. If agents are also unsure when draws are from the same distribution or not, my intuition is that that can do much the same work as limited memory. Is the slot machine in this casino the same as the slot machine in the casino I played in yesterday? Stuff like that.
Ah nice, good point! I had thought about having that sort of heterogeneity in there (it seems obviously true and helps with some other empirical predictions, in the paper), but hadn't realized it could do the work of limited memory. Nice! I'll think about whether there's a sensible way to incorporate that in the paper.
Wonderful!
Quick clarification question that i think is already addressed in the post but I'm being lazy. How much of this depends on resource--in particular, memory--limitations? Seems to me like even with perfect memory, causal uncertainty and limited data is enough to get the general shape of your results. Is the idea that with better memory you converge on steady (when it's true) more quickly, and so have less time to exhibit the gambler's fallacy?
Thanks Dan!
Yeah, that's right. Most of the graphs are with just limited data about the process, so really there are no "bounded-rationality" style constraints there—perfect Bayesians will do it. The main issue is that if they have perfect memory, they won't do it for *that long*: as the graph right above "second upshot" shows, once they've seen a sequence of 150 tosses, they already are only (on average) < 5% confident of Switchy, so will only commit the gambler's fallacy to a very small degree.
Limited memory explains how we can have the same effects for people who've seen tons of trials (eg people playing slots in a casino, or weirdos like me who've tossed lots of coins).
I wonder if a lot of the same work that can be done by limited memory could also be done by still more causal uncertainty. You let agents be uncertain about whether they're drawing from a steady, sticky or switchy distribution, but they're always sure they're drawing from the same distribution. If agents are also unsure when draws are from the same distribution or not, my intuition is that that can do much the same work as limited memory. Is the slot machine in this casino the same as the slot machine in the casino I played in yesterday? Stuff like that.
Ah nice, good point! I had thought about having that sort of heterogeneity in there (it seems obviously true and helps with some other empirical predictions, in the paper), but hadn't realized it could do the work of limited memory. Nice! I'll think about whether there's a sensible way to incorporate that in the paper.