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Elithrion comments on Buridan's ass and the psychological origins of objective probability - Less Wrong
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Okay, thanks for the explanation. It does seem that you're right*, and I especially like the needle example.
*Well, assuming you're allowed to move the hay around to keep the donkey confused (to prevent algorithms where he tilts more and more left or whatever from working). Not sure that was part of the original problem, but it's a good steelman.
You don't have to move the hay during the experiment. The donkey is the one that moves.
If he goes left as he gets hungry, you move the bale to his right a tad closer, and he'll slowly inch towards it. He'll slow down instead of speed up as he approaches it because he's also getting hungrier.
Does that really work for all (continuous? differentiable?) functions. For example, if his preference for the bigger/closer one is linear with size/closeness, but his preference for the left one increases quadratically with time, I'm not sure there's a stable solution where he doesn't move. I feel like if there's a strong time factor, either a) the ass will start walking right away and get to the size-preferred hay, or b) he'll start walking once enough time has past and get to the time-preferred hay. I could write down an equation for precision if I figure out what it's supposed to be in terms of, exactly...
Like I said, the hay doesn't move, but the donkey does. He starts walking right away to the bigger pile, but he'll slow down as time passes and he starts wanting the other one.
Interestingly, that trick does get the ass to walk to at least one bale in finite time, but it's still possible to get it to do silly things, like walk right up to one bale of hay, then ignore it and eat the other.
The solutions are almost certainly unstable. That is, once you find some ratio of bale sizes that will keep the donkey from eating, an arbitrarily small change can get it to eat eventually.
Okay, sure, but that seems like the problem is "solved" (i.e. the donkey ends up eating hay instead of starving).
It can also use the "always eat the left bale first" strategy, although that gets kind of odd if it does it with a bale of size zero.
There is a problem if you want to make it make an actual binary decision, like go to one bale and stay.