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Elithrion comments on Buridan's ass and the psychological origins of objective probability - Less Wrong
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The problem with the Problem is that it simultaneously assumes a high cost of thinking (gradual starvation) and an agent that completely ignores the cost of thinking. An agent who does not ignore this cost would solve the Problem as Vaniver says.
The Problem only assumes the universe is continuous. If you move a particle by a sufficiently small amount, you can guarantee an arbitrarily small change any finite distance in the future. Thanks to the butterfly effect, it has to be an absurdly tiny amount, but it's only necessary that it exists.
Also, it assumes that the Ass will eventually die, but that's really more for effect. The point is that it can't make the decision in bounded time.
I'm not convinced this is actually true for the same reason of continuity.
Its possible survival is not guaranteed by continuity. It is possible in real life, but it takes more than continuity to prove that.
I know. I was thinking that it might be possible for the ass to guarantee it won't die by having an interrupt based on how hungry it is.
If you could do an interrupt, you could just make it go to the left if it takes too long to decide.
You can make it so that it gets more left-biased as it gets hungrier, but this just means that the equilibrium has it slowly moving to the right thereby increasing the pull to the right enough to counter out the increased pull to the left from hunger.
My idea is the following:
As it stands the ass will (after an indefinite amount of time) wind up in one of three positions:
a) eating from the left bale,
b) eating from the right bale, or
c) dead.
I'm trying to arrange it so that it always winds up in one of (a) or (b).
If it can pick (a) or (b) then it can also pick something somewhere in between. The only way to get around this is to somehow define (a) and (b) so that they border on each other.
For example, if you talk about which bale it eats first, and it only needs to eat some of it, then you could have something where it walks to the right bale, is about to take a bite, but then changes its mind and goes to the left bale. If you change it by epsilon, it takes an epsilon-sized bite, and eats from the right bale first instead of the left bale.
Only if you assume bounded time. A ball unstably balanced on a one-dimensional hill will after an indefinite amount of time fall to one side or the other, even though the two equilibria aren't next to each other.
No. It almost certainly will fall eventually, but there is at least one possibility where it never does.
Sorry, I'm not sure I understand what you mean. What particle should we move to change the fact that the ass will eventually get hungry and choose to walk forward towards one of the piles at semi-random? It seems to me like you can move a particle to guarantee some arbitrarily small change, but you can't necessarily move one to guarantee the change you want (unless the particle in question happens to be in the brain of the ass).
If you slowly move the particles one at a time from one bale to the other, you know that once you've moved the entire bale the Ass will change its decision. At some point before that it won't be sure.
There might not actually be a choice where the Ass stands there until it starves. It might walk forward, or split in half down the middle and have half of it take one bale of hay and half take the other, or any number of other things. It's really more that there's a point where the Ass will eventually take a third option, even if you make sure all third options are worse than the first two.
Thanks (and I actually read the other new comments on the post before responding this time!) I still have two objections.
The first one (which is probably just a failure of my imagination and is in some way incorrect) is that I still don't see how some simple algorithms would fail. For example, the ass stares at the bales for 15 seconds, then it moves towards whichever one it estimates is larger (ignoring variance in estimates). If it turns out that they are exactly equal, it instead picks one at random. For simplicity, let's say it takes the first letter of the word under consideration (h), plugs the corresponding number (8) as a seed into a pseudorandom integer generator, and then picks option 1 if the result is even, option 2 if it's odd. It does seem like this might induce a discontinuity in decisions, but I don't see where it would fail (so I'd like someone to tell me =)).
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot). My very mediocre grasp of QM suggests to me that if you try to use continuity to break the ass's algorithm (and it's a sufficiently good algorithm), you'll just find the point where its decisions are dominated by quantum uncertainty and get it to make true random choices. Or something along those lines.
Your problem is that you're using an algorithm that can only be approximated on an analog computer. You can't do flow control like that. If you want it to do A if it has 0 as an input and B if it has 1 as an input, you can make it do A+(B-A)x where x is the input, but you can't just make it do A under one condition and B under another. If continuity is your only problem, you can make it do A+(B-A)f(x), where f(x)=0 for 0<=x<=0.49 and f(x)=1 for 0.51<=x<=1, but f(x) still has to come out to 1/2 when x is somewhere between 0.49<x<0.51.
If you tried to do your algorithm, after 15 seconds, there'd have to be some certainty level where the Ass will end up doing some combination of going left and choosing at random, which will keep it in the same spot if "random" was right. If "random" is instead left, then it stops if it's half way between that and right.
I'm not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.
That would make it so you can't purposely fool the Ass, but it won't keep that from happening on accident. For example, if you try to balance a needle on the tip outside when there's a little wind, you're (probably) not going to be able to do it by making it stand up perfectly straight. It's going to have to tilt a little so it leans into every gust of wind. But there's still some way to get it to balance indefinitely.
The Plank length is irrelevant but quantization isn't. Specifically, with with quantum mechanics it's possible to get the ass to be in a superposition of eating from one or the other (but not in the middle) in bounded time.
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.
See Daniel's comment here.