Lamport points out this substitutes another decision subject to the same limits: the agent must decide that it can’t decide.
When the Ass computes the expected utility of going each direction, it finds that they are equal. This is a decision subject that the Ass can decide in finite time, and furthermore that computation shows that it is obvious there is no value to be gained by spending further time deciding. It’s not even worth flipping a coin over- a right-hoofed Ass should go to the right bale, and a left-hoofed Ass should go to the left bale, or some other suitable default that saves cognitive resources.
Basically, it looks like a lot of the assumptions in Lamport’s argument are questionable,* and applying basic decision theory dissolves the problem immediately.
*Why does it take appreciably longer to visually measure larger bales of hay? All the Ass is doing is looking at them.
How is the ass to determine that they’re equal, as opposed to one being ε larger than the other in finite time?
Again, this is the reverse of how you should go about things. The amount of time to spend making a decision is proportional to the difference between the value of the actions- once you’re sure the difference is smaller than ε, you stop caring. You might as well flip a coin.
The amount of time to spend making a decision is proportional to the difference between the value of the actions- once you’re sure the difference is smaller than ε, you stop caring.
Aw, but how does one determine that the distance is smaller than ε? What if the difference is arbitrarily close to ε?
You assign a disutility to spending time t further evaluating, and when the disutility of further evaluation is equal to the expected increase in utility of the selected bale (because one of them is/may be ε larger than the other), you select randomly between them.
If you adjust the disutility of marginal time t to increase with the total amount of time spent deciding, you break the deadlock condition which exists if you are uncertain about whether the disutility of spending more time evaluating is greater or less than the expected utility increase of spending more time evaluating the choice; if that return-on-investment is ever within ε of zero, then some time later it must be greater than ε less than zero (because the disutility of the time t will have increased by more than 2ε).
I’m a bit confused, so please take this as exploratory rather than expository. What prevents the ass (or the decision process in general) from:
1) Having a pre-computed estimate of (a) how long it has before it would starve, (b) how the error of its size determinations depends on how long it spends observing, and (c) how much error in that estimates it cares about; and then,
2) Stop observing/deciding when the first limit is close (but far enough to still have time to eat!) or when the error of the difference between the two directions falls below the limit it cares about. (In the strictest interpretation of the question, this second step is not necessary.)
When I say “estimate” in step one, I mean a very wide pre-computed interval, not some precise computation. I don’t know exactly how long it’ll take me to die from hunger, but it’s clear that in a similar situation at some point I’d be hungry enough that I can anticipate without needing any complicated logic that I would die from not eating and stop comparing the choices. In that case you just need a way to distinguish them to pick one (i.e., left and right, not bigger and smaller), and you do so with any arbitrary rule (e.g., lexicographic ordering).
I effect, the ass has a not just the binary problem of choosing left and right, it has the ternary (meta-)problem of choosing between going left, going right, or searching for a better method of picking which direction is better. The first two may remain symmetrical, but at some point the third choice (“keep thinking”) will reach negative expected utility (trivially, when you anticipate to starve, but in real life you might also decide that spending another hungry hour deliberating over some very small amount of hay is not worth it).
I’m sure similar decision problems can be posed, where the tradeoffs are balanced such that you still have an issue, but this particular formulation seems almost as silly as claiming the ass will starve because of Zeno’s arrow paradox.
Run your decision procedure for a constant time. If it doesn’t halt, abort it and break the symmetry—e.g. by choosing the option that sorts first lexically.
The constant time part could work, but is hardly the only escape valve you should have. You have a utility estimate for each action- the estimates will have some variance, and you can run the procedure until either the variance is below a certain amount or the variance has decreased by less than some threshold in the last iteration or you’ve run out of time.
The Ass is not a digital computer. It’s an analog computer. It’s subject to continuity. That’s important.
If you look at the Ass’s center of mass five seconds after the experiment starts, and vary the relative sizes of the bales of hay continuously, the Ass’s position must also change continuously. If you found some ratio of hey where the Ass ends up at the left bale of hay, but if you add any amount, no matter how tiny, it ends up at the right bale of hay, the Ass is violating the laws of physics.
It gets a bit more complicated because you can’t add less than one particle to the bale of hay, but there are other things you can do, such as slowly move one piece of straw between the bales, or move the bales closer and further from the Ass.
When the Ass computes the expected utility of going each direction, it finds that they are equal. This is a decision subject that the Ass can decide in finite time, and furthermore that computation shows that it is obvious there is no value to be gained by spending further time deciding. It’s not even worth flipping a coin over- a right-hoofed Ass should go to the right bale, and a left-hoofed Ass should go to the left bale, or some other suitable default that saves cognitive resources.
Basically, it looks like a lot of the assumptions in Lamport’s argument are questionable,* and applying basic decision theory dissolves the problem immediately.
*Why does it take appreciably longer to visually measure larger bales of hay? All the Ass is doing is looking at them.
How is the ass to determine that they’re equal, as opposed to one being ε larger than the other in finite time?
Again, this is the reverse of how you should go about things. The amount of time to spend making a decision is proportional to the difference between the value of the actions- once you’re sure the difference is smaller than ε, you stop caring. You might as well flip a coin.
Aw, but how does one determine that the distance is smaller than ε? What if the difference is arbitrarily close to ε?
You assign a disutility to spending time t further evaluating, and when the disutility of further evaluation is equal to the expected increase in utility of the selected bale (because one of them is/may be ε larger than the other), you select randomly between them.
If you adjust the disutility of marginal time t to increase with the total amount of time spent deciding, you break the deadlock condition which exists if you are uncertain about whether the disutility of spending more time evaluating is greater or less than the expected utility increase of spending more time evaluating the choice; if that return-on-investment is ever within ε of zero, then some time later it must be greater than ε less than zero (because the disutility of the time t will have increased by more than 2ε).
I’m a bit confused, so please take this as exploratory rather than expository. What prevents the ass (or the decision process in general) from:
1) Having a pre-computed estimate of (a) how long it has before it would starve, (b) how the error of its size determinations depends on how long it spends observing, and (c) how much error in that estimates it cares about; and then,
2) Stop observing/deciding when the first limit is close (but far enough to still have time to eat!) or when the error of the difference between the two directions falls below the limit it cares about. (In the strictest interpretation of the question, this second step is not necessary.)
When I say “estimate” in step one, I mean a very wide pre-computed interval, not some precise computation. I don’t know exactly how long it’ll take me to die from hunger, but it’s clear that in a similar situation at some point I’d be hungry enough that I can anticipate without needing any complicated logic that I would die from not eating and stop comparing the choices. In that case you just need a way to distinguish them to pick one (i.e., left and right, not bigger and smaller), and you do so with any arbitrary rule (e.g., lexicographic ordering).
I effect, the ass has a not just the binary problem of choosing left and right, it has the ternary (meta-)problem of choosing between going left, going right, or searching for a better method of picking which direction is better. The first two may remain symmetrical, but at some point the third choice (“keep thinking”) will reach negative expected utility (trivially, when you anticipate to starve, but in real life you might also decide that spending another hungry hour deliberating over some very small amount of hay is not worth it).
I’m sure similar decision problems can be posed, where the tradeoffs are balanced such that you still have an issue, but this particular formulation seems almost as silly as claiming the ass will starve because of Zeno’s arrow paradox.
Run your decision procedure for a constant time. If it doesn’t halt, abort it and break the symmetry—e.g. by choosing the option that sorts first lexically.
The constant time part could work, but is hardly the only escape valve you should have. You have a utility estimate for each action- the estimates will have some variance, and you can run the procedure until either the variance is below a certain amount or the variance has decreased by less than some threshold in the last iteration or you’ve run out of time.
The Ass is not a digital computer. It’s an analog computer. It’s subject to continuity. That’s important.
If you look at the Ass’s center of mass five seconds after the experiment starts, and vary the relative sizes of the bales of hay continuously, the Ass’s position must also change continuously. If you found some ratio of hey where the Ass ends up at the left bale of hay, but if you add any amount, no matter how tiny, it ends up at the right bale of hay, the Ass is violating the laws of physics.
It gets a bit more complicated because you can’t add less than one particle to the bale of hay, but there are other things you can do, such as slowly move one piece of straw between the bales, or move the bales closer and further from the Ass.