Is there a way to phrase a problem of this type in a way that does not require such a state?
There is, and it is useful to look at such phrasings to allay those suspicions. However once we have looked at the issue enough to separate practical implications of imperfect knowledge from the core problem the simple version becomes more useful. It turns out that the trickiest part becomes unavoidable once we clear out the distractions!
When I was getting my head around the subject I made them up myself. I considered what the problem would look like if I took out the ‘absolute confidence’ stuff. For example—forget Omega, replace him with Patrick Jane. Say Jane has played this game 1,000 times before with other people and only got it wrong (and/or lied) 7 times.
I assume you can at least consider TV show entertainment level counterfactuals for the purpose of solving these problems. Analysing the behavior of fictional characters in TV shows is a legitimate use for decision theory.
Asking folks to hypothetically accept the unbelievable does not, IMHO, “clear out distractions”.
That would have made things difficult in high school science. Most example problems do exactly that. I distinctly remember considering planes and pulleys that were frictionless.* The only difference here is that the problem is harder (on our intuitions, if nothing else.)
* Did anyone else find it amusing when asked to consider frictionless ropes that were clearly fastened to the 200 kg weights with knots?
There is, and it is useful to look at such phrasings to allay those suspicions. However once we have looked at the issue enough to separate practical implications of imperfect knowledge from the core problem the simple version becomes more useful. It turns out that the trickiest part becomes unavoidable once we clear out the distractions!
And where, pray tell, might I look?
Asking folks to hypothetically accept the unbelievable does not, IMHO, “clear out distractions”.
When I was getting my head around the subject I made them up myself. I considered what the problem would look like if I took out the ‘absolute confidence’ stuff. For example—forget Omega, replace him with Patrick Jane. Say Jane has played this game 1,000 times before with other people and only got it wrong (and/or lied) 7 times.
I assume you can at least consider TV show entertainment level counterfactuals for the purpose of solving these problems. Analysing the behavior of fictional characters in TV shows is a legitimate use for decision theory.
That would have made things difficult in high school science. Most example problems do exactly that. I distinctly remember considering planes and pulleys that were frictionless.* The only difference here is that the problem is harder (on our intuitions, if nothing else.)
* Did anyone else find it amusing when asked to consider frictionless ropes that were clearly fastened to the 200 kg weights with knots?