Three envelope problems
There has been some discussion at The Excluded Middle about the three envelope problem (here and here). Here's the case. Sally has three envelopes in front of her. She believes the first envelope contains $200, and that one of the other two contains $300, and that the other contains nothing. But, she has no beliefs concerning which is which. Which one ought she choose? There has been discussion regarding the subjective vs. the objective ought. I don't think I want to get into that here. Let's agree that she subjectively ought to choose the first envelope, and that she objectively ought to choose the envelope that contains $300, whichever that is. I want to consider some more cases. I don't know exactly what is philosophically interesting about them right now, but I feel like something is. Here are some cases, with my intuitions to follow: (Assume Sally is solidly in the middle-class. She doesn't need more money to survive, but more would always be nice.)
(A) Sally believes the first envelope contains $200, and that one of the other envelopes contains $1 million, and that the other contains nothing. She doesn't have any beliefs about which is which. Which subjectively ought she choose?
(B) Sally believes the first envelope contains $10,000, and that one of the others contains $1 million, and that the other contains nothing. She doesn't have any beliefs about which is which (regarding the 2nd and 3rd). Which subjectively ought she choose?
(C) Sally believes the first envelope contains $1 million, and that one of the others contains $10 million, and that the other contains $1000. She doesn't have any beliefs about which is which (regarding the 2nd and 3rd). Which subjectively ought she choose?
(D) Sally believes the first envelope contains $5 million, that one of the others contains $20 million, and that the other contains $1 million. Same story, which subjectively ought she choose?
My intuitions:
(A) She ought to flip a coin to decide between the 2nd and 3rd. I think this answer is pretty non-controversial. $200 is nice, but worth risking for a 50% chance at 1 million.
(B) This one is maybe the hardest. I'm not sure what to say in this case.
(C) I think she ought to choose the first envelope.
(D) I think she flips a coin to decide between the 2nd and 3rd. $5 million is much more than $1 million, but $1 million is still nice, and the chance at $20 million is great.
Again, I don't know if you can draw anything interesting out of these cases. Maybe something about risk. But, they're fun to think about.
On a related note, if anyone can point me in the direction of some literature on these problems, or related ones, I would appreciate it. Thanks.
(A) Sally believes the first envelope contains $200, and that one of the other envelopes contains $1 million, and that the other contains nothing. She doesn't have any beliefs about which is which. Which subjectively ought she choose?
(B) Sally believes the first envelope contains $10,000, and that one of the others contains $1 million, and that the other contains nothing. She doesn't have any beliefs about which is which (regarding the 2nd and 3rd). Which subjectively ought she choose?
(C) Sally believes the first envelope contains $1 million, and that one of the others contains $10 million, and that the other contains $1000. She doesn't have any beliefs about which is which (regarding the 2nd and 3rd). Which subjectively ought she choose?
(D) Sally believes the first envelope contains $5 million, that one of the others contains $20 million, and that the other contains $1 million. Same story, which subjectively ought she choose?
My intuitions:
(A) She ought to flip a coin to decide between the 2nd and 3rd. I think this answer is pretty non-controversial. $200 is nice, but worth risking for a 50% chance at 1 million.
(B) This one is maybe the hardest. I'm not sure what to say in this case.
(C) I think she ought to choose the first envelope.
(D) I think she flips a coin to decide between the 2nd and 3rd. $5 million is much more than $1 million, but $1 million is still nice, and the chance at $20 million is great.
Again, I don't know if you can draw anything interesting out of these cases. Maybe something about risk. But, they're fun to think about.
On a related note, if anyone can point me in the direction of some literature on these problems, or related ones, I would appreciate it. Thanks.
Posted by Justin Snedegar on
08.11.2008 at 9:17pm
There's also a lot of literature that talks about the three-envelope type case. Schroeder talks about it in 'Means-End Coherence, Stringency, and Subjective Reasons.' Mark picks up the discussion from Jake Ross's dissertation. Ross got the case from John Broome's manuscript Reasoning, who got it from Parfit's On What Matters, who got it from Don Regan's Utilitarianism and Cooperation. Ralph Wedgwood also discusses it (and admits that it's a counterexample to the main thesis of his book) in The Nature of Normativity. Daniel Star and Stephen Kearns also discuss in a forthcoming paper in Ethics called 'Reasons: Explanation or Evidence?'
Niko Kolodny and John MacFarlane posted a new paper they've written solely dealing with Parfit's version of the case. It's called 'Ifs and Oughts,' and be found on either Niko's or John's websites.