This is a follow up to my pre-Christmas rant on the two-envelope paradox. A pretty large chunk of what I’ll say is basically transcribed from a conversation with Matt Weiner at the APA, but since Matt stubbornly refuses to start a blog I feel like someone may as well write up his ideas. And the rest of the good ideas are stolen from a conversation with Chris Meacham and Maya Eddon. (First disclaimer: Both of these conversations happened in the presence of impressively large amounts of alcohol, so it’s possible I’m radically mistranscribing ideas here. Second disclaimer: I’m writing this from a NyQuil-induced haze, so anything could happen. Third disclaimer: Because of the first two disclaimers, what was intended to be one big post will be serialised. Today’s installment is a pair of puzzles that motivates the problem I’m thinking about.)
God, being a generous sort, offers you the following sort of St Petersburg gift. He’ll toss a fair coin repeatedly until it lands heads. Letting n be the number of tosses it takes, he’ll then write you a check for 2n utils.
If you don’t believe in utils, as well you might not, all that matters here is that the gift structure God sets up is such that for any n, right now you’re indifferent between getting the gift he’ll give you if it takes n tosses, and a 1/2 chance of getting the gift he’ll give you if it takes n+1 tosses, and all of these gifts are worth more to you than nothing.
After God has done the coin tosses, but before he reveals the gift, God offers you the following deal. If you give him a dollar now, he’ll give you the gift he had planned plus two dollars. Two questions.
Should you make the deal?
If your answer is yes, can you give a coherent argument (or at least a valid argument using only plausible, consistent premises) for that conclusion?
I think those questions are quite hard. The second puzzle is just like the first, except that the coin is no longer perfectly fair – it has a 51% bias towards heads. Now ask the same two questions. Note that in the second case the expected utility of God’s gift is defined, and this might make a difference in how the two questions get answered.