Timeline for Two envelope (sub)-problem / related problem
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2016 at 10:32 | comment | added | John | It is not possible to apply a uniform probability distribution over an infinite set. Probabilities must add up to 1. If the probability distribution is uniform, let the chance of any amount be p. We would then have infinity x p = 1. This is impossible for any non-zero value of p. | |
| Mar 1, 2016 at 3:34 | comment | added | Fairly Nerdy | has no upper bound. After I open the envelope and see a value, I immediately know that whatever value I see is less than 1/2 of the value that you could have put in the envelope (which has no upper bound), so I am in the scenario where it is profitable to switch. This is of course assuming that I thought all numbers were equally likely, and therefore there was a 50% chance at this point to double, vs 50% chance to cut in half. If I thought there were some non-uniform probability distribution then I would have to weigh that into the computed value of switching | |
| Mar 1, 2016 at 3:30 | comment | added | Fairly Nerdy | John, You have done a good job of convincing me that you can put an IOU in the envelope with no upper bound. The other key part is that all of the numbers have an even probability distribution. I'm assuming that you can do this, although I have no idea how. (just flipping the coin isn't good enough to give the numbers an even distribution). & I would really like to see computer code (especially python) that can do this. In this case I would say, before I open the envelope there is no reason to switch because the average of all numbers with no upper bound for both envelopes | |
| Feb 29, 2016 at 10:24 | comment | added | John | Regarding finite and infinite amounts: in any particular game we can place an upper and lower bound on the values in the envelopes as soon as we open one. But mathematical consideration of the values from which the amounts could have been selected in the first place is a different point. I don't have to write zeros: I could write 'let X = Googolplex, then the amount in this envelope is (((X)^X)^X)^X + 53. I could write all sorts of things like this. You stated that there is a maximum amount that could be in any envelope. OK, what is it? | |
| Feb 29, 2016 at 10:12 | comment | added | John | I do not think that swapping carries an expectation of gain. But there is more than one way to argue this. One way, which I find convincing, is best put as the outsider's perspective: label the smaller amount in the pair X and therefore the larger amount is 2X. Swapping amounts to going up or down X, a fixed and given amount for any pair. This perspective is very hard for the player to accept. Nevertheless I think it is correct. Another way to argue that there is no expectation of gain is via mathematical arguments about the set from which the amounts were chosen. I find this less convincing. | |
| Feb 29, 2016 at 2:32 | comment | added | Fairly Nerdy | John, in the scenario you describe, what would be the fair value of purchasing one of those unopened envelopes ? I think the fair value has to be either a finite number (because you have finite money, or their is finite space to write 0's on the paper) or the fair value is infinity (because we assume you can think up a number infinitely large) My original answer focused on the finite money, but I would agree that the infinite money debate is also valid. If you say that envelope 1 has an infinite expected value, I would say why would I switch? I'm holding infinite money | |
| Feb 29, 2016 at 0:46 | comment | added | John | Consider a one off game. So one pair of envelopes. I fill one by just thinking of a number, then flip a coin to put double or half in the other. What is M? Why does M have to exist? Each real numbers is a finite number, but the set of real numbers is unbounded; in particular there is no upper bound, so no M. | |
| Feb 28, 2016 at 23:58 | comment | added | Fairly Nerdy | John, I'm not sure I understand the concept of a finite amount with no upper limit. I might not have the right math for it. In what way does that differ from infinity ? I completely agree that the infinite money scenario is a different one than the finite money scenario. In that scenario you end up with the EV of both envelopes being infinity. So even if envelope 2 = 1.25 envelope 1, that is just infinity = 1.25 infinity, or infinity = infinity. (I think, infinity has weird properties I don't completely understand) | |
| Feb 28, 2016 at 22:44 | comment | added | John | I'm not sure about your first bullet point (finite resources). The puzzle doesn't have to be restricted to sums less than a fixed amount. As a mathematical exercise we can work with the idea that any amount is a possibility for the sum in the envelopes. Clearly, 'infinity' isn't an amount or money. But then there is something between less than a fixed amount and infinite, namely a fine amount with no upper limit. | |
| Feb 28, 2016 at 19:56 | history | made wiki | Post Made Community Wiki by whuber♦ | ||
| Feb 27, 2016 at 21:37 | review | Late answers | |||
| Feb 27, 2016 at 21:37 | |||||
| Feb 27, 2016 at 21:17 | history | answered | Fairly Nerdy | CC BY-SA 3.0 |