Two envelopes paradox

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Author	Topic: Two envelopes paradox
Ash Lael E Equals MC What!!!! Member # 5491	written Thursday, June 23 2005 17:27 Profile Homepage #25 Look at it this way. As a contestant, you don't know if the 50K is the double or the half. There's a 50% chance of either. So let's do the 100 people simulation again. Half go into the 50K - 100K game, half go into the 25K - 50K game. What do the maths work out to now? 50 get the 50K, 25 get 100K, 25 get 25K. Now they all swap... total money given away is still the same. But the 50K bunch are better off. If you ARE the top limit, obviously there's no point swapping. If you DON'T KNOW if you're the upper or lower limit, it's in your favour to swap. Or something. I dunno, this is confusing. EDIT: quote: Originally written by Thuryl: Okay, new analysis. Suppose there are 200 players. 100 decide to adopt an "always switch" strategy (let's call them switchers or group S), 100 decide to adopt a "never switch" strategy (let's call them non-switchers or group NS). 50 S and 50 NS are randomly assigned to a group where the envelopes contain $25000 and $50000 -- let's call them S-A and NS-A respectively. 50 S and 50 NS are randomly assigned to a group where the envelopes contain $50000 and $100000 -- let's call them S-B and NS-B respectively. The first round of handing out envelopes occurs. 25 S-A get $25000, 25 NS-A get $25000, 25 S-A get $50000, 25 NS-B get $50000. 25 S-B get $50000, 25 NS-B get $50000, 25 S-B get $100000, 25 NS-B get $100000. Obviously, after switching, the switchers as a group are not any better off -- the switchers who got the smaller amount have simply exchanged places with the ones who got the larger amount, meaning that they're in exactly the same situation as the non-switchers. It seems obvious that switching is not really a better strategy than non-switching. Oh, damn. EDIT 2: quote: Originally written by Manhood Typing Kelandon: Wait, how did the no-switchers get anything but $50,000? You start out by knowing that you got 50K. If you don't switch, you should invariably get 50K. Yay! My poor brain. [ Thursday, June 23, 2005 17:30: Message edited by: Ash Lael ] -------------------- Sex is easier than love. Posts: 1861 \| Registered: Friday, February 11 2005 08:00
Kelandon Off With Their Heads Member # 4045	written Thursday, June 23 2005 17:28 Profile Homepage #26 Wait, how did the no-switchers get anything but $50,000? You start out by knowing that you got 50K. If you don't switch, you should invariably get 50K. EDIT: The correct analysis is this: 50 switchers get 25K. 50 switchers get 100K. 100 non-switchers get 50K. [ Thursday, June 23, 2005 17:32: Message edited by: Manhood Typing Kelandon ] -------------------- Arancaytar: Every time you ask people to compare TM and Kel, you endanger the poor, fluffy kittens. Kelandon's Pink and Pretty Page!!: the authorized location for all things by me The Archive of all released BoE scenarios ever Posts: 7968 \| Registered: Saturday, February 28 2004 08:00
Thuryl ...b10010b... Member # 869	written Thursday, June 23 2005 17:46 Profile Homepage #27 Here is an analysis of the problem. Essentially, the "paradox" rests on the flawed assumption that the envelopes are equally likely to contain any amount of money. This is impossible, because there is no probability distribution that is uniform across an infinite range. Since the envelopes can't contain any amount of money with equal probability, we have to take into account what amounts of money the game show organisers are actually likely to put in the envelopes. If there really is a 50% chance they put $100000 in the other envelope, then it's in your best interests to switch (actually, this is complicated by the non-linear relation between money and utility, but that's an issue for another time). But this condition won't hold true for every possible dollar value that could be in the envelopes, so always switching regardless of the amount of money isn't actually an optimal strategy. It is possible to apply strategy to the game, but only by finding out the probability distribution used to generate the particular sets of dollar amounts used, which requires playing it repeatedly. [ Thursday, June 23, 2005 17:54: Message edited by: Thuryl ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 \| Registered: Saturday, March 30 2002 08:00
Or else o'erleap. Law Bringer Member # 335	written Thursday, June 23 2005 18:01 Profile Homepage #28 quote: Originally written by Ash Lael: Hey, could you theoretically use the three doors trick to give yourself a slight advantage in Who Wants To Be a Millionaire with the 50/50 lifeline? You increase your chance to 1/2 instead of 1/4, but unless the lifeline works exactly like the doors (you pick, then two of the choices you didn't pick are eliminated) it's not as good as the door trick. That one relies on the non-randomness of the information you get. If you could pick one answer and then reveal two of the three others as false, you'd have a 3/4 chance of getting the question right by switching. That's the difference between this gameshow paradox and the three doors game, which isn't a paradox. The latter relies on gaining new information on probability. The former gives you no new information, so no matter how you crunch the numbers you're still guessing. —Alorael, who will nonetheless use these letters are proof that the grass really is greener on the other side of the fence. Probability does not lie! Please ignore the garbage in and garbage out. Posts: 14579 \| Registered: Saturday, December 1 2001 08:00
*i The Establishment Member # 6	written Thursday, June 23 2005 18:22 Profile #29 Yes, you do need to factor in the underlying distribution. A equal distribution everywhere is impossible because the integral over the range of the distribution must be 1. The integral over a constant to infinity 'blows up' and cannot be 1. Suppose we have a variable distribution. The probability of making money for a given value k is p, the probability of losing money for a given value k is 1-p. The breakeven value of p would be where the difference in the two options is zero. Option 1: Keep the envelope. Get k. Option 2: Trade in the envelope. Has a probability of: (1-p)(k/2) + p(2k) The net gain (equal to zero) is: (1-p)(k/2) + p(2k) - k = 0 (1-p)(1/2) + 2p - 1 = 0 1/2 - p/2 + 2p - 1 = 0 (3/2)p = 1/2 p = 1/3 So if the probability of making money is greater than 1/3, you should pick the other envelope. If the probability if less than or equal to, you should stay with what you have. -------------------- Your flower power is no match for my glower power!! Posts: 3726 \| Registered: Tuesday, September 18 2001 07:00
Jumpin Salmon ? Man, ? Amazing Member # 5755	written Thursday, June 23 2005 18:24 Profile #30 The way I see it is that the matter was decided back when Trinity announced me as the "right." Everything after that (except of course for any blather from mods or admins) was fluff that added nothing to my rightness. It is likely that bob barker would not have 100 thou to give away. it is possible that a company would have that amount. what would make the puzzle far more interesting would be if instead of 50% it was 10%. 50,000 or 500,000. or 5,000. Edit - stareye added something while i was writing that involves math and stuff. read at your own peril. this message funded by the price is right, but they aren't, so that is wrong [ Thursday, June 23, 2005 18:26: Message edited by: Jumpin' Salmon ] Posts: 4114 \| Registered: Monday, April 25 2005 07:00
*i The Establishment Member # 6	written Thursday, June 23 2005 18:31 Profile #31 What it basically says is that if your probability of making money is greater than 33.3%, then you should choose the other envelope. If it is less (say 10% in your example), then you should keep what you have. This assumes that we know p for every value of k, of course, which in and of itself would be very difficult to measure. To do this with complete rigor we would need to know p(k) and use that in there. -------------------- Your flower power is no match for my glower power!! Posts: 3726 \| Registered: Tuesday, September 18 2001 07:00
Jumpin Salmon ? Man, ? Amazing Member # 5755	written Thursday, June 23 2005 18:41 Profile #32 So an even better option might be to demand cash instead of check, punch Bob Barker in the gut, grab both envelopes, and run off stage, hopefully finding an exit before the wimpy security guards wipe their mouths and come looking for their phones so they can call the real police. this message sponsored by low blood sugars everywhere Posts: 4114 \| Registered: Monday, April 25 2005 07:00
*i The Establishment Member # 6	written Thursday, June 23 2005 18:52 Profile #33 If we have even more rigor to the problem as suggested. Suppose we have a multiplicative factor n. This determines how many times more money we could make. We pick an envelope with an amount k. The other envelope either has: k/n or nk inside of it. Note: in our previous example n was equal to 2. Now we can let this be arbitrary. If we do the same analysis as before, we get that the breakeven value of p is: p = 1/(1+n) So for n = 10, we should pick the other envelope if the probability of making money (p) is greater than 1/11 or about 9 percent. [ Thursday, June 23, 2005 18:53: Message edited by: i ] -------------------- Your flower power is no match for my glower power!! Posts: 3726* \| Registered: Tuesday, September 18 2001 07:00
Archmage Alex Skip to My Lou Member # 40	written Thursday, June 23 2005 19:02 Profile Homepage #34 Simple answer: You picked the first envelope at random. The introduction of this supremely random element makes any mathmatical/statistical solution dubious at best. The chances that the other envelope is better is completely random. Salmon has the only answer: analyze what you know of the show. Just for fun: "The laws of mathematics, as far as they refer to reality, are not certain, and as far as they are certain, do not refer to reality." -Albert Einstein -------------------- Take the Personality Test! INTJ 100% 75% 100% 44% Huzzah for the Masterminds! www.Keirsey.com for personality information. The Sloganizer! "Swing your Archmage Alex." Deep down, you wish you were a stick figure. Posts: 1629 \| Registered: Wednesday, October 3 2001 07:00
Spring Shaper Member # 5450	written Thursday, June 23 2005 21:03 Profile Homepage #35 I would switch.. Although $50,000 is a lot of money, if you win, you get double. If you lose, you get half. 50-50 chance. $100,000 is 4 times the amount of 25,000 so it would mean 4 times as much money spent. EDIT: I just done a two post paradox. [ Thursday, June 23, 2005 21:04: Message edited by: Spring ] -------------------- I'll put a Spring in your step. Polaris Posts: 2396 \| Registered: Saturday, January 29 2005 08:00
Spring Shaper Member # 5450	written Thursday, June 23 2005 21:04 Profile Homepage #36 I would switch.. Although $50,000 is a lot of money, if you win, you get double. If you lose, you get half. 50-50 chance. $100,000 is 4 times the amount of 25,000 so it would mean 4 times as much money spent. Pretty stupid gameshow, it must only go for about 10 minutes. It would be shorter if there was no selection process for the player. -------------------- I'll put a Spring in your step. Polaris Posts: 2396 \| Registered: Saturday, January 29 2005 08:00
Ash Lael E Equals MC What!!!! Member # 5491	written Thursday, June 23 2005 21:11 Profile Homepage #37 quote: Originally written by Evil Autocrat: You increase your chance to 1/2 instead of 1/4, but unless the lifeline works exactly like the doors (you pick, then two of the choices you didn't pick are eliminated) it's not as good as the door trick. That one relies on the non-randomness of the information you get. If you could pick one answer and then reveal two of the three others as false, you'd have a 3/4 chance of getting the question right by switching. Say you mentally pick one option randomly, then use the 50/50. There is a 50% chance that the one you picked won't get eliminated. In that event, you have a 75% chance of being right if you switch to the other one. If the one you picked is eliminated, you're back to 50/50. So the 3 door trick will increase your chances, marginally. :) -------------------- Sex is easier than love. Posts: 1861 \| Registered: Friday, February 11 2005 08:00
Zeviz Nuke and Pave Member # 24	written Thursday, June 23 2005 22:21 Profile Homepage #38 quote: Originally written by i: If we have even more rigor to the problem as suggested. Suppose we have a multiplicative factor n. This determines how many times more money we could make. We pick an envelope with an amount k. The other envelope either has: k/n or nk inside of it. Note: in our previous example n was equal to 2. Now we can let this be arbitrary. If we do the same analysis as before, we get that the breakeven value of p is: p = 1/(1+n)* So for n = 10, we should pick the other envelope if the probability of making money (p) is greater than 1/11 or about 9 percent. This is the correct calculation from purely statistical point of view. However, if we are dealing with such huge values of n, non-linear utility of money that Thuryl mentioned becomes a very large factor. The most extreme examples is: If n = 1000 and k = 1 million dollars. If we use statistics alone, we'd have to say that switching is good as long as p > 0.1%. However, in reality most people wouldn't want to risk not getting 1 million dollars for a 0.1% chance to win 1 billion dollars. (Most people wouldn't even know how to spend a billion dollars.) Another factor to consider is that people are generally risk-averse. And since larger n means larger risk, people would be even less likely to take statistically correct action. Let's test this: 1. A question for those who said that they would keep 50,000$: Would you switch if the other enevelope had a 50/50 chance of containing either 45,000$ or 55,000$ ? 2. A question for those who said that they would switch: Would you still switch if the other envelope had a 50/50 chance of containing either 9,000$ or 100,000$ ? [ Thursday, June 23, 2005 22:31: Message edited by: Zeviz ] -------------------- Be careful with a word, as you would with a sword, For it too has the power to kill. However well placed word, unlike a well placed sword, Can also have the power to heal. Posts: 2649 \| Registered: Wednesday, October 3 2001 07:00
Dolphin Shaper Member # 5437	written Thursday, June 23 2005 22:31 Profile #39 quote: 1. A question for those who said that they would keep 50,000$: Would you switch if the other enevelope had a 50/50 chance of containing either 45,000$ or 55,000$ ? In that case I might consider it depending on mood since the gain/loss in not as great as it was previously. [ Thursday, June 23, 2005 22:32: Message edited by: Dolphin. ] -------------------- Nena Posts: 2032 \| Registered: Wednesday, January 26 2005 08:00
Thuryl ...b10010b... Member # 869	written Thursday, June 23 2005 22:58 Profile Homepage #40 quote: Originally written by Ash Lael: Say you mentally pick one option randomly, then use the 50/50. There is a 50% chance that the one you picked won't get eliminated. In that event, you have a 75% chance of being right if you switch to the other one. If the one you picked is eliminated, you're back to 50/50. So the 3 door trick will increase your chances, marginally. :) Not really. The 3-door trick only works because there's a guarantee that the option you pick won't be the one eliminated. -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 \| Registered: Saturday, March 30 2002 08:00
Ash Lael E Equals MC What!!!! Member # 5491	written Friday, June 24 2005 03:24 Profile Homepage #41 But there's a 50% chance in the Millionaire situation... so sometimes you're still stuck with a 50/50, but sometimes you get a 75% chance... right? -------------------- Sex is easier than love. Posts: 1861 \| Registered: Friday, February 11 2005 08:00
Thuryl ...b10010b... Member # 869	written Friday, June 24 2005 03:31 Profile Homepage #42 No, because unlike in the 3-doors situation, the options that are removed are completely independent of the option you chose. The chance of picking a wrong answer which is not then removed on your first guess (and thus being right by switching) is exactly the same as the chance of picking a right answer on your first guess (and thus being wrong by switching), since only one wrong answer isn't removed and which wrong answer isn't removed is random. [ Friday, June 24, 2005 03:36: Message edited by: Thuryl ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 \| Registered: Saturday, March 30 2002 08:00
Ash Lael E Equals MC What!!!! Member # 5491	written Friday, June 24 2005 03:38 Profile Homepage #43 ... Damn you. I should steer away from maths. It toys with my emotions. One moment you feel invulnerable, the next you're scared and confused and wondering what's real in the world. -------------------- Sex is easier than love. Posts: 1861 \| Registered: Friday, February 11 2005 08:00
Thuryl ...b10010b... Member # 869	written Friday, June 24 2005 03:42 Profile Homepage #44 At least probability theory is a relatively easy and practical area of mathematics. This is a lot more fun than integration, at any rate. -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 \| Registered: Saturday, March 30 2002 08:00
Ash Lael E Equals MC What!!!! Member # 5491	written Friday, June 24 2005 03:53 Profile Homepage #45 Just so long as we don't get into the whole 0.9999... = 1 thing. ;) -------------------- Sex is easier than love. Posts: 1861 \| Registered: Friday, February 11 2005 08:00
Drakefyre Triad Mage Member # 7	written Friday, June 24 2005 04:02 Profile Homepage #46 But that makes perfect sense! I would not switch if it was 9000 or 100000. When you don't know the contents of the envelope you've chosen, the calculations are different from when you know that you have 50,000 straight out there. -------------------- "At times discretion should be thrown aside, and with the foolish we should play the fool." - Menander ==== Drakefyre's Demesne - Happy Happy Joy Joy desperance.net - We're Everywhere ==== You can take my Mac when you pry my cold, dead fingers off the mouse! Posts: 9436 \| Registered: Wednesday, September 19 2001 07:00
Drew By Committee Member # 4233	written Friday, June 24 2005 06:30 Profile #47 Not knowing about the historical size of prizes on the show, but knowing that I held $50k in my hand, I would take the other envelope. Why? Because this prize is over and above what I've budgetted to live my life on (a.k.a. my earnings and regular investments). Given that it's completely superfluous, I may as well go for as much as possible, and not worry if I lose half of what I've got. After all, it would still be $25,000 more than I had two miutes before, at essentially no cost to me in time or effort. Posts: 2242 \| Registered: Saturday, April 10 2004 07:00
Student of Trinity Electric Sheep One Member # 3431	written Friday, June 24 2005 06:51 Profile #48 Glad to see people having fun with this. How about another? Do people know the unexpected hanging? It's a bit older, so people who have heard it before could maybe hold off for a while, while others have their say. Prisoner is sentenced to death in a state where judges have arbitrary power. Judge's sentence is as follows: "You will be executed at dawn on one of the seven days of next week, but you will not know which day until noon on the day before." Prisoner sits in cell a couple of days before the fatal week begins, and reflects on the sentence. He notes that if he is so lucky as to survive through the first six dawns, he will then know at dawn on day 6 that day 7 will be his doom. But this would be contrary to the sentence, since he would know before noon. So the prisoner concludes that he cannot actually be executed on day 7. Prisoner then concludes that day 6 is out as well, for with day 7 eliminated, the same reasoning as before would now apply to day 6. And so on: with each last day eliminated by the perverse wording of the sentence, all the days are eliminated in order. Prisoner concludes that he cannot be executed at all, because the judge's perverse sentence is actually self-contradictory. Yahoo! At noon on day 3, the prisoner is informed that he will be executed at dawn on day 4. And so he is. The seemingly self-contradictory sentence was carried out flawlessly. What was wrong with the prisoner's analysis? -------------------- It is not enough to discover how things seem to seem. We must discover how things really seem. Posts: 3335 \| Registered: Thursday, September 4 2003 07:00
Dolphin Shaper Member # 5437	written Friday, June 24 2005 07:35 Profile #49 The prisoner assumed he would be hanged either Friday or Saturday. And if it is not on Friday, it must be on Saturday. And eliminated the possible days based on the previous assumption, which he has no rational basis to assume. Is that even close to right? [ Friday, June 24, 2005 07:36: Message edited by: Dolphin. ] -------------------- Nena Posts: 2032 \| Registered: Wednesday, January 26 2005 08:00