Two envelopes paradox
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Electric Sheep One
Member # 3431
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written Thursday, June 23 2005 09:54
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I am bored with electioneering, so. The game show host has two enevelopes. One contains a check for an unknown amount of money. The other contains a check for twice that amount, whatever it is. You, the player, choose one of the two, and open it to discover a check for $50,000. The host then asks whether you would like to keep that check, or exchange it for the check in the other envelope. Should you switch? (The paradox comes not so much in deciding what to do, as in justifying your choice.) -------------------- 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 |
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Guardian
Member # 3521
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written Thursday, June 23 2005 10:00
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I'd keep the $50,000 check. Greed is not a major vice of mine, and I've always liked a sure thing. I generally don't take major risks in life and although that manner of living may be a weakness, it's kept me out of physical and emotional harm, for the most part. I'm not a gambling man. Give me that $50,000 and let me get out of here. -------------------- Stughalf "Delusion arises from anger. The mind is bewildered by delusion. Reasoning is destroyed when the mind is bewildered. One falls down when reasoning is destroyed."- The Bhagavad Gita. Posts: 1798 | Registered: Sunday, October 5 2003 07:00 |
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Shaper
Member # 5437
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written Thursday, June 23 2005 10:05
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I would keep the original check for $50,000. Why: Being that the first check is a substantial amount of money I see no need to risk what may be in the mystery envelope. What besides curiosity or desire would make it necessary to risk losing all. That's not to say I don't take risks, but this particular case seems unnecessary. [ Thursday, June 23, 2005 10:06: Message edited by: Dolphin. ] -------------------- Nena Posts: 2032 | Registered: Wednesday, January 26 2005 08:00 |
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Shaper
Member # 73
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written Thursday, June 23 2005 10:08
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You wouldn't risk losing all. You'd risk losing half. I'd keep it. -------------------- The Lyceum - The Headquarters of the Blades designing community The Louvre - The Blades of Avernum graphics database Alexandria - The Blades of Exile Scenario database BoE Webring - Self explanatory Polaris - Free porn here Odd Todd - Fun for the unemployed (and everyone else too) Famous Last Words - A local pop-punk band They Might Be Giants - Four websites for one of the greatest bands in existance -------------------- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Posts: 2957 | Registered: Thursday, October 4 2001 07:00 |
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? Man, ? Amazing
Member # 5755
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written Thursday, June 23 2005 10:09
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I think the paradox is that you can use simple statistics to determine that you should always have the other envelope. I think it comes out to a probable 25% gain if you swap. But it doesn't matter in the end, because those numbers are wrong. My solution? What gameshow? What is the history of the gameshow? If you think that $50,000 is a lot to them, keep it. If it seems like not a lot to them, swap. The paradox here is remembering it how much they are willing to risk, not how greedy the contestant is likely to get. *this message sponsored by the half is better than nothing ad hoc support group* Posts: 4114 | Registered: Monday, April 25 2005 07:00 |
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Veteran*
Member # 5
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written Thursday, June 23 2005 10:09
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I'd go for the 100 grand. Posts: 455 | Registered: Tuesday, May 17 2005 07:00 |
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Triad Mage
Member # 7
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written Thursday, June 23 2005 10:09
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I would exchange it - the possible gain is $50,000, while the possible loss is only $25,000. $25,000 is still a lot of money, so getting the smaller envelope is no problem. And statistically speaking, the average amount of money gained from switching envelopes is 12,500. EDIT: ... added coherency [ Thursday, June 23, 2005 10:18: Message edited by: Drakefyre ] -------------------- "At times discretion should be thrown aside, and with the foolish we should play the fool." - Menander ==== Drakefyre's Demesne - Happy Happy Joy Joy desperance.net - We're Everywhere ==== You can take my Mac when you pry my cold, dead fingers off the mouse! Posts: 9436 | Registered: Wednesday, September 19 2001 07:00 |
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Nuke and Pave
Member # 24
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written Thursday, June 23 2005 10:13
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This is similar to the 3-door paradox. (where you choose a door, host opens one of the others and asks you to stay or switch) In our case, if you chose the other check and it's smaller, you lose 25,000$. If it's larger, you'll gain 50,000$. There is a 50% chance of either outcome. So the expected value of switching is -25,000*.5 + 50,000*.5 = 12,500$. This means that from statistics point of view, you are much better off switching. So I would switch, because the reward/risk ratio in this situation is better than almost any other investment with similar rewards. -------------------- 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 |
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? Man, ? Amazing
Member # 5755
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written Thursday, June 23 2005 10:23
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The $12,500 is wrong. If you don't open the envelope, and you are offered the other one instead, you should take it for the same reasons. But!!! Those reasons still exist when you are holding envelope 2 and glancing over at envelope 1. That is the paradox. As for the 3 door thing, that is different, and always switch your choice. Posts: 4114 | Registered: Monday, April 25 2005 07:00 |
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Nuke and Pave
Member # 24
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written Thursday, June 23 2005 10:31
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12,500 is your expected gain if you switch: Option 1: You keep 50,000. Option 2: 50% chance of getting 25,000. 50% chance of getting 100,000. Average payoff is 50% * 25,000 + 50% * 100,000 = 62,500. So your expected gain if you chose option 2 is: 62,500 - 50,000 = 12,500$. (In this context, "expected gain" means the average of gains you would expect if you repeated the situation "very many" times.) [ Thursday, June 23, 2005 10:32: 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 |
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? Man, ? Amazing
Member # 5755
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written Thursday, June 23 2005 10:56
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Imposible! Pretend that you don't get to look in the envelopes, that they are defined as holding amounts K and 2K. You do your math and figure out that you stand to gain K/4 if you swap. So you swap. If you swap again you statistically stand to gain K/4 again. By this logic, you are always losing if you don't take that other envelope. That is why I say $12,500 is garbage logic. Swap if you like, but I stand behind my decision to evaluate the resources of the game show. Edit- removed words in excess of what is necessary, retained vowels. *this message sponsored by jack and diane* [ Thursday, June 23, 2005 11:18: Message edited by: Jumpin' Salmon ] Posts: 4114 | Registered: Monday, April 25 2005 07:00 |
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Mongolian Barbeque
Member # 1528
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written Thursday, June 23 2005 11:14
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Of course, the true paradox is "How the heck did a moron like me to get on this show to begin with?" :eek: In all seriousness, though, this sounds like a mind-twisting version of The Price is Right. In a situation like this, you merely ask the audience, and stand there for five minutes looking desperately from one shouting face to another trying to decide who's yelling more loudly. :confused: Personally, if I found myself in this situation I'd shrug my shoulders, walk off stage, and try to get on a reasonable program like Jeopardy. -------------------- The A.E. van Vogt Information Site My Tribute to the Greatest Writer of the Science Fiction Golden Age Posts: 907 | Registered: Monday, July 15 2002 07:00 |
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Master
Member # 4614
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written Thursday, June 23 2005 11:27
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Well, the amounts of 50,000 and 100,000 are more "common", if you know what I mean in their grouping. However, this doesn't stop the other envelope from being 25,000. So, I think I'd keep the 50,000. At least you know exactly what you're getting and what you will get. After all, even as is, you'll still be lucky to have 25,000 of it after taxes. :P -------------------- -ben4808 For those who love to spam: CSM Forums RIFQ Posts: 3360 | Registered: Friday, June 25 2004 07:00 |
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Electric Sheep One
Member # 3431
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written Thursday, June 23 2005 12:14
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As far as I understand the problem myself, Jumpin' Salmon is exactly right. . The paradox here is that the analysis given by Drakefyre implies that you should always switch; and yet this analysis does not use any information about the amount of the first check. So it seems to imply that you should switch even if you hadn't seen the amount, which would be tantamount to just taking the other envelope in the first place. But everyone agrees that the initial choice is just random, and there is no way to give yourself any advantage in it. The resolution of the paradox that I know starts from JS's scheme to base your choice on your knowledge of the probable range of prizes. If you knew about the probably range of prizes, finding the $50K would change the probability of the other check being greater or less, away from being 50/50, to being something else. So the subtle point is that, even if you don't know the probable range of prizes, finding the $50K changes the probability that you must assume away from 50/50, to something unknown. So if you don't know enough about the game show, logic does not dictate that the smartest move is always to switch. Logic says that in this case, you're on your own: go with your gut. In effect, this paradox is a vindication of Donald Rumsfeld's distinction between known and unknown unknowns. -------------------- 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 |
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Warrior
Member # 3629
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written Thursday, June 23 2005 15:48
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The problem with Draykfire's logic is that you are already garaunteed the lesser amount no matter what. You just don't know what that lesser amount is. Hence, if you do decide to swap, you stand to lose everything you've gained (if the 50K was the greater amount) or you stand to double (if the 50K was the lesser). Posts: 51 | Registered: Friday, October 31 2003 08:00 |
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The Establishment
Member # 6
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written Thursday, June 23 2005 16:19
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quote:Yes, I believe that is exactly what Zeviz is saying. You are always, statistically speaking, "losing" if you keep the original envelope. This is how game theory and risk analysis work, valid sciences used in finance and industry today. Let's assume for the argument that the game show value of k is expected and could go either way with equal probability. All other factors that could be concocted average out such that there is an equal probablility of being higher or lower. With this, if you did the envelope experiment an infinite number of times and selected the other envelope, you would come out k/4 dollars ahead on average for each selection. -------------------- Your flower power is no match for my glower power!! Posts: 3726 | Registered: Tuesday, September 18 2001 07:00 |
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Warrior
Member # 3629
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written Thursday, June 23 2005 16:42
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quote:Then why not just choose the other envelope first? Posts: 51 | Registered: Friday, October 31 2003 08:00 |
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...b10010b...
Member # 869
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written Thursday, June 23 2005 16:55
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quote:The above analysis cannot possibly be right. Consider 100 different people all on the show with the same prize -- they don't know about each other, so they don't know that the prizes are (for the sake of argument) $25000 and $50000. Initially, we expect 50 of them to get $25000 and 50 to get $50000. They all switch, because that, supposedly, is the logical thing to do. After everyone switches, 50 have $50000 and 50 have $25000. Clearly, switching every time does not increase the amount of money you can expect to win. -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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E Equals MC What!!!!
Member # 5491
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written Thursday, June 23 2005 16:59
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My brain just exploded. That's awesome. -------------------- Sex is easier than love. Posts: 1861 | Registered: Friday, February 11 2005 08:00 |
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Off With Their Heads
Member # 4045
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written Thursday, June 23 2005 17:05
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The reason is that we don't know how much money is at stake. Stareye's analysis lets the dollar values be $25,000, $50,000, or $100,000. Thuryl's lets the dollar values only be $25,000 or $50,000. -------------------- 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 |
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Master
Member # 1046
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written Thursday, June 23 2005 17:11
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I'd keep it. $50 000 is already a lot of money - just enough to pay for tuition, public transit and lunch. -------------------- Polaris - Weather balloons, ninjas, and your big daddy Wise Man. What more could you want? Undead Theories - Don't Ask, Don't Tell Posts: 3323 | Registered: Thursday, April 25 2002 07:00 |
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E Equals MC What!!!!
Member # 5491
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written Thursday, June 23 2005 17:13
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quote:Ah, I get it now. Still pretty cool. And I'd swap, since the maths has proved it right. ;) EDIT: 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? [ Thursday, June 23, 2005 17:16: Message edited by: Ash Lael ] -------------------- Sex is easier than love. Posts: 1861 | Registered: Friday, February 11 2005 08:00 |
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...b10010b...
Member # 869
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written Thursday, June 23 2005 17:13
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That's the thing, though -- the values are what they are. My example works exactly the same if you assume the values are $50000 and $100000; in that case, half the people playing the game will lose $50000 and half will gain $50000, and still nobody is better off. What you can't do is say there's a 50% probability that the envelopes contain $25000 and $50000 and a 50% probability that the envelopes contain $50000 and $100000, because the value of what's in the envelopes is already decided. I'm pretty sure the source of the "paradox" is a misuse of probability theory. -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Off With Their Heads
Member # 4045
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written Thursday, June 23 2005 17:22
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But that's the thing: if you do this an infinite number of times, you're still not going to know, on any given iteration, whether the dollar amounts in the envelopes is 25K and 50K or 50K and 100K. I'm not sure if I can explain this clearly. But the key here is that in order to have exactly the same situation a second time, you have to let the dollar amount in either envelope stay variable. That is, you see the 50K the first time, trade envelopes, and get 25K. The second time, when you see the 50K and trade envelopes, you could get 100K. That's why it's not the same. -------------------- 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 |
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...b10010b...
Member # 869
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written Thursday, June 23 2005 17:25
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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. quote:Ah, but there's the rub. The second time you play, you could just as easily see $100,000 the first time. Since you have no way of knowing that the values are $100000 and $200000 instead of $50000 and $100000 (since the strategy relies on one set of envelopes being independent of the previous set), you switch and get $50000. [ Thursday, June 23, 2005 17:43: Message edited by: Thuryl ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
