The Nation Has Problems 6
Posted by GreekHouse on Monday, November 26th, 2007 at 8:00 pm
TNHP went on vacation last week for Thanksgiving and I'm probably going to put it on hold for a couple more weeks while I finish up school for the term. In the meantime, hopefully this will hold you over.
Rather than my standard discussion+problems, I will do something different today. I will first present a classic paradox in probability theory and then have a competition.
The Paradox
There are two envelopes each containing a check. The first check is for m dollars and the second is for 2m dollars. Each player then looks inside and sees how much money they have. The players are then offered the opportunity to switch envelopes.
The first player sees that he has x dollars. By not switching, his expectation is clearly x dollars. If he picked the larger of the two envelopes and switches, he will then end up with x/2 dollars after the switch. On the other hand, if he chose the smaller of the two envelopes, he will end up with 2x dollars after the switch. He then reasons that since he had a 1/2 probability of picking either envelope, his expectation from switching is
1/2*x/2 + 1/2*2x = 5x/4
Since 5x/4 > x he will profit from switching.
The paradox is that the second player has also made the same calculation and come to the same conclusion. However, both players cannot possibly profit from the switch since switching is a zero sum game.
The Competition
In honor of my stickandballguy.com email account finally working, I'm having a contest which involves anyone who is interested sending me an email there (username: greekhouse). The winner of the contest wins 500 SBG bucks and will have their name used in a future TNHP problem.
Each person who enters will send me a positive integer. Make sure you know what a positive integer is. For those of you who don't, it's a number which is both positive and an integer. Surprisingly, I did this with my Calc II students last Spring and many of them wrong down things that weren't positive integers. Most notably, people wrote down 0 and things like 0.0000000001x10^-999999999.
The person who writes down the smallest positive integer that is not written down by another person is the winner. For example, if the entries are 1,2,3, then the winner is the person who sent me 1. If the entries are 1,1, 999, the winner is the person who sent me 999.
To be honest, I'm not sure what an optimal strategy is for this game. It is possible that it depends on the number of players and might be different if there is an unknown number of players (which is the case in this instance). It is also possible that an optimal strategy doesn't exist in either case. I'd be curious to hear explanations for people's choices in addition to their numbers, so feel free to include this in the email. You may also post strategic ideas in the comments if you do not care that other people will be stealing your ideas.
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For the record. SBG bucks have no cash value.
You mean you can't buy any of that crap on AG's site with SBG bucks? Awww, man.
Sorry to disappoint, but I don't have any money to give away and I'm pretty sure the boss doesn't want me giving away his money either.
Not even an extra five minutes of lunch?
Sure, why not? In fact, take as long as you want.
Do SBG Bucks come with a converstion rate to GH Nickles?
It's times like this when I wish I had A Beautiful Mind. I wouldn't even mind the delusions, since I get lonely from time to time.
This is an actual conversation Q and I had once:
Me: Damnit! This sudoku sucks. I'm just gonna pull a John Nash and stare at it till it solves itself!
Q: John Nash? There are a few differences.
Me: Like?
Q: Well...he has a Nobel Prize.
Me: I'm working on it. First step, spontaneous puzzle solving.
It's times like this when I wish I had A Beautiful Mind.
Not to mention the 10M kroner you could spend on yourself and taking out the delusions for lunch.
Hmmm...I'm not sure if this really counts as a solution or anything, but I guess I'll go ahead and white it out anyway.
My phone's got a camera, it's built right in
But it's hard to keep the dirt and grease off the lens
The last time you were happy since so long ago now
I tried to take a picture but it didn't come out
And the messages sent are almost as blurred
My cryptic printstyles dials to songs no one’s heard
If we keep this up, things will never get better
When we disagree we fight in capital letters
I have to type eleven numbers into my cell phone
Just to make it spell ‘love’ so I usually don’t
And it takes up fifteen digits to spell out ‘goodbye’
But if I leave out the ‘good’ I can save us some time
I'm not sure exactly what goes wrong with the calculation as you presented it, but here's my stab at an explanation. Although it is true that the contestant (I will use the first person for brevity) will have either x/2 or 2x dollars after the switch, one of those amounts is a "fictitious" amount that isn't really an option. For instance, if I chose the envelope with m dollars, the probability of the other player having m/2 dollars in his envelope is zero, because the only possibilities are m and 2m. (I feel like this explanation is lacking something, but can't put my finger on it.)
So, I'm not 100% sure what is wrong with calculating the expectation the way that you did, but it seems to me that the proper way to calculate the expectation is to say that there is a 1/2 chance that I lose m dollars in the switch and a 1/2 chance that I gain m dollars in the switch, for a total expectation of 0.
I guess another way to state this would be to say that there's a 50% chance that x=m. In that case, by making the switch, there is a 100% chance that I gain m dollars. There's a 50% chance that x=2m. In that case, by making the switch, there is a 100% chance that I would lose m dollars.
55566688833
No, m is an unknown quantity.
Of course m is an unknown quantity. But I still know that I have m or 2m, and the other contestant has 2m or m, and I still still know that m-m=0. None of that requires actually knowing what m is.
Yes, this would be correct (and it is a valid way to compute the EV of switching), but it doesn't explain the paradox in the way described above which seems to be valid.
My love is as a fever, longing still
For that which longer nurseth the disease,
Feeding on that which doth preserve the ill,
The uncertain sickly appetite to please.
My reason, the physician to my love,
Angry that his prescriptions are not kept,
Hath left me, and I desperate now approve
Desire is death, which physic did except.
Past cure I am, now reason is past care,
And frantic-mad with evermore unrest;
My thoughts and my discourse as madmen's are,
At random from the truth vainly express'd;
For I have sworn thee fair and thought thee bright,
Who art as black as hell, as dark as night.
I said that it's not valid because you're including possibilities in your expectation value calculation that can never occur, but you didn't comment on that part of my initial response. (If x=m, then x/2 is an invalid amount, and if x=2m, then 2x is an invalid amount, but regardless of whether x=m or x=2m, we know that one of the amounts in your EV calculation is invalid.)
Perhaps a better way to explain this is to say that we know that the unkown envelope can have either have m or 2m in it. Therefore, the ratio of the potential possibilities had better be 2. By saying that our possible outcomes are x/2 and 2x, you are claiming that the ratio of the potential outcomes is 4, which would seem to contractict the assumption that the two envelopes either have m or 2m in them.
At any rate, we know that any answer that depends on x is going to be wrong. Knowing the amount in one envelope gives us no information about whether or not to make the switch. The EV is 3m/2 regardless of whether you switch or not, because your choices are 50% m and 50% 2m until you know two of the three of the following: the amount in the first envelope, the amount in the second envelope, and m.
To clarify my response here. Ubelmann's answer makes the assumption that m is a known quantity.
To make an analogy, suppose you're dealt the 3 of clubs from a deck of cards and your goal is to get as big a card as possible. You're then offered a chance to swap your card with the top card on the deck. In terms of deciding whether or not you should switch, you treat the value of the top card as unknown and proceed from there. Of course, if you know what the top card on the deck is, then the problem is easy. You switch if it's higher and don't switch otherwise.
What ubelmann's answer does is assume you know what m is. This is akin to assuming you know what the top card is in the above example and is not really in the spirit of the problem.
In no place in my solution did I assume that I know what m is. I know that there are two envelopes. One envelope has m and the other envelope has 2m, where m is some value that I do not know. Therefore, I know that the difference between the two envelopes is either +m or -m, where m is still some value that I do not know. I know that by switching I will either increase my wealth by m or decrease my wealth by m, even though I still have no idea what the value of m is. Therefore, my expectation is that by switching I have a 50% chance to increase by m (an unknown amount) and a 50% chance to decrease by m (an unknown amount), so that my total expectation is zero, independent of m and independent of x.
Oops, I misread your solution the first time.
The other comment above is valid though. We're not looking to compute the solution in terms of m, we're trying to figure out what is wrong with the above analysis.
Moss can't recall how to white-out, so don't read on if you don't want any of your fun spoiled!
In the Paradox, it may be true that both come up with the indefinite "5x/4" expectation for switching. However, the two quantities 5x/4 are NOT equal, because the two contestants have a different x in their envelopes.
So to the extent the "Paradox" lies in the fact that both contestants have the same expectation in switching, they do not. The contestant with the higher initial value has an apparent expectation that is 2x the apparent expectation of the other; however, neither knows what the expectation of the other is.
The better way to look at it is that each is making a bet of 1/2 x with a 50/50 chance of winning x in return. [He can't end up with less than 1/2 of x, but has a 50/50 chance at getting 2x.] So in actuality, it is a good bet to make (regardless of x). Each player should willingly make the switch, in fact.
And that is where the paradox lies. Of course, it's a really good bet when you're playing with house money, which seems to be the case...
SBG to the rescue.
Shall I compare thee to a summer's day?
Thou art more lovely and more temperate:
Rough winds do shake the darling buds of May,
And summer's lease hath all too short a date:
Sometime too hot the eye of heaven shines,
And often is his gold complexion dimm'd;
And every fair from fair sometime declines,
By chance or nature's changing course untrimm'd;
But thy eternal summer shall not fade
Nor lose possession of that fair thou owest;
Nor shall Death brag thou wander'st in his shade,
When in eternal lines to time thou growest:
So long as men can breathe or eyes can see,
So long lives this and this gives life to thee.
So in actuality, it is a good bet to make (regardless of x).
False. It's not a bad bet to make, but it is not a good bet, either.
If I pick an envelope at random, don't examine its value, and trade it for the other envelope, my EV is 3m/2. (The envelope I get is m with 50% probability and 2m with 50% probability.)
If you pick an envelope at random (from a completely independent pair of envelopes), do examine its value, and trade it for the other envelope, your EV is 3m/2, not 15m/8.
You and I get the same envelope regardless of whether or not we look at the value inside, so our EV's must necessarily be the same. Knowing x can't help you in your decision-making process unless you also know what m is or you also know what the value in the other envelope is. Any answer that depends on x is therefore going to be wrong.
I fight for the unconventional
My right, and its unconditional
I can only, be as real as I can
The disadvantage is
I never knew the plan
This isn't just the way to be a martyr
I can't, walk alone any longer
I fight, for the ones who can't fight
And if I lose, at least I tried
We are the new diabolic
We are the bitter bucolic
If I have to give my life you can have it
We are the pulse of the maggots
To be honest, I don't really expect anybody to get this one completely right. What ubelmann says is correct. Your EV should be 3m/2 in this case. The problem is that neither person knows what m is and they want to try and figure out their EV without knowing m.
It is possible in some cases to compute your EV based on x. For instance, if you know the value of m, then you already know if you have m or 2m. At that point, you know your EV from switching. If x=m then your EV from switching is 2x. If x=2m then your EV from switching is x/2.
This is the simplest case, but there are other ways you can compute your EV without knowing exactly what m is. There are basically two things that can go wrong with the above argument. The first is that the person made incorrect assumptions when computing his EV. The second will remain a mystery for now. The problem above is stated informally. If the problem is stated formally, then there is no contradiction--one of the two things must go wrong.
I won't-be the inconsequential
I won't-be the wasted potential
I can make it-as severe as I can
Until you realize
You'll never take a stand
It isn't, just a one-sided version
We've dealt, with a manic subversion
I won't, let the truth be perverted
And I won't leave another victim deserted
We-we are the new diabolic
We-we are the bitter bucolic
If I have to give my life you can have it
We-we are the pulse of the maggots
Do you understand?
Bah. I remember why I didn't like ubelmann's initial answer now. The problem is quite confusing and I managed to confuse myself.
The main problem is that there was an assumption that m is fixed. If m is fixed, then we can essentially treat it as a known value. This is fine if you want to take an omnipotent view of the problem.
The problem is that the person doing the calculation can't say that his expectation is 3m/2 because m is still unknown and can take on 1 of 2 possible values. I don't want to say too much more than that because it would give away the solution. I have a weird example where it is possible for it to be +EV for both players to switch.
I've made a mess out of all of my explanations here, so I'm sorry. I will leave the problem open for now, but will come back with a thorough breakdown later.
I wrote her off for the tenth time today
And practiced all the things I would say
When she came over I lost my nerve
I took her back and made her dessert
Now I know I'm being used
But that's okay man cause I like the abuse
I know she's playing with me
But that's okay cause I've got no self esteem
The main problem is that there was an assumption that m is fixed. If m is fixed, then we can essentially treat it as a known value. This is fine if you want to take an omnipotent view of the problem.
I don't see where there is any omnipotence required.
The only reason the contestant might possibly think that the potential outcomes are x/2 or 2x is that he knows that the ratio between the two amounts in the envelopes is 2. If he knows that that ratio is two, then he can deduce without any omnipotence that the amount in one envelope is m and the amount in the other envelope is 2m.
If you don't like the m, just give the two envelopes arbitrary values A and B. There's a 50% chance that the contestant gains (A-B) and a 50% chance that the contestant loses (A-B). Thus, the EV of the switch is (A-B)-(A-B)=0. There's nothing mystical or omnipotent about knowing there are only two values.
With the arbitrary values A and B, the apparently paradoxical EV result becomes 1/2*(x*A/B+x*B/A). This is a silly thing to write, though, because you know there are only two amounts (A, B) and not three amounts (x, x*A/B, x*B/A). Therefore, if you are going to multiply x by A/B, you know that x=B. If you are going to multiply x by B/A, you know that x=A. That's a simple deduction from the information given to you at the outset of the problem (that there are two envelopes), not omnipotence.
I think "omniscient" is the word, not "omnipotent".
Yes, omniscient is the word I should have used.
And I know that you tried to hide
To center yourself, but
WAS I THE BEAST THAT SUCKED INTO YOU?
A REAL DARK BITCH DOWN INSIDE OF YOU?
We got a problem, it's plain to see
Bitch, we got a problem
Ok, I think it's time for me to just clear this up a little bit. It is hard for me to explain why ubelmann's idea isn't what I want without formalizing the problem a little bit.
Both x and m are random variables (although clearly not independent). The expectation of the other envelope is also another random variable. So the real question being asked is "What is the expectation of the other envelope given x?" So we can use conditional probabilities and our expectation is not always 0. Of course, the sum (or integral) over all possible situations will be 0.
For example, suppose that m=2,4,or 8 all with probability 1/3. We then look at our envelope and see $2, we know that we should attempt to switch because our expectation from switching is $4. Likewise, if we see $16 we should not switch since our expectation is $8. Now suppose we have $8. Then m=4 or m=8 each with equal probability. Then our expectation from switching is 1/2*(4+16)=$10 and so we should switch.
However, we could really be better with our analysis in the last step. The EV calculation was assuming that our opponent would switch no matter what. Obviously our opponent is not going to switch if he has $16 unless he's a complete idiot, so our real expectation from switching is $4. Thus, we shouldn't switch. If you look at the above example when m=4, then it should at least appear superficially that both players have a +EV move from switching.
In our initial problem, we have no knowledge of the distribution of m, but our player made the assumption that it was equally likely that he either has m or 2m dollars, which is not correct.
False. It's not a bad bet to make, but it is not a good bet, either.
With all due respect, ubes, Moss thinks you are wrong.
(Moss is making the following assumption, which actually isn't stated in the problem: the players know that one envelope holds twice as much as the other. Moss takes it to be a good assumption, because otherwise the player is simply trading one value for another, with no knowledge of expectation or anything.)
Once you open the envelope, you know how much you stand to win and to lose by switching. What you don't know is the total amount of money on the table. The bet is whether the total amount is 3x or 3x/2. You can put up x/2 with an even chance of winning back x in return, while your other x/2 is protected.
If someone gave you the opportunity to bet $1 on a coin flip, with the chance to win back $2 (in addition to the $1 you bet), would you take the bet? (Moss hopes you would.) That is what the situation is here, as explained above.
And that doesn't depend on x. X only comes into play because the envelope's been opened -- it puts a real number on what you can win or lose, but the soundness of the bet doesn't depend on x. You have the choice of putting up 1/2 for a 50/50 chance of winning 1. That's a good bet.
The feature that makes the problem unique is that the two players are not putting up the same bet. Obviously one player is "putting up" twice as much as the other, but neither knows which is doing what. Of course, neither player is actually putting up anything, because the outcome is already determined -- but neither player knows what the outcome is.
Now here is a problem with the original statement of the Paradox: "Since 5x/4 > x he will profit from switching."
In actuality, a more correct statement is that "on average, he will profit from switching." Obviously not both players can profit. That's why it is significant that one player's 5x/4 is different from the other player's 5x/4.
Or better yet, 5x/4 is > 0, so on average he profits...because he is playing with house money in the first place.
All you have to do is try the experiment, Moss. You will not benefit by switching.
Moss would benefit 1/2 of the time. And the potential benefit is 2x the potential loss.
It seems like you've already forgotten the paradox. If you're playing against me and I make the same calculation, then I can benefit too! But it's impossible that we can both benefit at the same time.
Here's another example. Suppose neither of us look in the envelopes. If your envelope contains x dollars, then you do the switch for an expectation of 5x/4 dollars. Then we are offered the chance to switch again, which we gladly do. Our expectation is now 25x/16 dollars. Of course, we're now sitting there with the same envelope that contains x dollars.
GH may be a mathematician, but I am a physicist, so I will attempt to convince you through an experiment.
In the first column is the envelopes that you open up. The second column (cleverly hidden by the powers of html) represents the envelope that you do not have. Import the data into your favorite spreadsheet program, and total the first column. This is x--it is the amount that you would have if you never switched envelopes. Then total the second column--the amount that you get by switching every time--and let me know whether it is closer to x or 5x/4. If this is not enough evelopes to convince you, the OpenOffice random number generator is happy to supply you with more envelopes.
(Also, if I keep my physics hat on for a second, I will say that it is clear by symmetry that switching will not help you. First I give out the envelopes to the two players and no one is allowed to look at their values. Clearly neither set is preferable to the other at this point. Now you get to look at the values in your envelope, but the other player does not. Nothing has changed--the total amount that each player has is the same as before you looked in your envelopes. The information you have gained is not useful--there is still no reason to believe that either set of envelopes is worth more than the other. Switching is, on average, a neutral proposition.)
71.8 143.6
79.5 159.0
40.3 80.6
53.5 107.0
61.2 122.4
19.4 9.7
152.0 76.0
51.2 102.4
24.5 49.0
10.6 21.2
38.8 19.4
86.8 43.4
70.2 140.4
66.8 133.6
140.0 70.0
112.0 56.0
81.6 163.2
99.1 198.2
153.0 76.5
37.6 75.2
56.2 28.1
13.3 26.6
79.4 158.8
42.0 21.0
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97.6 48.8
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49.6 24.8
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185.8 92.9
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172.0 86.0
1.8 3.6
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124.0 62.0
106.0 53.0
67.5 135.0
145.2 72.6
72.3 144.6
21.8 10.9
42.8 85.6
78.2 39.1
80.8 161.6
197.6 98.8
41.4 20.7
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150.8 75.4
116.6 58.3
16.2 8.1
88.6 177.2
75.6 37.8
29.1 58.2
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144.6 72.3
83.7 167.4
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33.4 66.8
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79.8 159.6
74.0 148.0
9.9 19.8
65.8 131.6
151.6 75.8
52.8 105.6
55.6 27.8
28.8 57.6
20.3 40.6
0.4 0.8
1.1 2.2
31.2 62.4
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40.2 80.4
24.0 12.0
84.0 42.0
60.9 121.8
92.0 184.0
34.8 17.4
19.3 38.6
0.2 0.1
6.0 12.0
114.0 57.0
29.2 58.4
64.9 129.8
43.6 87.2
3.0 1.5
48.6 97.2
90.9 181.8
69.8 34.9
25.3 50.6
70.7 141.4
17.8 8.9
70.4 35.2
36.5 73.0
84.6 169.2
176.2 88.1
128.6 64.3
71.2 142.4
71.6 143.2
41.2 20.6
179.2 89.6
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58.2 29.1
181.0 90.5
82.4 41.2
77.0 154.0
165.2 82.6
68.4 34.2
12.2 6.1
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138.4 69.2
3.2 6.4
49.0 24.5
34.6 17.3
101.4 50.7
66.4 132.8
21.0 10.5
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48.8 97.6
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11.4 22.8
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17.2 8.6
1.8 0.9
33.4 16.7
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92.1 184.2
113.2 56.6
90.2 180.4
36.5 73.0
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184.0 92.0
43.3 86.6
29.4 58.8
58.8 117.6
11.3 22.6
29.4 58.8
49.6 99.2
36.4 72.8
186.4 93.2
24.3 48.6
164.8 82.4
47.4 94.8
84.0 42.0
70.0 35.0
97.1 194.2
5.0 10.0
48.1 96.2
28.0 14.0
62.8 125.6
95.0 47.5
62.6 125.2
27.6 13.8
93.3 186.6
98.2 196.4
93.2 186.4
66.3 132.6
62.7 125.4
72.1 144.2
82.5 165.0
127.6 63.8
39.3 78.6
91.2 182.4
40.0 80.0
172.6 86.3
32.4 16.2
1.1 2.2
134.4 67.2
8.7 17.4
152.4 76.2
47.0 94.0
22.8 45.6
22.8 45.6
48.7 97.4
83.2 166.4
84.4 42.2
13.1 26.2
100.6 50.3
74.0 37.0
90.2 45.1
153.8 76.9
80.8 40.4
112.8 56.4
24.2 12.1
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159.6 79.8
186.0 93.0
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23.4 46.8
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37.2 18.6
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35.5 71.0
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145.8 72.9
129.0 64.5
157.8 78.9
27.2 54.4
105.4 52.7
117.2 58.6
27.1 54.2
28.9 57.8
99.5 199.0
86.6 43.3
36.7 73.4
88.3 176.6
92.4 184.8
150.0 75.0
79.8 159.6
28.2 14.1
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1.6 3.2
120.6 60.3
19.4 38.8
64.4 128.8
67.1 134.2
94.7 189.4
32.0 64.0
53.2 26.6
7.0 3.5
22.6 11.3
62.8 125.6
13.6 6.8
150.6 75.3
67.5 135.0
119.0 59.5
8.8 17.6
158.6 79.3
17.2 8.6
170.0 85.0
12.1 24.2
91.6 45.8
30.1 60.2
119.0 59.5
79.6 39.8
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37.2 74.4
90.0 45.0
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194.2 97.1
93.1 186.2
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199.8 99.9
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43.6 21.8
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42.8 85.6
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137.6 68.8
29.4 14.7
113.6 56.8
104.2 52.1
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24.8 49.6
5.6 2.8
166.2 83.1
22.0 44.0
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31.5 63.0
48.0 96.0
32.9 65.8
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5.2 2.6
72.6 145.2
66.4 33.2
60.6 30.3
49.0 24.5
2.4 1.2
41.0 82.0
60.6 121.2
2.4 4.8
4.7 9.4
52.7 105.4
33.5 67.0
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49.1 98.2
76.3 152.6
68.8 137.6
83.3 166.6
31.0 62.0
135.8 67.9
50.7 101.4
37.4 74.8
194.6 97.3
13.5 27.0
96.6 48.3
26.0 52.0
16.7 33.4
198.2 99.1
109.2 54.6
39.7 79.4
5.2 2.6
27.7 55.4
51.9 103.8
136.0 68.0
10.7 21.4
72.2 36.1
1.2 2.4
63.7 127.4
92.1 184.2
62.2 31.1
7.0 3.5
146.2 73.1
24.4 12.2
65.4 130.8
83.0 166.0
84.3 168.6
66.4 33.2
53.0 106.0
112.0 56.0
59.8 29.9
43.4 86.8
8.6 4.3
11.7 23.4
27.6 13.8
183.2 91.6
52.0 26.0
187.6 93.8
96.7 193.4
180.8 90.4
96.8 48.4
81.8 163.6
141.0 70.5
174.6 87.3
14.3 28.6
80.0 160.0
154.4 77.2
114.6 57.3
37.2 74.4
99.1 198.2
67.4 134.8
12.8 6.4
58.6 29.3
125.6 62.8
10.9 21.8
28.8 57.6
6.4 3.2
101.6 50.8
28.6 57.2
23.0 46.0
79.8 39.9
99.8 199.6
39.6 19.8
52.7 105.4
89.2 44.6
178.4 89.2
123.8 61.9
19.8 9.9
144.2 72.1
67.8 135.6
140.0 70.0
71.7 143.4
69.2 138.4
60.4 30.2
190.8 95.4
4.5 9.0
198.8 99.4
80.8 40.4
26.4 13.2
179.8 89.9
160.2 80.1
17.8 35.6
85.3 170.6
66.6 33.3
0.6 0.3
43.1 86.2
161.6 80.8
83.6 41.8
62.8 125.6
136.0 68.0
162.6 81.3
98.3 196.6
159.2 79.6
52.2 26.1
90.9 181.8
114.2 57.1
6.7 13.4
68.6 34.3
24.9 49.8
116.6 58.3
140.0 70.0
6.8 3.4
71.4 142.8
48.7 97.4
92.5 185.0
187.8 93.9
45.8 91.6
95.0 190.0
61.4 122.8
59.1 118.2
169.8 84.9
83.0 41.5
153.8 76.9
140.4 70.2
74.8 149.6
77.2 154.4
13.3 26.6
111.2 55.6
19.0 38.0
152.2 76.1
23.5 47.0
0.3 0.6
74.4 148.8
3.2 6.4
26.4 52.8
65.3 130.6
60.3 120.6
33.1 66.2
99.6 199.2
170.6 85.3
182.8 91.4
69.6 139.2
177.4 88.7
62.8 125.6
36.8 18.4
81.1 162.2
56.8 113.6
64.2 128.4
76.1 152.2
18.2 36.4
46.6 23.3
61.0 122.0
59.6 119.2
0.4 0.2
31.2 62.4
34.4 68.8
77.4 154.8
44.4 88.8
180.0 90.0
192.8 96.4
20.5 41.0
27.0 13.5
96.7 193.4
189.8 94.9
16.7 33.4
46.0 23.0
120.4 60.2
154.0 77.0
112.2 56.1
59.8 119.6
124.6 62.3
47.5 95.0
29.4 58.8
51.0 102.0
20.8 10.4
47.8 95.6
32.1 64.2
134.4 67.2
12.0 24.0
8.2 16.4
170.6 85.3
70.4 35.2
69.2 138.4
45.0 90.0
70.8 35.4
0.8 0.4
158.8 79.4
12.8 25.6
89.6 44.8
138.8 69.4
9.2 18.4
130.6 65.3
82.9 165.8
5.9 11.8
120.4 60.2
99.6 199.2
28.9 57.8
20.3 40.6
76.6 153.2
85.0 170.0
80.1 160.2
77.8 38.9
32.6 65.2
19.0 9.5
89.9 179.8
85.8 42.9
114.6 57.3
44.0 22.0
10.1 20.2
I've got a winning proposition for ubelman's list. Assuming my opponents will acquiesce any time I want a trade I will employ the following 3 rules:
Actually, #3 on the list doesn't do what I wanted it to do, so it can probably be ignored. Also, any time I don't fall into one of the cases on the list, I can choose to trade or not trade.
This reminded me of The Nation Has Problems.
Classic.
That means E-6 is sending in "Mr. Billion" as his competition entry, right?
Moss is goin' with 15 also.
Nothin' like good film noir. And you gotta love Calvin & Hobbes.
If m > 1 month mortgage payment, I don't switch envelopes.
touché
Except you don't know the amount you have is m or 2m. What you have is x, which you do know. If you switch, you could receive back x/2 or 2x. So, basically, you're risking x/2 to get x more. Sounds like a good bet to me for either side. Of course, I'm sure RR meant that if x>1 month mortgage, he doesn't risk it.
The contest will remain open until around midnight on Sunday. I will continue to bump this thing every day until then.
In SBG Nation, you choose number.
In Soviet Russia, number chooses you!