The envelope problem, and a possible solution

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
You asked for something that makes common sense. How much common sense does your statement make?

[/ QUOTE ]
Remember, I've said from the beginning that there is no common sense answer.

At this point, I think we should just agree that your common sense is broader than mine. You can accept probabilities that change when you look at them, as a Bayesian does, and also mathematical formulae that are different depending on whether X is a number or a variable, as a Frequentist does.

I'd rather reject both and live with having no rigorous theory of probability that can also answer useful questions. Many people agree with me, many others choose the less distasteful (to them) of the two alternatives above. But you are rare in being willing to embrace both. For you, the envelope problem is no problem at all.

That probably sounds sarcastic, and maybe I meant it that way a little, but I admit I have no argument against it. You might well be right, I just hope you aren't.

[/ QUOTE ]


So would you take part 1 and part 2 in the Proposition Bet below? If so why and if not why not?

Look at this from a Gambler's point of view. You are told about the two envelopes and asked to choose one. You are offered these Proposition Bets.

1. Given 3-2 odds, would you be willing to bet $10 that your envelope contains the smaller amount?

2. Given 3-2 odds, would you be willing to bet the amount in your Envelope that it is the smaller amount?


1 is a good bet for you while 2 is not. Do you see why there is a difference between the two?


PairTheBoard

[AaronBrown]

[ QUOTE ]
So would you take part 1 and part 2 in the Proposition Bet below? If so why and if not why not?

1. Given 3-2 odds, would you be willing to bet $10 that your envelope contains the smaller amount?

2. Given 3-2 odds, would you be willing to bet the amount in your Envelope that it is the smaller amount?

1 is a good bet for you while 2 is not. Do you see why there is a difference between the two?

[/ QUOTE ]
I'm not personally insulted at being asked this question, but asking it does seem to imply anyone who thinks the envelope paradox raises important questions doesn't understand simple probability concepts. Is that your intent?

There are people who are genuinely confused about whether or not to switch envelopes, who might think the two bets above are the same. But it takes only logic, not probability theory, to see that the bet in (2) can be converted to an obviously poor bet of winning half the total amounts in the two envelopes if you have the smaller amount and paying two thirds of the total amount if you have the larger amount. That's not a mystery to me or anyone who thinks through it clearly.

The challenge is to come up with a consistent way of computing which bets to take. It's a lot harder than it looks. The envelope paradox is simplified to make the basic contradictions clear, which makes it easy to "resolve" with ad hoc techniques. But those techniques are not much practical help in realistic problems.

[PairTheBoard]

[ QUOTE ]
There are people who are genuinely confused about whether or not to switch envelopes, who might think the two bets above are the same. But it takes only logic, not probability theory, to see that the bet in (2) can be converted to an obviously poor bet of winning half the total amounts in the two envelopes if you have the smaller amount and paying two thirds of the total amount if you have the larger amount. That's not a mystery to me or anyone who thinks through it clearly

[/ QUOTE ]

So there's no mystery with the proposition bet? It doesn't produce a true paradox with equally strong arguments on both sides for why it might be a good bet? All it takes is simple logic to see why it's not a good bet? Yet the Envelope Paradox is essentially the same situation. Only in the Envelope "Paradox" the Proposition asks if you want to bet half the amount in your envelope giving you 2-1 odds.


PairTheBoard

[AaronBrown]

[ QUOTE ]
So there's no mystery with the proposition bet? It doesn't produce a true paradox with equally strong arguments on both sides for why it might be a good bet? All it takes is simple logic to see why it's not a good bet? Yet the Envelope Paradox is essentially the same situation. Only in the Envelope "Paradox" the Proposition asks if you want to bet half the amount in your envelope giving you 2-1 odds.

[/ QUOTE ]
I'm not sure why you keep belaboring this. I know the envelope switch is not a good bet, that's not the point. I don't believe that a friend and I can make each other infinitely wealthy by passing two envelopes back and forth all day (I tried, it doesn't work). You can, however, have two people get positive EV from opposite sides of a zero sum bet, if you tradie options with people who keep score in different currencies. But that's not the aspect of the paradox we've been discussing.

I think the hard part is to come up with a rigorous theory that spells out exactly when switching makes sense, that is also flexible enough to use for practical decision problems. You don't think that's hard, because you will accept things that I feel cannot be true and that are devilishly hard to apply. You're not alone in that, many people agree with you (although few are quite as broadminded as you, most limit themselves to one hard-to-swallow pill).

None of this makes me think you don't know basic logic or math, I don't know why you think that of me. If I were to get ad hominem, I'd bet you haven't done a lot of real world statistical analysis. Not because I think you've shown yourself incapable of it, but because the day-to-day experience of trying to apply your principles leads many practitioners to become agnostic like me.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
So there's no mystery with the proposition bet? It doesn't produce a true paradox with equally strong arguments on both sides for why it might be a good bet? All it takes is simple logic to see why it's not a good bet? Yet the Envelope Paradox is essentially the same situation. Only in the Envelope "Paradox" the Proposition asks if you want to bet half the amount in your envelope giving you 2-1 odds.

[/ QUOTE ]
I'm not sure why you keep belaboring this. I know the envelope switch is not a good bet, that's not the point. I don't believe that a friend and I can make each other infinitely wealthy by passing two envelopes back and forth all day (I tried, it doesn't work). You can, however, have two people get positive EV from opposite sides of a zero sum bet, if you tradie options with people who keep score in different currencies. But that's not the aspect of the paradox we've been discussing.

I think the hard part is to come up with a rigorous theory that spells out exactly when switching makes sense, that is also flexible enough to use for practical decision problems. You don't think that's hard, because you will accept things that I feel cannot be true and that are devilishly hard to apply. You're not alone in that, many people agree with you (although few are quite as broadminded as you, most limit themselves to one hard-to-swallow pill).

None of this makes me think you don't know basic logic or math, I don't know why you think that of me. If I were to get ad hominem, I'd bet you haven't done a lot of real world statistical analysis. Not because I think you've shown yourself incapable of it, but because the day-to-day experience of trying to apply your principles leads many practitioners to become agnostic like me.

[/ QUOTE ]

I was not trying to ridicule you by asking you to address the 2E Proposition Bet Paradox. I think the 2E Proposition Bet Paradox is a better version of the 2E Paradox because it strips away a couple of irrelevant factors that make for psychological muddy waters. In the 2E paradox the fact that you are being given the amount in the envelope tends to hide the fact that you are simply making a bet when you switch. Also, in the 2E paradox, since switching is EV neutral, you tend to feel less psychologically compelled to make a decision about what is best.

The 2E Proposition Bet Paradox is right there in front of you. Even a frequentist will take the first bet.

1) Given 3-2 odds bet $10 your envelope contains the smaller amount.

This is clearly a +EV bet. Even after opening the envelope, assuming you don't know the envelope amounts. Although the frequentist will say the outcome has been determined he will still put his money on what amounts to a 50-50 proposition with 3-2 odds in his favor. He will do this in the same way he will bet that a randomly dealt card having been dealt face down is the Ace of Spades if you give him 80-1 odds.

But as soon as you change the bet to 2)

2) Given 3-2 odds bet the amount in the envelope that it is the smaller amount.

any gambler worth his salt knows the bet is no good. Yet the same Paradox arises. You are being given 3-2 odds on what you just said was a 50-50 bet. That's got to be +EV.

When confronted with the 2E Proposition Bet Paradox you respond,

[ QUOTE ]
But it takes only logic, not probability theory, to see that the bet in (2) can be converted to an obviously poor bet of winning half the total amounts in the two envelopes if you have the smaller amount and paying two thirds of the total amount if you have the larger amount. That's not a mystery to me or anyone who thinks through it clearly.


[/ QUOTE ]

But with the original - But equivalent - Two Envelope Paradox you insist

[ QUOTE ]
But there's the other side as well, the one that says the probability that you have the smaller amount is 50% so the expected value of switching is positive. You can't refute a paradox by strengthening one side, that just makes the paradox more puzzling. You have to show why one side is wrong.


[/ QUOTE ]

It's easy to see why that side is wrong in the 2E Proposition Bet Paradox. It's because you're being asked to bet the amount in the envelope. The amount you bet also determines whether you win or lose. Bet a little and win. Bet a lot and lose. Not +EV even with 3-2 odds.

The 2E Paradox is equivalent as a Paradox. In the 2E Paradox you are being asked to bet half the amount in the envelope and are given 2-1 odds. The amount you bet determines whether you win or lose. Bet a little and win. Bet a lot and lose.

In my opinion this Does give the reason "why one side is wrong." And I think it makes common sense.


PairTheBoard

[AaronBrown]

Now I'm going to reverse myself and recombine the threads.

I think both Jason1990 and PairThe Board have redefined the original paradox in opposite ways.

Jason1990 does a thorough job of showing that no consistent prior distribution on T justifies the idea that the probability of switching is always 50%. However, there are consistent prior distributions that justify always switching, but they have infinite expectation for T. That's certainly a strong argument against switching, if it is made before T is selected.

But after T and H have been selected, and we look at X, all of that seems like abstract machinery, irrelevant to the problem at hand. I'm looking at $100, considerations of infinite expected value for T seem silly. I have no idea how T was selected, arguing from the assumption I know its probability distribution recalls the "assume a can opener" joke; the Bayesians get around that by asserting that I must have some subjective belief on the subject, but that has its own problems.

I'm not saying Jason1990 is wrong here, just that he drags into the problem a lot of formalism that is hard to apply to real decisions. If we ask him whether we should switch or not, instead of whether it's rational to always switch, none of the stuff he's added to the problem is useful. This is what I mean by hair-splitting. It offends my common sense that a simple problem like switch or don't switch requires such abstraction to avoid inconsistency. My intuition says that there's something Jason1990 (and I) don't know about the nature of probability, and that's why he wants to fence it in with abstractions and I maintain a mystical faith in common sense.

PairTheBoard, on the other hand, wants to throw out all the prior stuff and just have two envelopes, and he wants me to make a choice without seeing the amount in mine. The envelope paradox is that before I see the amount in my envelope switching is obviously EV neutral, but once I know the amount (that is, once I make PairTheBoard's proposition bet impossible) switching seems to be EV positive.

I think you're both dodging the boundary issue. If you start before T and H are determined, Jason1990's argument seems unassailable. If you ignore how T is selected, determine H but don't tell me what it is, PairTheBoard has the answer. But when precisely does the probability change? Is it when T is determined? When H is determined? When I know X? And what precisely is the nature of the thing "probability" that changes?

[PairTheBoard]

[ QUOTE ]
PairTheBoard, on the other hand, wants to throw out all the prior stuff and just have two envelopes, and he wants me to make a choice without seeing the amount in mine. The envelope paradox is that before I see the amount in my envelope switching is obviously EV neutral, but once I know the amount (that is, once I make PairTheBoard's proposition bet impossible) switching seems to be EV positive.


[/ QUOTE ]

NO! The Proposition Bet is a one time offer. So looking at the amount in the envelope does not make the bet impossible. This is another reason why the Proposition Bet Paradox is better for focusing on the conundrum - why is this apparent 50-50 bet with better than even money odds not +EV? On a one time basis you can look at your envelope amount and decline the bet if you like. As the maker of the proposition I'm betting that people I offer it too will not improve on a coin flip type decision enough to beat the 3-2 odds I'm offering. You can go into gymnastics about how I might be choosing the amounts and how seeing the envelope amount might help you improve your decision if you want. But in doing so you will be avoiding the central conundrum. After looking at the envelope amount you are happy to bet $10. You see it as a 50-50 proposition getting 3-2 odds. That's +EV. Why aren't you just as happy betting the Envelope amount? Isn't it still 50-50? Aren't you still getting 3-2 odds? Why isn't it still +EV?

The answer is simple. The amount of your bet now determines whether you win or lose - even though you don't know which. That messes with your EV.



[ QUOTE ]
If you ignore how T is selected, determine H but don't tell me what it is, PairTheBoard has the answer. But when precisely does the probability change? ... When H is determined? When I know X? And what precisely is the nature of the thing "probability" that changes?

[/ QUOTE ]

I think we are finally getting to your point. What are your best answers to the questions you pose above?


PairTheBoard

[jason1990]

Here's the dubious argument from the original post that we've been debating.

[ QUOTE ]
Argument 1: It's +EV to switch. You had a 50/50 chance of picking the high or low envelope so there's a 50% chance that the other envelope is the high and a 50% chance it's the low. Therefore, EV of switch = 0.5*(+100) + 0.5*(-50) = +25.

[/ QUOTE ]
Aaron, here's some of your comments about it.

[ QUOTE ]
Aaron-
I claim the envelope paradox is a true paradox, two strong arguments that lead to opposite conclusions. Yes, people invent rigorous ways to resolve it, but only by torturing commonsense ideas of uncertainty.

[/ QUOTE ]

[ QUOTE ]
Aaron-
Before we open our envelope, call the amount inside X. We know that the other envelope has a 50% chance of holding 0.5*X and a 50% chance of holding 2*X. Its expected holding is 1.25*X ... When there's no reason to think A is better or worse than B, we assume there's a 50% chance that A is better. Then if told we have a 50% chance of losing 0.5*$X and a 50% chance of gaining $X, we calculate our expected gain is 0.25*$X. It's hard to come up with general principles that prevent us from making errors in real problems.

[/ QUOTE ]

[ QUOTE ]
Aaron-
[The Paradox] asks a common sense question that appears to have two opposing common sense answers. People layer the mathematics on to it. That's fine, but making technical assumptions to make one argument right and one wrong doesn't answer the fundamental question of the paradox.

[/ QUOTE ]

[ QUOTE ]
Aaron-
The envelope paradox is simplified to make the basic contradictions clear, which makes it easy to "resolve" with ad hoc techniques. But those techniques are not much practical help in realistic problems.

[/ QUOTE ]
So you claimed that in order to counter Argument 1, we need to layer on extraneous mathematical assumptions which are designed solely to avoid this Paradox, which are typically ignored by practicing statisticians, and which are of little help in real life problems.

I then claimed that the only assumptions we need in order to counter Argument 1 are the elementary principles of probability.

[ QUOTE ]
Jason-
<ul type="square">
[1] if A and B are events, then P(A | B) = P(A and B)/P(B),
[2] if A and B are independent events, then P(A | B) = P(A),
[3] if A and B are mutually exclusive events, then P(A or B) = P(A) + P(B),
[4] if A is an event, then 0 &lt;= P(A) &lt;= 1.[/list]As far as I can tell, if we assume these four things, then it cannot be the case that

P(Y = 2X | X = k) = 0.5

for all k.

[/ QUOTE ]
(For those who haven't been following the notation of this thread, in the quote above, the statement "P(Y=2X|X=k)=0.5 for all k" means "the probability you chose the smaller envelope, given that you found it to contain $k, is 50%, regardless of the value of k".)


[ QUOTE ]
Jason-
the only techniques we need to use in order to arrive at the "sensible" conclusions are the techniques of elementary probability, such as assumptions [1]-[4] of my previous post. These are consistent, they are certainly not ad hoc, and they are extremely helpful in a wide array of realistic problems. Are you denying this?

[/ QUOTE ]
Here are some of your replies.

[ QUOTE ]
Aaron-
I accept all four of your assumptions, and don't know anyone who does not. But how do they lead to the conclusion?

The only problem I know of with P(Y = 2X | X = k) = 0.5 for all k is that the expected value of X must be infinite. I understand there are technical complexities with dealing with distributions that allow this statement to be true, but that's not the same as proving it false. Even if you outlaw them entirely, that's only a formal resolution of the paradox, not a refutation of the force of its argument.

[/ QUOTE ]

[ QUOTE ]
Aaron-
I maintain the ways people deny the force of the always switch argument result in restrictions that are commonly ignored, and if not ignored would make statisticians useless.

[/ QUOTE ]
I then showed you, using the probability space that you yourself suggested, how those assumptions lead to the conclusion. In other words, I showed that Argument 1 violates the very foundations of probability. No ad hoc assumptions are needed. No layers of ignorable mathematics are necessary. Only the axioms of probability which are given above. I apparently showed you something you had never seen before, since you said the only counter to Argument 1 that you had ever seen involved infinite expectations. Here is part of your reply.

[ QUOTE ]
Aaron-
Jason1990 does a thorough job of showing that no consistent prior distribution on T justifies the idea that the probability of switching is always 50%.

[/ QUOTE ]
In other words, I did a thorough job of showing that the logic in Argument 1, the same logic that you repeatedly claimed could only be refuted with ad hoc assumptions, violates the four assumptions given above, which are the fundamental principles of probability theory.

[ QUOTE ]
Aaron-
However, there are consistent prior distributions that justify always switching, but they have infinite expectation for T.

[/ QUOTE ]
What??? Suddenly, this is what you've been talking about all along? Who cares if there are other ways to justify always switching besides Argument 1? We've been discussing Argument 1, and you've been making claims about Argument 1 this whole time. And all of a sudden, that's not what the Paradox is about? And then ...

[ QUOTE ]
Aaron-
I think both Jason1990 and PairThe Board have redefined the original paradox

[/ QUOTE ]
Priceless. Incidentally, I am in complete agreement with PTB, and his perspective is covered as a special case of the proof I gave you by taking T to be a constant random variable. The fact that you think he and I are talking about two different things shows me that you really don't understand what either of us are saying.

I originally thought I misunderstood you. Then I thought you were just confused. Now I think you may be intentionally trying to confuse the readers of this forum. Why else would you suddenly change your tune 85 posts into the thread?

Argument 1 logically contradicts the axioms of probability theory. That's a fact. This may violate your intuition. But so will a lot of other things in mathematics. In fact, as you probably know, so will a lot of things in "real life". If you want to maintain that Argument 1 is valid in any sense, then you must either reject the axioms of probability or be logically inconsistent. I only hope that my participation in this thread has made this clear to at least one person who would have otherwise been confused. If you wish to go beyond this simple fact and discuss additional topics such as infinite expectations, Bayesian philosophy, or baseball statistics, then have fun.

[jason1990]

[ QUOTE ]
it takes only logic ... to see that the bet in (2) can be converted to an obviously poor bet ... That's not a mystery to me or anyone who thinks through it clearly.

The challenge is to come up with a consistent way of computing which bets to take. It's a lot harder than it looks. The envelope paradox is simplified to make the basic contradictions clear, which makes it easy to "resolve" with ad hoc techniques. But those techniques are not much practical help in realistic problems.

[/ QUOTE ]
I am stubbornly sticking to my prior my belief about you, Aaron, which is that you are an educated and intelligent man who can communicate effectively via the written word. That's why I've got to believe that there is some gross misunderstanding going on here.

In both the Paradox and in PTB's Prop Bets, the only techniques we need to use in order to arrive at the "sensible" conclusions are the techniques of elementary probability, such as assumptions [1]-[4] of my previous post. These are consistent, they are certainly not ad hoc, and they are extremely helpful in a wide array of realistic problems. Are you denying this? What am I missing here?

[AaronBrown]

[ QUOTE ]
I am stubbornly sticking to my prior my belief about you, Aaron, which is that you are an educated and intelligent man who can communicate effectively via the written word. That's why I've got to believe that there is some gross misunderstanding going on here.

In both the Paradox and in PTB's Prop Bets, the only techniques we need to use in order to arrive at the "sensible" conclusions are the techniques of elementary probability, such as assumptions [1]-[4] of my previous post. These are consistent, they are certainly not ad hoc, and they are extremely helpful in a wide array of realistic problems. Are you denying this? What am I missing here?

[/ QUOTE ]
Thank you for your kind words, although you could have left out "stubbornly." At least you didn't say "against all rational evidence."

I think you keep repeating one side of the paradox, and wondering why I can't see something so simple. I do see it. But there's the other side as well, the one that says the probability that you have the smaller amount is 50% so the expected value of switching is positive. You can't refute a paradox by strengthening one side, that just makes the paradox more puzzling. You have to show why one side is wrong.

I maintain the ways people deny the force of the always switch argument result in restrictions that are commonly ignored, and if not ignored would make statisticians useless.