Monty Hall-esque question

[southerndog]

Why not just actually play the game?

[jason1990]

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Why not just actually play the game?

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Are you suggesting we build a computer simulation to repeatedly play the game and actually compute (i.e. estimate) these probabilities? Well then, you have to decide on the details. For example, which of the following rules do you incorporate in your game:

(1) With each play, the computer uses the same dollar amounts for the envelopes.
(2) With each play, the computer randomly selects new dollar amounts for the envelopes.

You make a choice of either Rule (1) or Rule (2) and then run the simulation. You then want to estimate the conditional probability that the higher envelope was chosen, given that chosen envelope contains some given number X. So you let N be the number of times the chosen envelope contains X and you let M be the number of those times that this is the higher envelope. If you had chosen Rule (1), then you will find M/N is either 0 or 1, depending on the value of X. If you had chosen Rule (2), then you will find M/N is converging to some number (not necessarily 0.5) which depends on the method by which the computer is randomly filling the envelopes. So you would find that the simulation confirms the resolutions to the paradox which are given in this thread, and there is nothing left to argue about...

Unless someone says, "Hey, the original Two Envelope Paradox is a one-time shot. You are not allowed to repeat it." Then you can throw your simulation in the trash, because it won't convince them. And now you're back to arguing.

[mykey1961]

S = ( 50 + 100) / 2
L = (100 + 200) / 2

EV(1) = (S + L) / 2 = 112.50

EV(2) = (50 + 200) / 2 = 125.00

The "paradox" is that in your first example, you only look at 50% of the possible outcomes.

Where you picked the larger amount when the amounts were $50, and $100, and the smaller amount when the amounts were $100 and $200.

You never consider when you pick $50, or $200 on the first decision.

instead of EV(2) = (50 + 200) / 2 = 125,

A = 50, switching from 100 to 50
B = 100, switching from 50 to 100
C = 200, switching from 100 to 200
D = 100, switching from 200 to 100

EV(2) = ((A + B + C + D))/4 = 112.50

EV(2) = EV(1)


In the second part:

S and L are unknown values.

The EV(1) of your first choice is (S + L) / 2

The EV(2) of switching is (L + S) / 2

EV(2) = EV(1)

[Xhad]

I did some digging and found this. I'm still not sure it makes sense to me though.

http://www.faculty.ucr.edu/~eschwitz...lopeSimple.htm

[pzhon]

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You open one of the envelopes, and you see a check for $100 [x]. Now, the host offers you the chance to switch to the other envelope. (Again, no game theoretical assumptions about whether or not it is more likely that the game show would have a $200 prize or a $50 prize.) Do you switch?

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This is the well-known envelope paradox. It is understood by many, but hard to explain. (I like PairTheBoard's attempt, but it may not convince people who won't take the trouble to follow it.) There are attempts to explain it in the rec.puzzles newsgroup archives, and there are articles on it in academic journals (see Richard Thaler's column in the American Economic Review). Mike Caro, a poker theorist, got it mostly right, but misanalyzed part of it. Here are some important points for following discussions:

[img]/images/graemlins/diamond.gif[/img] The conditional probability that the other envelope holds x/2, given that X=x, can't be assumed to be 1/2. Indeed, you can check that this probability is not 1/2 for simple distributions and many particular values of x.

[img]/images/graemlins/diamond.gif[/img] For any distribution for the smaller amount, the strategies of always switching and of never switching have the same (possibly infinite) expected value.

[img]/images/graemlins/diamond.gif[/img] There is no uniform distribution on the positive real numbers, but that does not resolve the paradox. There are random variables with positive probability on all intervals of positive length in R+, such as the one defined by (1/z)-1 where z is uniformly distributed on (0,1).

[img]/images/graemlins/diamond.gif[/img] There are strategies which are better than always switching or never switching. You can choose a threshold t, and switch if you see x<t, and don't switch if you see x>t. If t is between the values in the envelopes, you make the right choice. Otherwise, you break even, on average. If you choose t randomly to have positive probability on all intervals of R+, then you will choose correctly better than 50% of the time regardless of the distribution. For any distribution with a finite expected value, this strategy will have a greater expected value than that of always switching.

[img]/images/graemlins/diamond.gif[/img] There exist distributions such that no matter what you see, you should switch. For these distributions, the expected value of always staying is infinite, as is the expected value of always switching.

These have all been proven rigorously, but people still argue because the only way these are intuitive is if you have studied this before. Math is hard.

[AaronBrown]

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These have all been proven rigorously, but people still argue because the only way these are intuitive is if you have studied this before. Math is hard.

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I have great respect for phzon and PairTheBoard, but this one statement is not only dead wrong, but very dangerous.

Skill in math can be good, but it often induces a special kind of blindness. Problems are simplified so they fit in neat mathematical frameworks, then some mathematicians can only see the math, not the original problem. This leads to people insisting they have the only rational solution to a problem, when what they should say is, they have a rational solution to a simplified problem which may or may not give insight into the real problem.

In this case, it's very easy to come up with mathematical formulations that justify always switching, and formulations that argue you shouldn't always switch. It's easy to come up with real-world examples where each type of formulation is a good model. That's why we call this a paradox. Explaining one side or the other, either in rigorous math or exhaustive words, misses the point. The original problem already did that better.

People who try to "resolve" or "explain" paradoxes don't understand them. No one believes both sides are true simultaneously. The point is to understand both arguments, not to decide which one is right but to see the limits of each. They can't both be true all the time in full generality. To learn from this paradox, you have to understand the force of both sides, and think about which one to apply in different situations.

The paradox was invented by Belgian mathematician Maurice Kraitchik. He was not an idiot. He was good at hard math. Yet he still saw the force of both arguments. He introduced it because both arguments are used all the time both by statisticians and in informal reasoning. Both are valuable tools in some situations. But it took this example to get people to admit that neither argument is universal, that you have to be careful using either one.

Sure, you can redefine the problem to bring it into the realm where one or the other argument is stronger. That's easy. The hard part is to define the precise border between the realms.

[elindauer]

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I have great respect for phzon and PairTheBoard, but this one statement is not only dead wrong, but very dangerous.

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His statement that all of his points have been proven rigorously is dead wrong? I'm surprised to hear you say this, since when I read his post I thought... yup, that sums it up pretty well.

Note that his points still leave open the possibility that switching is always right... and he even defines the border between the strategies (switching is always right when the probability function for selection the amounts has infinite expected value).

Probably I'm just missing something, so I'm curious, which part of phzon's post was incorrect?


I am sincerely curious and hope this won't come off and some kind of attempt to poke at you. I bow before your phenomenal mathematical talents. In fact, if it was some random person, I'd probably just blow it off figuring "ok, that guy doesn't know what he's talking about", and it's precisely because it is you, and I can't possibly make that argument, that I post this question.


thanks,
eric

edit: perhaps you mean that "well understood mathematically" should not be mistaken for "knows what to do when presented with this problem in real life" or "not interesting any more". I personally read phzon to say only the first thing and not the second two, but maybe you read it differently?

[AaronBrown]

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His statement that all of his points have been proven rigorously is dead wrong?

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No, my objection is to what followed immediately afterwards, that:

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but people still argue because the only way these are intuitive is if you have studied this before. Math is hard.

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People do not argue only because they can't follow the math. That's the dangerous belief.

Let's take another paradox, where perhaps positions are not so rigid. Zeno told the story of a race between Achilles and the Tortoise. The Tortoise has a head start. Achilles cannot win, because by the time he gets to where the Tortoise started, the Tortoise has moved forward some amount. By the time Achilles gets to that point, the Tortoise has moved forward again. However many times you iterate, Achilles is still behind the Tortoise.

If Achilles moves at constant speed A, the Tortoise moves at constant speed T and the headstart is distance H, it takes time H/A for Achilles to get to the original start, by which time the Tortoise has moved forward T*H/A. It takes Achilles T*H/A^2 to get there, then T^2*H/A^3 and so on. The sum from i = 0 to infinity of T^i*H/A^(i-1) = H/(A - T), so it will take Achilles that long to catch up with the Tortoise. The key is the sum of an infinite number of terms can have a finite sum.

That's all true, and can be made rigorous. But it just verifies a calculation that anyone can do without infinite sums. Achilles gains on the Tortoise with speed A - T, and has to make up H, so it will take H/(A - T). Everyone knows that is what will really happen.

We have not addressed the other side of the paradox. The math requires assumptions about space and time that are not physically true. A deeper consideration of the paradox can lead to insights into important issues in physics and mathematics.

If you say everyone who continues to argue about Zeno's paradox doesn't know enough math to sum an infinite series, you go down a dangerous road. The same thing is true of the envelope paradox.

[elindauer]

I knew I'd be glad I asked. Thanks for your well considered response.

FYI, I'd be really interested to read more about this:

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The math requires assumptions about space and time that are not physically true. A deeper consideration of the paradox can lead to insights into important issues in physics and mathematics.

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For both the Envelope problem and the Achilles problem, your thoughts on the deeper insights available if you study the other side of the paradox sound pretty intiguing.

-eric

[pzhon]

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His statement that all of his points have been proven rigorously is dead wrong?

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No,

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Good. Note that the first point, that you can't assume the probability is 1/2, specifically negated one argument made in the paradox. Do you disagree with this point? Do you still think there is no way to conclude that one side is wrong when it relies on a fallacy?

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my objection is to what followed immediately afterwards, that:

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but people still argue because the only way these are intuitive is if you have studied this before. Math is hard.

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People do not argue only because they can't follow the math.

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First, "counterintuitive" is very different from "can't be followed."

Second, there is a big difference between saying, "People argue because the only way this is intuitive..." and "People only argue because..." One of those is what I said. You objected to the other. What is the real reason you are arguing?

I don't see any real objection to what I said. You are trying to put words in my mouth to say that I am making mistakes you would like me to make.

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Skill in math can be good, but it often induces a special kind of blindness...

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It is common for people to assume that any strength must be balanced by some weakness. This is wrong, and both tiring and offensive to me. See the past discussion David Sklansky and I had about mathematics majors in the SMP forum. Don't assume I made a mistake just because I was right about something else.