Spiral Illusion

its really just concentric circles

Thanks I needed to puke.

Damn.. First thing..

Whoa! Trying to follow one circle around is....hard.

That pic moves!!!

Sometimes our brains fail us completely even without visual illusions. My favorite paradox (still unresolved by the math and metamath gurus) is deceptively simple to state. It is called the "two envelope problem", and goes as follows.

You are told you can take your pick of two sealed envelopes each containing cash. You are also told that one envelope contains twice as much cash as the other.

You pick an envelope at random, and find $20 inside. You are offered the opportunity to switch envelopes. What should you do?

You reason (using expected value logic) that there is a 50% chance the other envelope contains $40, and a 50% chance it contains $10. Therefore the expected value of the other envelope is 1/2 x ($10) + 1/2 x ($40) = $25. Since $25 is greater than the $20 you have, the mathematical logic for switching is compelling. Of course, it is wrong. The debate on why it is wrong has been raging for several decades. The problem is still unresolved, IMO. BTW, do not confuse this problem with the "Monte Hall" problem where it is easy to show that switching "doors" is the correct strategy.

tmex wrote:Sometimes our brains fail us completely even without visual illusions. My favorite paradox (still unresolved by the math and metamath gurus) is deceptively simple to state. It is called the "two envelope problem", and goes as follows.

You are told you can take your pick of two sealed envelopes each containing cash. You are also told that one envelope contains twice as much cash as the other.

You pick an envelope at random, and find $20 inside. You are offered the opportunity to switch envelopes. What should you do?

You reason (using expected value logic) that there is a 50% chance the other envelope contains $40, and a 50% chance it contains $10. Therefore the expected value of the other envelope is 1/2 x ($10) + 1/2 x ($40) = $25. Since $25 is greater than the $20 you have, the mathematical logic for switching is compelling. Of course, it is wrong. The debate on why it is wrong has been raging for several decades. The problem is still unresolved, IMO. BTW, do not confuse this problem with the "Monte Hall" problem where it is easy to show that switching "doors" is the correct strategy.

Thanks, now this is going to bug the crap out of me.

Also, why isn't the Monty Hall problem the same thing?
Granted you have 3 doors and not 2 envelopes, but you end up with some information you didn't have prior to making the second choice.
Probably because one of the choices is potentially the same as your initial choice.

I think in the problem you posted the expected value of the second envelope should not be looked at, should look at expected net gain, or loss if you pick the
second envelope.  So, $20 - (1/2 x($10)) + $20 - (1/2 x $40)) = $15

Or maybe something like:

For arguments sake, you pick the first envelope and there is $20 inside.
At this time there are now two possibilities for the second envelope. $10 or $40

This adds up to three possibilities but with only two choices, so the expected value is really

1/3($10) + 1/3($40) = $16.67  which is less than the $20 you originally picked.

For any initial amount of money selected your expected return if you switched would be:

(1/3 x 1/2) + (1/3 x 2) = 0.83 of the initial value.

In the Monte Hall Problem you are getting two doors for your one door. Imagine if Hall offers you both unopened doors for your one door. You would switch. Making the offer after he reveals a bogus prize behind one of the doors does not change anything. You already knew a bogus prize was behind one of the two doors he was offering.

You have $20. Giving it up for the other envelope does not allow you to consider its expected value any further. It is gone. The expected value of the alternate envelop is 1/2 x $10 + 1/2 x $40 = $25.

After reading dozens of pages from mathematicians to metaphysicians to linguists, I have concluded that the correct answer lies in Edwin Jaynes' principle of "maximum ignorance" which he developed in response to Bertrand Paradox (wiki that one if you want something that will ruin your day). In a nutshell, the principle of maximum ignorance advises not to "overthink" a problem. Keep it simple.

Even though even expected value theory is correct and used all the time without much thought, there is no reason to apply it to the two envelope problem. Just because a theory is correct does not mean it does not run the risk of providing a nonsensical result from time to time. I know that sounds weird, but it is the mathematical equivalent of quantum mechanics. I have no other way to explain it. It all boils down to the intricacies of how a problem is posed, and it is a wonderful example to show how wrong you can be while standing in the path of righteousness.

The irony of that graphic is that is what my vision and balance do to me 24/7 now. My brain tries to compensate, but at a certain point the information being processed by my right eye becomes more than it can handle in that the right side is no longer balanced with the left side.

Basically at speed whether on a motorcycle, on the boat or in the car, my right eye wanders involuntarily trying to find a way to process the peripheral motion, which is sent back to me as "slow down or you are going to puke" message. I've learned to ignore the puke message as it is just an overload reaction if you will and just tough it out.

The only thing I still cant properly compensate against is a sudden head turn, then my whole world shifts. If I am driving, it means I leave my lane and usually end up getting honked at.

The "moon illusion" is another unexplained human perception failing. Why does the moon look bigger on the horizon that it does above the horizon even though it is closer to us when directly above us? It has never been explained by physicists, astronomers, medical doctors,...nada. Nobody knows the answer.

Oh my, a harvest moon looks HUGE at horizon, but astronomers are adamant that there is no difference. Possibly a bending or magnification of the light rays as viewed at that particular angle with relation to the horizon?

tmex wrote:After reading dozens of pages from mathematicians to metaphysicians to linguists, I have concluded that the correct answer lies in Edwin Jaynes' principle of "maximum ignorance" which he developed in response to Bertrand Paradox (wiki that one if you want something that will ruin your day). In a nutshell, the principle of maximum ignorance advises not to "overthink" a problem. Keep it simple.

Even though even expected value theory is correct and used all the time without much thought, there is no reason to apply it to the two envelope problem. Just because a theory is correct does not mean it does not run the risk of providing a nonsensical result from time to time. I know that sounds weird, but it is the mathematical equivalent of quantum mechanics. I have no other way to explain it. It all boils down to the intricacies of how a problem is posed, and it is a wonderful example to show how wrong you can be while standing in the path of righteousness.

So which is the correct choice to maximize earnings?
keep the first envelope or switch?

or is there not an answer?

It makes no difference. After you select an envelope you don't have any additional information (regarding which envelop has the 2X amount) you did not have at the start of the exercise. In a way, that is also the answer (maximum ignorance). With no new information you have nothing to justify any action.

I don't think anyone understands why the expected value math breaks down. At least, I have read nothing plausible. Most of the mental "heavy lifters" seem to think it has to do with the linguistics of how the problem is posed. Analogous to when someone says "this statement is false".

Wade, there is atmospheric refraction - the moon appears to rise slightly sooner than it really does due to the bending of light rays by the atmosphere. However, there is no lensing effect. No one has been able to provide an answer, strange as that seems. I always tease cosmologists when they lecture on the origin of the universe (big bang), but cannot explain why the moon looks bigger on the horizon.

Lol, that made me chuckle loud enough that Lynda asked what was so funny.

tmex wrote:It makes no difference. After you select an envelope you don't have any additional information (regarding which envelop has the 2X amount) you did not have at the start of the exercise. In a way, that is also the answer (maximum ignorance). With no new information you have nothing to justify any action.

I don't think anyone understands why the expected value math breaks down. At least, I have read nothing plausible. Most of the mental "heavy lifters" seem to think it has to do with the linguistics of how the problem is posed. Analogous to when someone says "this statement is false".

Not very satisfactory.

Yes, Hokie, I feel the same way. How do you know when you are screwing up? It is terribly unsettling to me, but I am weird, and few other people seem to be bothered by it. FWIW, I have been at this problem for almost two decades for the very reason I described briefly above. I use expected value all the time relative to investment activity, and I always wonder if I am jumping off of a cliff.

tmex wrote:Yes, Hokie, I feel the same way. How do you know when you are screwing up? It is terribly unsettling to me, but I am weird, and few other people seem to be bothered by it. FWIW, I have been at this problem for almost two decades for the very reason I described briefly above. I use expected value all the time relative to investment activity, and I always wonder if I am jumping off of a cliff.

Well, I will probably be thinking about it for the next twenty years....the more I think about it, the worse it gets.

Welcome to the club, Hokie. The reality is, we are not as smart as we think we are.

tmex wrote:Welcome to the club, Hokie. The reality is, we are not as smart as we think we are.

LOL. That seemingly simple problem has proven all my past math and engineering courses inadequate!

Yes, Hokie. It is humbling, but we need that from time to time. It helps to keep a perspective on what we do. You should really check out the Bertrand Paradox. It is another two decade thought process for me.