Indecision in the game (Menzhevanie)

Two Australian researchers have found a promising approach to an 80-year-old mystery whose solution could have implications for a host of theoretical and applied fields: from visualizing some of the paradoxes of thermodynamics and optimizing the performance of technical systems to improving electronic circuits and developing a winning strategy for playing the stock market.

This riddle is called the "Two Envelope Paradox". In various variations and formulations, it has been known to mathematicians since 1930, although it was only described in the form of two envelopes in the late 1980s.

So, let's play. You are offered two envelopes with money (weighing, feeling and X-raying them is obviously prohibited). You only know that one of them contains an amount exactly twice as big as the second, but which one and what amounts exactly are completely unknown. You are allowed to open any envelope of your choice and look at the money in it. After that, you must choose whether to take this envelope or exchange it for the second one (without looking).

The question is, what should you do to win (i.e. get a larger sum of money)? It seems that the chance of winning and losing is always the same (50%), regardless of whether you keep the open envelope or take the second one instead. After all, the probability of finding a larger sum in envelope A is initially the same as the probability that a larger sum of money is in envelope B. And opening one of the envelopes (A) does not tell you anything about whether you see the largest or smallest sum of the two offered. However, calculating the average expected "cost" of the second envelope says otherwise.



Let's say you saw $10. So the other envelope contains either $5 or $20 with a probability of 50 x 50. According to probability theory, the weighted average amount in envelope B is: 0.5 x $5 + 0.5 x $20 = $12.5. Of course, when you open the alternative envelope, you will not see this amount, but either $20 or $5, simply according to the rules of the game. But 12.5 is (according to calculations) what it seems to be the average winning amount per stake when playing a sufficiently large number of rounds, if you always change envelopes.

And this result does not depend on the initial amount of money. After all, different pairs can be used in different rounds (10 and 20, 120 and 60, 20 and 40, 120 and 240, and so on). That is, in general, if envelope A contains sum C, then the statistically expected sum in envelope B will be 0.5 x C/2 + 0.5 x 2C = 5/4 C.

Thus, the theory says that it is always advantageous to change your initial choice (12.5 is greater than 10), although in some rounds you will lose. But intuition rebels against such a conclusion, which simply screams about the fundamental equality of the envelopes. After all, by changing them, you can start all the reasoning from the beginning (without opening the second one) and change again.

Various scientists have repeatedly claimed to resolve this paradox. Moreover, there are even disputes about how to understand what the paradox itself is. But the mathematical community has not yet reached a consensus, so the problem remains open.

Discussion started by Shpilevoy , on March 12 04:48
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Avax, 03/15/2015 10:05
Avax
ABBABB

There are two long-term losing strategies, but there are wins in them.
If you arrange them correctly so that the stages of losses are skipped as much as possible, and the stages of winning remain, then in the end there will be a plus.
But no one knows this optimal ABBABB in advance.

That's why they suggest introducing a mystical M - suddenly it coincides, then THERE IS A CHANCE that there will be a plus. Otherwise there is not even a chance to win.
Avax, 15.03.2015 10:00
Avax
This article shows the absurdity of the classical theory of faith.
It is clear to anyone that it is 50/50. But according to the formulas, it turns out that they looked at one envelope and the probability is completely different. So scientists try to explain that the theory of probability, despite its absurdity, still calculates correctly.
Valery, 12.03.2015 18:58
Valery
I WROTE ABOUT TWO OR EVEN THREE GAMES AT THE SESSION AT THE SAME TIME! CAN SOMEONE EXPLAIN ABBABB WITH A SIMPLE EXAMPLE AS IT APPLIED TO ROULETTE? I DID NOT FINALLY UNDERSTAND THE ARTICLE, SO I ASK FOR EXPLANATION
Coin_8, 12.03.2015 18:37
Coin_8
It goes like this: "Given two (chance-based) games, each of which has a higher probability of losing than winning, one can construct a winning strategy by playing these games alternately."
... But here's the paradox: alternating games A and B allows you to build up your capital, despite the fact that both are losing! Yes, not every alternation leads to victory. But only some combinations, for example, this one - ABBABB and so on.

***
Two strategies in one session - an ideal field for experimentation!
Shpilevoy, 12.03.2015 16:51
Shpilevoy
In 2003, Abbott was working in Britain (his home country, by the way). And one day, while having lunch with Cover, he discussed the riddle of the two envelopes. Thomas came up with an original winning strategy that was even more effective than the "always change envelopes" rule.

It consists of the following. You need to change or not change the envelopes in each run randomly, but with a probability that depends on the amount seen in the first envelope. That is, the smaller the amount in envelope A, the more likely it is to change the envelope and vice versa, a slightly larger amount in A indicates that you should rather keep the first envelope for yourself.

Back then, in 2003, Derek considered his colleague's idea to be nonsense and refused to think through such a strategy. And the scientist can be understood: judge for yourself, the amount seen does not tell a person absolutely nothing about the intentions of, say, the leader (who lays out the money), because the player does not know what range his opponent is playing in. Maybe from 10 cents to 100 dollars, or maybe from 5 dollars to a hundred million. And $25 seen, for example, once can equally (within the entire game) turn out to be both a mere trifle and the largest amount at stake. And therefore it is unclear whether it is worth changing the envelope in this round of the game or not.

However, after thinking about it, Abbott saw a deep philosophical and even physical meaning behind the “Kover strategy” (as Australian mathematicians called this technique). “The apparent paradox arose because you can’t shake the feeling that opening an envelope and seeing $10 doesn’t really tell you anything. And so it seemed strange that the expected value of your win in the case of switching envelopes was $12.5,” Abbott explained. “But we explain this incident in terms of symmetry breaking. Before opening the envelopes, the situation is symmetrical, so it doesn’t matter whether you switch envelopes or not. However, after you open the envelope and use the Kover strategy, you break the symmetry (immediately after opening envelope A, both envelopes are no longer equivalent), and then switching envelopes allows you to benefit in the long run (with a large number of visits).”

All this reminds one of the situation with the "reduction" of Schrödinger's cat to one of two states (dead or alive), although before opening the box of poison he is in a superposition of possible states. This is the problem of the influence of the observer on the result of observation. Do you feel that we are approaching some foundations of Nature?

Now, over 20 million computer simulations conducted by McDonnell and Abbott have shown that Cover's strategy allows you to get more money in the envelope game than a simple exchange. And, Australian scientists have discovered, a predetermined exchange, when a player chooses an alternative envelope only if the amount seen in the first one is less than the value chosen in advance and at random by him (the player), also works. And this is also counterintuitive, since the player knows about the minimum "switching" bar, but not those who put money in the envelopes.

To thoroughly understand how this happens, you can look at the article by the authors of the study in the Proceedings of the Royal Society A. What is important for us is a general explanation of the mystery of this game. And here we will need to turn to analogies from the world of physics and beyond.

The first is the "Brownian ratchet", invented by the famous physicist Richard Feynman. This is a mental device, which is a special case of the no less famous Maxwell's Demon, charged with maliciously violating the second law of thermodynamics, that is, to produce useful work without a temperature difference between two sources, but only due to the internal (thermal) energy of a single object (a vessel with gas).

The Feynman ratchet is designed and operates as follows (see the picture above). There are two chambers (boxes) with gas molecules (they are shown as red circles). The chambers are connected by a miniature shaft (working without friction), at one end of which there is a wheel with blades (on the left), and at the opposite end - a gear with a ratchet mechanism (on the right). Between them on the shaft is a weight on a string.

The ratchet allows the shaft to rotate in one direction, but prohibits it from turning in the other. The Brownian motion of the molecules in the left chamber causes them to hit the blades chaotically, but since the blades can only move in one direction, these hits gradually shift the wheel, doing work to lift the load using only the thermal energy of the molecules in the first chamber.

"The trick with the Brownian ratchet is that it again exploits the idea of symmetry breaking," Abbott says. The device extracts (apparently) useful work from Brownian motion, just as a gambler "extracts" an increase in his wealth from a random exchange of symmetry-broken envelopes (via Carpet's principle). The unequal situation of winning and losing probabilities in the envelope paradox is analogous to the Feynman ratchet.

True, physically such a ratchet cannot exist, even if skilled nanotechnologists could build it. Feynman himself explained why. The ratchet mechanism latch itself must be small enough to move in response to the impacts of individual molecules on the blades of the "mill". And therefore the latch will oscillate no less well from its own Brownian motion, opening from time to time and allowing the shaft to retreat.

Feynman calculated that if the temperatures (T1 and T2) in the chambers are equal, the average sum of forward movements will be balanced by the average sum of backward movements, so that the sum will be zero. If T2 is less than T1, then it would indeed be possible to observe the forward movement of these wheels. But in this case, the energy will be extracted from the temperature gradient, in accordance with the laws of physics.

With money, everything is somewhat simpler. But the Brownian ratchet helps us understand the principle of the new "cheating" strategy, the envelopes problem. Even more interesting is the analogy of the two envelopes paradox with another mathematical phenomenon, the Parrondo's paradox.

Derek Abbott (pictured) is considered the leading researcher into Parrondo's paradox (photo from wikipedia.org).

It goes like this: "Given two (chance-based) games, each of which has a higher probability of losing than winning, one can construct a winning strategy by playing these games alternately."

Here's an example. Let's say we have some initial capital. Then we add $1 to it or subtract $1 step by step depending on the result of the coin toss (heads or tails, guessed right or not). But the coins are not ordinary, but asymmetrical, so the probability of one side landing is different from 50%.

Further, in the game with capital we actually have two games - A and B. Moreover, in game A coin 1 is used with the probability of our winning 0.5 - e, where e is slightly more than zero. It is clear that with a large number of throws game A is always a loss for us.

In game B there are two (also asymmetrical) coins (2 and 3), which differ significantly in the probability of our winning from each other: for example (1/10) - e and (3/4) - e. In addition, a randomly selected number M is entered in advance. And the rule: if the current capital is a multiple of M - in this round we throw coin 2, if not a multiple - coin 3.

The same Abbott showed earlier that with M = 3 and e = 0.005, game B is a losing game just like A. The analysis also shows that the probability of using a "bad" coin in the next round is roughly 0.6 versus 0.4 for a "good" one, hence the loss in the sum of many attempts. But here's the paradox: alternating games A and B allows you to build up your capital, despite the loss of both! Yes, not every alternation leads to victory. But only some combinations, for example, this one - ABBABB and so on.

To dispel the illusion of a paradox (and it is such only for our superficial judgments, in fact it is a natural result of probability theory, as demonstrated by models using complex principles of analysis) it should be understood that in a combination of two games both become linked. This almost mystical connection is organized precisely by the number M. After all, with its introduction the course of game B begins to depend on the course of game A. If there were no connection, any combination of games would still lead to a loss.

Here the light begins to dawn on the envelope problem. The two separate games with coins are losing only if the statistical distribution of the results of all the throws in the game differs from the one formed when these two games are combined. The introduction of the number M and the connection between the choice of coin and the capital (which, alone, decreases and increases both in game A and in game shifts the probability of the distribution of all the throws to a state in which a positive expectation (of the result) appears. And the "envelopes" and "Parrondo" are related paradoxes. Derek himself calls the solution to the problem of two envelopes a breakthrough in the field of analysis of the Parrondo paradox (which has many manifestations in life). And the main mistake of a number of Derek's predecessors is calculating the probability of certain events with independent initial variables that are not independent in fact.

And here it is time to move on to the third analogy - from the field of finance. "Volatility pumping" - "Pumping volatility". This is not a mythical "golden" program for playing on the stock exchange, but a simplified model showing some useful features of a winning strategy for playing with shares (commodities, bonds, etc.).

It is clear that if a player has information about the financial instruments being acquired (the company's condition, lawsuits against its managers, this year's orange harvest, or the discovery of a new oil field), he can consciously create his portfolio. But what if he knows nothing except the current price of a share (or other acquisition), and where the price is currently heading? Neither whether the price will continue to fall, or will it start to rise later? Neither whether the current price is the maximum, minimum, or whether there will be a huge collapse later.

How is this similar to choosing between two envelopes: is the second one worth more than the one you are holding in your hands, or is it less? The "volatility pump" involves fairly chaotic buying and selling of assets with a small lag (bought cheaper - sold more expensive), without any concern about whether you have received the biggest benefit from the deal at the moment or missed the chance to become even richer. And this is very similar to a random change of envelopes with some "gradient" depending on the size of the observed amount (again, the Cover strategy).

Mark McDonnell (pictured), like his research partner Abbott, believes that the patterns discovered in the course of “splitting” the two-envelope paradox will allow many interesting processes to be explained on a single mathematical basis, and this will give impetus to new research in various fields – from mathematics and information theory directly to physics and engineering (photo University of South Australia).

And this is also similar to the principle of the Brownian ratchet. And this same principle is similar to the situation when it is necessary to improve the performance of a technical system with incomplete data on the conditions of its operation. "What is surprising is that our analysis shows that it is always possible to increase the capital received (in the envelope game) using the Cover method, without knowing anything about the permissible limit of the amount in rounds, as well as about the statistical distribution of bills across rounds," says Derek.

But is it possible, for example, to apply the consequence of Parrondo's paradox (or the explanation of the envelope phenomenon) to the stock market, i.e. to obtain income by combining shares like the game ABBABB? Alas, the paradox requires that the yield from at least one instrument depend on the size of the current total capital (like the choice of coin on the multiple of the already won amount to the number M), and this is fiction. Or is it?

The ability to discern true connections between phenomena where there seem to be no connections is a very valuable quality for a scientist. It helps explain processes that seem incredible to a superficial observer. Thus, from the notorious game with two envelopes, the thread stretches to many other areas in which the interaction of objects with asymmetry of chance is manifested, no matter whether such asymmetry is generated by a ratchet mechanism, the opening of envelope A, or the laws of the market.

And it is not for nothing that, for example, Abbott is also known as a researcher of stochastic resonance - a paradoxical, at first glance, phenomenon of amplification of a useful (periodic) signal in nonlinear systems when adding white noise to it. This interesting phenomenon is now finding application in electronic systems.

Look, what a beautiful analogy. How does "nature" know which part of the impulse to amplify? It is as unknown as which of the two envelopes contains a larger sum of money. However, under a number of conditions, the probability of correct amplification turns out to be higher than the probability of suppression of the useful component by added interference. Just as the probability of winning in "envelopes" can be deliberately increased, in defiance of the seeming uncertainty of the outcome of this simple game. But what kind of toys are we talking about here?