The Two-Envelope Paradox (Column 288)

With God's help

Dedicated with love to my mathematician sons, Nachman and Yossi.

In the previous column there was a surprising surge of public interest, even though there too I dealt with a theoretical question (interesting why). That is exactly what encourages me to keep dealing with such questions, and I am quite certain that the storms in the comments will continue here as well.

At the beginning of Column 286 I mentioned that my son Nachman sent me an article by Gill (R. D. Gill) from Leiden University, which deals with the well-known envelope paradox, and in the language of Ishmael it is called "Anna Karenina and the two envelope problem". In that column I gave an overview of the Anna Karenina principle on the logical and probabilistic plane, and now I arrive at the envelope paradox. I will return at the end of the column to the connection between Anna Karenina and the paradox (if there is one at all).

Let me say in advance that this column is somewhat long and not easy. It contains several digressions for the sake of methodological insights that seemed important to me, especially for amateurs (professionals probably do not need most of them). And I should further preface this with a full disclosure: I am far from being an expert in probability. What you have before you are my reflections on the matter (with comments from both my sons). I feel there is value in them, but perhaps I am mistaken. So here, more than ever, I will be happy for any comment from anyone who takes the trouble to read. I think one can see here just how confusing the field of probability and statistics is, and that in itself is a very important lesson.

The envelope paradox

This paradox was first presented by Barry Nalebuff, a game-theory expert from Yale University, and since then it has been discussed in many articles in various fields (probability theory, mathematical economics, logic, and philosophy). A clear and pleasant formulation of the paradox appears in an article by Marius Cohen on YNET[1]. You are presented with two sealed, identical envelopes, each containing some amount of money. You are told that one contains twice the amount in the other. You of course do not know which of the two identical envelopes contains the larger amount. You are asked to choose an envelope, and since they are identical and you have no information, clearly each envelope is equally eligible to be your choice. You choose one of them at random. But now, once you are holding the envelope you chose, you are offered the chance to switch and take the other one. Would you prefer to do so, or are you better off staying with the one in your hand?

Calculation A: On the face of it, it seems there is no advantage to one over the other, and therefore no reason to switch. One can even calculate the expected payoff given that we are holding envelope A: the probability that it contains an amount X is 1/2, and the probability that it contains 2X is also 1/2. Therefore the expected payoff from choosing envelope A is: 0.5X+0.5_*2X=1.5X. But this is also the expected payoff if we choose the other envelope (B), so a priori it is clear that there is no reason to switch between them.

Calculation B: But one can also perform a different calculation here. Suppose I am holding an envelope whose contents are X. What is in the other envelope? Either 0.5X or 2X. I have two possibilities to compare: staying with the current envelope or switching. If I stay with it, I receive the amount X that is inside it. If I switch, I have a 1/2 chance of gaining X and a 1/2 chance of losing 0.5X. The expected value in such a case is:

0.5X-0.5*0.5X=0.25X, that is, the switch should yield me 0.25X. It is worthwhile to switch.

In passing, I note that one can think about this paradox also in a case where the player opens the envelope in his hand and sees what amount is inside, and only afterward decides whether to switch. At first glance, both of these calculations remain valid in that case as well. Even when he opens the first envelope and finds an amount X, the player still does not know whether this amount is the smaller or the larger of the two, and so apparently that information is irrelevant. Think through the calculation after opening the envelope, and you will see that it can be repeated in exactly the same way. The second envelope contains either 2X or 0.5X with equal probabilities, and therefore the higher expected payoff will still result if he switches envelopes. That is, opening the envelope does not really change anything here, and the paradox remains intact.

Two problems with the second calculation

What is wrong with the second calculation? It can be seen from two different angles: a. How does this result fit with the previous calculation, which said there is no difference which envelope we choose? It now turns out that specifically the other one would have been preferable. This does not necessarily point to a problem specifically in the second calculation; it only shows that there is a problem in one of the calculations. b. This is already a problem specifically with the second calculation. After the switch, we can do exactly the same calculation again, and once again get the result that it is better to switch the envelope and return to the first one. That is, it is not really better to choose the second. So why was it originally right to switch, if after the switch it turns out that this was an unnecessary move? Put differently: suppose there is another player holding the second envelope, and he too can decide whether to switch. According to this calculation, he too should want to switch and get the first one. But it cannot be that, on the basis of the same information and by the same line of reasoning, two opposite decisions would be reached, one preferring the first envelope and the other the second.

The Monty Hall problem

This picture calls to mind the Monty Hall problem. The player is presented with three doors. Behind one of them is a valuable prize (an expensive car), and behind each of the other two stands a goat. The player is supposed to choose a door and open it. If the car is behind it, he wins the car. If not (he finds a goat there), he goes home grieving and crestfallen. Suppose that person chose a door but did not open it. The host, who himself knows where the car is, opens a different door for him, one behind which there is a goat (the point is that he intentionally opens a door with a goat behind it, and not that he randomly opened one of the doors and that is what happened to come out)[2]. He now asks the player whether he would prefer to open the door he originally chose or switch and open the third door.

Again, apparently there is symmetry between the doors, but a simple calculation shows that this is a mistake. The probability that the car is behind the first door chosen is, of course, 1/3. After the host opens the second door, that obviously does not change the a priori probability, since whether the chosen door is the correct one or not, the host always has another goat door to open, and he opens it.[3] Therefore the opening of the second door does not change in the least the probability that the door initially chosen is correct. If so, even after the host opens a door, the probability that the car stands behind the first door chosen remains 1/3. But now that the second door has been opened, the player can move to the third door and choose it. His chance of winning in that case is 2/3, since it is already known that the second door is not the prize door. Therefore the prize is either behind the first chosen door or behind the third. Hence it is better for him to switch and choose the third door, since then his probability of winning is 2/3, and that is higher than 1/3 (which is the chance of winning if he stays with the first door).

It is important to understand that Monty Hall is not a "paradox" but a "problem". This is indeed the correct answer (he should switch doors). It is strange and interesting, because intuitively this does not look like the correct answer. But that is just a cognitive failure. There is no paradox here, only a counterintuitive result of a probabilistic calculation. There are quite a few such cases.[4]

The difference between the two situations

What is the difference between this problem and the envelope problem? Here there is no possibility of returning and offering him another switch, so here one cannot formulate the paradox. Moreover, unlike the envelope problem, here between the first stage (when he chose the first door) and the third stage (when he has to decide whether to switch his choice), new information has been added for him (that the second door is wrong). His second choice is made on the basis of fuller information than the first, and therefore it is not surprising that his chance of winning now increases. Therefore it is also no wonder that in such a case the symmetry between the doors is broken. By contrast, in the case of the envelopes there is no addition of information between the stage of choosing the first envelope and the moment when he has to decide whether to switch (even if he opens the envelope, it does not seem that any relevant information has been added. Still, see below on this). Therefore there cannot be any difference there between the envelopes.

The conclusion is that, apparently, the envelope paradox in both its versions (whether the player opens the first envelope or not) is indeed a paradox, whereas the Monty Hall problem is merely a counterintuitive result.

Is the envelope paradox really a paradox?

Above I concluded that the envelope problem is a paradox, unlike the Monty Hall problem. The reason is that there are two methods of calculation there that both seem correct, yet they yield different results (one teaches that there is symmetry between the envelopes and the other teaches that it is better to switch).

However, the following claim may be raised here. After all, it is obvious that one of the methods is incorrect. It cannot be that a mathematical problem has two different solutions and both are correct. Therefore, in truth, this is a problem and not a paradox. We may not necessarily know which of the two solutions is correct, but clearly it is only one of them.

Moreover, in the case of the envelopes we even know which of the two solutions is the correct one. Clearly the two envelopes are on equal footing and there is no reason to switch envelopes. We saw above that the second calculation is problematic and cannot be correct. Therefore it is clear that Calculation B, which gives an advantage to switching, is the erroneous calculation, and Calculation A is the correct one. But it is not clear at exactly which point the problem lies in Calculation B. If so, we have proof of which of the two is the erroneous calculation, but for the time being we still cannot point to the flaw in the erroneous calculation (B).

Is such a state of affairs paradoxical? In light of what I explained, it would seem at first glance that the envelopes too are only a problem and not a paradox.

What counts as solving a paradox: the example of Achilles and the tortoise

But that is not entirely precise. To understand this, let us take the paradox of Achilles and the tortoise as a parable. Suppose Achilles runs at a speed of 10 meters per second, and the tortoise crawls at a speed of 5 meters per second (this is a 4×4 tortoise). Because of the difference in athleticism, Achilles, the most generous of men, allows the tortoise to begin the race 10 meters ahead of him. Once the starting shot is heard, Achilles begins to close the distance, and after one second he is at the point from which the tortoise began the race. But in the meantime the tortoise has advanced another 5 meters. Fine, Achilles of course keeps running, and within half a second he is at the previous point where the tortoise had been, but by then it has moved another 2.5 meters away. And so after another quarter of a second Achilles reaches that point, but the tortoise has already advanced another 1.25 meters, and so on. It follows that Achilles never catches the tortoise.

Here too, someone might say that this is not a paradox, since it is obvious that Achilles does catch the tortoise. There is a simple calculation that shows this. But that is not a solution to the paradox. So long as you have not shown what is flawed in the calculation I presented here, you have not solved the paradox. The fact that you know what the correct answer is does not constitute a solution to the paradox. On the contrary, that is the reason this is a paradox: you know that the second calculation is incorrect, but you do not succeed in pointing out exactly where the mistake lies.[5] The solution must present the error in the second mode of calculation, that is, show where the mistake is. The same applies to the envelopes. Even if we know what the truth is, that is not a solution to the paradox. In order to solve it, we have to point to the error in the alternative calculation. So long as we have not done that, one might perhaps say that we know the truth, but not that we have solved the paradox.

By the way, with respect to Achilles and the tortoise, the error in the incorrect calculation I presented is simple. The calculation I presented is completely accurate, but the conclusion (that Achilles does not catch the tortoise) does not follow from it. If we do the correct calculation, we discover that Achilles catches the tortoise after 2 seconds, when he has covered 20 meters.[6] Now we can easily see that the description I gave above is nothing but a decomposition of the first 2 seconds of the race (until Achilles reaches the tortoise) into an infinite collection of ever smaller segments: one second plus half a second plus a quarter second plus an eighth of a second, and so on. If you sum that whole series, you get a total of 2 seconds.[7] We have discovered that this entire description refers only to the first two seconds of the race. And indeed, during the first two seconds of the race Achilles does not catch the tortoise. He catches it exactly when two seconds have passed, and then he passes it and races on. Note that now we not only know which of the two is the erroneous calculation, but also understand where the mistake lies in the alternative calculation. Now one can say that the paradox is solved.

Is there a similar solution to the envelope paradox?

First proposal for a solution

In his article, Marius Cohen describes a solution that at first glance looks very enchanting. In these two envelopes there are two given amounts: X and 2X. Now I choose one envelope, and the two possibilities are that it contains X and then the second contains 2X, or that it contains 2X and the second contains X. If you look at it this way, you discover that whether one switches or not, the expected payoff obtained in both situations is equal. Either I hold the envelope with 2X, in which case switching yields a loss of X, or I hold X, in which case switching yields a gain of X. That is, the two possibilities are either gaining X or losing X with probability 1/2, and therefore the expected gain from switching is zero.

Is this a solution?

Is this not a simple solution to the problem? At first glance, this solution has a certain charm and seems neat and natural, but something here is nevertheless troubling. On the face of it, Cohen is simply presenting a solution to a different problem from the one we defined above. He is not solving the paradox, but presenting a situation in which the problem simply does not arise. What exactly is different between the two situations? Calculation B above assumes a given amount in my envelope, and two possibilities regarding both envelopes. If so, the first problem assumes knowledge (though not explicit) of the amount in my envelope (X), and lack of knowledge regarding the amount in the other envelope (either 2X or 0.5X). The second problem assumes partial knowledge (not explicit) regarding both envelopes (one contains X and the other 2X, but it is not known which amount is in which envelope), and therefore there are two possibilities there regarding the amounts in the two envelopes.

But on a second look one sees that these are not two different situations but two ways of analyzing the same state of affairs. After all, the knowledge in the player’s possession is identical in both situations. Note that the amount in my envelope is not really known (it was designated as X). Therefore the only question is how it is more correct to analyze the situation, but Calculation B above and Cohen’s calculation are two modes of calculation dealing with the very same situation. If so, what Cohen showed is that there is yet another form of calculation, let us call it from here on Calculation C, which yields the correct result (remember that above we saw that A is the correct answer), but note that he does not point to the flaw in Calculation B. He does not explain why specifically that calculation is the correct one and Calculation B is not. I remind you that above I already noted that a solution to a paradox requires also identifying the flaw in the incorrect solution. In short, we are left with the question why specifically one should choose mode of analysis C, or what is wrong with Calculation B. That he did not explain to us, and therefore it is hard to see this as a solution. That was also my feeling on reading the solutions Gill proposed in his article.

Calculation D

To sharpen the difficulty, I will now present yet another possible mode of calculation, Calculation D in our count. I am holding one envelope and opposite me there is another identical one. Suppose the second envelope contains an amount X. Now there are two possibilities: either my envelope contains 2X or it contains 0.5X. The expected gain from switching is now negative, of course, and the conclusion is that I should not switch (on the contrary, another player holding that envelope would do well to switch with me). But again, this is not a different situation but a different calculation of the very same situation. This is already the fourth form of calculation, and it leads to a different conclusion: Calculation A leads to indifference with respect to switching. Calculation B leads to a preference for switching. Calculation C (Cohen’s) also leads to indifference (like Calculation A). And Calculation D leads to a preference for not switching (and not merely to indifference). Note that Calculation D leads us to a result we did not yet have. This shows that presenting an additional calculation does not solve the paradox, but perhaps intensifies it.

To sharpen this further, I should note that Cohen concludes his article by wondering whether, in the case where one opens the first envelope, the calculation would be different. There is a lively discussion of this in the literature dealing with this paradox, and I already mentioned it above. Note that Calculation D is no longer relevant to that situation. If I opened my envelope, there are no longer two possibilities regarding the amount in it. I now have before me only the first three calculations. Will the result in the case where the player opens the envelope be different with respect to them? Calculation A will of course give the same result. And so will Calculations B and C. The difference between the situations arises only in Calculation D. From the standpoint of the first three calculations, we would say that the problem in which a player opens the envelope is completely identical to the problem in which he does not open it, as I assumed above. But Calculation D (if it is correct, and as stated it probably is not correct) shows that there may be a difference between these cases.

Before we proceed, I will make a general remark about paradoxes.

Are there paradoxes in the world?

Many raise the question whether there are any genuine paradoxes in the world at all. In the final analysis, the truth is always one, and if we have two methods that seem correct to us and yet yield different results, necessarily only one of them is correct, except that we do not know which one. But as we saw above, even if there is one correct answer, and it now turns out that this is always the case, a paradox is a calculation known to be erroneous, but where we cannot identify its point of failure. That is precisely what a paradox is. A paradox is not a situation in which there are two different answers and both are correct (a pluralistic situation). Only one of them is ever correct, and the paradox is the inability to point to which one is the correct one, or what the problem is in the erroneous calculation.

Of course, this is not the case with respect to two ethical arguments that lead to different conclusions. Here a pluralistic situation may exist, and therefore this is not necessarily a paradox but perhaps simply different facets of ethics. One person may raise an argument in favor of a targeted killing of a terrorist that harms an innocent person together with him, on the grounds that the lives of our citizens take precedence, and at the same time also raise the contrary argument that one may not save oneself at the cost of another’s life (except in a case where he is a pursuer). Is there a contradiction between the arguments? Certainly not. They present two sides of the problem, and both are true together. In the bottom line, one must make a decision, that is, prefer one over the other, but even at the stage of decision it does not necessarily become clear that one of them was mistaken. In the ethical realm, two "calculations" that yield different results point to the many facets of the problem, and therefore this is a conflict (how to act in practice) and not a paradox (a contradiction). But with respect to facts, or to a mathematical result, the situation is different. Here there is no possibility of speaking about two faces or two different aspects of the matter. If there are two different solutions, we are dealing with a contradiction, and it is clear that in practice only one of them is correct.

In this sense, a paradox is a kind of riddle. We have two modes of calculation that yield different results (or two modes of thought that yield different conclusions). It is clear to us that one of them is erroneous, and in many cases it is even clear to us which one. The riddle is to put a finger on the erroneous calculation, and especially on the flaw in the other (erroneous) way. Indeed, it seems that there are no "real" paradoxes in existence (a pluralistic state of affairs), but there are paradoxes that constitute a challenging riddle,[8] and that is their main function.

An ill-defined problem

Yet there is nevertheless one possibility in which a pluralistic state of affairs (more than one correct solution) is created even with respect to facts or with respect to a mathematical problem, namely when the problem or the situation is not fully defined. For example, think of the following problem: given a first-degree equation with two unknowns: X+Y=5. Someone may propose the solution (1,4) or the solution (2,3), and of course there are many more solutions. Which of them is correct? Both are correct. How can it be that a mathematical problem has two different solutions and both are correct? Is it not true here that there are no real paradoxes? Note that these solutions contradict one another, and yet such a situation is familiar and known to all of us and does not seem to create any special problem. The reason is that this problem is not fully defined (it is not defined in a way that gives it a unique solution). If we were to add further conditions and define the problem completely, it would have a unique solution. For example, one could add another independent equation that binds those same two unknowns.[9]

To amuse you in the days of corona, think about the following problem. A person stands at some point on the surface of the earth. He walks one kilometer north, then one kilometer east, and finally returns one kilometer south, arriving back exactly at the starting point. What is that point? Before you read on, try for a moment to think about it yourself.

Well, you surely thought of the obvious solution, namely the South Pole. The person goes from there one kilometer north (in whatever direction he likes), then walks one kilometer east (all this takes place on a circle all of whose points are one kilometer from the South Pole), and then from whatever point he reaches, if he walks one kilometer south he will return exactly to the South Pole. Simple, right? But in truth there is another infinite collection of solutions, less intuitive ones. Let us pause for a moment, and try to think about it again.

All right, here they are. Think about circles of latitude a little below the North Pole. As one goes down southward from the pole, the circumference of the circle of latitude containing the point one reaches obviously grows. Now we can define the additional solutions. Let us go down from the North Pole southward (in whatever direction we choose) and stop at a point whose circle of latitude has a circumference of exactly one kilometer. Now stand on one of the points on that circle, and go one more kilometer south. The set of points obtained in this way is arranged on a circle that lies one kilometer below the previous circle, and it is easy to see that all of them satisfy the requirements of the riddle.

So this mathematical problem too, which clearly deals with reality, is not well defined (it has infinitely many solutions). If I am given the problem, I cannot give a single unequivocal answer. In order to define the problem fully, that is, so that it will have only one solution, one must add constraints to it, for example: that the point lies on a certain line of longitude, or that it lies south of the equator, and so on. So we see that even a problem dealing with reality (and one that is also mathematical) can have more than one solution if it is not fully defined.

This description may perhaps give us a direction for a possible solution to the envelope paradox. If we have seen that there are several methods of calculation that yield different results, and all of them seem correct (that is, we have found no flaw in any of them), then either we are mistaken, and if not, then the only possibility is that here too the problem is not well defined. The conclusion is that we should examine where there is a degree of freedom in our problem, and if we discover such a degree of freedom, perhaps we will be able to define it fully, and then we will discover that only one of these methods of calculation is correct. Alternatively, each of the methods of calculation assumes something different, that is, treats a different problem.

Where in the envelope problem could there be a gap requiring definition? Apparently everything is known and everything is well defined. The obvious candidate is the probability of getting a smaller or a larger amount in the second envelope. For some reason we assumed that these two probabilities are equal and each is 1/2. Does this 1/2 really necessarily follow from the definition of the problem? Here there may lie a degree of freedom that will help us solve the paradox. Why do we assume that the probability of the two amounts in the second envelope is equal? Presumably because in the absence of other knowledge that is the natural assumption. If so, there you have a possible gap (absence of information) on which the problem may hang. This is Gadi Alexandrovich’s claim in his post on the envelope problem (and below we shall see that it is not necessary for solving our problem).

Second proposal for a solution: the distribution

When we assign probabilities to different states, this is the result of some given distribution. The distribution describes the probability attached to each event in our event-space. From the distribution one can calculate the probability of any simple or complex event. For example, when tossing a die there are six possible outcomes. If the die is fair, then the distribution is uniform, meaning that there is the same probability of obtaining each of the six results. If, for example, we ask what the probability is of obtaining an even result, the answer is 1/2 (3 times 1/6). By contrast, if the die is not fair, for instance if the probability of obtaining 1 or 3 is 0.1, and the probability of obtaining each of the other possibilities is 0.2 (all of this sums to 1, of course), then this is a non-uniform distribution. In such a case, the probability of obtaining an even result is 0.6. Note the conclusion that follows from this. Suppose one asks you, with respect to a given die, what the probability is of obtaining an even result. The answer is not 0.5 but rather that the problem is not defined. To complete its definition, you need to be given the distribution (is the die fair, and if not, how does it behave?). Only given the distribution can one calculate unambiguously the probabilities of every event. Otherwise, there will be several calculations, all of which may be correct, meaning that the problem is not well defined.

This can be presented differently. The expected value of the game or lottery is defined as the average outcome we would reach if we repeated them again and again (infinitely many times, or the limit obtained as the number of games is increased as much as we like). The amount I have in hand per game (that is, the total amount toward which the results of all the games tend, divided by the number of games, as the number of games tends to infinity) is the expected value. Now we must ask ourselves what it means to repeat the coin toss or the envelope game again and again. In the context of the die, one must know the nature of the die. In the context of the envelopes, we must know how the contents of the envelopes before the player are determined each time. Clearly, in order to define this, we must fix the lottery according to which the sums inside the envelopes are determined and how the pair of envelopes is chosen for the game. For example, we determine that the amount X appears in this lottery with probability P, for each possible amount X, arrange two pairs of envelopes (one with the possibility that X is the larger amount and the other that it is the smaller), and then choose a pair of envelopes at random and play with it. This is exactly the definition of the distribution. The conclusion is that without the distribution one cannot speak of probabilities and therefore also not of expected value.[10]

If so, Gadi Alexandrovich argues, it is now natural to ask how the probabilities were calculated for obtaining this or that amount in the two envelopes. What is given to us is only the ratio between the amounts, but nothing whatever about the amounts themselves (the value of X). So there you have a gap in the problem that must be defined in order for the problem to be well defined: the probability distribution of obtaining amounts in the different envelopes. For example, if the distribution for receiving an amount declines as the amount rises, then it is not correct to assume equal probability for a switch that gives a higher amount and a switch that gives a lower amount. So here one of our assumptions in the paradox has already been broken (= equal probability for the two possibilities). Given a certain distribution of the probabilities of obtaining amount X in the envelopes, the problem is then well defined, and that gives the player a tool with which to calculate the expected payoff and whether it is worthwhile for him to switch envelopes. If that player does not know the distribution, then he lacks information and cannot answer the question. The problem is not fully defined. True, there is some distribution and he merely does not know what it is, but then again he is mistaken because he does not have all the information in hand. This is not a paradox. At most, it can only mean that such a player has no unambiguous winning strategy in the condition of ignorance in which he finds himself.

If you think about the case in which the player opens the envelope and sees the amount inside it, you can understand the problem more easily. If he found ₪200 there, he is supposed to use the distribution to calculate the probability of finding ₪100 or ₪400 in the second envelope, and there is no a priori reason to assume that the probabilities are equal. On the contrary, if the probability of a higher amount in an envelope decreases (as is likely in most practical cases), then the probability that the second envelope contains 400 is lower than the probability that it contains 100. The distribution will tell me what the ratio is between the probabilities, and that will determine the winning strategy. For example, if the probability of getting 400 is 10% and the probability of getting 100 is 90%,[11] then the expected gain from switching is:
60 – = 100*0.9 – 300*0.1
that is, it is not worthwhile for him to switch. But if the probability of getting 400 is 40%, then even though that probability is lower than the probability of losing 100, it would still be worthwhile for him to switch. The expected gain from switching in such a case is:
60 = 100*0.6 – 300*0.4.
You see that the distribution determines the result.

Gadi Alexandrovich argues that the solution to the problem lies in the fact that we did not define the distribution, and therefore the problem is not defined. But he adds and argues that even if one takes the distribution into account, the problem is still not solved. One can define a problem in which the distribution for every amount is uniform (the probability of every amount is the same). The result is that in such a state it really is worthwhile to switch envelopes, except that then it is worthwhile to switch again. Here we are led into a paradox even though the problem is well defined (there is a given distribution). But that is a mistake. It is important to understand that a uniform distribution over all amounts (that is, an infinite number of amounts) is not defined, since the probability of each amount is 0. Therefore such a problem is still not defined. If so, apparently we have solved the paradox. The problem is not well defined, and the assumption that the probability of each of the two outcomes is identical is an unfounded assumption.

But that too is not correct, since one can think of a state in which the probability really is equal (for example when there is not an infinite number of possibilities). Moreover, the four calculations above yielded different results even though they all assumed equal probability. That is, this thesis will not explain the contradiction between the calculations we made.

A variation on the problem

Alexandrovich there brings a different version of the problem in the name of Prof. Noga Alon. Its essence is a distribution that can indeed be defined and yet leaves the problem intact. This will sharpen for you what I wrote here, that the problem is not really solved even if one brings in the matter of the distribution.

Suppose the game is defined so that the amounts put into the envelopes can only be powers of 10: ₪10, ₪100, ₪1000, and so on. Now let us define the distribution in the following way: the probability of getting a pair of envelopes with amounts 10 and 100 is 1/2. The probability of getting 100 and 1000 is 1/4, and so on. More generally, the probability of getting the pair (10ⁿ , 10ⁿ⁺¹) is 1/2ⁿ. This is a legitimate distribution (since the sum of the probabilities of all the possibilities is 1). Now everything in the problem is already defined, and therefore the calculation should be unambiguous and its result should represent the correct conclusion.

Let us discuss the problem in which the player opens the first envelope and then must decide whether to switch. The player found in it an amount of 10ⁿ ₪. In the second envelope there may be either 10^n-1 or 10ⁿ⁺¹. The probabilities of these two outcomes are: 1/2^n-1 for the first outcome, and 1/2ⁿ for the second. In such a situation, the probability that the second envelope contains an amount ten times smaller than the amount in our envelope is double the probability that the second envelope contains a larger amount. So with probability 1/3 we will gain from switching envelopes, and with probability 2/3 we will lose.[12] The expected gain from switching in such a case can be calculated simply, and it of course comes out positive (because the larger amount is 10 times as large, while the probability of reaching it is only half the probability of reaching the smaller amount). In this case the calculation gives: 24*10^n-1, that is, if you found ₪100 in your envelope, the expected gain from switching is ₪240; the advantage of switching is significant, and this of course rises with the amount.

But now one may ask what is wrong with this result. Perhaps this really is the result, counterintuitive though it may be. Is this a paradox or merely a counterintuitive result? Here we must return to Calculations A, B, and C, and see that they can all be applied here (not D, because it is not relevant to the case where one opens the first envelope). In other words, there is symmetry between the envelopes (there is no reason to assume that I happened to open specifically the smaller of the two). Alternatively, my friend who is also holding his own envelope will reach the same conclusion, and he too will want to switch. It is not sensible that the rational conclusion is that each of us should switch envelopes. Therefore this is a paradox (and not merely a counterintuitive result).

Alexandrovich there proposes that one should not identify the notion "it is worthwhile to switch" with the difference between the expected payoffs of the two possibilities. He also shows that the expected-value calculations for these states (both for switching and for not switching) yield infinite results. But, as he himself argues, in the final analysis something here does not seem sensible, and the matter remains unresolved.

At the end of the previous section I already explained why this proposal cannot provide a solution to the paradox. Here I will add yet another problem with it. In order to arrive at his infinities, Alexandrovich performs expected-value calculations that include the probability of receiving an envelope with a certain amount, then a calculation of the expected gain from switching or not switching in that case, and then an averaging over all amounts. But it seems to me that the question under discussion here is different (conditional probability): given that we received some amount (let us assume for purposes of the discussion that we opened the envelope), is the expected gain from switching positive or negative? That calculation was already done above, and there is nothing infinite about it (neither in the two calculations nor in the difference between them).

In other words, the experiment he proposes to repeat infinitely many times is to choose a random pair of envelopes from a collection of pairs arranged with the frequency described by the distribution above, check whether switching is worthwhile, and then average over all the pairs. By contrast, I propose choosing a particular pair of envelopes and opening one of them. If I happened to get an amount of ₪100, I check the value of switching. If I did not get 100, I close the envelope again, return the two envelopes to the pile, shuffle, and choose another pair. I check only the cases in which we received ₪100 and average over all the gains from all those cases. To the best of my understanding, in the example he proposed the result will be a gain of ₪240 per case if one switches. Therefore his proposal does not really solve the problem.

Third proposal for a solution: back to the first proposal

So taking the distribution into account has not saved us. I find myself returning again to the proposal described in Marius Cohen’s article. If we succeed in explaining why Calculation C is correct and not B, that will constitute a solution to the paradox, even in our formulation.

It seems to me that the analysis I made at the end of the previous paragraph can explain this very well. Calculations B and D assume a certain amount in one envelope (either mine or the other one), and two possibilities for the amount in the second envelope. To repeat this experiment infinitely many times means to perform an experiment in which the amount in my envelope is given. But that is not the case with which we are dealing. On the contrary, as you can see at the end of the previous section, we draw a pair of envelopes, and out of that pair we choose one, and then wonder whether it is worthwhile to switch. If so, Marius Cohen’s calculation is indeed the one that describes the situation as it really is. His calculation assumes that we have a pair of envelopes containing two given amounts. What is open at the moment is the question which of the two I am holding (the larger or the smaller). Therefore Calculation C is precisely the relevant calculation for the case at hand. Calculations B and D refer to a situation in which I choose one envelope with a given amount, and then draw a second envelope containing an amount half as large or twice as large. But in such a case it really is worthwhile to switch, and there is no paradox here.

Think about the following game.[13] I have an envelope in my hand, I opened it, and it contains ₪100. I am now offered a lottery in which with probability 1/2 I will receive ₪200 and with probability 1/2 I will receive ₪50. Participation in the lottery requires me to pay ₪100 (the amount in my envelope). Is it worthwhile to pay that amount? The answer is of course yes. Here there is no dilemma and no paradox. In such a case the expected value is clear, and it is certainly worthwhile to participate in the lottery (that is, to switch envelopes). This is the situation in the case where the amount in my envelope is fixed and known. Here Calculation B is indeed the correct calculation. Calculations A and C deal with a different case, and therefore their result is different. Calculation D deals with a third case (the reverse one), in which the amount in the other envelope is kept fixed and the envelope in my hand is randomized.

Do we really need the distribution?

If I am right, then the root of the solution does not lie at all in the question of the distribution. The description of the game contains within it three different games, and we did not enter into the differences between them, but to each of them a different calculation is appropriate. For the game as we understood it, Calculation C is appropriate, and therefore in it one should not switch (the expected gain from switching is 0). In other games, like the one I described here, Calculation B is appropriate, in which case it is indeed worthwhile to switch (the expected gain from switching is positive), or Calculation D, in which case it is worthwhile not to switch (the expected gain from switching is negative).

So, in the bottom line, Alexandrovich is right that this really is an ill-defined problem. But what is missing is not exactly the distribution; rather, the description of the situation is incomplete. Of course, one can also see this as a lack in the distribution, since each of the situations has, of course, a different distribution of outcomes. Therefore I do not know whether my solution here is essentially different from Alexandrovich’s or not. I do think that at this point there is no need to remain unresolved, as happened in his case. And this, as we have seen, brings us back to Marius Cohen’s solution, except that now it is also clear why it is correct to choose Calculation C and not Calculation B. Therefore one can now see this as a solution to the paradox.

Fourth proposal for a solution: Gill

My son Nachman drew my attention to the fact that Gill, in section 1.3 of his aforementioned article, shows that the problem is not related to the distribution at all, and that Alexandrovich’s entire discussion (of the original problem, not Noga Alon’s) is fundamentally mistaken. The two-envelope problem that I described at the beginning speaks of two given envelopes whose amounts are completely known. They are not randomized in any way whatsoever. The only randomization in the process is the choice between the envelopes.

To see this, let us follow Gill’s formulation there. Think about two identical envelopes such that envelope A contains ₪100 and envelope B contains ₪200. I choose one of them at random (and I do not know whether this is envelope A or B), and now I have to decide whether to exchange it for the second envelope. Now we shall see that even in such a case, where there is no randomization or distribution of the amounts, one can formulate the paradox. To do so, let us denote the amount in the envelope we chose by S. If this is envelope A then S=100, and if it is B then S=200. Each of these events has probability 1/2 because the choice between the envelopes is random. Precisely for that reason there is no significance here to the distribution (the 1/2 probability for each possibility follows from the choice of the envelope and not from a lottery of the amounts in the envelopes).

The amount in the second envelope will be denoted by T. We know that there are two possibilities: T=0.5S (if S=200), or T=2S (if S=100). Here too the probability of each of these possibilities is 1/2 (because the choice of the first envelope is random).

Now, according to the formula for total expectation, we get:

E (T) = E (T | T = 2S) · P (A) + E (T | T = S/2) · P (B)

The amount T (that is, the amount in the envelope remaining before me) is calculated as the sum of two possibilities: the amount in that envelope if S=100 (that is, if I chose envelope A) plus the amount in that envelope if S=200 (that is, if I chose envelope B).

If we substitute the relevant quantities, we get:

E (T) = 2S · 1/2 + S/2 · 1/2 = S + S/4 = 5S/4

We have obtained that the amount in that envelope is larger by a quarter than the one in the envelope I chose. That is, it is worthwhile for me to switch.

But it is clear that even in this case such a result cannot be possible. There is symmetry between the envelopes, and therefore it cannot be that the other envelope yields a greater average gain than the one I chose. So there you have the paradox even without assuming any lottery over the amounts in the envelopes, and without any need to resort to distributions of the amounts.

Gill argues that the root of the problem is that we calculated the wrong quantity. What we calculated here is the expectation of T as such, whereas what we should have calculated is the conditional expectation E(T/S) . After all, we are dealing with the expectation of T given that it is known that the envelope in our hand now contains S, and that by definition is a conditional expectation. Gill in section 1.3 shows that the calculation of the conditional expectation gives:

E (T/S=a) = a

That is, there is no additional gain in switching. It is important to understand that the calculation here is nothing other than Calculation C, which was presented at the beginning of the column. The mistake was that we calculated an absolute expectation whereas we should have calculated a conditional expectation.

The calculation of E(T) that was done in the previous formula is nothing other than Calculation B, which is a calculation of the expected gain in the case presented by my son Yossi (one chooses an envelope and randomizes the amount in the second envelope). In such a case the expectation really is not conditional, because the randomization is independent, and therefore, as I explained, in such a case it really is worthwhile to switch. But the correct calculation for our case, in which one chooses an envelope and the amount in the second envelope is determined dependently on the envelope chosen, is the calculation of the conditional expectation, which is Calculation C, and as we saw its conclusion is that there is no additional gain in switching.

We see once again that the paradox can be presented even without resorting to distributions. There is no random lottery of the amounts in the envelopes here, and therefore Alexandrovich’s claim regarding this case (that a uniform distribution over an infinite number of amounts is undefined) is not relevant to the discussion. We also see here that in the end it can be shown that the paradox does not exist even without resorting to the distribution. In that sense, it seems to me that this solution resembles what I proposed in the previous section. The meaning of the solution is that, given the precise situation, one can present a consistent calculation even without resorting to the distribution, and even under the assumption that the probability of the two possibilities is indeed 1/2. By contrast, when dealing with the problem presented by Noga Alon, there one really does need to resort to the distribution, and therefore that is a different problem. It seems to me that our solution will work there.

What does all this have to do with Anna Karenina?

At the beginning of the column I mentioned the article I received from my son Nachman, which proposes solutions to the envelope paradox and connects them to the Anna Karenina principle (discussed in column 286). I tried to think about what the connection between these two really is.

Ultimately, as far as I understand, there is not much connection. Gill merely tried to show that there are many different proposals for solving the paradox, just as there are many different ways to be unhappy. That is, there are many incorrect solutions, but there is only one correct solution. This reminds me somewhat of Rabbi Medan’s charming remark that he knows 22 explanations for why we read the Book of Ruth on Shavuot, but only one explanation for why we read the Book of Esther on Purim. What he meant to say is that when there is a multiplicity of explanations, it is quite clear that there is something unsatisfactory in each of them. When the explanation is clear and correct, there is no need to go looking for more explanations.

But here, if I am right, the different solutions all revolve around the same point: the distribution and the incompleteness of the problem. One can approach this from different angles, and usually each proposal deals with a different case (and as we saw, it is correct with respect to that case), but at bottom there is something common to them all. As stated, there are no real paradoxes with respect to mathematics or facts. In reality there is always one correct answer. If we encounter different calculations that seem correct, then it appears that we are simply dealing with different real situations (the problem is not fully defined). We can now see that all the solutions I presented converge and complement one another. A family can be happy in a well-defined way, if it does so in the right way.

[1] The article is taken from Galileo.

[2] If he simply opened one of the other doors and it happened to turn out that there was a goat behind it, the analysis is different (in such a case the probability of winning is 1/2 in both options, whether he switches or not).

[3] Here enters the assumption I mentioned above, that the host’s opening is not random, but rather an intentional opening of a door with a goat behind it.

[4] It seems to me that Daniel Kahneman once wrote about the fascinating evolutionary question of how it is that our probabilistic thinking is so defective (this has been the source of most of his livelihood), despite the fact that these are very vital skills for our survival. Apparently he and Tversky were agents sent on behalf of the god of evolution in order to correct the situation. After them we will all ascend the evolutionary straight and narrow. Kahneman’s winning of the Nobel Prize is itself a distinctly evolutionary act, whose "purpose" (an illegal word in evolutionary discourse; the use is always borrowed) was to bring these failures to the awareness of us all and thereby improve our probabilistic thinking. Not for nothing, the worlds in which Kahneman and Tversky did not discover these failures and/or in which Kahneman did not win the Nobel Prize have gone extinct.

[5] Although, as I noted above, even if we did not know which of the calculations is the correct one, the very fact that there are two contradictory calculations is enough to define the problem as a paradox. Knowing which of them is correct only intensifies it.

[6] Achilles’ position at time t is described by 10t. The tortoise’s position at time t is 10+5t. When t=2, both have covered 20 meters, that is, that is the point at which Achilles passes the tortoise.

[7] And likewise if you sum the distances, 10 meters plus 5 plus 2.5 plus 1.25 and so on, which comes to 20 meters.

[8] Gadi Alexandrovich, in his post on the envelope paradox, defines this as a proof by contradiction, because he is referring to paradoxes created by bad definitions, and in such cases the paradox is a proof by contradiction that the definition is faulty or vague. But not all paradoxes are of that sort. For example, the envelope paradox, as he himself notes there, is not connected to the definition (later we shall see that it is connected to the lack of full definition of the problem, and not of one term or another), and therefore I prefer to relate to paradoxes as a riddle. That is a more general and more correct statement.

[9] Note that a similar state can exist even with one unknown. For example, a quadratic equation is a problem that is not fully defined. Take the equation: x²-5x+4=0 . It has two solutions: x=1,4, and of course both are correct. In order to define the problem completely (so that it will have a unique solution), one must add a constraint, for example: x>2. In such a case the only correct solution is x=4.

Of course, semantically one can view the problem as well defined, and its solution as a set of numbers rather than a single number (this is also the case with an inequality, such as: x>2).

[10] It seems to me that this may depend on the definition of probability, which is disputed among professionals (the philosophy of probability). Some view probability as a calculation of the frequency of outcomes, whereas others view it as a calculation defined mathematically for a single game.

[11] These are probabilities conditional on the fact that the envelope in my hand contains 200. Otherwise, the probabilities would not sum to 1, of course.

[12] The precise calculation is according to Bayes’ formula. I did not go into it because the result is obvious.

[13] In the course of a discussion with my son Yossi about the envelope paradox, he brought up this game.