The Two envelope Paradox: Philosophy Forums

Philosophy Forums » Logic and Philosophy of Math » The Two envelope Paradox » The Two envelope Paradox

Page: 1 2 3 4 5 6

The Two envelope Paradox

LauLuna
Aspirant

Usergroup: Members
Joined: Jul 10, 2008
Total Topics: 3
Total Posts: 43

Posted 07/23/08 - 05:02 AM:	#51
Death Monkey wrote: If you don't think that testing the coin tells you anything about the odds of it coming up heads or tails in future flips, (...). I've never denied such an obvious fact. I only add that the construction of the coin is not the unique causal factor able to provoke a systematic slant. Death Monkey wrote: As I pointed out, the flaw here is that if you do construct the experiment as you suggest, whereby some set of possible distributions is constructed, and then for each trial you use coin-flipping to choose one distribution for that trial, then it will absolutely not be the case that conditional probability of having opened the smaller of the two envelopes does not depend on the value of that envelope. No matter what the overal distribution of possible distributions you allow for the experiment is, there will be a resulting probability distribution for combinations of values in the envelopes, and that distribution will necessarily be one for which it is not always 50/50 for all observed values of the first envelope to be the smaller of the two values. Let me illustrate. Let's constrain the set of distributions as follows. We will consider all cases where the value of the smaller envelope ranges from $1 to $X, in $1 increments, with equal probabilities for all values. Now for each value of X, we have a possible distribution. We then use coin-flipping, as you suggested, to choose the value of X for a particular trial. We now need a distribution of X. Let's say that we set up our coin-flipping decision such that X is uniformly distribution from $10 to $10000, in $1 increments. I assume you meant 'from $1 to $10000'. Death Monkey wrote: This gives us a conditional probability distribution for whether the first envelope is the larger or smaller, that clearly does depend on the value seen. For example, in the case cited above, since $10 is pretty low in the overal range, it is more likely to be the smaller value. But if you saw something like $9000, it would be less likely. And for any value over $10000, you could be 100% sure that it is the larger envelope. No, not at all. When I see $10 in my envelope the sample space collapses into just two possibilities: $10/$20 and $10/$5. Now, I have been informed that all distributions are equiprobable. So, I have no reason to believe that $10/$20 is more likely than $10/$5. You've just proposed a version of the strengthened paradox that requires no infinite sample set. I think there are two main issues in our exchange: 1. Why should we be confident that a fair coin will behave according to 50%/50%, in spite of there being many other causal factors in the determination of the result? This has already been proposed in another thread ('Probability') on this forum. 2. Can the paradox be solved by resorting to the fact that the probability of the distributions of pairs of quantities into the envelopes need not be equiprobable and cannot be presumed to be so? The second is what the strengthened version addresses and attempts to give a negative answer. I propose to initiate a new thread about the 'finite' version of the strengthened paradox. Best

Death Monkey
Tenured Poster

Usergroup: Members
Joined: Sep 18, 2003
Location: Eindhoven, The Netherlands
Total Topics: 7
Total Posts: 2437

Posted 07/23/08 - 06:58 AM:	#52
LauLuna, I think there are two main issues in our exchange: 1. Why should we be confident that a fair coin will behave according to 50%/50%, in spite of there being many other causal factors in the determination of the result? This has already been proposed in another thread ('Probability') on this forum. 2. Can the paradox be solved by resorting to the fact that the probability of the distributions of pairs of quantities into the envelopes need not be equiprobable and cannot be presumed to be so? The second is what the strengthened version addresses and attempts to give a negative answer. I propose to initiate a new thread about the 'finite' version of the strengthened paradox. There is no paradox in the finite version. If you specify the distribution of possible combinations, then the statistics are all quite straightforward. The so-called paradox in the standard version is simply a result of incorrect math. As I have already pointed out in this thread, the calculations by which one obtains that they should be better off switching on average, are just doing the math wrong. This principle of assuming that the odds should be equal whenever you don't know what they are, is clearly not valid for this problem (and thus not valid in general). This is easily demonstrated by showing that if you apply this principle to this problem, you do not get the correct answer, where the correct answer is easily proven, by symmetry, to be that you are no better off switching. This clearly shows that the equaprobability assumption is not valid for this problem, and thus not valid in general. As for the coin tossing issue, I really don't see where you are having a problem. The bottom line is that either you believe that the coin is fair, and that external factors are not going to bias the coin flip, in which case you assume that the flips will be 50/50, or you do not believe this, in which case you do not assume that the flips will be 50/50. So this principle of equal probabilities you have suggested never even enters into it. It's not that a person says "Gee, I don't have any idea what the relative distribution of heads and tails will be, so I will just assume they are equal". That would clearly be a foolish assumption to make. A person decides, for whatever reasons, that he can reasonably expect the coin to come up heads just as often as tails. This can be because he observed that the coin is has symmetric geometry and mass distribution, and he understands physics well enough to realize that this means that external bias (other than just not flipping properly) is impossible. It could be because he has thoroughly tested the coin, and found it to give even odds under the conditions he is concerned with. It could be because somebody he trusts has told him that it gives even odds. It doesn't matter. The point is that one does not simply assume that since they don't know the odds, they are even. It is also easy to construct much simpler examples than the envelope problem, to demonstrate this. The issue of whether or not the belief that external factors won't play a role, is is actually a justified belief, is a completely seperate one. As an example of a simpler demonstration, consider this problem: Let's say that we know that if we reach into a bag, we can find a cat, a dog, a plate, a bowl, or a flower. We don't know the distribution though, so should we just assume 20% for each? How about the odds of getting an animal? Should we assume 50%, since we don't know how likely we are to get an animal or not an animal? How about getting a dish? Or getting something alive? As soon as we apply the principle of indifference to one of these cases, we get conditional probabilitie for all the others. But depending on which one we apply the principle to, we get different answers. Obviously there is a serious problem here. The problem, of course, is that we cannot just assume that just because we only know about one way to categorize the outcomes, that each category in that categorization is going to be equally likely. If you don't know the distribution, then you just don't. I cannot imagine any reason, other than simply misguided intuition, why in the absense of any information, one would think that a uniform distribution should be a better guess than any other possibility you could imagine. Certainly there is no mathematical justification for this conclusion. DM
Pseudoscience makes Baby Jesus cry.

LauLuna
Aspirant

Usergroup: Members
Joined: Jul 10, 2008
Total Topics: 3
Total Posts: 43

Posted 07/24/08 - 02:52 AM:	#53
Death Monkey wrote: There is no paradox in the finite version. If you specify the distribution of possible combinations, then the statistics are all quite straightforward. The so-called paradox in the standard version is simply a result of incorrect math. As I have already pointed out in this thread, the calculations by which one obtains that they should be better off switching on average, are just doing the math wrong. We tackle this in the new thread. Death Monkey wrote: The issue of whether or not the belief that external factors won't play a role, is is actually a justified belief, is a completely seperate one. Well, this has been my point all the time. Death Monkey wrote: As an example of a simpler demonstration, consider this problem: Let's say that we know that if we reach into a bag, we can find a cat, a dog, a plate, a bowl, or a flower. We don't know the distribution though, so should we just assume 20% for each? Why not?. What would you do if you had to bet on this without any additional information? The principle of indifference is simple: whenever you have no reason to believe that an event is likelier than any of its alternatives, assign all alternatives equal probability. It sounds sensible. Remember that we are not trying to find out what is actually happenning out there, we are just looking for the best way to harness the incomplete information we have. Death Monkey wrote: How about the odds of getting an animal? Should we assume 50%, since we don't know how likely we are to get an animal or not an animal? That would be silly since we do have reasons to believe otherwise, namely, we know that among the 5 possible findings only 2 are animals. Death Monkey wrote: If you don't know the distribution, then you just don't. I cannot imagine any reason, other than simply misguided intuition, why in the absense of any information, one would think that a uniform distribution should be a better guess than any other possibility you could imagine. Certainly there is no mathematical justification for this conclusion. When we ignore the distribution and have all the same to make a choice, assuming equiprobability is not irrational if we have no reasons to believe otherwise. And this is certainly what we do when we toss a fair coin: we ignore how the rest of the factors will behave but we have no reason to believe they will favor one side up instead of the other; then we apply indifference and assign 50% to each side. Indifference is a principle in the subjectivistic or Bayesian interpretation of probability. I guess it makes little sense in any other. Still it is what we resort to in many cases, for instance, in flipping a fair coin. But I grant it is not obvious why we should apply the principle. Why not use a meta-principle of indifference and assume that any other tentative assignment has equal odds to behave correctly? Why should we not infer that it is indifferent whether we apply the principle of indifference or we don't? Of course, the principle is not of mathematical nature. That's why sometimes I entertain fanciful thoughts and wonder whether there is some physical or ontological implicit assumption behind the principle. For example, the assumption that things tend to mix up into chaotic states, that is, something akin to entropy. Regards

Death Monkey
Tenured Poster

Usergroup: Members
Joined: Sep 18, 2003
Location: Eindhoven, The Netherlands
Total Topics: 7
Total Posts: 2437

Posted 07/24/08 - 06:31 AM:	#54
LauLuna, As an example of a simpler demonstration, consider this problem: Let's say that we know that if we reach into a bag, we can find a cat, a dog, a plate, a bowl, or a flower. We don't know the distribution though, so should we just assume 20% for each? Why not?. What would you do if you had to bet on this without any additional information? Then it would not matter. If you don't have any information to base your guess on, then any guess is as good as any other. The principle of indifference is simple: whenever you have no reason to believe that an event is likelier than any of its alternatives, assign all alternatives equal probability. As my example demonstrates, you can't do this consistently, because there will be multiple ways to categorize your "alternatives", and each arbitrary choice results in the principle of indifference giving different numbers. It sounds sensible. Remember that we are not trying to find out what is actually happenning out there, we are just looking for the best way to harness the incomplete information we have. And as the two envelope problem clearly illustrates, this isn't it. Simply assuming equal probabilities not only is no better a guess for the distribution than any other guess you could make, but in cases such as this one, it is a guess that cannot possibly be correct. How about the odds of getting an animal? Should we assume 50%, since we don't know how likely we are to get an animal or not an animal? That would be silly since we do have reasons to believe otherwise, namely, we know that among the 5 possible findings only 2 are animals. No, you do not have any reason to believe otherwise. The fact that only two of the possible five are animals only makes a difference if you have already assumed that those five options are equally probable. In other words, it depends on which categorization you start with. If you start by saying all five objects are equally probable, then you have to conclude that the animal vs non-animal options are not equally probable. But if you start by saying that the animal vs non-animal options are equally probable, then you have to conclude that the five objects are not equally probable. There are no logical grounds upon which you can decide which of the possible categorizations should be the one you apply the principle of indifference to. You are just intuitively deciding that it should be applied to the distinct object types, rather than to the other categorizations I mentioned. But there is no justification for this. You have not been provided with any information that would allow you to conclude that the selection of the objects that have been placed in the bag was made in a way that corresponds to your choice. It is entirely possible, for example, that the method by which they were chosen was to place 10 animals and 10 non-animals in the bag, and that the specific objects present just happen to be the ones that were randomly selected. You don't know. And simply guessing and then treating that guess as knowledge, does not allow you to draw valid conclusions. If you do assume that all five objects are equally probable, you must conclude that animals and non-animals are not equally probable. But you have absolutely no justification for drawing this conclusion. If you don't know the distribution, then you just don't. I cannot imagine any reason, other than simply misguided intuition, why in the absense of any information, one would think that a uniform distribution should be a better guess than any other possibility you could imagine. Certainly there is no mathematical justification for this conclusion. When we ignore the distribution and have all the same to make a choice, assuming equiprobability is not irrational if we have no reasons to believe otherwise. The issue is that it cannot be expected to be any better than any other guess we could make either. It is, in no way, the optimal or best guess to make. And this is certainly what we do when we toss a fair coin: we ignore how the rest of the factors will behave but we have no reason to believe they will favor one side up instead of the other; then we apply indifference and assign 50% to each side. Again, I very much disagree with this. When I don't know whether the coin will come up 50/50 or not, I most certainly do not assume that it will. Indifference is a principle in the subjectivistic or Bayesian interpretation of probability. I guess it makes little sense in any other. Still it is what we resort to in many cases, for instance, in flipping a fair coin. I think that you are thinking about the principle that is used of assigning equal probabilities to micro-states, when evaluating relative probabilities of macro-states. But even there, we require an unambiguous set of micro-states. But I grant it is not obvious why we should apply the principle. Why not use a meta-principle of indifference and assume that any other tentative assignment has equal odds to behave correctly? Why should we not infer that it is indifferent whether we apply the principle of indifference or we don't? Of course, the principle is not of mathematical nature. But that is a lot different. Saying that it is as good a guess as any other, is a lot different than saying that we should assume that it is correct. That's why sometimes I entertain fanciful thoughts and wonder whether there is some physical or ontological implicit assumption behind the principle. For example, the assumption that things tend to mix up into chaotic states, that is, something akin to entropy. Yes, this is what I mentioned above about micro-states. The point of the example I gave with the objects in the bag is that we don't know what the actual micro-states are. Your intuitive response that it would be silly to assign 50/50 odds to animal vs non-animal demonstrates that you have intuitively taken the objects themselves to be the micro-states, but of course this need not be the case. In situations where the micro-states are clear and unambiguous, such as in thermodynamics, we don't have this problem. But that is only because they are clear. Even then, in cases where it is not clear, we can run into problems. Simply treating the set of distinct outcomes you can have as thought they were eqauprobable microstates, is not valid. DM
Pseudoscience makes Baby Jesus cry.

LauLuna
Aspirant

Usergroup: Members
Joined: Jul 10, 2008
Total Topics: 3
Total Posts: 43

Posted 07/27/08 - 03:28 AM:	#55
Dear Death Monkey, let me know if I have understood you about micro/macro-states and entropy. Tell me if the following is an appropriate example. Let's say we have a system with two different elementary ingredients (like atomic propositions) and two possible values for each ingredient: 1/0. Now we take a micro-state to be a configuration of the ingredient/value table. There are 2^2 = 4 possible micro-states. We take a macro-state to be one of the following: A) all ingredients = 1 B) all ingredients = 0 C) one ingredient = 1 and one = 0 Macro-states A and B correspond to 1 micro-state each. C corresponds to 2 different micro-states. So the most probable macro-state is C, the thermodynamic equilibrium. But doesn't this assume that for each elementary ingredient the value 1 is just as probable as the value 0? Is this not again the principle of indifference? Regards

Death Monkey
Tenured Poster

Usergroup: Members
Joined: Sep 18, 2003
Location: Eindhoven, The Netherlands
Total Topics: 7
Total Posts: 2437

Posted 07/28/08 - 01:36 AM:	#56
LauLuna, But doesn't this assume that for each elementary ingredient the value 1 is just as probable as the value 0? Is this not again the principle of indifference? Sort of. They call this the equaprobability principle. But it should be noted that this is understood to be an empirically derived principle, that is only valid in some cases. It is not some universal logical or statistical principle. In fact, it is understood that in thermodynamics, there is an underlying physical reason for this principle to hold. Usually it is based on issues of symmetry. For example, in the thermodynamics of an ideal gas, the equaprobability principle follows directly from symmetry. In quantum systems, it arises from symmetries in quantum dynamics. And again, it does not always hold. in cases where it does not hold, the resulting thermodynamic statistics end up being a bit strange (though the laws of thermodynamics still hold). The reason I brought this up is to show that there are really two issues at stake. First, if we can establish a unique set of micro-states, we still need some justification for assuming that they will be equaprobible. And second, if we have done the above, we still need to be working with our microstates in order to apply the principle, and not with some arbitrarily selected set of macro-states. I should point out here that there is something that could be called a "principle of indifference", and which is in some ways intuitively similar to what you are describing. consider if I have a coin. We won't assume that it is a fair coin, or that it will be flipped properly. But we also won't assume any specific distribution either. As I have mentioned, it would not be valid to assume a 50/50 result. But nevertheless, it seems like it does not matter whether we choose heads or tails. We don't know, so it does not make any difference. This can be called the principle of indifference. And this is absolutely correct. But we have to be careful about what this means formally, and what its implications are. What it means is that if you flip the coid (without showing me), and I randomly guess either heads or tails, with equal probability, then I have a 50/50 chance of guessing correctly. So in that sense, it does not matter what I guess. No strategy I can choose can be expected to do any better than guessing at random, with equal probabilities. But it is important to note that this is because my guess is random. It does not say anything about the actual distribution. For example, this principle would apply even if the coin was 100% likely to come up heads every time. I think this is where the intuition behind the principle of indifference you are proposing comes from. There are a wide variety of situations where your principle and the one I have mentioned end up giving the same results. So it seems as though, in those situations, you are best off just assuming that all possibilities are equally likely. But this really isn't the case. You are just best off guessing at random, with equal probabilities. You can see this in the envelope problem. If we just consider one trial: You get two envelopes, and that is it. You open one envelope, and it has $100 in it. If you choose at random, with 50/50 odds, whether to keep it or open the other, then you are equally likely to end up with the larger envelope as with the smaller one. But that does not mean that it is equally likely that the other envelope has $50 or $200. That is where the difference between the valid version of the principle of indifference, and the invalid version, becomes apperant. When you weight the probabilities based on payoff, it suddenly makes a big difference whether you say that the two outcomes are equally likely (not true), or that you are equally likely to guess the correct outcome if you guess at random with 50/50 odds (true). DM
Pseudoscience makes Baby Jesus cry.

LauLuna
Aspirant

Usergroup: Members
Joined: Jul 10, 2008
Total Topics: 3
Total Posts: 43

Posted 07/28/08 - 05:08 AM:	#57
Death Monkey wrote: In fact, it is understood that in thermodynamics, there is an underlying physical reason for this principle to hold. Usually it is based on issues of symmetry. For example, in the thermodynamics of an ideal gas, the equaprobability principle follows directly from symmetry. In quantum systems, it arises from symmetries in quantum dynamics. And again, it does not always hold. in cases where it does not hold, the resulting thermodynamic statistics end up being a bit strange (though the laws of thermodynamics still hold). I wonder whether something of the kind is behind our expectation that, as the number of trials increases, the result tends to the expected distribution. Perhaps we are assuming the world has a penchant for disorder, to realize all possibilities, to spread over the whole possibility space. Death Monkey wrote: As I have mentioned, it would not be valid to assume a 50/50 result. But nevertheless, it seems like it does not matter whether we choose heads or tails. We don't know, so it does not make any difference. This can be called the principle of indifference. Perhaps, from a subjectivistic interpretation of probablility, there is ultimately no difference. It can be argued that I am 'rationally entitled' to choose at random iff the probability I must 'rationally assign' to each possible event is the same. Regards

Death Monkey
Tenured Poster

Usergroup: Members
Joined: Sep 18, 2003
Location: Eindhoven, The Netherlands
Total Topics: 7
Total Posts: 2437

Posted 07/28/08 - 05:38 AM:	#58
LauLuna, I wonder whether something of the kind is behind our expectation that, as the number of trials increases, the result tends to the expected distribution. Perhaps we are assuming the world has a penchant for disorder, to realize all possibilities, to spread over the whole possibility space. I don't know about that. The expectation you mention above follows directly from the assumption that the expected distribution is the distribution that the data is actually coming from. When that expectation is not met, that is a clear indication that our assumption was simply wrong. The distribution we expected was not the distribution our data actually came from. As I have mentioned, it would not be valid to assume a 50/50 result. But nevertheless, it seems like it does not matter whether we choose heads or tails. We don't know, so it does not make any difference. This can be called the principle of indifference. Perhaps, from a subjectivistic interpretation of probability, there is ultimately no difference. It can be argued that I am 'rationally entitled' to choose at random iff the probability I must 'rationally assign' to each possible event is the same. Maybe I am misunderstanding you, but if I am not, then the whole point is that this is not the case. What this (and other) thought experiments show is that they are not the same. You are rationally entitled to choose at random if you do not possess any information that would allow you to logically determine that some other strategy would be better. For example, if you know that the coin is more likely to come up heads, you are clearly better off choosing heads every time than you would be to choose at random (50/50). You are, however, only rationally entitled to assign equal probabilities to each possibility if you have some justification for doing so, such as empirical evidence, arguments from symmetry, and so on... Again, it should be pointed out as I mentioned earlier in the thread that the fundamental issue in subjective probability is that you cannot directly apply probability theory to real-world problems where the outcome is already determined. In such cases, you must first formulate a probabilistic model (or thought experiment, if you prefer), and then apply probability theory to that. Any conclusions you draw are then dependant on how well the model you have constructed actually represents the real problem. The two variations on the "principle of indifference" that I mentioned in my last post are quite similar intuitively, but represent very different probabilistic models. In your version, the probabilistic model is of the form where you have some experiment that can produce one of several possible outcomes, and if you repeat the experiment many many times, then on average each outcome will happen just as often as every other outcome. The version I mentioned is of the form where you have some experiment that can produce one of several possible outcomes, and each time you repeat the experiment you randomly choose one of the possible outcomes from an equaprobable distribution. In this case, the conclusion that after many many repetitions of the experiment, you will have been correct 1/Nth of the time , where N is the number of possibilities, does not depend at all on the relative probabilities of the possible outcomes of the experiment. The outcomes of the experiment need not even be random at all. DM
Pseudoscience makes Baby Jesus cry.

Page: 1 2 3 4 5 6

You don't have permission to post.

Please login or register.