Two envelopes problem

3809:

switch) are incorrect. He analyses Smullyan's arguments in detail, showing that intermediate steps are being taken, and pinpointing exactly where an incorrect inference is made according to his formalization of counterfactual inference. An important difference with Chase's analysis is that he does not take account of the part of the story where we are told that the envelope called envelope A is decided completely at random. Thus, Chase puts probability back into the problem description in order to conclude that arguments 1 and 3 are incorrect, argument 2 is correct, while Yi keeps "two envelope problem without probability" completely free of probability and comes to the conclusion that there are no reasons to prefer any action. This corresponds to the view of Albers et al., that without a probability ingredient, there is no way to argue that one action is better than another, anyway.

1428:

envelope contains the amount A, he differentiates the actual situation in two different games: The first game would be played with the amounts (A, 2A) and the second game with the amounts (A/2, A). Only one of them is actually played but we don't know which one. These two games need to be treated differently. If the player wants to compute his/her expected return (profit or loss) in case of exchange, he/she should weigh the return derived from each game by the average amount in the two envelopes in that particular game. In the first case the profit would be A with an average amount of 3A/2, whereas in the second case the loss would be A/2 with an average amount of 3A/4. So the formula of the expected return in case of exchange, seen as a proportion of the total amount in the two envelopes, is:

1789:

amounts, the "equally likely" assumption starts to appear a bit unreasonable. Suppose we stop, just with these ten equally likely possibilities for the smaller amount in the two envelopes. In that case, the reasoning in steps 6 and 7 was entirely correct if envelope A happened to contain any of the amounts 2, 4, ... 512: switching envelopes would give an expected (average) gain of 25%. If envelope A happened to contain the amount 1, then the expected gain is actually 100%. But if it happened to contain the amount 1024, a massive loss of 50% (of a rather large amount) would have been incurred. That only happens once in twenty times, but it is exactly enough to balance the expected gains in the other 19 out of 20 times.

3805:

in which two amounts of money are put in the two envelopes first, and only after that is one chosen arbitrarily and given to the player. In the first argument, however, we consider the amount of money in the envelope first given to the player as fixed and consider the situations where the second envelope contains either half or twice that amount. This would only be a reasonable counterfactual world if in reality the envelopes had been filled as follows: first, some amount of money is placed in the specific envelope that will be given to the player; and secondly, by some arbitrary process, the other envelope is filled (arbitrarily or randomly) either with double or with half of that amount of money.

1700:. However, many mathematicians and statisticians interpret the argument as an attempt to calculate the expected amount in Envelope B, given a real or hypothetical amount "A" in Envelope A. One does not need to look in the envelope to see how much is in there, in order to do the calculation. If the result of the calculation is an advice to switch envelopes, whatever amount might be in there, then it would appear that one should switch anyway, without looking. In this case, at Steps 6, 7 and 8 of the reasoning, "A" is any fixed possible value of the amount of money in the first envelope. 1704:(which seems to have started with Nalebuff) in which the owner of envelope A does actually look in his envelope before deciding whether or not to switch; though Nalebuff does also emphasize that there is no need to have the owner of envelope A look in his envelope. If he imagines looking in it, and if for any amount which he can imagine being in there, he has an argument to switch, then he will decide to switch anyway. Finally, this interpretation was also the core of earlier versions of the two envelopes problem (Littlewood's, Schrödinger's, and Kraitchik's switching paradoxes); see 1172: 1850:). The originally proposed problem does not make clear exactly how the smaller of the two sums is determined, what values it could possibly take and, in particular, whether there is a minimum or a maximum sum it might contain. However, if we are using the Bayesian interpretation of probability, then we start by expressing our prior beliefs as to the smaller amount in the two envelopes through a probability distribution. Lack of knowledge can also be expressed in terms of probability. 856: 20: 3813:

the envelopes contained $ 5.00 and $ 10.00 respectively, a player who opened the $ 10.00 envelope would expect the possibility of a $ 20.00 payout that simply does not exist. Were that player to open the $ 5.00 envelope instead, he would believe in the possibility of a $ 2.50 payout, which constitutes a smaller deviation from the true value; this results in the paradoxical discrepancy.

2800:, and the total expected value of switching converges to 0. In addition, if an ongoing distribution with a probability factor greater than one-half is made finite by, after any number of terms, establishing a final term with "all the remaining probability," that is, 1 minus the probability of all previous terms, the expected value of switching with respect to the probability that 1167:{\displaystyle {\begin{aligned}\operatorname {E} (B)&=\operatorname {E} (B\mid A<B)P(A<B)+\operatorname {E} (B\mid A>B)P(A>B)\\&=\operatorname {E} (2A\mid A<B){\frac {1}{2}}+\operatorname {E} \left({\frac {1}{2}}A\mid A>B\right){\frac {1}{2}}\\&=\operatorname {E} (A\mid A<B)+{\frac {1}{4}}\operatorname {E} (A\mid A>B)\end{aligned}}} 816:) is that, in the scenario provided, the mathematics use relative values of A and B (that is, it assumes that one would gain more money if A is less than B than one would lose if the opposite were true). However, the two values of money are fixed (one envelope contains, say, $ 20 and the other $ 40). If the values of the envelopes are restated as 2376: 1749:, and then seems to believe that given that information, the other envelope would be equally likely to contain twice or half that amount. That assumption can only be correct, if prior to knowing what was in Envelope A, the writer would have considered the following two pairs of values for both envelopes equally likely: the amounts 3907:/4. The first paper just presents the two problems. The second discusses many solutions to both of them. The second of his two problems is nowadays the more common, and is presented in this article. According to this version, the two envelopes are filled first, then one is chosen at random and called Envelope A. 3870:

Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two

3733:

However, Clark and Shackel argue that this blaming it all on "the strange behavior of infinity" does not resolve the paradox at all; neither in the single case nor the averaged case. They provide a simple example of a pair of random variables both having infinite mean but where it is clearly sensible

2808:

will exactly negate the sum of the positive expected values that came before, and again the total expected value of switching drops to 0 (this is the general case of setting out an equal probability of a finite set of values in the envelopes described above). Thus, the only distributions that seem to

1741:

one is supposed to assign equal probabilities when one has no knowledge at all concerning the possible values of some quantity. Thus the fact that we are not told anything about how the envelopes are filled can already be converted into probability statements about these amounts. No information means

1275:

is the expected value of the amount in Envelope B. By the same calculation it is also the expected value of the amount in Envelope A. They are the same hence there is no reason to prefer one envelope to the other. This conclusion was, of course, obvious in advance; the point is that we identified the

89:

A person is given two indistinguishable envelopes, each of which contains a sum of money. One envelope contains twice as much as the other. The person may pick one envelope and keep whatever amount it contains. They pick one envelope at random but before they open it they are given the chance to take

75:

Since the situation is symmetric, it seems obvious that there is no point in switching envelopes. On the other hand, a simple calculation using expected values suggests the opposite conclusion, that it is always beneficial to swap envelopes, since the person stands to gain twice as much money if they

3856:, where it concerns a pack of cards, each card has two numbers written on it, the player gets to see a random side of a random card, and the question is whether one should turn over the card. Littlewood's pack of cards is infinitely large and his paradox is a paradox of improper prior distributions. 3842:

concerning two equally rich men who meet and compare their beautiful neckties, presents from their wives, wondering which tie actually cost more money. He also introduces a variant in which the two men compare the contents of their purses. He assumes that each purse is equally likely to contain 1 up

3799:

James Chase argues that the second argument is correct because it does correspond to the way to align two situations (one in which we gain, the other in which we lose), which is preferably indicated by the problem description. Also, Bernard Katz and Doris Olin argue this point of view. In the second

1833:

The first two resolutions discussed above (the "simple resolution" and the "Bayesian resolution") correspond to two possible interpretations of what is going on in step 6 of the argument. They both assume that step 6 indeed is "the bad step". But the description in step 6 is ambiguous. Is the author

1732:

of probability, which means that probability reasoning is not only applied to truly random events like the random pick of an envelope, but also to our knowledge (or lack of knowledge) about things which are fixed but unknown, like the two amounts originally placed in the two envelopes, before one is

785:

It is pointed out that the 'A' in the first part of the formula is the expected value, given that envelope A contains less than envelope B, but the 'A', in the second part of the formula is the expected value in A, given that envelope A contains more than envelope B. The flaw in the argument is that

303:

There have been many solutions proposed, and commonly one writer proposes a solution to the problem as stated, after which another writer shows that altering the problem slightly revives the paradox. Such sequences of discussions have produced a family of closely related formulations of the problem,

3804:

in which the player, counterfactually, got the other envelope to the one he was actually (factually) given is a highly meaningful counterfactual world and hence the comparison between gains and losses in the two worlds is meaningful. This comparison is uniquely indicated by the problem description,

1655:

The mechanism by which the amounts of the two envelopes are determined is crucial for the decision of the player to switch her envelope. Suppose that the amounts in the two envelopes A and B were not determined by first fixing the contents of two envelopes E1 and E2, and then naming them A and B at

1176:

A will be larger when A is larger than B, than when it is smaller than B. So its average values (expectation values) in those two cases are different. And the average value of A is not the same as A itself, anyway. Two mistakes are being made: the writer forgot he was taking expectation values, and

3825:

As an extension to the problem, consider the case where the player is allowed to look in envelope A before deciding whether to switch. In this "conditional switching" problem, it is often possible to generate a gain over the "never switching" strategy", depending on the probability distribution of

3812:

Bliss argues that the source of the paradox is that when one mistakenly believes in the possibility of a larger payoff that does not, in actuality, exist, one is mistaken by a larger margin than when one believes in the possibility of a smaller payoff that does not actually exist. If, for example,

1853:

A first variant within the Bayesian version is to come up with a proper prior probability distribution of the smaller amount of money in the two envelopes, such that when Step 6 is performed properly, the advice is still to prefer Envelope B, whatever might be in Envelope A. So though the specific

2038:

Imagine what might be in the first envelope. A sensible strategy would certainly be to swap when the first envelope contains 1, as the other must then contain 2. Suppose on the other hand the first envelope contains 2. In that case, there are two possibilities: the envelope pair in front of us is

1804:

Many authors have also pointed out that if a maximum sum that can be put in the envelope with the smaller amount exists, then it is very easy to see that Step 6 breaks down, since if the player holds more than the maximum sum that can be put into the "smaller" envelope they must hold the envelope

1703:

This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced in its present-day form, Gardner (1989) and Nalebuff (1988).) It is common in the more mathematical literature on the problem. It also applies to the modification of the problem

1557:

We could actually open our envelope before deciding on switching or not and the above formula would still give us the correct expected return. For example, if we opened our envelope and saw that it contained 100 euros then we would set A=100 in the above formula and the expected return in case of

3816:

Albers, Kooi, and Schaafsma consider that without adding probability (or other) ingredients to the problem, Smullyan's arguments do not give any reason to swap or not to swap, in any case. Thus, there is no paradox. This dismissive attitude is common among writers from probability and economics:

649:

The famous mystification is evoked by confusing the situation where the total amount in the two envelopes is fixed with the situation where the amount in one envelope is fixed and the other can be either double or half that amount. The so-called paradox presents two already appointed and already

3808:

Byeong-Uk Yi, on the other hand, argues that comparing the amount you would gain if you would gain by switching with the amount you would lose if you would lose by switching is a meaningless exercise from the outset. According to his analysis, all three implications (switch, indifferent, do not

1918:

Though Bayesian probability theory can resolve the first mathematical interpretation of the paradox above, it turns out that examples can be found of proper probability distributions, such that the expected value of the amount in the second envelope, conditioned on the amount in the first, does

1788:

amount happened to be in Envelope A, we apparently believe in advance that all the following ten amounts are all equally likely to be the smaller of the two amounts in the two envelopes: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 (equally likely powers of 2). But going to even larger or even smaller

3795:

reasoning. We want to compare the amount that we would gain by switching if we would gain by switching, with the amount we would lose by switching if we would indeed lose by switching. However, we cannot both gain and lose by switching at the same time. We are asked to compare two incompatible

1422: 3847:

of pennies, the total number of pennies minted to date. The men do not look in their purses but each reason that they should switch. He does not explain what is the error in their reasoning. It is not clear whether the puzzle already appeared in an earlier 1942 edition of his book. It is also

1427:

Tsikogiannopoulos presented a different way to do these calculations. It is by definition correct to assign equal probabilities to the events that the other envelope contains double or half that amount in envelope A. So the "switching argument" is correct up to step 6. Given that the player's

3895:

Gardner confessed that though, like Kraitchik, he could give a sound analysis leading to the right answer (there is no point in switching), he could not clearly put his finger on what was wrong with the reasoning for switching, and Kraitchik did not give any help in this direction, either.

1808:

But the problem can also be resolved mathematically without assuming a maximum amount. Nalebuff, Christensen and Utts, Falk and Konold, Blachman, Christensen and Utts, Nickerson and Falk, pointed out that if the amounts of money in the two envelopes have any proper probability distribution

2048: 1854:

calculation performed in step 6 was incorrect (there is no proper prior distribution such that, given what is in the first envelope A, the other envelope is always equally likely to be larger or smaller) a correct calculation, depending on what prior we are using, does lead to the result

650:

locked envelopes, where one envelope is already locked with twice the amount of the other already locked envelope. Whereas step 6 boldly claims "Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2", in the given situation, that claim can never apply to

1672:, then the switching argument is correct and she is recommended to switch envelopes. This version of the problem was introduced by Nalebuff (1988) and is often called the Ali-Baba problem. Notice that there is no need to look in envelope A in order to decide whether or not to switch. 1805:

containing the larger sum, and are thus certain to lose by switching. This may not occur often, but when it does, the heavy loss the player incurs means that, on average, there is no advantage in switching. Some writers consider that this resolves all practical cases of the problem.

1845:

A large literature has developed concerning variants of the problem. The standard assumption about the way the envelopes are set up is that a sum of money is in one envelope, and twice that sum is in another envelope. One of the two envelopes is randomly given to the player

3915:. Barry Nalebuff's asymmetric variant, often known as the Ali Baba problem, has one envelope filled first, called Envelope A, and given to Ali. Then a fair coin is tossed to decide whether Envelope B should contain half or twice that amount, and only then given to Baba. 1656:

random (for instance, by the toss of a fair coin). Instead, we start right at the beginning by putting some amount in envelope A and then fill B in a way which depends both on chance (the toss of a coin) and on what we put in A. Suppose that first of all the amount

307:

No proposed solution is widely accepted as definitive. Despite this, it is common for authors to claim that the solution to the problem is easy, even elementary. Upon investigating these elementary solutions, however, they often differ from one author to the next.

71:, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch? 1550: 1646: 1284: 3746:

questioned if the paradox has anything to do with probabilities at all. He did this by expressing the problem in a way that does not involve probabilities. The following plainly logical arguments lead to conflicting conclusions:

1267: 644: 249: 2588: 1687:

The simple resolution above assumed that the person who invented the argument for switching was trying to calculate the expectation value of the amount in Envelope A, thinking of the two amounts in the envelopes as fixed

3796:

situations. Only one of them can factually occur, the other is a counterfactual situation—somehow imaginary. To compare them at all, we must somehow "align" the two situations, providing some definite points in common.

786:

the same symbol is used with two different meanings in both parts of the same calculation but is assumed to have the same value in both cases. This line of argument is introduced by McGrew, Shier and Silverstein (1997).

1711:

This kind of interpretation is often called "Bayesian" because it assumes the writer is also incorporating a prior probability distribution of possible amounts of money in the two envelopes in the switching argument.

4089:

McDonnell, Mark D; Grant, Alex J; Land, Ingmar; Vellambi, Badri N; Abbott, Derek; Lever, Ken (2011). "Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching".

570: 1792:

Alternatively, we do go on ad infinitum but now we are working with a quite ludicrous assumption, implying for instance, that it is infinitely more likely for the amount in envelope A to be smaller than 1,

2371:{\displaystyle {\begin{aligned}P(\{1,2\}\mid 2)&={\frac {P(\{1,2\})/2}{P(\{1,2\})/2+P(\{2,4\})/2}}\\&={\frac {P(\{1,2\})}{P(\{1,2\})+P(\{2,4\})}}\\&={\frac {1/3}{1/3+2/9}}=3/5,\end{aligned}}} 775:

A widely-discussed way to resolve the paradox, both in popular literature and part of the academic literature, especially in philosophy, is to assume that the 'A' in step 7 is intended to be the

2053: 1660:

in envelope A is fixed in some way or other, and then the amount in Envelope B is fixed, dependent on what is already in A, according to the outcome of a fair coin. If the coin fell Heads then 2

861: 3800:

argument, we consider the amounts of money in the two envelopes as being fixed; what varies is which one is first given to the player. Because that was an arbitrary and physical choice, the

3472: 3718:

producing this variant of the paradox must have an infinite mean. So before the player opens an envelope the expected gain from switching is "∞ − ∞", which is not defined. In the words of

5025: 1433: 735: 3879:. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man? 764: 699: 1563: 3256: 291:

that step is not correct, to be sure not to make this mistake in a situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. The puzzle is

3734:

to prefer one to the other, both conditionally and on average. They argue that decision theory should be extended so as to allow infinite expectation values in some situations.

2516:> 1, and though the second envelope is more likely to be smaller than larger, its conditionally expected amount is larger: the conditionally expected amount in Envelope B is 3661: 3605: 3349: 3137: 3087: 2896: 1842:

which might be in envelope A? Thus, there are two main interpretations of the intention of the composer of the paradoxical argument for switching, and two main resolutions.

3392: 2743:

before it) the mean is infinite, but when the probability factor is less than one-half, the mean converges. In the cases where the probability factor is less than one-half,

1720:

The simple resolution depended on a particular interpretation of what the writer of the argument is trying to calculate: namely, it assumed he was after the (unconditional)

4755: 3002: 2790: 2658: 1901: 1189: 3700: 2842: 2717: 2480: 2427: 3791:

A number of solutions have been put forward. Careful analyses have been made by some logicians. Though solutions differ, they all pinpoint semantic issues concerned with

2719:. That is, only if the mean of all possible values of money in the envelopes is infinite. To see why, compare the series described above in which the probability of each 2035:= 0, 1, 2, ... These probabilities sum to 1, hence the distribution is a proper prior (for subjectivists) and a completely decent probability law also for frequentists. 2596:. This means that the player who looks in envelope A would decide to switch whatever he saw there. Hence there is no need to look in envelope A to make that decision. 1668:/2 is put in Envelope B. If the player was aware of this mechanism, and knows that she holds Envelope A, but do not know the outcome of the coin toss, and do not know 168: 3875:

in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2

3545: 3510: 2521: 3323: 3177: 343: 3930:. The literature contains dozens of commentaries on the problem, much of which observes that a distribution of finite values can have an infinite expected value. 3286: 2953: 2599:

This conclusion is just as clearly wrong as it was in the preceding interpretations of the Two Envelopes Problem. But now the flaws noted above do not apply; the

4962: 3903:

presented two different two-envelope problems, each with one envelope containing twice what is in the other, and each with computation of the expectation value 5

449: 386: 2678: 469: 426: 406: 363: 2603:

in the expected value calculation is a constant and the conditional probabilities in the formula are obtained from a specified and proper prior distribution.

578: 1417:{\displaystyle \operatorname {E} (B)=\operatorname {E} (A\mid A<B)+{\frac {1}{4}}\operatorname {E} (A\mid A>B)=x+{\frac {1}{4}}2x={\frac {3}{2}}x} 2929:), with the same probability distribution and infinite expected value. However, if we do look into the first envelope, then for all values observed ( 3139:, one would have to replace expected value as the decision criterion, thereby employing a more sophisticated argument from mathematical economics. 2512:

So either the first envelope contains 1, in which case the conditional expected amount in the other envelope is 2, or the first envelope contains

1919:

exceed the amount in the first, whatever it might be. The first such example was already given by Nalebuff. See also Christensen and Utts (1992).

5032: 2921:

If we do not look into the first envelope, then clearly there is no reason to switch, since we would be exchanging one unknown amount of money (

3726:

generally breaks down when confronted with games having a diverging expectation, and compares it with the situation generated by the classical

1784:

Suppose for the sake of argument, we start by imagining an amount of 32 in Envelope A. In order that the reasoning in steps 6 and 7 is correct

1728:

value (conditional on what might be in Envelope A). To solve this and related interpretations or versions of the problem, most authors use the

4227:

Tsikogiannopoulos, Panagiotis (2012). "Παραλλαγές του προβλήματος της ανταλλαγής φακέλων" [Variations on the Two Envelopes Problem].

1989:, the smaller of the two amounts of money, such that this bad conclusion is still true. One example is analyzed in more detail, in a moment. 1724:

of what's in Envelope B. In the mathematical literature on Two Envelopes Problem, a different interpretation is more common, involving the

4888: 474: 1809:

representing the player's prior beliefs about the amounts of money in the two envelopes, then it is impossible that whatever the amount

2611:

Most writers think that the new paradox can be defused, although the resolution requires concepts from mathematical economics. Suppose

767:. No "average amount A" can ever form any initial basis for any expected value, as this does not get to the heart of the problem. 1910:

In these cases, it can be shown that the expected sum in both envelopes is infinite. There is no gain, on average, in swapping.

2380:

and consequently the probability it's the {2, 4} pair is 2/5, since these are the only two possibilities. In this derivation,

1769:

in odds form: posterior odds equal prior odds times likelihood ratio). But now we can apply the same reasoning, imagining not

792:

Expected value in B = 1/2 ((Expected value in B, given A is larger than B) + (Expected value in B, given A is smaller than B))

5168: 4945: 4043: 2735:. When the probability of each subsequent term is greater than one-half of the probability of the term before it (and each 1834:

after the unconditional (overall) expectation value of what is in envelope B (perhaps - conditional on the smaller amount,

1276:

false step in the argument for switching by explaining exactly where the calculation being made there went off the rails.

795:

If we then take the sum in one envelope to be x and the sum in the other to be 2x the expected value calculations become:

3945: 2486:(again) envelope A happens to contain 2. Those are the only two ways that envelope A can end up containing the amount 2. 5173: 5093: 4092: 1813:

in the first envelope might be, it would be equally likely, according to these prior beliefs, that the second contains

3359:

provides necessary and sufficient conditions for the utility function to resolve the paradox, and notes that neither

2031:

Suppose for example that the envelope with the smaller amount actually contains 2 dollars with probability 2/3 where

283:

The puzzle is to find the flaw in the line of reasoning in the switching argument. This includes determining exactly

3722:, this is "just another example of a familiar phenomenon, the strange behavior of infinity". Chalmers suggests that 3547:

are bounded simply because the amount of money in an envelope is bounded by the total amount of money in the world (

3397: 1738: 1554:

This result means yet again that the player has to expect neither profit nor loss by exchanging his/her envelope.

706:

contain A/2 but can contain A/2 only in the very specific instance where envelope A, by chance, actually contains

273:

As it is more rational to just open an envelope than to swap indefinitely, the player arrives at a contradiction.

710: 1943: 742: 677: 575:

So on this supposition that the total amount is fixed, swapping is not better than keeping. The expected value

4708:

A famous example of a proper probability distribution of the amounts of money in the two envelopes, for which

5178: 3182: 3663:. From this perspective, the second paradox is resolved because the postulated probability distribution for 4510:

Albers, Casper J; Kooi, Barteld P; Schaafsma, Willem (2005), "Trying to resolve the two-envelope problem",

669:

contain 2A, but can contain 2A only in the very specific instance where envelope A, by chance contains the

4379: 1745:

In steps 6 and 7 of the switching argument, the writer imagines that envelope A contains a certain amount

1733:

picked at random and called "Envelope A". Moreover, according to a long tradition going back at least to

1279:

We could also continue from the correct but difficult to interpret result of the development in line 7:

3610: 3554: 3328: 3092: 3042: 2851: 3918:

Broome in 1995 called a probability distribution 'paradoxical' if for any given first-envelope amount

1828: 4685: 5056: 4172: 3817:

Smullyan's paradox arises precisely because he takes no account whatever of probability or utility.

3362: 2489:

It turns out that these proportions hold in general unless the first envelope contains 1. Denote by

1781:. And similarly, ad infinitum, repeatedly halving or repeatedly doubling as many times as you like. 738:, but nowhere else. The difference between the two already appointed and locked envelopes is always 295:

solved by finding another way to calculate the probabilities that does not lead to a contradiction.

4711: 2958: 2746: 2614: 1857: 1545:{\displaystyle E={\frac {1}{2}}\cdot {\frac {+A}{3A/2}}+{\frac {1}{2}}\cdot {\frac {-A/2}{3A/4}}=0} 4658:

Blachman, NM; Christensen, R; Utts, JM (1996), "Letter with corrections to the original article",

3670: 2812: 2687: 2436: 2383: 1838:), or is he after the conditional expectation of what is in envelope B, given any possible amount 1797:

infinitely more likely to be larger than 1024, than between those two values. This is a so-called

4846:"Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems" 3980: 3849: 3848:

mentioned in a 1953 book on elementary mathematics and mathematical puzzles by the mathematician

3792: 2040: 1725: 1641:{\displaystyle E={\frac {1}{2}}\cdot {\frac {+100}{150}}+{\frac {1}{2}}\cdot {\frac {-50}{75}}=0} 779:

in envelope A and that we intended to write down a formula for the expected value in envelope B.

4042:

A complete list of published and unpublished sources in chronological order can be found in the

2493:

the amount we imagine finding in Envelope A, if we were to open that envelope, and suppose that

3985: 3727: 3703: 3355:

increased toward infinity (a common assumption in mathematical economics and decision theory).

4937: 4926: 5043: 1180:

It would have been easier to compute E(B) directly. Denoting the lower of the two amounts by

3515: 3480: 5102: 4242: 4101: 3970: 3940: 1985:. Now, it turns out that one can quite easily invent proper probability distributions for 1729: 48: 3853: 3299: 3153: 319: 8: 4289:"The exchange paradox: Probabilistic and cognitive analysis of a psychological conundrum" 3975: 3950: 3702:) cannot arise in a real-life situation. Similar arguments are often used to resolve the 3265: 2932: 832:

by switching. One does not gain a greater amount of money by switching because the total

5106: 5091:

McDonnell, M. D.; Abott, D. (2009). "Randomized switching in the two-envelope problem".

4246: 4105: 431: 368: 5145: 5071: 4690: 4543: 4316: 4232: 4025: 3960: 2663: 2043:

that we are dealing with the {1, 2} pair, given that the first envelope contains 2, is

454: 411: 391: 348: 40: 4526:

Falk, Ruma; Nickerson, Raymond (2009), "An inside look at the two envelopes paradox",

1424:

so (of course) different routes to calculate the same thing all give the same answer.

1262:{\displaystyle \operatorname {E} (B)={\frac {1}{2}}2x+{\frac {1}{2}}x={\frac {3}{2}}x} 4941: 4547: 4539: 4320: 4308: 4029: 4021: 3835: 3356: 1798: 1721: 5149: 1825:, is a non-sequitur, also when there is no maximum to the amounts in the envelopes. 5137: 5110: 5006: 4977: 4921: 4903: 4867: 4857: 4811: 4784: 4780: 4699: 4671: 4667: 4644: 4640: 4617: 4613: 4586: 4535: 4477: 4473: 4450: 4446: 4414: 4347: 4336:"Puzzles: Cider in Your Ear, Continuing Dilemma, The Last Shall Be First, and More" 4300: 4184: 4109: 4074: 4069: 4057: 4017: 3743: 3143: 1829:

Introduction to further developments in connection with Bayesian probability theory

661:

This claim is never correct for the situation presented; this claim applies to the

639:{\displaystyle \operatorname {E} ={\frac {1}{2}}2x+{\frac {1}{2}}x={\frac {3}{2}}x} 4405:

Nalebuff, Barry (1989), "Puzzles: The Other Person's Envelope is Always Greener",

4933: 4437:

Christensen, R; Utts, J (1992), "Bayesian Resolution of the "Exchange Paradox"",

3965: 3955: 3723: 813: 56: 44: 244:{\displaystyle {1 \over 2}(2A)+{1 \over 2}\left({A \over 2}\right)={5 \over 4}A} 4997: 4907: 4261: 4150: 4128: 3908: 3900: 3884: 3859: 3719: 3009: 2583:{\displaystyle {\frac {3}{5}}{\frac {a}{2}}+{\frac {2}{5}}2a={\frac {11}{10}}a} 1766: 776: 162: 23:

The problem concerns two envelopes, each containing an unknown amount of money.

5141: 4981: 4815: 4491:

Albers, Casper (March 2003), "2. Trying to resolve the two-envelope problem",

4304: 2680:. It can be shown that this is possible for some probability distributions of 1938:

be the larger. Notice that once we have fixed a probability distribution for

267:

The person concludes that the most rational thing to do is to swap back again.

5162: 4312: 4204:"The Two Envelope Paradox and Using Variables Within the Expectation Formula" 4188: 3834:

The envelope paradox dates back at least to 1953, when Belgian mathematician

1801:: probability calculus breaks down; expectation values are not even defined. 4703: 4590: 4288: 4203: 3296:). Although this is not true for all utility functions, it would be true if 5114: 4113: 1177:

he forgot he was taking expectation values under two different conditions.

114:

is the smaller amount is 1/2, and that it is the larger amount is also 1/2.

5010: 2925:), whose expected value is infinite, for another unknown amount of money ( 4862: 4845: 52: 4872: 4419: 4352: 4335: 665:

only (see below). In the situation presented, the other envelope cannot

4464:

Blachman, NM; Christensen, R; Utts, J (1996). "Letters to the Editor".

257:

so, on average, the person reasons that they stand to gain by swapping.

60: 4802:

Chalmers, David J. (2002). "The St. Petersburg Two-Envelope Paradox".

3763:/2. So the potential gain is strictly greater than the potential loss. 2606: 1675:

Many more variants of the problem have been introduced. Nickerson and

3012:, this problem can be described as a failure of dominance reasoning. 1676: 2809:

point to a positive expected value for switching are those in which

1981:, and hence to the recommendation to switch, whether or not we know 565:{\displaystyle G={1 \over 2}(x)+{1 \over 2}(-x)={1 \over 2}(x-x)=0} 68: 5076: 5070:

Bliss (2012). "A Concise Resolution to the Two Envelope Paradox".

4237: 3477:

Some writers would prefer to argue that in a real-life situation,

824:, it's much easier to see that, if A were greater, one would lose 646:

is the same for both the envelopes. Thus no contradiction exists.

4360:

Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix.

1734: 1696:). The only uncertainty is which envelope has the smaller amount 76:

switch, while the only risk is halving what they currently have.

36: 2906:

have identical probability distributions, by symmetry, and both

19: 2482:

is the probability that the envelope pair is the pair 2 and 4,

2429:

is the probability that the envelope pair is the pair 1 and 2,

4995:

Katz, Bernard; Olin, Doris (2007). "A tale of two envelopes".

851:

Line 7 should have been worked out more carefully as follows:

316:

Suppose that the total amount in both envelopes is a constant

3913:

Penrose Tiles to Trapdoor Ciphers and the Return of Dr Matrix

2039:

either {1, 2} or {2, 4}. All other pairs are impossible. The

3015:

Under dominance reasoning, the fact that we strictly prefer

782:

Step 7 states that the expected value in B = 1/2(2A + A/2).

270:

The person will thus end up swapping envelopes indefinitely.

4928:

Satan, Cantor, and infinity and other mind-boggling puzzles

3911:

independently mentioned this same version in his 1989 book

2684:(the smaller amount of money in the two envelopes) only if 1973:

6 in the "always switching" argument led us to the finding

1184:, and taking it to be fixed (even if unknown) we find that 4171:

McGrew, Timothy; Shier, David; Silverstein, Harry (1997).

4088: 1922:

Denote again the amount of money in the first envelope by

132:

is the smaller amount, then the other envelope contains 2

5128:

Syverson, Paul (1 April 2010). "Opening Two Envelopes".

4563:

The Puzzle of the Two-Envelope Puzzle—a Logical Approach

4657: 4463: 4008:

Falk, Ruma (2008). "The Unrelenting Exchange Paradox".

3922:, the expectation of the other envelope conditional on 3782:. So the potential gain is equal to the potential loss. 3751:

Let the amount in the envelope chosen by the player be

3737: 143:

is the larger amount, then the other envelope contains

59:. The problem is typically introduced by formulating a 4170: 3142:

For example, we could assume the decision maker is an

3039:; however, as already shown, that is not true because 4714: 3673: 3613: 3557: 3518: 3483: 3400: 3365: 3331: 3302: 3268: 3185: 3156: 3095: 3045: 2961: 2935: 2854: 2815: 2749: 2690: 2666: 2617: 2524: 2439: 2386: 2051: 1992:

As mentioned before, it cannot be true that whatever

1860: 1715: 1566: 1436: 1287: 1192: 859: 745: 713: 680: 581: 477: 457: 434: 414: 394: 371: 351: 322: 298: 171: 4604:

Binder, DA (1993), "Letter to editor and response",

828:

by switching and, if B were greater, one would gain

4509: 4222: 4220: 2607:

Proposed resolutions through mathematical economics

304:resulting in voluminous literature on the subject. 55:. It is a variant of an older problem known as the 4925: 4749: 3694: 3655: 3599: 3539: 3504: 3466: 3386: 3343: 3317: 3280: 3250: 3171: 3131: 3081: 2996: 2947: 2890: 2836: 2784: 2711: 2672: 2652: 2582: 2474: 2421: 2370: 1895: 1664:is put in Envelope B, if the coin fell Tails then 1640: 1544: 1416: 1261: 1166: 758: 729: 693: 638: 564: 463: 443: 420: 400: 380: 357: 337: 243: 4577:Broome, John (1995), "The Two-envelope Paradox", 4226: 3709: 5160: 4389:– via Mathematical Association of America. 4287:Nickerson, Raymond S.; Falk, Ruma (2006-05-01). 4217: 4201: 3862:popularized Kraitchik's puzzle in his 1982 book 3089:. To salvage dominance reasoning while allowing 812:In non-technical language, what goes wrong (see 4771:Binder, D. A. (1993). "Letters to the Editor". 4436: 4058:"A Simple Solution to the Two Envelope Problem" 264:and reason in exactly the same manner as above. 5090: 5026:"The Two-envelope Paradox With No Probability" 4631:Ross (1994), "Letter to editor and response", 4493:Distributional Inference: The Limits of Reason 3467:{\displaystyle E(u(W+A))=E(u(W+B))<\infty } 2501:≥ 1. In that case the other envelope contains 1913: 1821:. Thus step 6 of the argument, which leads to 1650: 702:, but nowhere else. The other envelope cannot 388:in the other. If you select the envelope with 4525: 4286: 4259: 4148: 4126: 428:by swapping. If you select the envelope with 107:the amount in the player's selected envelope. 5023: 4963:"The Non-Probabilistic Two Envelope Paradox" 4886: 4797: 4795: 2458: 2446: 2405: 2393: 2284: 2272: 2257: 2245: 2231: 2219: 2183: 2171: 2148: 2136: 2114: 2102: 2074: 2062: 1962:each with probability 1/2, independently of 1705: 770: 98:Now suppose the person reasons as follows: 4377: 4269:Dialogues, Logics and Other Strange Things 4158:Dialogues, Logics and Other Strange Things 4136:Dialogues, Logics and Other Strange Things 2727:with one in which the probability of each 730:{\displaystyle 2{\frac {\text{Total}}{3}}} 5075: 4871: 4861: 4792: 4418: 4351: 4236: 4073: 4055: 3829: 1706:the concluding section, on history of TEP 809:which is equal to the expected sum in A. 759:{\displaystyle {\frac {\text{Total}}{3}}} 694:{\displaystyle {\frac {\text{Total}}{3}}} 260:After the switch, denote that content by 93: 5127: 4994: 4920: 4801: 4404: 4380:"The Psychology of Learning Probability" 4333: 3820: 18: 4828: 4514:, vol. 145, no. 1, p. 91 4387:Statistics for the Twenty-first Century 4202:Schwitzgebe, Eric; Dever, Josh (2008), 3992: 3786: 3774:. Now by swapping, the player may gain 3251:{\displaystyle E(u(W+B)|A=a)<u(W+a)} 1682: 840:) remains the same, and the difference 117:The other envelope may contain either 2 63:challenge like the following example: 5161: 4843: 4789:Comment on Christensen and Utts (1992) 4770: 4684: 4603: 4576: 4490: 4260:Priest, Graham; Restall, Greg (2007), 4149:Priest, Graham; Restall, Greg (2007), 4127:Priest, Graham; Restall, Greg (2007), 3288:is strictly preferred to switching to 2731:is only 1/3 as likely as the previous 1934:be the smaller of the two amounts and 165:of the money in the other envelope is 5069: 4960: 4505: 4503: 4432: 4430: 4400: 4398: 4396: 4378:Falk, Ruma; Konold, Clifford (1992). 4373: 4371: 4369: 4367: 4282: 4280: 4278: 3027:should imply that we strictly prefer 1930:. We think of these as random. Let 311: 5084: 4688:(1995). "The Two-envelope Paradox". 4630: 4007: 4003: 4001: 3766:Let the amounts in the envelopes be 3738:Smullyan's non-probabilistic variant 1679:systematically survey a total of 8. 471:by swapping. So you gain on average 67:Imagine you are given two identical 4173:"The Two-Envelope Problem Resolved" 3871:men can reason: "I have the amount 3852:, who credited it to the physicist 3755:. By swapping, the player may gain 2012:, but it can be true that whatever 1777:in Envelope A. And similarly, for 2 13: 5094:Proceedings of the Royal Society A 4500: 4427: 4393: 4364: 4275: 4093:Proceedings of the Royal Society A 3689: 3461: 3338: 3126: 3076: 2955:) we would want to switch because 2885: 2831: 2706: 1716:Simple form of Bayesian resolution 1346: 1306: 1288: 1193: 1133: 1093: 1032: 989: 934: 886: 864: 582: 299:Multiplicity of proposed solutions 150:Thus the other envelope contains 2 14: 5190: 3998: 3656:{\displaystyle u(W+B)\leq u(W+M)} 3600:{\displaystyle u(W+A)\leq u(W+M)} 3344:{\displaystyle \beta <\infty } 3132:{\displaystyle E(B)=E(A)=\infty } 3082:{\displaystyle E(B)=E(A)=\infty } 3023:for all possible observed values 2891:{\displaystyle E(B)=E(A)=\infty } 2723:is 2/3 as likely as the previous 2433:envelope A happens to contain 2; 2024:is larger in expected value than 4560: 4540:10.1111/j.1467-9639.2009.00346.x 4407:Journal of Economic Perspectives 4340:Journal of Economic Perspectives 4022:10.1111/j.1467-9639.2008.00318.x 3866:, in the form of a wallet game: 1739:principle of insufficient reason 789:A correct calculation would be: 5121: 5063: 5017: 4988: 4954: 4914: 4887:Clark, M.; Shackel, N. (2000). 4880: 4837: 4822: 4764: 4678: 4651: 4624: 4597: 4570: 4554: 4519: 4484: 4457: 4334:Nalebuff, Barry (Spring 1988). 4327: 4271:, College Publications: 135–140 4253: 4160:, College Publications: 135–140 4138:, College Publications: 135–140 2796:other than the first, smallest 79: 43:. It is of special interest in 16:Puzzle in logic and mathematics 4785:10.1080/00031305.1993.10475966 4738: 4725: 4718: 4672:10.1080/00031305.1996.10473551 4645:10.1080/00031305.1994.10476075 4618:10.1080/00031305.1991.10475791 4565:(online ed.), p. 274 4478:10.1080/00031305.1996.10473551 4451:10.1080/00031305.1992.10475902 4195: 4164: 4142: 4120: 4082: 4075:10.5840/logos-episteme20112318 4049: 4036: 3838:proposed a puzzle in his book 3710:Controversy among philosophers 3683: 3677: 3650: 3638: 3629: 3617: 3594: 3582: 3573: 3561: 3534: 3522: 3499: 3487: 3455: 3452: 3440: 3434: 3425: 3422: 3410: 3404: 3387:{\displaystyle u(w)<\beta } 3375: 3369: 3312: 3306: 3245: 3233: 3224: 3211: 3207: 3195: 3189: 3166: 3160: 3146:maximizer with initial wealth 3120: 3114: 3105: 3099: 3070: 3064: 3055: 3049: 2985: 2972: 2965: 2879: 2873: 2864: 2858: 2825: 2819: 2804:is equal to the last, largest 2773: 2760: 2753: 2700: 2694: 2641: 2628: 2621: 2461: 2443: 2408: 2390: 2287: 2269: 2260: 2242: 2234: 2216: 2186: 2168: 2151: 2133: 2117: 2099: 2083: 2059: 1944:joint probability distribution 1884: 1871: 1864: 1742:that probabilities are equal. 1370: 1352: 1330: 1312: 1300: 1294: 1205: 1199: 1157: 1139: 1117: 1099: 1016: 995: 976: 964: 958: 940: 928: 916: 910: 892: 876: 870: 553: 541: 525: 516: 500: 494: 191: 182: 1: 4831:Optimal Statistical Decisions 4750:{\displaystyle E(B|A=a)>a} 2997:{\displaystyle E(B|A=a)>a} 2914:are greater than or equal to 2785:{\displaystyle E(B|A=a)<a} 2653:{\displaystyle E(B|A=a)>a} 2505:/2 with probability 3/5 and 2 1896:{\displaystyle E(B|A=a)>a} 278: 5169:Probability theory paradoxes 4358:and Gardner, Martin (1989) 4262:"Envelopes and Indifference" 4151:"Envelopes and Indifference" 4129:"Envelopes and Indifference" 3695:{\displaystyle E(X)=\infty } 3258:for at least some values of 2837:{\displaystyle E(X)=\infty } 2712:{\displaystyle E(X)=\infty } 2475:{\displaystyle P(\{2,4\})/2} 2422:{\displaystyle P(\{1,2\})/2} 90:the other envelope instead. 7: 4844:Powers, Michael R. (2015). 4833:. McGraw-Hill. p. 109. 4829:DeGroot, Morris H. (1970). 3933: 1914:Second mathematical variant 1903:for all possible values of 1799:improper prior distribution 1651:Nalebuff asymmetric variant 798:Expected value in B = 1/2 ( 663:Nalebuff asymmetric variant 10: 5195: 4889:"The Two-Envelope Paradox" 1926:and that in the second by 451:first you lose the amount 408:first you gain the amount 84: 5174:Decision-making paradoxes 5142:10.1007/s12136-010-0096-7 4773:The American Statistician 4660:The American Statistician 4633:The American Statistician 4606:The American Statistician 4466:The American Statistician 4439:The American Statistician 4305:10.1080/13576500500200049 154:with probability 1/2 and 4908:10.1093/mind/109.435.415 4293:Thinking & Reasoning 3840:Recreational Mathematics 3150:whose utility function, 2004:is equally likely to be 771:Other simple resolutions 158:/2 with probability 1/2. 4982:10.1093/analys/62.2.157 4816:10.1093/analys/62.2.155 4056:Markosian, Ned (2011). 3981:Sleeping Beauty problem 3850:John Edensor Littlewood 3262:(that is, holding onto 3179:, is chosen to satisfy 2041:conditional probability 1730:Bayesian interpretation 1726:conditional expectation 49:Bayesian interpretation 5115:10.1098/rspa.2009.0312 5051:Cite journal requires 4751: 4189:10.1093/analys/57.1.28 4114:10.1098/rspa.2010.0541 3986:St. Petersburg paradox 3893: 3830:History of the paradox 3728:St. Petersburg paradox 3704:St. Petersburg paradox 3696: 3657: 3601: 3541: 3540:{\displaystyle u(W+B)} 3506: 3505:{\displaystyle u(W+A)} 3468: 3388: 3345: 3319: 3282: 3252: 3173: 3133: 3083: 2998: 2949: 2892: 2838: 2786: 2713: 2674: 2654: 2584: 2509:with probability 2/5. 2476: 2423: 2372: 1897: 1642: 1546: 1418: 1263: 1168: 760: 731: 695: 640: 566: 465: 445: 422: 402: 382: 359: 339: 245: 94:The switching argument 73: 24: 5024:Byeong-Uk Yi (2009). 4961:Chase, James (2002). 4752: 4704:10.1093/analys/55.1.6 4591:10.1093/analys/55.1.6 3868: 3843:to some large number 3821:Conditional switching 3697: 3658: 3602: 3542: 3507: 3469: 3389: 3346: 3320: 3283: 3253: 3174: 3134: 3084: 2999: 2950: 2893: 2839: 2787: 2739:is twice that of the 2714: 2675: 2655: 2585: 2477: 2424: 2373: 1898: 1765:. (This follows from 1643: 1547: 1419: 1264: 1169: 761: 732: 696: 641: 567: 466: 446: 423: 403: 383: 360: 340: 253:This is greater than 246: 110:The probability that 65: 29:two envelopes problem 22: 5179:Probability problems 4863:10.3390/risks3010026 4712: 4229:Mathematical Reviews 4062:Logos & Episteme 3993:Notes and references 3941:Bayesian probability 3802:counterfactual world 3787:Proposed resolutions 3714:As mentioned above, 3671: 3611: 3555: 3516: 3481: 3398: 3363: 3329: 3325:had an upper bound, 3318:{\displaystyle u(w)} 3300: 3266: 3183: 3172:{\displaystyle u(w)} 3154: 3093: 3043: 2959: 2933: 2852: 2813: 2747: 2688: 2664: 2615: 2522: 2437: 2384: 2049: 1858: 1683:Bayesian resolutions 1564: 1558:switching would be: 1434: 1285: 1190: 857: 743: 711: 678: 579: 475: 455: 432: 412: 392: 369: 365:in one envelope and 349: 338:{\displaystyle c=3x} 320: 169: 31:, also known as the 5107:2009RSPSA.465.3309M 5101:(2111): 3309–3322. 5011:10.1093/mind/fzm903 4528:Teaching Statistics 4420:10.1257/jep.3.1.171 4353:10.1257/jep.2.2.149 4247:2014arXiv1411.2823T 4106:2011RSPSA.467.2825M 4100:(2134): 2825–2851. 4010:Teaching Statistics 3951:Boy or Girl paradox 3281:{\displaystyle A=a} 2948:{\displaystyle A=a} 2592:which is more than 4747: 3961:Monty Hall problem 3946:Bertrand's paradox 3899:In 1988 and 1989, 3692: 3653: 3597: 3537: 3502: 3464: 3384: 3341: 3315: 3278: 3248: 3169: 3129: 3079: 3035:without observing 2994: 2945: 2888: 2848:, it follows that 2834: 2782: 2709: 2670: 2650: 2580: 2472: 2419: 2368: 2366: 1893: 1757:; and the amounts 1638: 1542: 1414: 1259: 1164: 1162: 756: 727: 691: 636: 562: 461: 444:{\displaystyle 2x} 441: 418: 398: 381:{\displaystyle 2x} 378: 355: 335: 312:Example resolution 241: 53:probability theory 41:probability theory 25: 4947:978-0-679-40688-4 4922:Smullyan, Raymond 3971:Newcomb's paradox 3854:Erwin Schrödinger 3836:Maurice Kraitchik 3357:Michael R. Powers 2844:. Averaging over 2673:{\displaystyle a} 2575: 2556: 2543: 2533: 2345: 2291: 2198: 1722:expectation value 1630: 1612: 1599: 1581: 1534: 1495: 1482: 1451: 1409: 1390: 1344: 1271:We learn that 1.5 1254: 1238: 1219: 1131: 1081: 1051: 1027: 766: 754: 750: 737: 725: 721: 701: 689: 685: 631: 615: 596: 539: 514: 492: 464:{\displaystyle x} 421:{\displaystyle x} 401:{\displaystyle x} 358:{\displaystyle x} 236: 219: 205: 180: 5186: 5154: 5153: 5125: 5119: 5118: 5088: 5082: 5081: 5079: 5067: 5061: 5060: 5054: 5049: 5047: 5039: 5037: 5031:. Archived from 5030: 5021: 5015: 5014: 5005:(464): 903–926. 4992: 4986: 4985: 4967: 4958: 4952: 4951: 4931: 4918: 4912: 4911: 4902:(435): 415–442. 4893: 4884: 4878: 4877: 4875: 4865: 4841: 4835: 4834: 4826: 4820: 4819: 4799: 4790: 4788: 4768: 4762: 4756: 4754: 4753: 4748: 4728: 4707: 4682: 4676: 4674: 4655: 4649: 4647: 4628: 4622: 4620: 4601: 4595: 4593: 4574: 4568: 4566: 4558: 4552: 4550: 4523: 4517: 4515: 4507: 4498: 4496: 4488: 4482: 4481: 4461: 4455: 4453: 4434: 4425: 4423: 4422: 4402: 4391: 4390: 4384: 4375: 4362: 4357: 4355: 4331: 4325: 4324: 4284: 4273: 4272: 4266: 4257: 4251: 4250: 4240: 4224: 4215: 4214: 4208: 4199: 4193: 4192: 4168: 4162: 4161: 4155: 4146: 4140: 4139: 4133: 4124: 4118: 4117: 4086: 4080: 4079: 4077: 4053: 4047: 4040: 4034: 4033: 4005: 3976:Siegel's paradox 3926:is greater than 3891: 3744:Raymond Smullyan 3716:any distribution 3701: 3699: 3698: 3693: 3662: 3660: 3659: 3654: 3606: 3604: 3603: 3598: 3546: 3544: 3543: 3538: 3511: 3509: 3508: 3503: 3473: 3471: 3470: 3465: 3393: 3391: 3390: 3385: 3350: 3348: 3347: 3342: 3324: 3322: 3321: 3316: 3287: 3285: 3284: 3279: 3257: 3255: 3254: 3249: 3214: 3178: 3176: 3175: 3170: 3144:expected utility 3138: 3136: 3135: 3130: 3088: 3086: 3085: 3080: 3003: 3001: 3000: 2995: 2975: 2954: 2952: 2951: 2946: 2897: 2895: 2894: 2889: 2843: 2841: 2840: 2835: 2791: 2789: 2788: 2783: 2763: 2718: 2716: 2715: 2710: 2679: 2677: 2676: 2671: 2659: 2657: 2656: 2651: 2631: 2589: 2587: 2586: 2581: 2576: 2568: 2557: 2549: 2544: 2536: 2534: 2526: 2481: 2479: 2478: 2473: 2468: 2428: 2426: 2425: 2420: 2415: 2377: 2375: 2374: 2369: 2367: 2357: 2346: 2344: 2340: 2326: 2317: 2313: 2304: 2296: 2292: 2290: 2237: 2211: 2203: 2199: 2197: 2193: 2158: 2128: 2124: 2094: 1950:is fixed, since 1902: 1900: 1899: 1894: 1874: 1823:always switching 1647: 1645: 1644: 1639: 1631: 1626: 1618: 1613: 1605: 1600: 1595: 1587: 1582: 1574: 1551: 1549: 1548: 1543: 1535: 1533: 1529: 1517: 1513: 1501: 1496: 1488: 1483: 1481: 1477: 1465: 1457: 1452: 1444: 1423: 1421: 1420: 1415: 1410: 1402: 1391: 1383: 1345: 1337: 1268: 1266: 1265: 1260: 1255: 1247: 1239: 1231: 1220: 1212: 1173: 1171: 1170: 1165: 1163: 1132: 1124: 1086: 1082: 1074: 1072: 1068: 1052: 1044: 1028: 1020: 982: 765: 763: 762: 757: 755: 748: 747: 739: 736: 734: 733: 728: 726: 719: 718: 707: 700: 698: 697: 692: 690: 683: 682: 674: 645: 643: 642: 637: 632: 624: 616: 608: 597: 589: 571: 569: 568: 563: 540: 532: 515: 507: 493: 485: 470: 468: 467: 462: 450: 448: 447: 442: 427: 425: 424: 419: 407: 405: 404: 399: 387: 385: 384: 379: 364: 362: 361: 356: 344: 342: 341: 336: 250: 248: 247: 242: 237: 229: 224: 220: 212: 206: 198: 181: 173: 33:exchange paradox 5194: 5193: 5189: 5188: 5187: 5185: 5184: 5183: 5159: 5158: 5157: 5126: 5122: 5089: 5085: 5068: 5064: 5052: 5050: 5041: 5040: 5035: 5028: 5022: 5018: 4993: 4989: 4965: 4959: 4955: 4948: 4934:Alfred A. Knopf 4919: 4915: 4891: 4885: 4881: 4842: 4838: 4827: 4823: 4800: 4793: 4769: 4765: 4724: 4713: 4710: 4709: 4683: 4679: 4656: 4652: 4629: 4625: 4602: 4598: 4575: 4571: 4559: 4555: 4524: 4520: 4508: 4501: 4489: 4485: 4462: 4458: 4435: 4428: 4403: 4394: 4382: 4376: 4365: 4332: 4328: 4285: 4276: 4264: 4258: 4254: 4225: 4218: 4206: 4200: 4196: 4169: 4165: 4153: 4147: 4143: 4131: 4125: 4121: 4087: 4083: 4054: 4050: 4041: 4037: 4006: 3999: 3995: 3990: 3966:Necktie paradox 3956:Decision theory 3936: 3892: 3883: 3832: 3826:the envelopes. 3823: 3789: 3740: 3724:decision theory 3712: 3672: 3669: 3668: 3612: 3609: 3608: 3556: 3553: 3552: 3517: 3514: 3513: 3482: 3479: 3478: 3399: 3396: 3395: 3364: 3361: 3360: 3330: 3327: 3326: 3301: 3298: 3297: 3267: 3264: 3263: 3210: 3184: 3181: 3180: 3155: 3152: 3151: 3094: 3091: 3090: 3044: 3041: 3040: 2971: 2960: 2957: 2956: 2934: 2931: 2930: 2853: 2850: 2849: 2814: 2811: 2810: 2759: 2748: 2745: 2744: 2689: 2686: 2685: 2665: 2662: 2661: 2627: 2616: 2613: 2612: 2609: 2567: 2548: 2535: 2525: 2523: 2520: 2519: 2464: 2438: 2435: 2434: 2411: 2385: 2382: 2381: 2365: 2364: 2353: 2336: 2322: 2318: 2309: 2305: 2303: 2294: 2293: 2238: 2212: 2210: 2201: 2200: 2189: 2154: 2129: 2120: 2095: 2093: 2086: 2052: 2050: 2047: 2046: 1916: 1870: 1859: 1856: 1855: 1831: 1718: 1685: 1653: 1619: 1617: 1604: 1588: 1586: 1573: 1565: 1562: 1561: 1525: 1518: 1509: 1502: 1500: 1487: 1473: 1466: 1458: 1456: 1443: 1435: 1432: 1431: 1401: 1382: 1336: 1286: 1283: 1282: 1246: 1230: 1211: 1191: 1188: 1187: 1161: 1160: 1123: 1084: 1083: 1073: 1043: 1042: 1038: 1019: 980: 979: 879: 860: 858: 855: 854: 814:Necktie paradox 807: 793: 773: 746: 744: 741: 740: 717: 712: 709: 708: 681: 679: 676: 675: 623: 607: 588: 580: 577: 576: 531: 506: 484: 476: 473: 472: 456: 453: 452: 433: 430: 429: 413: 410: 409: 393: 390: 389: 370: 367: 366: 350: 347: 346: 321: 318: 317: 314: 301: 289:what conditions 281: 276: 228: 211: 207: 197: 172: 170: 167: 166: 96: 87: 82: 57:necktie paradox 45:decision theory 17: 12: 11: 5: 5192: 5182: 5181: 5176: 5171: 5156: 5155: 5136:(4): 479–498. 5130:Acta Analytica 5120: 5083: 5062: 5053:|journal= 5038:on 2011-09-29. 5016: 4987: 4976:(2): 157–160. 4953: 4946: 4913: 4879: 4836: 4821: 4810:(2): 155–157. 4791: 4779:(2): 157–163. 4763: 4746: 4743: 4740: 4737: 4734: 4731: 4727: 4723: 4720: 4717: 4677: 4650: 4639:(3): 267–269, 4623: 4596: 4569: 4553: 4518: 4499: 4483: 4456: 4426: 4413:(1): 171–181, 4392: 4363: 4346:(2): 149–156. 4326: 4299:(2): 181–213. 4274: 4252: 4216: 4194: 4163: 4141: 4119: 4081: 4048: 4035: 3996: 3994: 3991: 3989: 3988: 3983: 3978: 3973: 3968: 3963: 3958: 3953: 3948: 3943: 3937: 3935: 3932: 3909:Martin Gardner 3901:Barry Nalebuff 3885:Martin Gardner 3881: 3860:Martin Gardner 3831: 3828: 3822: 3819: 3793:counterfactual 3788: 3785: 3784: 3783: 3764: 3739: 3736: 3720:David Chalmers 3711: 3708: 3691: 3688: 3685: 3682: 3679: 3676: 3652: 3649: 3646: 3643: 3640: 3637: 3634: 3631: 3628: 3625: 3622: 3619: 3616: 3596: 3593: 3590: 3587: 3584: 3581: 3578: 3575: 3572: 3569: 3566: 3563: 3560: 3536: 3533: 3530: 3527: 3524: 3521: 3501: 3498: 3495: 3492: 3489: 3486: 3463: 3460: 3457: 3454: 3451: 3448: 3445: 3442: 3439: 3436: 3433: 3430: 3427: 3424: 3421: 3418: 3415: 3412: 3409: 3406: 3403: 3383: 3380: 3377: 3374: 3371: 3368: 3340: 3337: 3334: 3314: 3311: 3308: 3305: 3277: 3274: 3271: 3247: 3244: 3241: 3238: 3235: 3232: 3229: 3226: 3223: 3220: 3217: 3213: 3209: 3206: 3203: 3200: 3197: 3194: 3191: 3188: 3168: 3165: 3162: 3159: 3128: 3125: 3122: 3119: 3116: 3113: 3110: 3107: 3104: 3101: 3098: 3078: 3075: 3072: 3069: 3066: 3063: 3060: 3057: 3054: 3051: 3048: 3010:David Chalmers 3008:. As noted by 2993: 2990: 2987: 2984: 2981: 2978: 2974: 2970: 2967: 2964: 2944: 2941: 2938: 2887: 2884: 2881: 2878: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2833: 2830: 2827: 2824: 2821: 2818: 2781: 2778: 2775: 2772: 2769: 2766: 2762: 2758: 2755: 2752: 2708: 2705: 2702: 2699: 2696: 2693: 2669: 2649: 2646: 2643: 2640: 2637: 2634: 2630: 2626: 2623: 2620: 2608: 2605: 2579: 2574: 2571: 2566: 2563: 2560: 2555: 2552: 2547: 2542: 2539: 2532: 2529: 2471: 2467: 2463: 2460: 2457: 2454: 2451: 2448: 2445: 2442: 2418: 2414: 2410: 2407: 2404: 2401: 2398: 2395: 2392: 2389: 2363: 2360: 2356: 2352: 2349: 2343: 2339: 2335: 2332: 2329: 2325: 2321: 2316: 2312: 2308: 2302: 2299: 2297: 2295: 2289: 2286: 2283: 2280: 2277: 2274: 2271: 2268: 2265: 2262: 2259: 2256: 2253: 2250: 2247: 2244: 2241: 2236: 2233: 2230: 2227: 2224: 2221: 2218: 2215: 2209: 2206: 2204: 2202: 2196: 2192: 2188: 2185: 2182: 2179: 2176: 2173: 2170: 2167: 2164: 2161: 2157: 2153: 2150: 2147: 2144: 2141: 2138: 2135: 2132: 2127: 2123: 2119: 2116: 2113: 2110: 2107: 2104: 2101: 2098: 2092: 2089: 2087: 2085: 2082: 2079: 2076: 2073: 2070: 2067: 2064: 2061: 2058: 2055: 2054: 1915: 1912: 1892: 1889: 1886: 1883: 1880: 1877: 1873: 1869: 1866: 1863: 1830: 1827: 1717: 1714: 1684: 1681: 1652: 1649: 1637: 1634: 1629: 1625: 1622: 1616: 1611: 1608: 1603: 1598: 1594: 1591: 1585: 1580: 1577: 1572: 1569: 1541: 1538: 1532: 1528: 1524: 1521: 1516: 1512: 1508: 1505: 1499: 1494: 1491: 1486: 1480: 1476: 1472: 1469: 1464: 1461: 1455: 1450: 1447: 1442: 1439: 1413: 1408: 1405: 1400: 1397: 1394: 1389: 1386: 1381: 1378: 1375: 1372: 1369: 1366: 1363: 1360: 1357: 1354: 1351: 1348: 1343: 1340: 1335: 1332: 1329: 1326: 1323: 1320: 1317: 1314: 1311: 1308: 1305: 1302: 1299: 1296: 1293: 1290: 1258: 1253: 1250: 1245: 1242: 1237: 1234: 1229: 1226: 1223: 1218: 1215: 1210: 1207: 1204: 1201: 1198: 1195: 1159: 1156: 1153: 1150: 1147: 1144: 1141: 1138: 1135: 1130: 1127: 1122: 1119: 1116: 1113: 1110: 1107: 1104: 1101: 1098: 1095: 1092: 1089: 1087: 1085: 1080: 1077: 1071: 1067: 1064: 1061: 1058: 1055: 1050: 1047: 1041: 1037: 1034: 1031: 1026: 1023: 1018: 1015: 1012: 1009: 1006: 1003: 1000: 997: 994: 991: 988: 985: 983: 981: 978: 975: 972: 969: 966: 963: 960: 957: 954: 951: 948: 945: 942: 939: 936: 933: 930: 927: 924: 921: 918: 915: 912: 909: 906: 903: 900: 897: 894: 891: 888: 885: 882: 880: 878: 875: 872: 869: 866: 863: 862: 797: 791: 777:expected value 772: 769: 753: 724: 716: 688: 635: 630: 627: 622: 619: 614: 611: 606: 603: 600: 595: 592: 587: 584: 561: 558: 555: 552: 549: 546: 543: 538: 535: 530: 527: 524: 521: 518: 513: 510: 505: 502: 499: 496: 491: 488: 483: 480: 460: 440: 437: 417: 397: 377: 374: 354: 334: 331: 328: 325: 313: 310: 300: 297: 280: 277: 275: 274: 271: 268: 265: 258: 251: 240: 235: 232: 227: 223: 218: 215: 210: 204: 201: 196: 193: 190: 187: 184: 179: 176: 163:expected value 159: 148: 137: 126: 115: 108: 100: 95: 92: 86: 83: 81: 78: 15: 9: 6: 4: 3: 2: 5191: 5180: 5177: 5175: 5172: 5170: 5167: 5166: 5164: 5151: 5147: 5143: 5139: 5135: 5131: 5124: 5116: 5112: 5108: 5104: 5100: 5096: 5095: 5087: 5078: 5073: 5066: 5058: 5045: 5034: 5027: 5020: 5012: 5008: 5004: 5000: 4999: 4991: 4983: 4979: 4975: 4971: 4964: 4957: 4949: 4943: 4939: 4935: 4930: 4929: 4923: 4917: 4909: 4905: 4901: 4897: 4890: 4883: 4874: 4869: 4864: 4859: 4855: 4851: 4847: 4840: 4832: 4825: 4817: 4813: 4809: 4805: 4798: 4796: 4786: 4782: 4778: 4774: 4767: 4760: 4744: 4741: 4735: 4732: 4729: 4721: 4715: 4705: 4701: 4697: 4693: 4692: 4687: 4681: 4673: 4669: 4665: 4661: 4654: 4646: 4642: 4638: 4634: 4627: 4619: 4615: 4611: 4607: 4600: 4592: 4588: 4584: 4580: 4573: 4564: 4557: 4549: 4545: 4541: 4537: 4533: 4529: 4522: 4513: 4506: 4504: 4494: 4487: 4479: 4475: 4471: 4467: 4460: 4452: 4448: 4445:(4): 274–76, 4444: 4440: 4433: 4431: 4421: 4416: 4412: 4408: 4401: 4399: 4397: 4388: 4381: 4374: 4372: 4370: 4368: 4361: 4354: 4349: 4345: 4341: 4337: 4330: 4322: 4318: 4314: 4310: 4306: 4302: 4298: 4294: 4290: 4283: 4281: 4279: 4270: 4263: 4256: 4248: 4244: 4239: 4234: 4230: 4223: 4221: 4212: 4205: 4198: 4190: 4186: 4182: 4178: 4174: 4167: 4159: 4152: 4145: 4137: 4130: 4123: 4115: 4111: 4107: 4103: 4099: 4095: 4094: 4085: 4076: 4071: 4068:(3): 347–57. 4067: 4063: 4059: 4052: 4045: 4039: 4031: 4027: 4023: 4019: 4015: 4011: 4004: 4002: 3997: 3987: 3984: 3982: 3979: 3977: 3974: 3972: 3969: 3967: 3964: 3962: 3959: 3957: 3954: 3952: 3949: 3947: 3944: 3942: 3939: 3938: 3931: 3929: 3925: 3921: 3916: 3914: 3910: 3906: 3902: 3897: 3890: 3886: 3880: 3878: 3874: 3867: 3865: 3861: 3857: 3855: 3851: 3846: 3841: 3837: 3827: 3818: 3814: 3810: 3806: 3803: 3797: 3794: 3781: 3777: 3773: 3769: 3765: 3762: 3758: 3754: 3750: 3749: 3748: 3745: 3742:The logician 3735: 3731: 3729: 3725: 3721: 3717: 3707: 3705: 3686: 3680: 3674: 3666: 3647: 3644: 3641: 3635: 3632: 3626: 3623: 3620: 3614: 3591: 3588: 3585: 3579: 3576: 3570: 3567: 3564: 3558: 3550: 3531: 3528: 3525: 3519: 3496: 3493: 3490: 3484: 3475: 3474:is required. 3458: 3449: 3446: 3443: 3437: 3431: 3428: 3419: 3416: 3413: 3407: 3401: 3381: 3378: 3372: 3366: 3358: 3354: 3335: 3332: 3309: 3303: 3295: 3291: 3275: 3272: 3269: 3261: 3242: 3239: 3236: 3230: 3227: 3221: 3218: 3215: 3204: 3201: 3198: 3192: 3186: 3163: 3157: 3149: 3145: 3140: 3123: 3117: 3111: 3108: 3102: 3096: 3073: 3067: 3061: 3058: 3052: 3046: 3038: 3034: 3030: 3026: 3022: 3018: 3013: 3011: 3007: 2991: 2988: 2982: 2979: 2976: 2968: 2962: 2942: 2939: 2936: 2928: 2924: 2919: 2917: 2913: 2909: 2905: 2901: 2882: 2876: 2870: 2867: 2861: 2855: 2847: 2828: 2822: 2816: 2807: 2803: 2799: 2795: 2779: 2776: 2770: 2767: 2764: 2756: 2750: 2742: 2738: 2734: 2730: 2726: 2722: 2703: 2697: 2691: 2683: 2667: 2647: 2644: 2638: 2635: 2632: 2624: 2618: 2604: 2602: 2597: 2595: 2590: 2577: 2572: 2569: 2564: 2561: 2558: 2553: 2550: 2545: 2540: 2537: 2530: 2527: 2517: 2515: 2510: 2508: 2504: 2500: 2497:= 2 for some 2496: 2492: 2487: 2485: 2469: 2465: 2455: 2452: 2449: 2440: 2432: 2416: 2412: 2402: 2399: 2396: 2387: 2378: 2361: 2358: 2354: 2350: 2347: 2341: 2337: 2333: 2330: 2327: 2323: 2319: 2314: 2310: 2306: 2300: 2298: 2281: 2278: 2275: 2266: 2263: 2254: 2251: 2248: 2239: 2228: 2225: 2222: 2213: 2207: 2205: 2194: 2190: 2180: 2177: 2174: 2165: 2162: 2159: 2155: 2145: 2142: 2139: 2130: 2125: 2121: 2111: 2108: 2105: 2096: 2090: 2088: 2080: 2077: 2071: 2068: 2065: 2056: 2044: 2042: 2036: 2034: 2029: 2027: 2023: 2019: 2015: 2011: 2007: 2003: 1999: 1995: 1990: 1988: 1984: 1980: 1976: 1975:E(B|A=a)>a 1972: 1967: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1937: 1933: 1929: 1925: 1920: 1911: 1908: 1906: 1890: 1887: 1881: 1878: 1875: 1867: 1861: 1851: 1849: 1843: 1841: 1837: 1826: 1824: 1820: 1816: 1812: 1806: 1802: 1800: 1796: 1790: 1787: 1782: 1780: 1776: 1772: 1768: 1764: 1760: 1756: 1752: 1748: 1743: 1740: 1736: 1731: 1727: 1723: 1713: 1709: 1707: 1701: 1699: 1695: 1691: 1680: 1678: 1673: 1671: 1667: 1663: 1659: 1648: 1635: 1632: 1627: 1623: 1620: 1614: 1609: 1606: 1601: 1596: 1592: 1589: 1583: 1578: 1575: 1570: 1567: 1559: 1555: 1552: 1539: 1536: 1530: 1526: 1522: 1519: 1514: 1510: 1506: 1503: 1497: 1492: 1489: 1484: 1478: 1474: 1470: 1467: 1462: 1459: 1453: 1448: 1445: 1440: 1437: 1429: 1425: 1411: 1406: 1403: 1398: 1395: 1392: 1387: 1384: 1379: 1376: 1373: 1367: 1364: 1361: 1358: 1355: 1349: 1341: 1338: 1333: 1327: 1324: 1321: 1318: 1315: 1309: 1303: 1297: 1291: 1280: 1277: 1274: 1269: 1256: 1251: 1248: 1243: 1240: 1235: 1232: 1227: 1224: 1221: 1216: 1213: 1208: 1202: 1196: 1185: 1183: 1178: 1174: 1154: 1151: 1148: 1145: 1142: 1136: 1128: 1125: 1120: 1114: 1111: 1108: 1105: 1102: 1096: 1090: 1088: 1078: 1075: 1069: 1065: 1062: 1059: 1056: 1053: 1048: 1045: 1039: 1035: 1029: 1024: 1021: 1013: 1010: 1007: 1004: 1001: 998: 992: 986: 984: 973: 970: 967: 961: 955: 952: 949: 946: 943: 937: 931: 925: 922: 919: 913: 907: 904: 901: 898: 895: 889: 883: 881: 873: 867: 852: 849: 847: 843: 839: 836:of A and B (3 835: 831: 827: 823: 819: 815: 810: 805: 801: 796: 790: 787: 783: 780: 778: 768: 751: 722: 714: 705: 686: 672: 668: 664: 659: 657: 656:any average A 653: 647: 633: 628: 625: 620: 617: 612: 609: 604: 601: 598: 593: 590: 585: 573: 572:by swapping. 559: 556: 550: 547: 544: 536: 533: 528: 522: 519: 511: 508: 503: 497: 489: 486: 481: 478: 458: 438: 435: 415: 395: 375: 372: 352: 332: 329: 326: 323: 309: 305: 296: 294: 290: 286: 272: 269: 266: 263: 259: 256: 252: 238: 233: 230: 225: 221: 216: 213: 208: 202: 199: 194: 188: 185: 177: 174: 164: 160: 157: 153: 149: 146: 142: 138: 135: 131: 127: 124: 120: 116: 113: 109: 106: 102: 101: 99: 91: 77: 72: 70: 64: 62: 58: 54: 50: 46: 42: 38: 34: 30: 21: 5133: 5129: 5123: 5098: 5092: 5086: 5065: 5044:cite journal 5033:the original 5019: 5002: 4996: 4990: 4973: 4969: 4956: 4927: 4916: 4899: 4895: 4882: 4873:10419/167837 4856:(1): 26–34. 4853: 4849: 4839: 4830: 4824: 4807: 4803: 4776: 4772: 4766: 4758: 4695: 4689: 4686:Broome, John 4680: 4666:(1): 98–99, 4663: 4659: 4653: 4636: 4632: 4626: 4609: 4605: 4599: 4582: 4578: 4572: 4562: 4561:Chen, Jeff, 4556: 4534:(2): 39–41, 4531: 4527: 4521: 4511: 4492: 4486: 4472:(1): 98–99. 4469: 4465: 4459: 4442: 4438: 4410: 4406: 4386: 4359: 4343: 4339: 4329: 4296: 4292: 4268: 4255: 4231:(in Greek). 4228: 4210: 4197: 4183:(1): 28–33. 4180: 4176: 4166: 4157: 4144: 4135: 4122: 4097: 4091: 4084: 4065: 4061: 4051: 4038: 4016:(3): 86–88. 4013: 4009: 3927: 3923: 3919: 3917: 3912: 3904: 3898: 3894: 3888: 3876: 3872: 3869: 3863: 3858: 3844: 3839: 3833: 3824: 3815: 3811: 3807: 3801: 3798: 3790: 3779: 3775: 3771: 3767: 3760: 3756: 3752: 3741: 3732: 3715: 3713: 3664: 3551:), implying 3548: 3476: 3352: 3293: 3289: 3259: 3147: 3141: 3036: 3032: 3028: 3024: 3020: 3016: 3014: 3005: 2926: 2922: 2920: 2915: 2911: 2907: 2903: 2899: 2845: 2805: 2801: 2797: 2793: 2740: 2736: 2732: 2728: 2724: 2720: 2681: 2610: 2600: 2598: 2593: 2591: 2518: 2513: 2511: 2506: 2502: 2498: 2494: 2490: 2488: 2483: 2430: 2379: 2045: 2037: 2032: 2030: 2025: 2021: 2017: 2013: 2009: 2005: 2001: 1997: 1993: 1991: 1986: 1982: 1978: 1974: 1970: 1968: 1963: 1959: 1955: 1951: 1947: 1939: 1935: 1931: 1927: 1923: 1921: 1917: 1909: 1904: 1852: 1847: 1844: 1839: 1835: 1832: 1822: 1818: 1814: 1810: 1807: 1803: 1794: 1791: 1785: 1783: 1778: 1774: 1770: 1762: 1758: 1754: 1750: 1746: 1744: 1719: 1710: 1702: 1697: 1693: 1689: 1686: 1674: 1669: 1665: 1661: 1657: 1654: 1560: 1556: 1553: 1430: 1426: 1281: 1278: 1272: 1270: 1186: 1181: 1179: 1175: 853: 850: 845: 844:is fixed to 841: 837: 833: 829: 825: 821: 817: 811: 808: 803: 799: 794: 788: 784: 781: 774: 703: 670: 666: 662: 660: 655: 651: 648: 574: 315: 306: 302: 292: 288: 284: 282: 261: 254: 155: 151: 144: 140: 133: 129: 122: 118: 111: 104: 97: 88: 80:Introduction 74: 66: 61:hypothetical 47:and for the 32: 28: 26: 4936:. pp. 4698:(1): 6–11. 4585:(1): 6–11, 3889:Aha! Gotcha 3864:Aha! Gotcha 1767:Bayes' rule 5163:Categories 4612:(2): 160, 1848:envelope A 673:amount of 287:and under 279:The puzzle 103:Denote by 5077:1202.4669 4548:122078010 4321:143472998 4313:1354-6783 4238:1411.2823 4213:: 135–140 4044:talk page 4030:120397860 3690:∞ 3633:≤ 3577:≤ 3462:∞ 3382:β 3339:∞ 3333:β 3292:for some 3127:∞ 3077:∞ 2898:(because 2886:∞ 2832:∞ 2707:∞ 2078:∣ 1942:then the 1621:− 1615:⋅ 1584:⋅ 1504:− 1498:⋅ 1454:⋅ 1359:∣ 1350:⁡ 1319:∣ 1310:⁡ 1292:⁡ 1197:⁡ 1146:∣ 1137:⁡ 1106:∣ 1097:⁡ 1057:∣ 1036:⁡ 1005:∣ 993:⁡ 947:∣ 938:⁡ 899:∣ 890:⁡ 868:⁡ 704:generally 667:generally 548:− 520:− 69:envelopes 5150:12344371 4970:Analysis 4924:(1992). 4804:Analysis 4757:for all 4691:Analysis 4579:Analysis 4512:Synthese 4495:(thesis) 4177:Analysis 3934:See also 3882:— 3778:or lose 3759:or lose 3004:for all 2792:for all 2660:for all 2016:, given 1996:, given 1977:for all 1971:bad step 1786:whatever 1737:and his 5103:Bibcode 4938:189–192 4243:Bibcode 4211:Sorites 4102:Bibcode 2008:/2 or 2 1817:/2 or 2 1753:/2 and 1735:Laplace 671:smaller 654:nor to 345:, with 161:So the 85:Problem 37:paradox 35:, is a 5148: 4944: 4546: 4319: 4311: 4028: 3667:(with 5146:S2CID 5072:arXiv 5036:(PDF) 5029:(PDF) 4966:(PDF) 4892:(PDF) 4850:Risks 4544:S2CID 4383:(PDF) 4317:S2CID 4265:(PDF) 4233:arXiv 4207:(PDF) 4154:(PDF) 4132:(PDF) 4026:S2CID 3770:and 2 3351:, as 1761:and 2 1692:and 2 820:and 2 749:Total 720:Total 684:Total 652:any A 5057:help 4998:Mind 4942:ISBN 4896:Mind 4742:> 4309:ISSN 3607:and 3512:and 3459:< 3394:nor 3379:< 3336:< 3228:< 2989:> 2910:and 2902:and 2777:< 2645:> 1969:The 1964:X, Y 1960:Y, X 1956:X, Y 1952:A, B 1948:A, B 1936:Y=2X 1888:> 1773:but 1677:Falk 1365:> 1325:< 1152:> 1112:< 1063:> 1011:< 971:> 953:> 923:< 905:< 27:The 5138:doi 5111:doi 5099:465 5007:doi 5003:116 4978:doi 4904:doi 4900:109 4868:hdl 4858:doi 4812:doi 4781:doi 4700:doi 4668:doi 4641:doi 4614:doi 4587:doi 4536:doi 4474:doi 4447:doi 4415:doi 4348:doi 4301:doi 4185:doi 4110:doi 4098:467 4070:doi 4018:doi 3031:to 3019:to 2918:). 2484:and 2431:and 2018:A=a 1998:A=a 1958:or 1946:of 1811:A=a 1795:and 1775:a/2 1597:150 1593:100 846:T/3 802:+ 2 293:not 285:why 147:/2. 139:If 128:If 125:/2. 121:or 51:of 39:in 5165:: 5144:. 5134:25 5132:. 5109:. 5097:. 5048:: 5046:}} 5042:{{ 5001:. 4974:62 4972:. 4968:. 4940:. 4932:. 4898:. 4894:. 4866:. 4852:. 4848:. 4808:62 4806:. 4794:^ 4777:47 4775:. 4696:55 4694:. 4664:50 4662:, 4637:48 4635:, 4610:47 4608:, 4583:55 4581:, 4542:, 4532:31 4530:, 4502:^ 4470:50 4468:. 4443:46 4441:, 4429:^ 4409:, 4395:^ 4385:. 4366:^ 4342:. 4338:. 4315:. 4307:. 4297:12 4295:. 4291:. 4277:^ 4267:, 4241:. 4219:^ 4209:, 4181:57 4179:. 4175:. 4156:, 4134:, 4108:. 4096:. 4066:II 4064:. 4060:. 4024:. 4014:30 4012:. 4000:^ 3887:, 3730:. 3706:. 2573:10 2570:11 2028:. 2020:, 2000:, 1966:. 1954:= 1907:. 1708:. 1628:75 1624:50 848:. 658:. 5152:. 5140:: 5117:. 5113:: 5105:: 5080:. 5074:: 5059:) 5055:( 5013:. 5009:: 4984:. 4980:: 4950:. 4910:. 4906:: 4876:. 4870:: 4860:: 4854:3 4818:. 4814:: 4787:. 4783:: 4761:. 4759:a 4745:a 4739:) 4736:a 4733:= 4730:A 4726:| 4722:B 4719:( 4716:E 4706:. 4702:: 4675:. 4670:: 4648:. 4643:: 4621:. 4616:: 4594:. 4589:: 4567:. 4551:. 4538:: 4516:. 4497:. 4480:. 4476:: 4454:. 4449:: 4424:. 4417:: 4411:3 4356:. 4350:: 4344:2 4323:. 4303:: 4249:. 4245:: 4235:: 4191:. 4187:: 4116:. 4112:: 4104:: 4078:. 4072:: 4046:. 4032:. 4020:: 3928:x 3924:x 3920:x 3905:A 3877:A 3873:A 3845:x 3780:X 3776:X 3772:X 3768:X 3761:A 3757:A 3753:A 3687:= 3684:) 3681:X 3678:( 3675:E 3665:X 3651:) 3648:M 3645:+ 3642:W 3639:( 3636:u 3630:) 3627:B 3624:+ 3621:W 3618:( 3615:u 3595:) 3592:M 3589:+ 3586:W 3583:( 3580:u 3574:) 3571:A 3568:+ 3565:W 3562:( 3559:u 3549:M 3535:) 3532:B 3529:+ 3526:W 3523:( 3520:u 3500:) 3497:A 3494:+ 3491:W 3488:( 3485:u 3456:) 3453:) 3450:B 3447:+ 3444:W 3441:( 3438:u 3435:( 3432:E 3429:= 3426:) 3423:) 3420:A 3417:+ 3414:W 3411:( 3408:u 3405:( 3402:E 3376:) 3373:w 3370:( 3367:u 3353:w 3313:) 3310:w 3307:( 3304:u 3294:a 3290:B 3276:a 3273:= 3270:A 3260:a 3246:) 3243:a 3240:+ 3237:W 3234:( 3231:u 3225:) 3222:a 3219:= 3216:A 3212:| 3208:) 3205:B 3202:+ 3199:W 3196:( 3193:u 3190:( 3187:E 3167:) 3164:w 3161:( 3158:u 3148:W 3124:= 3121:) 3118:A 3115:( 3112:E 3109:= 3106:) 3103:B 3100:( 3097:E 3074:= 3071:) 3068:A 3065:( 3062:E 3059:= 3056:) 3053:B 3050:( 3047:E 3037:a 3033:B 3029:A 3025:a 3021:B 3017:A 3006:a 2992:a 2986:) 2983:a 2980:= 2977:A 2973:| 2969:B 2966:( 2963:E 2943:a 2940:= 2937:A 2927:B 2923:A 2916:X 2912:B 2908:A 2904:B 2900:A 2883:= 2880:) 2877:A 2874:( 2871:E 2868:= 2865:) 2862:B 2859:( 2856:E 2846:a 2829:= 2826:) 2823:X 2820:( 2817:E 2806:a 2802:A 2798:a 2794:a 2780:a 2774:) 2771:a 2768:= 2765:A 2761:| 2757:B 2754:( 2751:E 2741:X 2737:X 2733:X 2729:X 2725:X 2721:X 2704:= 2701:) 2698:X 2695:( 2692:E 2682:X 2668:a 2648:a 2642:) 2639:a 2636:= 2633:A 2629:| 2625:B 2622:( 2619:E 2601:a 2594:a 2578:a 2565:= 2562:a 2559:2 2554:5 2551:2 2546:+ 2541:2 2538:a 2531:5 2528:3 2514:a 2507:a 2503:a 2499:n 2495:a 2491:a 2470:2 2466:/ 2462:) 2459:} 2456:4 2453:, 2450:2 2447:{ 2444:( 2441:P 2417:2 2413:/ 2409:) 2406:} 2403:2 2400:, 2397:1 2394:{ 2391:( 2388:P 2362:, 2359:5 2355:/ 2351:3 2348:= 2342:9 2338:/ 2334:2 2331:+ 2328:3 2324:/ 2320:1 2315:3 2311:/ 2307:1 2301:= 2288:) 2285:} 2282:4 2279:, 2276:2 2273:{ 2270:( 2267:P 2264:+ 2261:) 2258:} 2255:2 2252:, 2249:1 2246:{ 2243:( 2240:P 2235:) 2232:} 2229:2 2226:, 2223:1 2220:{ 2217:( 2214:P 2208:= 2195:2 2191:/ 2187:) 2184:} 2181:4 2178:, 2175:2 2172:{ 2169:( 2166:P 2163:+ 2160:2 2156:/ 2152:) 2149:} 2146:2 2143:, 2140:1 2137:{ 2134:( 2131:P 2126:2 2122:/ 2118:) 2115:} 2112:2 2109:, 2106:1 2103:{ 2100:( 2097:P 2091:= 2084:) 2081:2 2075:} 2072:2 2069:, 2066:1 2063:{ 2060:( 2057:P 2033:n 2026:a 2022:B 2014:a 2010:a 2006:a 2002:B 1994:a 1987:X 1983:a 1979:a 1940:X 1932:X 1928:B 1924:A 1905:a 1891:a 1885:) 1882:a 1879:= 1876:A 1872:| 1868:B 1865:( 1862:E 1846:( 1840:a 1836:x 1819:a 1815:a 1779:a 1771:a 1763:a 1759:a 1755:a 1751:a 1747:a 1698:x 1694:x 1690:x 1688:( 1670:a 1666:a 1662:a 1658:a 1636:0 1633:= 1610:2 1607:1 1602:+ 1590:+ 1579:2 1576:1 1571:= 1568:E 1540:0 1537:= 1531:4 1527:/ 1523:A 1520:3 1515:2 1511:/ 1507:A 1493:2 1490:1 1485:+ 1479:2 1475:/ 1471:A 1468:3 1463:A 1460:+ 1449:2 1446:1 1441:= 1438:E 1412:x 1407:2 1404:3 1399:= 1396:x 1393:2 1388:4 1385:1 1380:+ 1377:x 1374:= 1371:) 1368:B 1362:A 1356:A 1353:( 1347:E 1342:4 1339:1 1334:+ 1331:) 1328:B 1322:A 1316:A 1313:( 1307:E 1304:= 1301:) 1298:B 1295:( 1289:E 1273:x 1257:x 1252:2 1249:3 1244:= 1241:x 1236:2 1233:1 1228:+ 1225:x 1222:2 1217:2 1214:1 1209:= 1206:) 1203:B 1200:( 1194:E 1182:x 1158:) 1155:B 1149:A 1143:A 1140:( 1134:E 1129:4 1126:1 1121:+ 1118:) 1115:B 1109:A 1103:A 1100:( 1094:E 1091:= 1079:2 1076:1 1070:) 1066:B 1060:A 1054:A 1049:2 1046:1 1040:( 1033:E 1030:+ 1025:2 1022:1 1017:) 1014:B 1008:A 1002:A 999:2 996:( 990:E 987:= 977:) 974:B 968:A 965:( 962:P 959:) 956:B 950:A 944:B 941:( 935:E 932:+ 929:) 926:B 920:A 917:( 914:P 911:) 908:B 902:A 896:B 893:( 887:E 884:= 877:) 874:B 871:( 865:E 842:x 838:x 834:T 830:x 826:x 822:x 818:x 806:) 804:x 800:x 752:3 723:3 715:2 687:3 634:x 629:2 626:3 621:= 618:x 613:2 610:1 605:+ 602:x 599:2 594:2 591:1 586:= 583:E 560:0 557:= 554:) 551:x 545:x 542:( 537:2 534:1 529:= 526:) 523:x 517:( 512:2 509:1 504:+ 501:) 498:x 495:( 490:2 487:1 482:= 479:G 459:x 439:x 436:2 416:x 396:x 376:x 373:2 353:x 333:x 330:3 327:= 324:c 262:B 255:A 239:A 234:4 231:5 226:= 222:) 217:2 214:A 209:( 203:2 200:1 195:+ 192:) 189:A 186:2 183:( 178:2 175:1 156:A 152:A 145:A 141:A 136:. 134:A 130:A 123:A 119:A 112:A 105:A

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge

Two envelopes problem

Index