Talk:Necktie paradox

Pruning

Deleted: paradox?

Reasoning: outdated (2006), not WP:OR (source provided), term "paradox" used correctly (quoting Wikipedia itself: "A paradox … despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion" (irrespective of anyone calling it also misrepresentation or gambling)). Necktie paradox is also listed under List of paradoxes. Response was reshaped into separate section to preserve content.

Deleted: I fail to see the paradox

Reasoning: Misplaced and assumed that entering any bet is always in one's interest. Response was reshaped into separate section to preserve content.

Deleted: Related to exchange paradox?

Reasoning: Relation to envelope (exchange) paradox is now clearly stated.

Deleted: Added a source

Reasoning: Nice but does not enrich the talk page.

Deleted: a mistake?

Reasoning: Erroneous statement was since removed and only the more expensive tie is mentioned as the prize.

Deleted: Intuitive explanation

Reasoning: Not relevant mention of new information in Two envelopes problem.

Deleted: Fixed Dollar Amounts

Reasoning: Values/errors described (2012) were replaced since.

Deleted: Minor edit.

Reasoning: Hasty edit described here was based on some confusion and soon reverted by other editor. Gain of $40 tie is clearly not a gain of $20.

Cleaned subsections

• Simple explanation

• A Detailed Analysis

The Flaw got slightly shortened, but it seems pointless with such a behemoth which may be simply removed at a point.

SafetyControl (talk) 21:12, 11 February 2018 (UTC)

"Solutions"

Article nails it, while solutions remaining on this talk page often miss the point or introduce some new tiny errors and unnecessary complexity or are just too long for such a simple paradox.

Bold was added at two places. Example: [It is important] "to point out the precise fallacy of a particular statement in the paradox" Fontwell

SafetyControl (talk) 21:12, 11 February 2018 (UTC)

RE:paradox?

ALL paradoxes by their very nature involve a certain amount of misrepresentation and that is what makes them often a paradox. If you care to look at many of the other examples of paradoxes in wikipedia, you will see they are not significantly different. xs935 17:49, 17 December 2006

Clarification

I don't believe the paradox was correctly stated:

The first man considers this : The probability of me winning or losing is 50:50. If I lose my necktie, then I lose the value of the cheaper necktie. If I win, then I win the value of a more expensive necktie. By winning, I more than double the value of my neckties. In other words, I can bet x and have a 50% chance of winning more than x. Therefore it is definitely in my interest to make the wager.

Specifically, the statement "If I lose my necktie, then I lose the value of the cheaper necktie" was incorrect. Also, the statement "By winning, I more than double the value of my neckties" is only true if he currently owns excactly one necktie, and in any case it is irrelevant to the paradox.

I rewrote the third paragraph thus:

The first man considers this: The probability of me winning or losing is 50:50. If I lose, then I lose the value of my necktie. If I win, then I win more than the value of my necktie. In other words, I can bet x and have a 50% chance of winning more than x. Therefore it is definitely in my interest to make the wager.

Nasorenga 14:32, 1 February 2007 (UTC)

The paradox lies in the fact that

it is assumed that there is a 50% chance that your tie is worth more, meaning you should be indifferent to the wager (50% you have more expensive tie vs. 50% you have less expensive tie). However, due to the expectation calcualation (the heart of the paradox), you always want to wager (as does your opponent) In most "real-life" wagers, the wager comes about due to differing expected values (sort of like the probability an event will occur, only with payoffs factored in). In retrospect, that may be the answer to the paradox. From our perspective, we see a 50/50 game, whereas they see the expected value from their perspective. However, we don't know any more than them, so I don't know why this would matter Akshayaj 18:46, 17 July 2007 (UTC)

To my mind, not realy solved.

There seem to be broadly speaking two classes of paradox. One is where the paradox statements are basically correct but lead to a counter-intuitive result eg the Twins Paradox (speed of light). The other is where the paradox statements seem to be correct at every stage but never-the-less result in a final statement that is clearly and actually wrong.

Here, we are dealing with the second kind. Now, the "solution" to this sort of paradox is NOT to show how to arrive at the correct answer by another method. The whole point is that we can tell what the correct answer is anyway but the paradox gives us a wrong answer. Rather, the solution is to point out the precise fallacy of a particular statement in the paradox.

As an example, the Two Envelopes Paradox (which is very similar to this one) has a wiki page that suggests several problematic parts of the paradox - such as the assumed 50:50 probability implying a uniform non zero probability distributed to infinity (impossible) or the idea of counterfactuals (which is the most instinctively correct solution to my mind).

So I think most of the solution part has to go.

One part of the solution does refer to their "flawed reasoning", which is good. However, it goes on to say "each is considering his tie to be both the more expensive tie and the less expensive tie at the same time, while it can only be one or the other".

The thing is, it is not obviously flawed to do that (i.e. consider his tie to be both the more expensive tie and the less expensive tie at the same time). That is what people do with probabilities all the time. We don't know if it is one or the other, so we consider both. Typically this line of reasoning isn't a problem at all e.g.

If the coin lands heads I win $1, if it lands tails I win $2, therefore my expected win is $1.50 therefore I would be happy to stump up $1.25 to play the game.

The above reasoning is all correct even though the coin can only be "one thing [heads] or the other [tails]". So there has to be some more explanation as to why this "one or the other" doesn't apply in this particular case - if, indeed, it doesn't.

Fontwell (talk) 11:47, 5 September 2008 (UTC)

I've made some changes which I believe cover off the criticisms made by Fontwell - hopefully the article now does a better job of explaining exactly what the flaw in the reasoning consists of.--Nabav (talk) 13:16, 16 May 2010 (UTC)

A Detailed Analysis

At the time I made these comments, the discussion page of this Wikipedia article contained a comment in the section under the heading “Clarification” that questioned the correct statement of the paradox. Obviously, to have any meaningful discussion about the Necktie Paradox it is important to have that discussion in reference to a correctly stated version of the paradox. Although I didn’t go back through and trace the revision history of that discussion, it appears that the previous statement of the paradox in the main article was subsequently revised to amend the problems in question. Appropriately, my comments below are in reference to the paradox as stated and the article as worded on January 18, 2009.

My plan is to post the following discussion on the discussion page for feedback. Depending on the feedback I would expect to proceed with one of the following options: (1) revise the discussion, (2) post the solution in a section of the main article under the heading “Solution,” (3) create a new article titled “Solution to The Necktie Paradox,” or (4) abandon the discussion.

First of all, I have no background in classical paradoxes, so I have no insight into the historical or proper statement of the Necktie Paradox. Accordingly, my comments are based solely on a logical analysis of the description of the paradox as stated in the Wikipedia article.

Second, the Necktie Paradox as stated appears to be a paradox because (or in the sense that) it appears that both men have an advantage in the described bet when, in fact, they do not.

Third, if the Necktie Paradox is a paradox, then there must be some flaw in the statements or in the logic of the paradox.

Fourth, although the Necktie Paradox appears to be a paradox as stated in the article, it is possible to revise or interpret the conditions to create cases where either one man or the other does, in fact, have an advantage. However, it is not possible to create such sets of conditions that provide an advantage to both men at the same time. Accordingly, any conditions appearing to do this would thereby constitute another version of the paradox.

Fifth, in analyzing the Necktie Paradox it is primarily important to remember that the outcome of described wager was decided at the time the second tie was purchased thereby establishing the winner of the wager before it is made. Accordingly, although the outcome of the wager is determined by a real event, the basis of the wager is the perception and assessment of that event by the two men making the wager. An important implication in this is that because the outcome is determined prior to the bet, additional information has the potential of either altering or only appearing to alter the perception of the event.

And sixth, although the statement of the wager appears to represent a wager that could easily be enacted in real life, further analysis reveals the following: (1) Unless clearly defined, what is assumed or considered to be random may be open to interpretation. (2) It is insufficient in the description of the wager to simply imply that the process used for the necktie selection is random. (3) And, the probability of the outcome of the wager is dependent upon the interpretation of what is a random process for necktie selection.

The Flaw

Basically, the flaw in the Necktie Paradox revolves around getting different expected outcomes from different interpretations of what appears to be the same random event. Specifically, one of the difficulties in finding and understanding the flaw in the paradox is that there is not just one single flaw, but that there are basically three different flaws, each is specific to a different set of conditions arising from the different interpretations of the necktie selection process being a random event. It makes the logic of the paradox more convincing and more confusing to understand and invalidate. In order to understand each of the three flaws, it is first necessary to define and understand at least four sets of different sets of conditions that are open to interpretation.

Anatomy of the Paradox

Basically, the logical structure of the paradox can be divided into four parts. The first part is the narrative describing the origin of the ties and the terms of the wager. It is stated as follows: “Two men are each given a necktie by their respective wives as a Christmas present. Over drinks they start arguing over who has the more expensive necktie. They agree to have a wager over it. They will consult their wives and find out which necktie is the more expensive. The terms of the bet are that the man with the more expensive necktie has to give it to the other as the prize.” For reference in the following discussion this will be referred to as “the narrative.”

The second part of the logical structure is a characterization of the reasoning made by the first man stated in five brief sentences. The five statements are: (1) “The probability of me winning or losing is 50:50.” (2) “If I lose, then I lose the value of my necktie.” (3) “If I win, then I win more than the value of my necktie.” (4) “In other words, I can bet x and have a 50% chance of winning more than x.” (5) “Therefore it is definitely in my interest to make the wager.” For reference in the following discussion this will be referred to as “the first man’s reasoning.”

The third part of the logical structure is the application of the reasoning of the first man to the second man. It is simply stated in the first clause of the subsequent statement. The first clause is stated as follows: “The second man can consider the wager in exactly the same way.” For reference in the following discussion this will be referred to as “the symmetry of reasoning.”

The fourth part of the logical structure of the paradox is in concluding that the two men cannot both have the subject advantage. It is simply stated in the second clause of the aforementioned statement. The second clause is stated as follows: “therefore, paradoxically, it seems both men have the advantage in the bet.” For reference in the following discussion this will be referred to as “the paradoxical conclusion.”

Case 1

Case 1 is simply my designation for conditions in which the selection of the neckties has a random equivalence to the following process: (1) The wife of the first man selects her husband’s tie first. (2) The wife of the second man then selects her husband’s tie with a 50/50 probability of having a cost greater or less than the first man’s tie.

In Case 1, the stated reasoning of the first man is correct. By the definition of Case 1 regarding what is considered random tie selection, the first man has a 50 percent probability of losing x and a 50 percent probability of winning more than x. However, unless he is made aware that the conditions of Case 1 were exclusively used in the necktie selection process, he would be unaware of his advantage (presuming that the remaining cases do not also provide an advantage). In contrast, the same reasoning is not correct for the second man. The second man has a 50 percent chance of winning x and a 50 percent chance of losing more than x. Accordingly, the flaw in the paradox under the conditions of Case 1 is in concluding that the reasoning of the first man was applicable to the second man. As a result, there is no paradox in the outcome of Case 1. In brief, “the narrative” is valid, “the first man’s reasoning” is valid, “the symmetry of reasoning” is invalid, and accordingly, “the paradoxical conclusion” is also invalid.

A real life scenario of Case 1 is as follows: Suppose one wife purchases her husband’s necktie. Subsequently, the other wife narrows her selection to two neckties, one having cost more and one having cost less. Following the suggestion of the store clerk she simply flips a coin to determine which tie to purchase. For the purpose of an example, let’s suppose that the cost of first wife’s tie was $40 and the costs of the second wife’s selection were $20 and $60. Now, when we apply the reasoning “I can bet x and have a 50% chance of winning more than x,” it becomes fairly obvious that the first man can win $60 or lose $40, and that the second man can win $40 or lose $60, thereby confirming by example the advantage to the first man and the disadvantage to the other.

Because Case 1 provides an obvious cost-related bias in the necktie selection process, one could argue that this case is thereby not completely random and should therefore be excluded as a case in proving or disproving the paradox. However, the wording of the wager in “the narrative” implies nothing more than a 50/50 chance of winning.

Case 2

In a similar manner, Case 2 is simply my designation for conditions in which the selection of the neckties has a random equivalence to a second process as follows: (1) The wives of both men mutually select two ties. (2) The wives then flip a coin to select which wife will purchase which tie. Worth noting is that this selection process is equivalent to the process of two wives selecting ties without bias related to cost. The coin flip simply removes any cost bias that might arise from the wives not being equivalent in that respect. It also ties the winner and loser of the bet to the coin flip rather than to the tie selection, which makes it easier for the reader to eliminate cost bias as an element that might clutter the reader’s reasoning process.

In Case 2, the reasoning of the first man as stated in the paradox is incorrect. And, because his reasoning is incorrect, the application of his reasoning to the second man is also incorrect. To uncover the faulty logic we merely have to associate the cost of either tie with either man, and then evaluate the validity of the possible outcomes.

Accordingly, presuming that x is the cost of the first man’s tie (consistent with the paradox) and y is the cost of the second man’s tie, if y is greater than x, the first man wins y, while the second man loses y. In contrast, if x is greater than y, the second man wins x, while the first man loses x. In both cases the amount won by one man is equal to amount lost by the other. And, in both cases, the amount won and the amount lost is equal to the greater amount. Accordingly, the flaw in Case 2 is in the first man reasoning that his fourth statement (“in other words, I can bet x and have a 50% chance of winning more than x”) is correct. It then follows that the subsequent conclusion (“therefore it is definitely in my interest to make the wager”) is also incorrect. As a result, there is no paradox in the outcome of Case 2. However, also note that the flaw is different than in Case 1. In brief, “the narrative” is valid, “the first man’s reasoning” is invalid, “the symmetry of reasoning” is valid, and accordingly, “the paradoxical conclusion” is also invalid.

An obvious point of confusion in analyzing the paradox under Case 2 is the above-illustrated ambiguity in correctly assessing the amount of the bet. The ambiguity results from not having a clear distinction between what I will call “the apparent amount of the wager” and “the actual amount of a wager.” And, although the two amounts can be (and normally are) the same, confusion can result when the amount of the wager is a variable or an unknown.

By definition, the actual amount of a wager is the amount at risk (that which can be lost). Accordingly, in Case 2 with y greater than x the apparent bet for the first man is x, and the apparent bet for the second man is y. However, the actual bet for either man is actually y, the greater of x and y. Accordingly, when y is greater than x, it only appears that the first man can only lose x. This is true because he won y (the greater amount) in the bet as a result of having received x (the lesser amount) as a gift. Because the two events are linked in this respect, the bet was actually won as a consequence of the necktie selection. And, it was in the necktie selection that the first man was at risk of losing y. Furthermore, because the option to engage in the bet occurs after the necktie selection, the existence of the option appears to mask the aforementioned direct link between the necktie selection and the bet. And, because the first man can choose to bet or not to bet while physically being in possession of x, it provides the appearance that he has the option to bet x.

A real life scenario of Case 2 is as follows: Suppose both wives are shopping in the same store at the same time and both narrow their selection to the same two ties. Following the suggestion of the store clerk they simply flip a coin to determine which wife purchases which tie. For the purpose of this discussion let’s suppose that the cost of one tie is $40 and the cost of the other is $60. Now, when we apply the reasoning represented in the statement, “I can bet x and have a 50% chance of winning more than x,” it becomes more evident to someone aware of the coin flip that either man has a 50 percent chance of winning or losing only the more expensive tie, in this case the $60 tie.

A predominate point of confusion in being drawn into the paradox is that Case 2 violates the reasoning of the statement “in other words, I can bet x and have a 50% chance of winning more than x,” while Case 1 violates reasoning that there is symmetry in the reasoning between the two men. In terms of Case 1 and Case 2, the “the first man’s reasoning” is easier to associate with the context of Case 1 (for which it is true) than in the context of Case 2 (for which it is not true). By comparison, in terms of Case 1 and Case 2, “the symmetry of reasoning” is easier to associate with the context of Case 2 (for which it is true) than in the context of Case 1 (for which it is not true). Also, as mentioned in Case 1, if either man is explicitly made aware of the necktie selection process, he will be more likely to follow the correct reasoning when attempting to resolve the paradox.

Case 3

In a similar manner, Case 3 is simply my designation for conditions in which the selection of the neckties has a random equivalence to a third process as follows: (1) The wives of the both men mutually select two ties. (2) The wives then flip a coin to select which wife will purchase which tie. (3) The wives then reveal some information to their husbands regarding the cost of either one or both of the neckties. The possible combinations appear to be as follows: (a) the cost of one tie without identifying the owner, (b) the cost of both ties without identifying either owner, (c) the cost of one tie revealed to the respective owner, (d) the cost of one tie revealed to the respective rival, (e) the cost of each man’s respective tie revealed only to himself, (f) the cost of each man’s respective tie revealed only to his rival, and (g) the cost of one man’s tie revealed to both men. For reference in the following discussion, each respective condition will be referred to as Cases 3a through 3g.

In Case 3 the issue is whether either or both men having knowledge of cost-related information provides an advantage in the wager. The answer to that question is dependent on whether or not the knowledge of the amount reveals any information that affects the probability of the expected outcome. Regarding the seven cases listed above, none provide any such advantage, though the latter five have the appearance of being cases that are represented by Case 1 (though they are not). Furthermore, because Case 3 provides information related to the amount of the bet and because the amount of the bet is an essential part of the reasoning within the paradox, it thereby alters the configuration of the flaw in the paradox. It is for this reason that Case 3 was included in the analysis.

Cases 3a through 3g

Regarding Cases 3a and 3b, if the two men know the cost of either or both ties but have no information indicating who owns which, the condition is identical to Case 2. Accordingly, not only is it fairly easy to understand this condition, but because it is typically easier to follow a train of actual numbers than to follow a train of numbers represented by variables, assigning amounts to the costs of the ties likely increases the understandability of the paradox.

Regarding Cases 3c through 3g, basically they are all simply different combinations of what either one man or both men should be able to reason. That is, one man knowing only one amount provides no advantage. However, as mentioned above, by assigning an amount it appears to enforce the representation of Case 1. For a more detailed analysis let’s consider Case 3c, where the cost of just one man’s tie is revealed to only himself.

The Details of Case 3c

In Case 3c, if the first man knows the cost of his tie, that knowledge specifically establishes the actual amount of his bet and accordingly provides a different interpretation to the statement “In other words, I can bet x and have a 50% chance of winning more than x.” For example, suppose the first man is informed that the cost of his tie was $40. The above statement can now be restated as “In other words, I can bet $40 and have a 50% chance of winning more than $40.” However, in this case the statement “I can bet $40” is valid, and the statement I “have a 50% chance of winning more than $40” is invalid. This is in direct contrast to Case 2 where the statement “I can bet x” is invalid, but the statement I “have a 50% chance of winning more than x” is valid. Accordingly, when x is treated as an unknown amount, the reasoning process is simply different than when x is treated as a known amount. And although the outcome of either method of reason should yield the same expected outcome, the interpretation of the amount being bet and won is different.

More specifically, the condition of knowing the cost of only one necktie does not affect the actual probability of winning or losing. However, it does affect the validity of statements and the reasoning that characterizes and represent the related probabilities. To illustrate this distinction more clearly it may helpful to consider a more carefully constructed example. Suppose (1) x and y are used to represent the costs of the two neckties that were gifted to the first and second man respectively, (2) L and G represent a pair of necktie costs where L is the lesser and G is the greater, and (3) there is an equal probability that x is equal to L or G. Accordingly, if x equals L, the first man wins G, and if x equals G, the first man loses G. And, the true amount of the first man’s bet is G and not x. It is specifically not equal to x, because x can also be equal to L.

Next, suppose that F is used to represent a fixed value of x in cases where x becomes known at a point in time subsequent to when x and y have become equal to either L and G or G and L respectively. Now however, if x equals F and F equals L, the first man still wins G as before, but now G can be reasoned to be greater than F. Similarly, if x equals F and F equals G, again the first man still loses G as before, but now G can be reasoned to be equal to F, and although the amount of his bet is still G, G is now a lesser value than the value of G when he wins. Accordingly, his actual bet is now F, and when he wins, the value of G is greater than the value of G when he loses. Furthermore, once x becomes known (or treated as being known), L and G can no longer be represented to be only a single pair.

Accordingly, when x is not fixed, L and G can be represent all possible pairs where one is greater than the other. However, when x becomes fixed, the required representation is no longer of all possible pairs, but only pairs including the F as a member of the pair. Accordingly, when F equals G, L must be less than F, thereby limiting possible necktie pairs to those represented by F and L(<F), where L(<F) represents that L is less than F. Similarly, when F equals L, G is greater than F, thereby limiting possible necktie pairs to those represented by F and G(>F), where G(>F) represents that G is greater than F. Accordingly, although the pair L and G can represent the lesser and greater of a fixed pair under all conditions, it requires three variables (F, L(<F), and G(>F)) to represent the lesser and greater of a pair when one is known or treated as having a fixed value. Furthermore, because the relationship between x and y when equal to L and G is different than when equal to F and L(<F) or when equal to F and L(>F), the meaning and validity of “I can bet x and have a 50% chance of winning more than x” is simply not the same.

Accordingly, if the pair of neckties is represented to be F and L(<F), then the first man loses F. Similarly, if the pair of neckties is represented to be F and G(>F), then the first man wins G(>F). Accordingly, treating x as fixed or not fixed alters the truth of the statement “I can bet x,” which is significant to identifying and labeling the flaw in the paradox. Also, note that it is not required for the value of x to be known for x to be fixed. However, if x is known, then x is also fixed. Furthermore, although it is possible to consider either man’s bet from the reference of his tie having a fixed or known cost, it is not possible to consider this possibility for the cost of both ties to be known at the same time without exposing the winner. Obviously, exposing the winner precludes having a basis for a wager.

Furthermore, because the cost of the first man’s tie is known in Case 3c, it is easy to falsely presume the conditions of Case 1, which represents the condition that the first man’s wife had purchased her husband’s tie before the occurrence of necktie selection process. However, because Case 3c has the same necktie selection process as Case 2 (by definition), the first man should realize that although he has a perceived 50/50 probability of winning when x is unknown, knowing x alters that probability.

Because of the presumed 50/50 probability (of either necktie having a cost greater or less than the other), it would be logical to conclude that for any x-and-y pair there must be a 50/50 chance of y being greater or less than x. And, while this is true, this subject 50/50 probability does not lead to the logical conclusion that for any specific x that there is a 50/50 probability of x being greater or less than y. In fact, it would be expected that the greater the cost of the first man’s tie, the lesser the probability that he would win the bet. For example, if the cost of the first man’s tie was $1000, the probability of him winning the bet would likely be much less than 50/50, and if the cost of the first man’s tie was $1, the probability of him winning would likely be much greater than 50/50.

This conclusion is basically an inevitable consequence of any real distribution. Accordingly, from a real and practical sense, the selected neckties must come from a real distribution of neckties having some degree of practicality of cost. Accordingly, the amount of advantage in the bet is related to the expected distribution of necktie costs. If neckties could cost any amount with equal probability, then there would be no such advantage. However, in the real world, neckties have a greater probability of being within a range, and knowing that position in the range affects the probability of winning or losing the subject bet.

Another approach to analyzing Case 3c is to view the bet from a statistical point of view. Accordingly, if the bet were repeated many times, a pattern would emerge that would statistically indicate that although each man was winning and losing a statistically equal number of times, each man would also discover that his winnings and losses were statistically balanced or equal as well. Accordingly, the first man would discover that the average value of x will have been greater in the bets where he loses x than in bets where he wins so that the average value of x equals the average value of y. However, in complete agreement with the statement, “I can bet x and have a 50% chance of winning more than x,” the first man will lose x 50 percent of the time and will win more than x 50 percent of the time. For example, if all ties had a cost of either $20 or $40, the first man would lose a $40 tie 50 percent of the time and win a $40 tie 50 percent of the time. Furthermore, because the Necktie Paradox only represents the bet being performed one time as a single event, it is easy to overlook the less encumbered reasoning when viewing the wagering as being performed a statistically significant number of times.

This same basic reasoning applies to Cases 3d through 3g, though it applies respectively for either or both men depending on the case.

Case 4

It is also possible to create a fourth case, Case 4, by the same derivation used to derive Case 3 from Case 2. However, Case 4 conditions would always favor the first man by the definition of Case 1. As discussed in Case 1, the specific flaw in the paradox depends on each man’s knowledge of the whether he knows that the selection process is based on the conditions of Case 1 or Case 2. Also, as in Case 3 the flaws in the paradox are simply different combinations of those discussed in Case 3. For example, if two ties were known by both men to have costs of $20 and $40, it would appear that each man has the same probability to have either tie. However, if we made a statistical analysis of the bet repeated many times, we would see that although the first man won 50 percent of the time, overall he would win more expensive ties than he lost. This is consistent with the statement “In other words, I can bet x and have a 50% chance of winning more than x.” In this case having knowledge of the amounts irrelevant, “the first man’s reasoning” is valid, but “the symmetry of reasoning” is invalid.

Summary

In Case1 “the narrative” is valid, “the first man’s reasoning” is valid, and “the symmetry of reasoning” is invalid. Accordingly, “the paradoxical conclusion” is also invalid.

In Cases 2, 3a, and 3b “the narrative” is valid, “the first man’s reasoning” is invalid, and “the symmetry of reasoning” is valid. Accordingly, “the paradoxical conclusion” is also invalid. In these cases “the first man’s reasoning” is invalid because of the words “I can bet x” are invalid.

In Case 3c “the narrative” is valid, “the first man’s reasoning” is invalid, and “the symmetry of reasoning” is valid. Accordingly, “the paradoxical conclusion” is also invalid. In Case 3c “the first man’s reasoning” is invalid because the words “have a 50% chance of winning more than x” are invalid.

In Cases 3d through 3g “the narrative” is valid, while “the first man’s reasoning” and “the symmetry of reasoning” are valid or invalid on a mutually exclusive basis depending on which man has knowledge of which condition. Accordingly, “the paradoxical conclusion” is invalid.

Case 4 is the same as Case 1. However, the cause for the invalidity of “the symmetry of reasoning” is dependent on which man has knowledge of which condition

Bill Wolf (talk) 23:20, 16 February 2009 (UTC)

Still don't get it

With respect to Bill Wolf's thorough explaination, I still don't understand in a laymen's basis what the logical fallacy is here.

I understand this is the same premise as the Wallet problem so I hope someone can explain simpler: I have no clue what is in my wallet today. A stranger's wallet also contains an unknown amount (to both of us). Whoever has more money forfiets it to the other. I don't understand why it's a fallacy to assume I will either lose what I have, or win more than what I have. It is a fact that, whatever I have, the other person has either less than me, or more than me, right?. If he has less than me. I lose what I have. If it he has more than me, I gain more than what I have.

Similarly, I either win what the other guy has, or I lose MORE than what the other guy has. This is all just speaking it out in colloquial terms, not with variables and math... have I said anything false yet?

plugging in example numbers (just to illustrate the problem I have understanding this issue): Say it TURNS OUT I have 20 bucks in my wallet. I can either lose my 20 bucks if he has less than me, or I can win 20+ bucks if he has more than me.

I'm not clear on whether it changes things if I know what's in my own wallet or not (whether I presume my wallet contains "X" or if I know it contains 20 bucks). Is the problem with the presumption that there is a 50% chance of either scenario? I don't get it. TheHYPO (talk) 08:43, 4 December 2009 (UTC)

Yes the "50% chance of either scenario" is a presumption that must be condemned. I don't know "what I have" in my wallet. The more I actually have, the more likely "what I have" is the greater of two, and the more likely I must lose "what I have". The less I actually have, the more likely I win "more than what I have".--P64 (talk) 18:07, 10 May 2011 (UTC)

Very well said, P64! This is the kind of talk we also need on the two envelopes page. Richard Gill (talk) 07:19, 11 August 2011 (UTC)

Two envelopes

There is much work in progress at Two envelopes problem. One natural outcome is that Exchange paradox will become a redirect to that article. Probably this stub will become a redirect to Two envelopes problem or progress on that article will naturally yield a much improved short article here.

Some above talk concerns whether there is any Necktie Paradox with a distinct career in print. So far the answer seems to be "barely so", in company with the Wallet Game.--P64 (talk) 18:08, 10 May 2011 (UTC)

The Two envelopes problem has made much progress recently and I suggest that it is now time to redirect this article there. Any objections? Martin Hogbin (talk) 21:08, 10 August 2011 (UTC)

Not quite time, since first we should add a section on the necktie paradox to the two envelopes article. A mathematical description of each of the two problems is different: the gains and losses, the two amounts. Probability assumptions about the two values of the two neckties are different from those about the values in then envelopes. Symmetry is not so evident, and each person may actually know the value of their necktie. However I believe that the error(s) in reasoning are the same. Any resolution of one problem can be translated to a resolution of the other. But remember the Anna Karenina principle: there are many ways in which one can rightly imagine that an informal logical or mathematical argument goes wrong. Because formal assumptions are not made and a formal framework is not specified, we can imagine the author of the necktie argument as having all kinds of different ideas in mind, guiding his "argument". This is precisely the point made in the long post above "a detailed analysis". Read it, folks! Of course we have a Wikipedia specific problem: has a so-called "reliable source" said this before? Then we can use it. If not, we can't. But we could publish our analysis elsewhere and hope later editors will find it useful. See [1] for a first try. Comments welcome. Richard Gill (talk) 07:09, 11 August 2011 (UTC)

The history is that the neckties evolved into wallets and then into envelopes. On the way, the generic name "exchange paradox", was applied to all. So a merge is the natural thing, but a simple redirect is not in order, till the TEP page also does the two neckties version justice. Richard Gill (talk) 07:12, 11 August 2011 (UTC)

Another solution is to create a disambiguation page called "Exchange paradox" and from there point to the "Necktie paradox," "Two envelope problem" and maybe even a "Wallet game" page still to be created. iNic (talk) 13:08, 11 August 2011 (UTC)

Two neckties as two envelopes: the math

Let me express the two necktie paradox in a similar mathematical notation to a popular notation for the two envelopes. Remember that the two rich men who have met in a bar don't know how much each of their neckties, presents from their wives, cost. We do know that they each believe the other to be as rich as themselves and I suppose they also imagine their wives are similar, too. Denote by A and B the prices which were paid by their wives for the two neckties. Let the probability measure P express the subjective beliefs of the first man about these two prices. I could use Q for the other. For instance, P(B<100) is the first rich man's personal probability that the other guy's tie cost less than $100. Now we are told that each person's beliefs about the prices of both ties are symmetric on exchange of the two ties. For instance, P(B<100)=P(A<100), the first guy is equally sure about his tie costing less than $100 as he is that the other guy's tie cost less than $100. The other guy has equal certainty too, but maybe his beliefs are a bit different. So we also have Q(B<100)=Q(A<100), but there's no reason why Q-probabilities and P-probabilities should be the same.

By the way, it is possible that the two ties turn out to be worth the same amount, but if so, the game is called off. So more precisely, I'll take P to represent the first guy's beliefs about the game conditional on the event that A and B are different. From now on, P(A=B)=0.

First Guy reasons as follows. If I accept the wager, and if A < B, then I end up with A+B > 2A, but if A>B then I end up with 0. By symmetry of P these two possibilities are equally likely. I have probability 1/2 of losing A, and probability 1/2 of gaining more than A. Hence I'll take the wager.

Of course, Second Guy reasons exactly the same, using his prior beliefs Q (conditioned on the prices of the two ties being different).

Neither First Guy nor Second Guy are good at doing probability. They both want to compute the expected net gain, if they accept the wager. It seems that they want to do it by splitting up the expectation value according to the two complementary possibilities, A < B and B < A respectively. This is how that should be done, from First Guy's point of view; thus the symbol E for expectation value should actually have a subscript P attached, to indicate that expectations are being taken according to First Guy's subjective probabilities P.

P-Expectation net gain = E(-A|B < A)P(B < A)+E(B|A < B)P(A < B) = ( E(B|A < B)-E(A|B < A) )/2 = 0

At the first step (first equality sign) I used the standard rule for computing an expectation value by averaging over conditional expectations in two complementary situations, knowing that in situation B < A, First Guy ends up with A less than what he started with, while in the situation A < B, First Guy ends up with B more than what he started with. Next I substitute the subjective probabilities P(A < B)=P(B < A)=1/2, by symmetry of First Guy's beliefs about the prices of the two neckties, also when given that they are different from one another. Finally I use symmetry again of First Guy's prior beliefs to show that his expectation value of the price of the other guy's tie when it is larger than his is equal to his expectation value of the price of his own tie when it is larger than the other.

Of course we knew in advance that First Guy should find the wager uninteresting. Arguing directly he could have said that with probability half I'll end up with A+B and with probability half I'll end up with 0, because by the symmetry of my beliefs about A and B together, I'm completely neutral as to who has the more valuable tie, even when you tell me the price of both of them added up together.

The more important thing is to show what went wrong with First Guy's faulty reasoning. Well, by comparison with the correct logical steps which parallel his own obviously wrong steps, we see that he is confusing expectation values, conditional expectation values, and the values themselves. He wants to compare the loss and the gain parts and show that the gain part is larger. He could correctly write the gain part E(B|A < B) as E(A+(B-A)|A < B) = E(A|A <B )+E(B-A|A < B) where the second term is certainly positive. So he wants to cancel out the two most similar parts of the loss and the gain part, E(A|B < A) and E(A|A < B). But according to any symmetric beliefs P, E(A|B < A) > E(A|A < B). In fact the difference is precisely, by using symmetry to exchange A and B, and then combining the difference of two conditional expectations given the same condition as the conditional expectation of the difference, of course, E(B-A|A < B) .

Conclusion. Necktie paradox is indeed essentially the same as Two Envelope Paradox, it relies on shortening the probability calculations so that the reader doesn't notice that actually the writer is using conditional expectations and that the expected value of one of two amounts when it is the smaller of the two, is smaller than when it is the larger of the two.

Kraitchik's original "explanation" is hardly an explanation, he just does a careful calculation, in the situation that P makes A and B independent and uniformly distributed over 1$, 2$, ..., 100$, of the actual expected gain by First Guy and shows it is zero by symmetry. Personally I prefer my general proof by symmetry of P that P-expected net gain equals zero. But others might prefer to work through a small example just to see how the symmetry kicks in.

My analysis of "what went wrong" is that Kraitchik's rich man does not realise that the expected value of either tie when it is the least costly of the two is less than when it is the more costly of the two. I don't see how he got to be so rich. Probably just married an heiress.

Richard Gill (talk) 16:21, 11 August 2011 (UTC)

Simple explanation

There are two ties, Tx and Ty, worth x and y respectively. Assume x>y. For each guy, the probability of each tie is 1/2.

Correct reasoning:

Profit in case of having Ty (the cheaper one), is x. In case of having Tx, profit is -x. Expected profit: (1/2)x - (1/2)x = 0.

Mistaken reasoning:

My profit, if I have the cheaper tie Ty, is x (since y<x and I win the bet).

My profit, if my tie Ty is the more expensive one, is -y (since y>x and I lose the bet). My expected profit: (1/2)x-(1/2)y>0

His mistake was to change x>y to y>x when calculating his loss rather than changing Ty to Tx. — Preceding unsigned comment added by 131.215.248.17 (talk) 11:29, 10 May 2013 (UTC)

This idea is excellent and could be made even simpler.

Let the ties of guy1 and guy2 worth x and y respectively.

Let M be the maximum of x and y. If x < y then guy1 wins M. If x > y then guy1 looses M.

Paradox resolved? --41.216.54.68 (talk) 18:27, 7 June 2013 (UTC)

The Actual Solution

The "resolution" discussed on the main page and at length on this talk page doesn't actual explain where each man's logic breaks down; it just lay out a different logical argument and says "well, they should have just thought about it this way."

In fact, each man's logic is perfectly sound, given their assumed probability distribution. The "resolution" comes along and changes the probability distribution without bothering to tell anyone, and turns around and criticizes the two men for using the one the paradox lays out.

Each man's logic hinges on the assumption that, whatever the value of their necktie, there is a 50% chance that the other man's is more expensive. This is why their logic, and anyone who came along saying something like "but what if their tie costs $100", is correct. Half the time they will win more than they wagered. The paradox arises because both neckties cannot simultaneously have this probability distribution (assuming they each have different prices). But, if each man believes his does, he is acting logically.

The proposed "resolution" assumes a different distribution. While the men fix the values of their tie, the "resolution" makes it variable. More specifically, it compares a scenario of one man winning when he has a $20 dollar tie to losing when he has a $30 dollar tie. This method implicitly assumes that having a more expensive tie makes a man more likely to lose. This assumption resolves the paradox, but only because it changes the original assumption of each man,- that, regardless of tie price, they have a 50% chance of winning or losing. It replaces these mutually incompatible distributions with one that is consistent, but functions differently.

In conclusion: both men are proceeding logically from conflicting premises, which is why they both seem to "win" (at least one of them is just wrong about his chances). The current "resolution" quietly changes the nature of the probability assumptions to avoid this incompatibility without actually explaining it, and provides the resulting (different) logical outcome without explaining why the men's solution was actually wrong (which it wasn't).

-Xensity — Preceding unsigned comment added by Xensity (talk • contribs) 19:51, 24 March 2014 (UTC)

A more expensive tie does increase chance to lose. There's a probability distribution for the price and so on, and since they are in disagreement, we should assume that they have the same expected cost. It gets interesting when the price of a tie diverges from its value, or the cost of one tie is centered around the actual cost of the other. 23.121.191.18 (talk) 04:14, 15 October 2017 (UTC)

A source where this isn't a paradox

I remember reading a book - fairly confident the title is "The Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers: A Definitive Collection of the Best Puzzles Ever Devised" by Erwin Brecher, but can't be certain - which said that this is a very oddly constructed unbalanced bet wherein it is indeed possible for both sides to feel they are advantaged. Therefore to see the current article treat this paradox as resolved is very surprising. I am not an expert on this subject, but an explanation of how to reconcile Erwin Brecher's conclusion with the article would be welcome. Banedon (talk) 05:48, 21 June 2016 (UTC)