# The Necktie Paradox: A Different Perspective

• BenVitale
In summary, the conversation is about the 'necktie paradox' and its various perspectives and solutions. The paradox involves two men betting on the value of their respective neckties, with both thinking they have the advantage. However, the flaw lies in their calculation of relative gain instead of absolute gain. The paradox is solved by considering the two possible outcomes for each man, resulting in a 50% chance of either winning or losing an expensive tie. There is also discussion on a variation of the paradox involving two envelopes and a new strategy for solving it. The conversation also touches on creating a similar problem with three people and three neckties.
BenVitale
There's discussion on the 'necktie paradox' on this blog, where I'm a regular visitor.

I don't agree with the perspectives of those who have responded on that blog.

In the wikipedia, The necktie paradox states that each stands to either win or lose an expensive tie, each at 50% probability, so the game has no advantage to either man.

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, and 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.

The first man reasons as follows: 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. The second man can consider the wager in exactly the same way; therefore, paradoxically, it seems both men have the advantage in the bet.

Is there a problem here?

But, I look at this problem differently:

Say, the cheaper necktie has value y, and
the other one has the value (y + z) with z > 0.

Let's assume that both men have an equal chance of being correct,
the expected value in winnings for either man is,

(0.50)(y + z) – (0.50)(y) = (0.50)z

Both men are expected to make money if they bet.
So both men are correct in choosing to bet.

Is this correct?

Last edited:
BenVitale said:
Say, the cheaper necktie has value y, and
the other one has the value (y + z) with z > 0.

Let's assume that both men have an equal chance of being correct,
the expected value in winnings for either man is,

(0.50)(y + z) – (0.50)(y) = (0.50)z

Both men are expected to make money if they bet.
So both men are correct in choosing to bet.

Is this correct?

No, since you assume that z > 0, you have that y + z > y and will thus win by a probability 100% (not 50%)

The correct way of calculating expected gain is:

for player with necktie A (worth A dollars)
P(B>A)*B - P(A>B)*A = 0.5*(B-A)

for player with necktie B (worth B dollars)
P(A>B)*A - P(B>A)*B = 0.5*(A-B)

but since A>B or B>A with equal probability, both have equal chance of winning / losing the difference

If you keep reading that wikipedia article, it answers your question.

The paradox is solved because the men's reasoning is flawed: 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 two outcomes of the game each should consider are:
* If I have the more expensive tie, and I make the bet, I lose my more expensive tie.
* If I have the less expensive tie, and I make the bet, I win a more expensive tie.
Thus each stands to either win or lose an expensive tie, each at 50% probability, so the game has no advantage to either man.

farful said:
If you keep reading that wikipedia article, it answers your question.

I read the whole wikipedia article, but i wasn't convinced... i needed to do the math, and see what others may say.

I thought since we have 2 neckties of different $values, the cheaper one is y and the other is (y+z), with z > 0, etc. Now, I see that my reasoning was flawed. I understand this: winterfors said: No, since you assume that z > 0, you have that y + z > y and will thus win by a probability 100% (not 50%) The correct way of calculating expected gain is: for player with necktie A (worth A dollars) P(B>A)*B - P(A>B)*A = 0.5*(B-A) for player with necktie B (worth B dollars) P(A>B)*A - P(B>A)*B = 0.5*(A-B) but since A>B or B>A with equal probability, both have equal chance of winning / losing the difference BenVitale said: Say, the cheaper necktie has value y, and the other one has the value (y + z) with z > 0. Let's assume that both men have an equal chance of being correct, the expected value in winnings for either man is, (0.50)(y + z) – (0.50)(y) = (0.50)z Only the more expensive tie is won or lost, so the expected winnings are zero: (0.50)(y + z) – (0.50)(y+z) = 0 What do you, guys, think of the variation of the "necktie paradox": http://wapedia.mobi/en/Two_envelopes_problem the expected value of the money in the other envelope is: (0.50) *1/2 * x + (0.50) * 2x = 5x/4 Or the expected value in either of the envelopes 1.5 C, C being the lower of the 2 amounts Read about the new strategy for the two-envelope paradox: http://www.physorg.com/news169811689.html Last edited by a moderator: In the paradox, the essential flaw is calculating relative gain rather than absolute gain. A$100 tie is 10 times more expensive than a $10 tie, but you can't say you expect to gain 900% or lose 90%, since the basis for the two is different. Better to say you stand to gain or lose$90 either way.

Thanks to all of you for the feedback and help.

Imagine if we had 3 people exchanging 3 neckties A1, A2 and A3

Would you help me create an interesting problem on this basis?

## 1. What is the Necktie Paradox?

The Necktie Paradox is a thought experiment that questions the traditional understanding of cause and effect. It suggests that a person can travel back in time and give their younger self a necktie, creating a paradox because the younger self would not have owned the necktie originally.

## 2. What is the different perspective offered by the Necktie Paradox?

The Necktie Paradox offers the perspective that the concept of cause and effect may not be as straightforward as we believe. It suggests that the past, present, and future are not necessarily linear and that events can influence each other in unexpected ways.

## 3. Is the Necktie Paradox possible in reality?

As a thought experiment, the Necktie Paradox is not intended to be a literal scenario. It is meant to challenge our understanding of time and causality. In reality, the laws of physics and the concept of causality make the Necktie Paradox impossible.

## 4. How does the Necktie Paradox relate to other paradoxes?

The Necktie Paradox is similar to other paradoxes, such as the grandfather paradox, which also questions the implications of time travel. However, the Necktie Paradox offers a unique perspective by focusing on the concept of cause and effect rather than the potential consequences of time travel.

## 5. What are the implications of the Necktie Paradox?

The Necktie Paradox challenges our understanding of the universe and the concept of causality. It suggests that the laws of physics may not be as straightforward as we believe and that events can have unexpected consequences. It also raises philosophical questions about the nature of time and the possibility of changing the past.

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