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A Necktie Paradox


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9 replies to this topic

#1 BMAD

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Posted 09 July 2013 - 12:01 AM

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?


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#2 Nins_Leprechaun

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Posted 09 July 2013 - 04:06 AM

the first and biggest problem is that neither realizes that their wife will kill them if they give away their christmas present
also...

Spoiler for


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#3 bonanova

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Posted 09 July 2013 - 08:53 AM

Spoiler for sounds like

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#4 gavinksong

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Posted 09 July 2013 - 10:20 AM

Spoiler for


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#5 witzar

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Posted 09 July 2013 - 04:11 PM

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.

Spoiler for


Edited by witzar, 09 July 2013 - 04:12 PM.

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#6 bonanova

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Posted 10 July 2013 - 07:09 AM

Spoiler for

 

Spoiler for because


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#7 plasmid

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Posted 11 July 2013 - 03:49 AM

The most intuitive way (well, only intuitive way) I can think of to explain why the logic "I have a 50/50 chance of winning/losing" and "if I win then I gain more than if I lose" is faulty is because, particularly if this were to happen in the real world, the wives would not spend some amount of money on a necktie that could like anywhere within the range of positive real numbers. There would have to be some probability distribution of how much money they could spend on a tie. As soon as you create a probability distribution from which the two ties are drawn, it becomes clear that the more valuable your tie is, the more likely you are to lose the bet.


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#8 BMAD

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Posted 11 July 2013 - 04:06 AM

then intuitively would

 

The most intuitive way (well, only intuitive way) I can think of to explain why the logic "I have a 50/50 chance of winning/losing" and "if I win then I gain more than if I lose" is faulty is because, particularly if this were to happen in the real world, the wives would not spend some amount of money on a necktie that could like anywhere within the range of positive real numbers. There would have to be some probability distribution of how much money they could spend on a tie. As soon as you create a probability distribution from which the two ties are drawn, it becomes clear that the more valuable your tie is, the more likely you are to lose the bet.

then intuitively, wouldn't the same logic apply in reverse?  A husband knowing the frugal nature of his wife believes with very much certainty that his necktie was bought heavily discounted and thus has a great chance of winning


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#9 plasmid

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Posted 11 July 2013 - 04:19 AM

Yes, you're more likely to win if your tie is cheap and more likely to lose if your tie is expensive.

 

That's why, even if you go into the problem thinking you have a 50/50 chance of winning if you don't have any prior knowledge of the values of the ties, you in fact have a greater chance of losing if your tie is expensive than if it's cheap. So the outcomes are really not equal and are dependent on the value of your tie.


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#10 BMAD

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Posted 11 July 2013 - 05:16 AM

All are winners!


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