Necktie Paradox
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11 members have voted
After thinking a moment, you conclude the bet must have a positive expected value. This is because, if you win, you will win a tie worth more money than your wager.
However, both of you could accept the bet under the same logic, and it can't be positive EV for both of you.
The question for the poll is would you take the bet (multiple votes allowed)? The question for the forum is where is the flaw in the positive EV argument?
If you win, you win a tie worth more than the wager, but if you lose, you lose more than what the other guy wagered.
Quote: TigerWuI don't understand what the paradox is...
The paradox is it seems to be a positive EV bet, but that betrays the logic both people could make the same argument and a bet can never be good for both sides.
Quote: WizardThe paradox is it seems to be a positive EV bet, but that betrays the logic both people could make the same argument and a bet can never be good for both sides.
Oh. Well, the flaw is that they're not taking into account that they stand to lose more than they're wagering if they're wrong.
Eg. if my tie cost $10,000, then I know my chances of winning the bet are much lower than 50%. If my tie cost $1, than much higher than 50%.
Quote: WizardOver drinks, you and a friend argue over who has the more expensive necktie. Your friend proposes a wager where the one with the necktie worth the least wins both ties. You assume a 50% chance of winning and losing. Fortunately, both of you have saved receipts to verify the values.
After thinking a moment, you conclude the bet must have a positive expected value. This is because, if you win, you will win a tie worth more money than your wager.
However, both of you could accept the bet under the same logic, and it can't be positive EV for both of you.
The question for the poll is would you take the bet (multiple votes allowed)? The question for the forum is where is the flaw in the positive EV argument?
Quote: michael99000You stated that you feel you have a 50% chance of winning/losing. But wouldn’t that only be the case if you knew the value of your tie was exactly equal to the median cost of a tie?
Eg. if my tie cost $10,000, then I know my chances of winning the bet are much lower than 50%. If my tie cost $1, than much higher than 50%.
Let's just say you eyeball the other tie and feel it is approximately of the same value, but not exactly.
Quote: WizardAfter thinking a moment, you conclude the bet must have a positive expected value. This is because, if you win, you will win a tie worth more money than your wager.
We could use thinking this to say that any bet must have a positive expected value, because “if you win . . .”
Expected value also has to take into account if you lose
Quote: WizardLet's just say you eyeball the other tie and feel it is approximately of the same value, but not exactly.
I still think the flaw in the logic is that you have a 50% chance of winning.
You’re not risking x to win what you know is more than x with a 50% chance of being successful. If you were, that would be the positive EV good bet.
You MIGHT be risking the more expensive tie.
Quote: MaxPenAlso have to take into account whether or not the tie will have to be deloused or not.
It does. You have a 50% chance of losing your tie and a 50% chance of winning a better tie.
Quote: TomGExpected value also has to take into account if you lose
Let's say I offer you a bet. We flip a fair coin. You wager $1. If you win the flip, I give you at least $1 in winnings. Would you take the bet?
Quote: WizardIt does. You have a 50% chance of losing your tie and a 50% chance of winning a better tie.
Let's say I offer you a bet. We flip a fair coin. You wager $1. If you win the flip, I give you at least $1 in winnings. Would you take the bet?
The answer is we don’t know enough information. We know we have a 50% chance of winning. If we win, we win more than we risk. If we lose, we lose more than we stood to gain. But you really need to know the expected distribution of tie prices that our opponent may be wearing.
For example, if I am wearing a $50 tie and my opponent is either wearing a $25 tie or $75 tie with equal probability, then it’s a fair wager. Yes, if I win, I risked $50 to gain $75 on a 50%. But if I lose, I stood to gain only $25 vs my $50.
If however, I’m wearing a $25 tie and my opponent’s tie is either $10 or $100 with 50% probability, then I should take the bet. It’s +EV.
Quote: unJonThe answer is we don’t know enough information. We know we have a 50% chance of winning. If we win, we win more than we risk. If we lose, we lose more than we stood to gain. But you really need to know the expected distribution of tie prices that our opponent may be wearing.
For example, if I am wearing a $50 tie and my opponent is either wearing a $25 tie or $75 tie with equal probability, then it’s a fair wager. Yes, if I win, I risked $50 to gain $75 on a 50%. But if I lose, I stood to gain only $25 vs my $50.
If however, I’m wearing a $25 tie and my opponent’s tie is either $10 or $100 with 50% probability, then I should take the bet. It’s +EV.
This is the best explanation yet. I would compare it to the two-envelope paradox. There's another similar one where Bill Gates and Warren buffet bet over who has more money in his wallet.
Quote: WizardOver drinks, you and a friend argue over who has the more expensive necktie. Your friend proposes a wager where the one with the necktie worth the least wins both ties. You assume a 50% chance of winning and losing. Fortunately, both of you have saved receipts to verify the values.
After thinking a moment, you conclude the bet must have a positive expected value. This is because, if you win, you will win a tie worth more money than your wager.
However, both of you could accept the bet under the same logic, and it can't be positive EV for both of you.
The question for the poll is would you take the bet (multiple votes allowed)? The question for the forum is where is the flaw in the positive EV argument?
Here is my answer (don't know if it correct or not).
Both of you know this, but neither of you can remember who has the $20 tie and who has the $19 tie.
50% of the time you will win a "$20 tie" and 50% you will lose a "$20 tie".
Therefore, $0 is the EV. In other words, the bet has no value to you.
Also,
IMO, shouldn't it be "...you and a friend argue over who has the LEAST expensive necktie." (I only thought this, because of the 2nd sentence "Your friend proposes a wager where the one with the necktie worth the least wins both ties.")
Note: I am not trying to split hairs here, but I thought I would write about this because, it may have something to do with working out an answer differently (possibly?)
----
Update (about 210 pm):
Darn it : ) , I took too long to write this (when I started writing this reply, the post below was the previous one in this thread):
Quote: Wizard
December 16th, 2019 at 1:10:38 PM
(snip)
Let's say I offer you a bet. We flip a fair coin. You wager $1. If you win the flip, I give you at least $1 in winnings. Would you take the bet?
Quote: WizardIt does. You have a 50% chance of losing your tie and a 50% chance of winning a better tie.
Except that this isn't true. Either you have the cheaper tie, or you don't. The result is not up to chance.
You can't really specify an EV as the assumption that you have a 50% chance of winning if you accept the bet is false.
Quote: ThatDonGuyExcept that this isn't true. Eit mi her you have the cheaper tie, or you don't. The result is not up to chance.
You can't really specify an EV as the assumption that you have a 50% chance of winning if you accept the bet is false.
I agree with this. The Wizard seems to be assuming that just because there’s 2 possible outcomes, that the odds are therefore 50/50.
You can only assume you have a 50% chance of winning if you know that the value of your tie is such that there’s an equal number of neckties in the world that are more expensive than yours as there are less expensive than yours.
The only information you have in regards to estimating your odds of winning is the knowledge of your own ties value
The Wiz meant the 50% to be exogenous for the hypothetical. You can assume you put both neckties in a bag and randomly draw one of them out. Doesn’t change the point the “paradox” is meant to illuminate.Quote: michael99000I agree with this. The Wizard seems to be assuming that just because there’s 2 possible outcomes, that the odds are therefore 50/50.
You can only assume you have a 50% chance of winning if you know that the value of your tie is such that there’s an equal number of neckties in the world that are more expensive than yours as there are less expensive than yours.
The only information you have in regards to estimating your odds of winning is the knowledge of your own ties value
I also disagree that the only information you have is of your own ties value. You could also have knowledge about the type of person your friend is, what types of ties he would buy. You also can see your friend’s tie so could have information about whether it’s an old, new, silk, plaid, cheap or expensive tie. Factoring in all that information, the hypothetical says you conclude there’s a 50% chance your tie is the cheaper one.
What about those Glad plastic bag ties? Or Hefty trashbag ties?
Quote: WizardThe paradox is it seems to be a positive EV bet, but that betrays the logic both people could make the same argument and a bet can never be good for both sides.
For this example, it's not +EV because you didn't include the "if you lose you lose more than you stand to win" thing.
However, I'd say it's possible in certain conditions for some wager X to be +EV for one individual and -EV for another individual, based on the information they have.
Quote: WizardLet's say I offer you a bet. We flip a fair coin. You wager $1. If you win the flip, I give you at least $1 in winnings. Would you take the bet?
Absolutely. Because that would be positive expected value for me. It would be negative expected value for you, as you are risking more than $1 to win only $1.
In the necktie problem, let the value of my necktie = N, let the value of the other guys tie = T. I am risking N, to win T. The expected value is .5T - .5N.
Let's say neckties can cost only $1 or $2. There is a 50% chance my necktie costs $1 and a 50% chance it costs $2. Under these conditions, N = (.5 x $1) + (.5 x $2). Which is exactly the same for T. Which means under these conditions, the expected value is zero.
Now things get a little bit messier: Ties don't cost only $1 or $2. They could cost virtually any number (so long as there are no more than two digits after the decimal, but with sales tax, it might be even more than that). But in our example, there are only two values, N and T. Let's say that if N is greater than T, then my necktie cost x. If N is less than T, then my necktie cost y. Because we know there is a 50% chance N is greater than T, and a 50% chance N is less than T, that means N = .5x + .5y, which must be the exact same value for T. The expected value is 0.
Short answer. There is a 50% chance you win the more expensive tie and a 50% chance you lose the more expensive tie.
Quote: michael99000I agree with this. The Wizard seems to be assuming that just because there’s 2 possible outcomes, that the odds are therefore 50/50.
I'm not trying to defend switching, but playing the devil's advocate to find the flaw in my logic.
Quote: TomGAbsolutely. Because that would be positive expected value for me. It would be negative expected value for you, as you are risking more than $1 to win only $1.
I just want to emphasize that I'm just playing around here and challenging the forum to find the flaw in an erroneous argument. I'm not truly defending the decision to switch.
I'll be more careful and cautious with my wording next time.
Let's try to lower the temperature some holiday cheer:
Direct: https://www.youtube.com/watch?v=4IZJGB6RLq0
Neckties are -EV no matter how you slice it.Quote: WizardOver drinks, you and a friend argue over who has the more expensive necktie. Your friend proposes a wager where the one with the necktie worth the least wins both ties. You assume a 50% chance of winning and losing. Fortunately, both of you have saved receipts to verify the values.
After thinking a moment, you conclude the bet must have a positive expected value. This is because, if you win, you will win a tie worth more money than your wager.
However, both of you could accept the bet under the same logic, and it can't be positive EV for both of you.
The question for the poll is would you take the bet (multiple votes allowed)? The question for the forum is where is the flaw in the positive EV argument?
Is that too simple?
Scenario 1 : Alfred offers the bet but hasn't yet bought the tie. Your tie is, say, $50. If you take the bet Alfred is going to randomly buy a tie worth $10, $20, $30, $40, $60, ... $90 (i.e. 4 lower priced, 4 higher priced).
Your expected value is +EV. So what about Alfred's.
He will land up with your tie if he buys 10 ,20, 30 or 40, so he will win $50, $50, $50, $50.
He will lose his tie ifhe buys 60,70,80,90; so he will lose $60, $70, $80, $90.
So Alfred is being very generous and you should take the bet.
Scenario 2 : Bert also hasn't bought the tie but gets cashback when makiing purchases. He's clever and knows your tie is worth $50, so agrees to either buy a tie at $1, or $51 with equal probability. However after cashback the $51 tie costs him $49.
So Bert has a +EV, you have a +EV, the bank lost $2.
this is a scheme for mandatory 'risk of loss' in return for an illusory gain? What is the winner going to do? Wear both?
Quote: FleaStiffWhat is the winner going to do? Wear both?
Given that they are wearing ties to a bar implies they probably wear them a lot. People who must wear ties tend to have lots of them, to mix things up. Think about it -- for business attire ties are about the only way men can be a little creative.
Personally, I own about 20 ties, but most are quite ugly. Anybody need one?
I'll bet you both of my ties are worth less than all your ties. I'll give you 2 to 1 on your money.Quote: WizardGiven that they are wearing ties to a bar implies they probably wear them a lot. People who must wear ties tend to have lots of them, to mix things up. Think about it -- for business attire ties are about the only way men can be a little creative.
Personally, I own about 20 ties, but most are quite ugly. Anybody need one?
Quote: AxelWolfI'll bet you both of my ties are worth less than all your ties. I'll give you 2 to 1 on your money.
I would need way more odds than that.
Of course. They probably wear pants, belts, shoes, etc., but they don't obsess about them. If you show up at certain functions, the bar will supply a tie or a suit-jacket for the event. Its no big deal, but no one focusses on creativity or price. It is an archaic relic of days gone by. The only .functional 'value' to a necktie is in tabulating what I've eaten.Quote: WizardGiven that they are wearing ties to a bar implies they probably wear them a lot.
Cuffs are relics of sword-hilts, not items of value, so a bet involving such items is about as meaningless as the item itself.
Quote: FleaStiffOf course. They probably wear pants, belts, shoes, etc., but they don't obsess about them. If you show up at certain functions, the bar will supply a tie or a suit-jacket for the event. Its no big deal, but no one focusses on creativity or price. It is an archaic relic of days gone by. The only .functional 'value' to a necktie is in tabulating what I've eaten.
Cuffs are relics of sword-hilts, not items of value, so a bet involving such items is about as meaningless as the item itself.
And yet, everyone knows to wear their best suit and tie when going to see the bank loan officer to get that much needed loan.
I'ze wouldn't know, my loans always came bankers with a vowel on the end of their name whose offices were in bars.Quote: unJonAnd yet, everyone knows to wear their best suit and tie when going to see the bank loan officer to get that much needed loan.
Quote: FleaStiffI'ze wouldn't know, my loans always came bankers with a vowel on the end of their name whose offices were in bars.
And look how well that turned out.
