Posers and Puzzles
09 Apr 05
09 Apr 05
Suppose someone comes up to you and a stranger on the street and asks you if you want to play a game. He explains that the game is very simple; you and the stranger both take out your wallets. Whoever has the least money, gets to keep the contents of both wallets. Assuming you didn't suspect any kind of fraud or trickery, would you play the game?
You could reason that, if I have more than the stranger, he'll win just what I have. But if he has more, I'll win more than I have. Therefore, the game favors me. And the stranger can reason the same way. But they can't both have the advantage. So what's wrong with their reasoning?
09 Apr 05
Originally posted by ark13If I've got $7, I will end up with $14.01 at most, or zero at the least. It's double or nothing in that case.
Suppose someone comes up to you and a stranger on the street and asks you if you want to play a game. He explains that the game is very simple; you and the stranger both take out your wallets. Whoever has the least money, gets to keep the contents of both wallets. Assuming you didn't suspect any kind of fraud or trickery, would you play the game?
You co ...[text shortened]... the same way. But they can't both have the advantage. So what's wrong with their reasoning?
The flaw is thinking that it is better to gain money than to lose money, just because the quantity of the gain will be greater (even if only by a penny) than the quantity lost. This is why people buy lottery tickets.
09 Apr 05
Originally posted by BigDoggProblemYou may have misunderstood. The one with the least money gets it all. So if you've got $7 and I've got $100, you'd end up with $107. You could and probably would end up with more than $14.01
If I've got $7, I will end up with $14.01 at most, or zero at the least. It's double or nothing in that case.
The flaw is thinking that it is better to gain money than to lose money, just because the quantity of the gain will be greater (even if only by a penny) than the quantity lost. This is why people buy lottery tickets.
09 Apr 05
Originally posted by ark13I think the fallacy in the thinking lies in the fact that the above reasoning does not really consider expected payoff properly -- i think it is making some kind of implicit assumption that there is roughly equal probability of my winning or losing the game, but this should depend on how much i am carrying in my wallet (which is known to me).
Suppose someone comes up to you and a stranger on the street and asks you if you want to play a game. He explains that the game is very simple; you and the stranger both take out your wallets. Whoever has the least money, gets to keep the contents of both wallets. Assuming you didn't suspect any kind of fraud or trickery, would you play the game?
You co ...[text shortened]... the same way. But they can't both have the advantage. So what's wrong with their reasoning?
consider the case in which I am (for some random reason) carrying 5000 dollars in my wallet. Then the reasoning that I am always better off to play the game will fail miserably because not too many people carry that kind of cash on them.
that is sort of an extreme case, but the idea is that your expected payoff should depend on how much cash you are carrying and on how much the average person you meet on the streets would be expected to be carrying.
of course, if your wallet is empty, like mine too often, then you'd be crazy to not play...
09 Apr 05
Originally posted by ark13No, I did not.
You may have misunderstood. The one with the least money gets it all. So if you've got $7 and I've got $100, you'd end up with $107. You could and probably would end up with more than $14.01
I repeat: the flaw in the reasoning is that the quantity of potential gain vs. potential loss does not matter. The only thing that matters is that you have a 50% chance of winning.
09 Apr 05
Originally posted by BigDoggProblemThis is simply not true.
The only thing that matters is that you have a 50% chance of winning.
Generally, I have about 100% chance of winning or at least drawing because I seldom
carry any cash with me.
My father, on the other hand, is literally a walking bank. The average person wouldn't
have 1/10 the money he typically carries around, and so the chances of his winning are
very, very slim.
In order to understand the odds, one would have to know what the average amount of
money a person carries on them and then make a determination.
Nemesio
09 Apr 05
Originally posted by NemesioThe problem does not mention how much money you or a stranger might have. The "you" in this case is only hypothetical.
This is simply not true.
Generally, I have about 100% chance of winning or at least drawing because I seldom
carry any cash with me.
My father, on the other hand, is literally a walking bank. The average person wouldn't
have 1/10 the money he typically carries around, and so the chances of his winning are
very, very slim.
In order to understand ...[text shortened]... the average amount of
money a person carries on them and then make a determination.
Nemesio
It is just as likely that a stranger has no money and you have a few pennies; therefore, even in your case personally, it's still 50/50. You may know alot about yourself, but you know nothing of the stranger.
This is much like the old "no more children after one female is born" problem. The odds stay 50/50 unless you are willing to introduce the nuances of human society into the equation as probability. Such equations would be waaaaay too complicated to make for a good puzzle.
The wording of the original puzzle does not mean "you" in the literal sense; I believe it to be hypothetical, especially in light of the final question: Why do both players think they have an advantage, when it is not possible for both to have the advantage simultaneously?
09 Apr 05
Originally posted by BigDoggProblemOk ok. I was taking this literally, in which case I would generally have an advantage.
The problem does not mention how much money you or a stranger might have. The "you" in this case is only hypothetical.
It is just as likely that a stranger has no money and you have a few pennies; therefore, even in your case personally, it's [b]still 50/50. You may know alot about yourself, but you know nothing of the stranger.
This is m ...[text shortened]... have an advantage, when it is not possible for both to have the advantage simultaneously?[/b]
Let me see if I understand what you are saying.
Let's say there are 10 people in the world with wallets. Person A has 1 dollar, Person
B has two dollars...Person J has 10 dollars.
As such, Person A will win 9 of 9 times, Person B will win 8 of 9 times...and Person J
will win 0 of 9 times.
Of 90 trials, there are 45 winners and 45 losers.
Is this what you mean?
If so, then I don't see that either player has a statistical advantage (it's even odds), so
when either player claims to have it, they are in error.
Did I follow?
Nemesio
09 Apr 05
1 edit
Originally posted by BigDoggProblemI think I get the problem in the spirit it was intended now, but I disagree that
It is just as likely that a stranger has no money and you have a few pennies; therefore, even in your case personally, it's [b]still 50/50. You may know alot about yourself, but you know nothing of the stranger.
...The odds s ...[text shortened]... ations would be waaaaay too complicated to make for a good puzzle.[/b]
the knowledge about myself is not helpful.
If I know generally that people tend to carry 50 bucks on them at a time (and
I wouldn't call this a 'nuance'😉 and I know precisely that I have 2 bucks, then
it's a good call for me to play the game. Through experience, I know that most
people carry more cash than I do and, since it costs me relatively little to play
and the potential payoff is good for me, I would be foolish to say that my odds
are 50/50. (The same goes for my father, who might have 1000 bucks on him.)
Nemesio
Edit: that is to say, I have complete knowledge about me and general knowledge
about the rest of the world. As such, I can make a more sober determination to
get a better sense of whether I am really looking at a 50/50 situation or better.
09 Apr 05
Originally posted by NemesioI agree that knowledge about yourself is helpful if you were to play the game in real life, but I don't think the 'real-life' version of this would work well as a puzzle. That's mainly what I'm trying to say.
I think I get the problem in the spirit it was intended now, but I disagree that
the knowledge about myself is not helpful.
If I know generally that people tend to carry 50 bucks on them at a time (and
I wouldn't call this a 'nuance'😉 and I know precisely that I have 2 bucks, then
it's a good call for me to play the game. Through experience, I k ...[text shortened]... nation to
get a better sense of whether I am really looking at a 50/50 situation or better.
09 Apr 05
Originally posted by ark13Well, first of all, there will be some average amount of money people carry. If a person has more than the average amount, he'll most likely lose. If a person has less, he'll most likely win.
Suppose someone comes up to you and a stranger on the street and asks you if you want to play a game. He explains that the game is very simple; you and the stranger both take out your wallets. Whoever has the least money, gets to keep the contents of both wallets. Assuming you didn't suspect any kind of fraud or trickery, would you play the game?
You co ...[text shortened]... the same way. But they can't both have the advantage. So what's wrong with their reasoning?
However this is how one analyzes the chances if you don't know what the other guy's got. There's no true chance involved; one side will win and one will lose 100%. So if both have less than the average, both don't have the advantage - they only think they have the advantage based on their limited knowledge.
09 Apr 05
1 edit
Originally posted by BigDoggProblemThis is similar to the problem with two envelopes, one of which contains twice as much as the other. No question involving probabilities makes sense unless you specify a distribution, even if it's just a guess on the part of the players (ie you're studying a Bayesian game). And in case you're thinking 'uniform is a natural enough distribution' or 'uniform is the Bayesian way of modelling the absence of knowledge', there is no such thing as a uniform distribution over an unbounded range. If you're trying to do this without any kind of assumption as to the probability that a person has a certain amount in their wallet, any answers you get are likely to be nonsense.
This is much like the old "no more children after one female is born" problem. The odds stay 50/50 unless you are willing to introduce the nuances of human society into the equation as probability. Such equations would be waaaaay too complicated to make for a good puzzle.
09 Apr 05
Originally posted by BigDoggProblemI'm a bit puzzled as to why the person being offered the deal considers themselves to have the advantage. To actually offer the deal, said stanger in all probability would have less than average in their wallet (it would be daft of him to intiate the deal with a huge amount of money in it).
The problem does not mention how much money you or a stranger might have. The "you" in this case is only hypothetical.
It is just as likely that a stranger has no money and you have a few pennies; therefore, even in your case personally, it's [b]still 50/50. You may know alot about yourself, but you know nothing of the stranger.
This is m ...[text shortened]... have an advantage, when it is not possible for both to have the advantage simultaneously?[/b]