bet your wallet

[phillip1882]

two rich men who have no idea how much money they currently have in their wallets meet. rich man A says to rich man B: "i'll bet you that i have more money in my wallet than you have in yours. In fact, let's switch wallets." should rich man B accept the wager?

[Joatmon]

two rich men who have no idea how much money they currently have in their wallets meet. rich man A says to rich man B: "i'll bet you that i have more money in my wallet than you have in yours. In fact, let's switch wallets." should rich man B accept the wager?

 

I'd like to meet one of these men and hope he would give me the chance!

 

 

 

[phillip1882]

here's another one of my favorites, in a similar vien.

 

there are two closed boxes. the host informs you that one box has 3 times the amount of money of the other.

you may open one box, and then based on the amount you see, switch if you so choose.

would it always be in your benefit to switch?

[imatfaal]

here's another one of my favorites, in a similar vien.

 

there are two closed boxes. the host informs you that one box has 3 times the amount of money of the other.

you may open one box, and then based on the amount you see, switch if you so choose.

would it always be in your benefit to switch?

 

I have never been at home with these sort of problems. But my feeling would be along the lines of

A. I open a box and it has £300 in it. I can switch and have

B. 50% chance of £100 and 50% chance of £900 - that seems to have an expectation value of £500.

C. unless I cannot risk the £100 payout (ie if an employer did thisone might be forced to decline - you gotta pay mortgage and bills that are more than 1/3 of the value) then it seems like you always choose to switch

[Phi for All]

two rich men who have no idea how much money they currently have in their wallets meet. rich man A says to rich man B: "i'll bet you that i have more money in my wallet than you have in yours. In fact, let's switch wallets." should rich man B accept the wager?

Does switching the wallets mean that now man A's wallet belongs to man B? If so, then switching completely changes the focus of the original bet.

 

And if switching means man B no longer has access to the other things in his wallet besides money, then he shouldn't take the bet. He needs his ID.

 

Also, man A specified "money", which is different than "cash". A platinum credit card or a bank debit card represents a lot of "money", technically. And riddles sometimes depend on technicalities.

[D H]

Does switching the wallets mean that now man A's wallet belongs to man B? If so, then switching completely changes the focus of the original bet.

 

And if switching means man B no longer has access to the other things in his wallet besides money, then he shouldn't take the bet. He needs his ID.

 

Also, man A specified "money", which is different than "cash". A platinum credit card or a bank debit card represents a lot of "money", technically. And riddles sometimes depend on technicalities.

You are reading too much into the problem. Forget IDs, credit cards, and all that extraneous stuff. The participants swap the cash contents of their wallets.

 

This is a variant of the two envelope problem, as is phillip1882's problem in post #3.

 

See http://en.wikipedia.org/wiki/Two_envelopes_problem.

[Phi for All]

You are reading too much into the problem.

I read exactly what was given to me. Without all the extraneous stuff, you're left with no clear decision at all.

 

Man A first bets he has more money. This is a straightforward bet, assuming the winner gets to keep the cash from both wallets.

 

But then he changes things by offering to swap wallets. Is man A betting that his wallet holds more cash than man B's? If this is true, why would he want to switch? And in this instance, the winner is only getting the difference between the two amounts of money. The bet has changed from it's original form.

[khaled]

This sounds silly, but I've written an informal algorithm on the Envelope Problem (that make use of luck and luckiness):

 

1. Alpha = how lucky are you in general %

2. Beta = how lucky do you feel at the moment %

2. make an initial decision (set a bet on one of the envelops)

3. IF Alpha > 50% AND Beta > Alpha THEN Finalize your bet

-- ELSE IF Alpha > 50% AND Beta < Alpha THEN Swap & Finalize

-- ELSE (as preconfigured) Someone else (whose Beta > Alpha > 50%) will make the Swap for you

Edited March 29, 2012 by khaled

[Bignose]

This sounds silly, but I've written an informal algorithm on the Envelope Problem (that make use of luck and luckiness):

 

1. Alpha = how lucky are you in general %

2. Beta = how lucky do you feel at the moment %

2. make an initial decision (set a bet on one of the envelops)

3. IF Alpha > 50% AND Beta > Alpha THEN Finalize your bet

-- ELSE IF Alpha > 50% AND Beta < Alpha THEN Swap & Finalize

-- ELSE (as preconfigured) Someone else (whose Beta > Alpha > 50%) will make the Swap for you

 

Alpha and Beta are how casinos make money... -or- trying to reason using them doesn't really make any sense.

[D H]

I read exactly what was given to me. Without all the extraneous stuff, you're left with no clear decision at all.

That's kinda the point of all of these versions of the two envelope problem.

 

Man A first bets he has more money. This is a straightforward bet, assuming the winner gets to keep the cash from both wallets.

Why is this anymore straightforward than the two participants swapping the cash contents of their wallets?

 

 

To me, the problem with all of these variations of the two envelope problem is pretty simple: They implicitly assume a discrete uniform distribution over an infinite number of items, or a continuous uniform distribution over an infinitely long interval. The resolution is pretty simple, too: Get rid of that erroneous assumption.

[Phi for All]

Why is this anymore straightforward than the two participants swapping the cash contents of their wallets?

The first bet man A makes is that he has more money in his wallet than man B. The implied wager is that the winner gets to keep the loser's cash. The second bet is to switch wallets. The implied wager is that both men keep the cash in the other man's wallet.

 

If one man has $1000 and the other man has $2000, in the first bet the winner (the man who had more money) gets all $3000. In the second bet, the winner is whoever had the lesser amount and traded for the higher amount. No matter who it is, the winner of the second bet ends up with $2000 but the loser gets $1000. The net loss is the same with either bet for the loser, but the net gain is better for the winner in the first bet.

[D H]

The first bet man A makes is that he has more money in his wallet than man B. The implied wager is that the winner gets to keep the loser's cash. The second bet is to switch wallets. The implied wager is that both men keep the cash in the other man's wallet.

 

If one man has $1000 and the other man has $2000, in the first bet the winner (the man who had more money) gets all $3000. In the second bet, the winner is whoever had the lesser amount and traded for the higher amount. No matter who it is, the winner of the second bet ends up with $2000 but the loser gets $1000. The net loss is the same with either bet for the loser, but the net gain is better for the winner in the first bet.

Close.

 

In the winner-take-all formulation of the wallet paradox, it is the man with the lesser amount of cash in his wallet (not the greater amount) who wins everything. Each man will think "I have X dollars in my wallet, so that is what I stand to lose. If I win, I'll keep that original amount and I will win more than that. Therefore it is to my advantage to play." In the swap formulation of the problem, each man will think "I have X dollars in my wallet, so that is the most I stand to lose. That other guy might have ten times that, or a hundred times that. so my potential winnings are unbounded. Therefore it is to my advantage to play."

 

It really doesn't matter which version of the problem you choose to analyze. Phrase the problem correctly and it will appear to each man that playing the game is advantageous to him. That is the paradox: How can each person see a positive outcome when this is clearly a zero sum game?

 

The answer is that a probability distribution cannot be both unbounded and uniform. The arguments for why it is advantageous to play the game implicitly assume such an unbounded, uniform distribution. The supposed paradox is a simple matter of garbage in, garbage out.

[Phi for All]

In the winner-take-all formulation of the wallet paradox, it is the man with the lesser amount of cash in his wallet (not the greater amount) who wins everything.

I don't see how this is possible. The first bet is "I'll bet you that i have more money in my wallet than you have in yours." If man A has $2000, he wins man B's $1000. If man A has $1000, man B wins that money because he has $2000. How can the man with the lesser amount win everything when he loses the first bet?

[D H]

I don't see how this is possible.

One simple possibility: The OP phrased this oldie but goodie incorrectly.

 

Google "wallet paradox" and you'll get tons of hits. Use Google scholar and you'll still get quite a few hits. Here's just one such result:

http://academic.scranton.edu/faculty/carrollm1/wallet.pdf

In this paper, Carroll, Jones, and Rykken analyze the wallet game from the perspective of Nash equilibrium. They find that there is none.

 

While you're googling "wallet paradox", you might also want to do the same for "necktie paradox" and "two envelope paradox".

[Purephysics]

This is almost like the problem of probability most people miss in the 'one in three' box trick:

 

There are three boxes; one has £1000 in it, the other two are empty.

You can pick any one you like A, B, or C.

Once you pick a box, one of the other (empty) boxes will be removed. Leaving you with the £1000 box and an empty one.

 

The question is; should you stay with your original selection or switch?

 

Many people think "it makes no difference, I'll just stick with my first choice." But you were selecting from a 1:3 chance originally, when one box is removed if you stick, you still have a 1:3 chance. If you switch, you have a 1:2 chance, which is better.

So many people I have done this little experiment on just stick with their original choice. I've never quite understood why, it always seemed obvious to me.

[Joatmon]

This is almost like the problem of probability most people miss in the 'one in three' box trick:

 

There are three boxes; one has £1000 in it, the other two are empty.

You can pick any one you like A, B, or C.

Once you pick a box, one of the other (empty) boxes will be removed. Leaving you with the £1000 box and an empty one.

 

The question is; should you stay with your original selection or switch?

 

Many people think "it makes no difference, I'll just stick with my first choice." But you were selecting from a 1:3 chance originally, when one box is removed if you stick, you still have a 1:3 chance. If you switch, you have a 1:2 chance, which is better.

So many people I have done this little experiment on just stick with their original choice. I've never quite understood why, it always seemed obvious to me.

This seems like a variation of the Monty Hall problem as discussed recently in the topic "Why is the Monty Hall Problem so controversial". I think that by swapping you increase your chances to 2:3.

If you originally chose the box with money in it and swap you must lose.

If you pick a box with nothing in it you must win.( assuming the other empty box is removed)

You will pick an empty box twice as often as the box containing money. So you will win two times out of three if you swap.

Edited April 20, 2012 by Joatmon

[imatfaal]

Purephysics

 

We have been discussing that in another thread

http://www.scienceforums.net/topic/65398-why-is-the-monty-hall-problem-so-controversial/

 

Monty Hall is the common name for it from a US television show.

[Purephysics]

Are you see over here in the UK it doesn't seem to have a name per sé.

 

And in my experience, you can apply as much math as you want; sometime you just get lucky.