Alex and Davey - who wins this time?

[bonanova]

It was quiet at Morty's last night ...

Until Matt the Mathematician proposed a game.

Alex and Davey would both put their billfolds on the table,

and Matt would count the money in each.

Whoever had more would forfeit his money to the other player.

"Think it over," Matt said, and he ordered another frosty one.

Alex thought to himself,

"There's no reason to believe I have more money than Davey

has, or less, for that matter; so my chances of winning are 50%.

Now if I lose, I'll lose whatever's in my wallet.

But if I win, it'll be because Davey has more money than I.

So on a tossup bet, my winnings are more than my losses."

"I'm in," he finally announced.

And Davey thought the same.

He figured his chances of winning were as good as Alex's,

and he'd either lose the amount in his wallet or win

an amount greater than that.

"You're on," said Davey with a smile.

If the boys were thinking clearly, how could the game favor both of them?

Or if not, where is the flaw in their reasoning?

[Guest]

Ooh, this is a good one!

Looking forward to see what responses this gets as good arguments can be made for more than one answer.

[unreality]

This is classic! bonanova are you a Martin Gardner fan too?

[Guest]

Here is my reasoning as Alex:

It is safe to assume that each wallet contains about an average amount of money. So if that is the case, then Davey probably has either marginally more or marginally less than me. If it is marginally less, I lose my average amount of money (chances 1:2). If it is more, I win and get double the average amount of money (chance 1:2). This makes the expected value of the bet (0*avg * .5 + 2*avg * .5) = avg.

In other words, the expected value of the bet is equal to the amount of money I already have. Why risk my money on a bet with the same expected value as the cost?

However, if I know I have less than the average amount of money to be found in wallet, then I know I have an advantage and the odds are greater in my favour.

Now, if I see Davey jump in and take the bet, I will know that he has less than the average amount in his billfold and I should not bet. If I see him hesitate, then I can assume that he has an average amount in his wallet and is waiting to see if I jump in to tell if I think I have the advantage, in which case, he will not bet. If he has an large amount in his wallet, he will not take the bet.

No matter how I slice it - I should not bet and neither should Davey. Instead, we should split on a pizza and tell Matt to figure out how much pizza he's going to get if Davey and I are splitting the bill between us.

[unreality]

Typo: the second chances are 2:1 not 1:2, I think thats what you meant (cuz (1:2 + 2:1) /2 = 1:1)

anyway, I would just slip all my money out of my wallet as I set it on the table I win the contents of both of our wallets ;D

[Guest]

Typo: the second chances are 2:1 not 1:2, I think thats what you meant (cuz (1:2 + 2:1) /2 = 1:1)

No, the chance of my losing is 1:2 and the chance of my winning is 1:2. The expected value if I lose is $0.00 and the expected value if I win is 2 * the average amount that can be expected to be in a wallet. So the total expected value (before the contents of either billfold are inspected - and before my recall of the amount in my own wallet) is 1:2 * $0.00 + 1:2 * 2avg = .5 * 0 + .5 * 2avg = avg.

[Guest]

but they don't take the bet because they think they have better chances

they do it because their winnings if they win will be larger than their losings if they lose.

Which is true no matter what.

[Guest]

Either one of them should take all the money out of their wallet and they can't lose, only win if one of them was dumb enough to leave their money in their wallet.

[bonanova]

Either one of them should take all the money out of their wallet and they can't lose, only win if one of them was dumb enough to leave their money in their wallet.

A good lateral-thinking game strategy, if a bit underhanded.

Bravo.

But the questions in the OP were about the analysis of the game.

If the boys were thinking clearly, how could the game favor both of them?

Or if not, where is the flaw in their reasoning?

Can a game favor both participants [if the rules are followed]?

Its symmetry seems to favor neither. If that's true, supply an analysis leading to that conclusion.

It may not be correct to analyze for very nearly equal amounts.

Only that neither has reason to think one has more.

For example, one be willing to bet the other visited a cash machine more recently.

[Guest]

I know I'm bringing this back from several months ago, but I'm interested to know if there is more of a resolution to this problem. Is there an optimal strategy (short of cheating)? Anyone else have an idea?

It was quiet at Morty's last night ...

Until Matt the Mathematician proposed a game.

Alex and Davey would both put their billfolds on the table,

and Matt would count the money in each.

Whoever had more would forfeit his money to the other player.

"Think it over," Matt said, and he ordered another frosty one.

Alex thought to himself,

"There's no reason to believe I have more money than Davey

has, or less, for that matter; so my chances of winning are 50%.

Now if I lose, I'll lose whatever's in my wallet.

But if I win, it'll be because Davey has more money than I.

So on a tossup bet, my winnings are more than my losses."

"I'm in," he finally announced.

And Davey thought the same.

He figured his chances of winning were as good as Alex's,

and he'd either lose the amount in his wallet or win

an amount greater than that.

"You're on," said Davey with a smile.

If the boys were thinking clearly, how could the game favor both of them?

Or if not, where is the flaw in their reasoning?

[Guest]

No, the chance of my losing is 1:2 and the chance of my winning is 1:2. The expected value if I lose is $0.00 and the expected value if I win is 2 * the average amount that can be expected to be in a wallet. So the total expected value (before the contents of either billfold are inspected - and before my recall of the amount in my own wallet) is 1:2 * $0.00 + 1:2 * 2avg = .5 * 0 + .5 * 2avg = avg.

I think this was on the right track: the game technically favors both players because the winnings will always be greater than the losings. For example, if you have x dollars in your wallet, you would lose that $x if you were to lose. However, if you were to win, this is because the other player has more than $x, and you win that amount.

The only thing I would have to include is that being the second person to accept the bet should give you a decent hint as to whether it would make sense to play. If the other player has consented to playing the game, he likely thinks that he has less than or equal to the average amount of money in his wallet (so that there is a worthwhile chance of winning), and therefore it may not be smart to accept playing. Remember, also, that if the first person accepting to play has an empty wallet, he will accept to play the game every time!

[Guest]

I know I'm bringing this back from several months ago, but I'm interested to know if there is more of a resolution to this problem. Is there an optimal strategy (short of cheating)? Anyone else have an idea?

The beauty of the solution is that there was no cheating involved.

Both participants are allowed to remove the money from their wallets, because there was no explicit rule made to the contrary.

So, assuming both men had the common sense to realize that fact, makes the odds of winning nil.

But both men were really betting on whether or not the other was dumb enough to leave the money in their wallet, thereby forfeiting its contents.

What I wonder what would have happened if one of them had an IOU slip in their wallet?

[Guest]

Just realized I quoted the wrong reply above, I meant to quote this one:

but they don't take the bet because they think they have better chances

they do it because their winnings if they win will be larger than their losings if they lose.

Which is true no matter what.

My answer above also assumes that removing your money would be considered cheating and neither will do it. I'm not sure if this interpretation was intended or not.

Edited May 19, 2008 by frotorious

[Guest]

put one one dollar note on the table - what sort of loss is that?

Likely to be a tie - what happens then?

[Guest]

I'll use the elliptical logic - Generally one might assume that you're only carrying enough cash for your current needs. If Alex and Davey are aware of each other's drinking habits, they can calculate approximately how much they each might have brought to the bar (this assumes that Morty's is a bar). If they've kept track of how much each has spent, each can have a decent guestimate as to the other's wallet, while knowing what they have in their own. Unfortunately only one of them brought cabfare. 8•P

Alright - the other elliptical logic. If one has more than the other, then the person who has more can afford to lose it, and the person who has less needs it more. The flaw is in thinking that the person who has more on this particular evening has more in general.

[Guest]

hmmmm... okay new thinking. The thought process becomes that of playing the stock market. "I am investing $20 to possibly have a 250% return." The odds favor each of them because they are each theoretically risking less, while reaping a large return for small investment. Flaw flaw flaw - Uh - the flaw is that if one loses their money, they have in fact risked a larger amount than the other.

[Guest]

Using Texas Holdem as an analogy...If I am holding a pair of Kings pre-flop, and I have 1 opponent. It is perfectly normal for me to assume that this hand favors me, and that I will show a profit if I play it. I will win unless my opponent out-draws me or he is holding a pair of Aces and I cannot out-draw him.

If my opponent is holding a similiar hand (say AA or QQ), then he can make the same favorable assumptions and be correct as well.

The actual winning %, however, can be entirely different. If I am holding KK, I can have as little as 18% to win if my opponent is holding AA. But there is nothing wrong with my assumption.

[Guest]

I think there's 2 ways of looking at this.

1) Informed:

Naturally you might assume that Alex and Davey have some idea of how much is in their wallets. Presumably there is some sort of statistical average of how much a typical bloke has in his wallet, and the nearer they both are to that, the closer the odds get to 50% and the closer the winnings are to the amount bet (so no advantage to either). But this does not allow for the fact that they have a choice:

If Alex has more money than average, the odds are against him and he should not bet.

If Alex has an average amount of money, the odds are still against him because if Davey had more money he probably would not have bet. So Alex should not bet.

If Alex knows he has very little money, the odds may favour him, though if Davey chooses to bet it suggests that Davey thinks the same thing, so in general there is no particular advantage to Alex.

The chance of winning depends on the amount of money in your wallet. The less money in your wallet, the better your chances. Alex should not think he has a 50% chance of winning regardless of what is in his wallet. If 2 people are playing poker, on average they will have equal chances, but if you bet on any hand thinking it has a 50% chance of winning, you will lose because your opponent will only bet when he has a good hand. Considering this, poker hands have on average a less than 50% chance of being successful, which is why you must sometimes fold. Similarly, anything more than a tiny amount in Alex's wallet has a less than 50% chance of success.

2) Uninformed:

Let's say Alex and Davy both have no idea what is in their wallets. So their decision is uninformed and they both have a 50% chance of winning. If one wins, the amount he wins will exceed the amount he risked. Where's the flaw in that? The relationship between amount bet and odds of winning is disguised, but still exists.

By way of illustration, let's say one wallet has €20 and the other €100 but you don't know which is which. If you walk away and don't bet, you get what is in your own wallet, a 50% chance of €20, 50% chance of €100, on average €60. If you bet, you'll get either nothing or €120. Still €60 on average.

[Guest]

Do credit cards count? I know you cant "physically" count money on a card but plausible right? or how about after both guys agree to the bet they agree to split the cash with whom ever wins, that way their only out the difference of what they both have, i.e I have 100 bucks Davey has 75, so 175 divided by two is 87.50 so I'm only out 12.50 which in some bars thats 3 pitchers : )

It was quiet at Morty's last night ...

Until Matt the Mathematician proposed a game.

Alex and Davey would both put their billfolds on the table,

and Matt would count the money in each.

Whoever had more would forfeit his money to the other player.

"Think it over," Matt said, and he ordered another frosty one.

Alex thought to himself,

"There's no reason to believe I have more money than Davey

has, or less, for that matter; so my chances of winning are 50%.

Now if I lose, I'll lose whatever's in my wallet.

But if I win, it'll be because Davey has more money than I.

So on a tossup bet, my winnings are more than my losses."

"I'm in," he finally announced.

And Davey thought the same.

He figured his chances of winning were as good as Alex's,

and he'd either lose the amount in his wallet or win

an amount greater than that.

"You're on," said Davey with a smile.

If the boys were thinking clearly, how could the game favor both of them?

Or if not, where is the flaw in their reasoning?

Edited May 21, 2008 by Barron