1. Imagine two equally rich people who can't remember how much money they have in their wallets are trying to decide if they should play the following game: Each person takes out his wallet and counts the money inside. Whoever has the least money wins all of the other person's money. Since they are equally rich and have no idea how much money is in their wallets, each has a 50% chance of winning the game. The first person would probably reason that he should play, since he stands to wins more than he could lose; any time you have a 50% chance of winning a bet that pays off more than you could lose, it's a statistically good bet.

The problem is, the same reasoning applies to both people. It would seem that each one has the advantage over the other!

2.

3. Originally Posted by Scifor Refugee
Since they are equally rich and have no idea how much money is in their wallets, each has a 50% chance of winning the game. The first person would probably reason that he should play, since he stands to wins more than he could lose; any time you have a 50% chance of winning a bet that pays off more than you could lose, it's a statistically good bet.
I'm not sure why you claim that the players stand to win more than they could lose. They have a 50/50 chance of either losing all their money, or doubling their money.

4. Originally Posted by Neutrino
I'm not sure why you claim that the players stand to win more than they could lose. They have a 50/50 chance of either losing all their money, or doubling their money.
But they only win if they have less money than the other person. They have a 50% chance of winning more than they are risking.

5. Originally Posted by Scifor Refugee
Originally Posted by Neutrino
I'm not sure why you claim that the players stand to win more than they could lose. They have a 50/50 chance of either losing all their money, or doubling their money.
But they only win if they have less money than the other person. They have a 50% chance of winning more than they are risking.
Now wait a minute you said they would win all the other person's money or did you only mean the money they have in their wallet. If you really mean all their money and not just what is in their wallet then since you said that they were equally rich then the expectation is not positive but zero as Neutrino said.

So I would assume you meant only the money in the wallet, but I do not see any paradox here even if you were right but you are not because the expectation is still zero. They each have an equal chance of winning or losing the larger amount. It is not as if you would only win the larger amount and only lose the smaller amount, which it seems your puzzle is trying to trick people into thinking. Whether you win or lose it is the larger amount you win or lose.

6. Originally Posted by mitchellmckain
Now wait a minute you said they would win all the other person's money or did you only mean the money they have in their wallet.
I was wondering this at first too, but when I re-read the OP and see

Originally Posted by Scifor Refugee
Whoever has the least money wins all of the other person's money
it seems to clearly be saying that they win ALL of the other person's money.

But I agree, no matter what the case is, I don't see any kind of paradox. If they win what's in the other guy's wallet, then they have a 50/50 shot of winning or losing a random amount of money, versus a set amount of money. They still could lose ALL their money, or double their money in both cases.

7. and if there both equaly rich why would they care about losing?
but thats not the point as statisticly there is realy just a 50/50 chance of winning or losing.

8. i think you have to put everything into perspective - since they both have an equal advantage over one another then they're back to a 50-50 state. It's like in a video game, you and your enemy both have weak guns that would with which you'd need to fire a lot of bullets to kill your enemy. Now, if you're both given super strong one hit killing guns, then you'll both an advantage over the other, but since it's still evenly balanced, there's still only a 50-50 chance of one winning.

9. This thread reminds me of a conversation I had with a Kenyan, who was defending lion hunting on the grounds that the lion had a 50-50 chance.
"Either you hit it, or you miss it. That's 50-50."

10. hi

that original poster has a good point...

if you were told that you had to risk / stake X amount, if you lose, you lose X, if you win you will always get an amount >2X
The outcome of the contest is decided on a 50/50 chance.... then people should always take the bet as they are risking X amount in order to win an amount >X (>2X in total)

For example, if someone offered you odds of greater than 2:1 on a fair coin toss and was prepared to play for as long as you wanted that would be a great money spinner.

In the original posters scenario the odds also appear to be in your favour as you are risking X to win an amount greater than X or lose X..

In the coin toss scenario the person offering greater than 2:1 odds will definitely lose money over time, in the original posters scenario the odds would appear to be even... yet it is only the nature of the contest that is different, the apparent odds of greater than 2:1 are in both games.... what I mean is that the bet seems good until you hear about how it is to be decided...

(the problem is that the amount of money present in the wallet isn't a good way of deciding a winner as people quickly learn the correct strategy, have no money in your wallet, at best you win, at worst you draw.... there doesn't seem to be a way around this and so the game can't be played for many rounds)

Cheers

11. The first person would probably reason that he should play, since he stands to wins more than he could lose; any time you have a 50% chance of winning a bet that pays off more than you could lose, it's a statistically good bet.

The problem is, the same reasoning applies to both people. It would seem that each one has the advantage over the other!
I'm afraid I don't get the paradox. The fact that the same argument for how likely they are to win applies to both participants is surely the incentive for them to play, isn't it? Neither of them is playing a loaded game, otherwise the guy who is loaded against has the least incentive. There is no "first" person or "second" person in this game.

The only paradox is in the application of that rule as to whether one should participate or not to a straight no-House, two people against each other bet! It's only a rule-of-thumb after all, and doesn't in itself guarantee an outcome.

This is why it's called "gambling", after all.

Also, I think you're getting what you think of as a paradox, because the thing being used to determine the outcome is the money that they have, which one of them incidentally is going to win. But the money as determinant is nothing more than that. They're not really staking the money in their wallet at the point that they throw it down on the table. They might as well throw a dice, if the outcome is that one of them gets the contents of the other's wallet.

12. Originally Posted by Scifor Refugee
Originally Posted by Neutrino
I'm not sure why you claim that the players stand to win more than they could lose. They have a 50/50 chance of either losing all their money, or doubling their money.
But they only win if they have less money than the other person. They have a 50% chance of winning more than they are risking.
well, it is still a 50/50 bet. the amount you win doesnt change a thing. if i flip a coin, tails i give up \$1000, heads i get \$2000, it is still 50 percent i get tails.

13. Wow, I started this thread a long time ago and promptly forgot all about it. I’ll try to address a few points that people have raised here, and explain why it appears to be a paradox to me.

In trying to decide if a game of chance is statistically favorable or not one must look at the odds of winning and the potential payout. If the odds of winning x payout is greater than .5, it is statistically favorable to play. For example, in a game where there is a 50% chance of winning it will always be in your favor to play if the payoff is better than 1:1. If you win 50% of the time and win more than you have to bet, you will always make money in the long run if you play the game many times.

In my original post I described a situation with two specific conditions:

1. The odds of winning are 50%
2. If the player wins, he will win more money than he had to risk.

Therefore, it appears that it will always be in the player’s favor to play, just as it would always be in your favor to play a game where you lose \$1 if a coin lands “heads” but win more than \$1 if a coin lands “tails”.

The paradox here is that this reasoning applies equally to both players. It’s impossible for both players to simultaneously have an advantage over the other. To put it another way, if two people played the game a large number of times it would be expected that they would both come out ahead, which is impossible.

(the problem is that the amount of money present in the wallet isn't a good way of deciding a winner as people quickly learn the correct strategy, have no money in your wallet, at best you win, at worst you draw.... there doesn't seem to be a way around this and so the game can't be played for many rounds)
It's assumed that the amount of money in each wallet is unknown and random. If one player knows that he only has a small amount of money, obviously the paradox doesn't apply.

They might as well throw a dice, if the outcome is that one of them gets the contents of the other's wallet.
No, it's critical that the winner is decided by seeing who has the least amount of money. If you simply determined the winner by by rolling a dice or something, then both players would simply have a 50% chance of winning or losing a random amount of money. This goes to point 2 above; if you win, you will win more than you initially bet.

14. I think the paradox lies with the idea that you assume each player has an expected value for the difference in there wallet contents of zero (whichs gives the 50-50 split) and then argue that they thus have an edge since the player with the least amount of money wins. If on average they have the same amount of money in their wallets then neither one has an advantage in the first place.

Does that make sense?

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