Let's play a game.

[Timminz]

On the one hand, I agree with john, in that the expected value of the game (using weighted averages) is infinite. On the other hand, I would not pay an infinite amount to play that game only once. At the moment I can't give a rational reason why, but practically, you're most likely to win only $2, so paying a large sum seems like a bad bet.

[Timminz]

i just thought of a modified version of the problem:

suppose that, for each failed coin toss, you had to split your winnings with someone. for example, if you flipped tails twice, and then flipped heads the third time, you would split your winnings with two other people.

how much money would a rational person pay to play this game, assuming that they are only seeking to maximize their own winnings?

That would depend on what the initial prize was. I suspect paying much more than half the initial prize would be a bad idea.

[Haggis_McMutton]

if i HAD to pay a fee for this game, i would pay one cent, as that is the smallest common non-zero amount of currency which can qualify as a "fee"

BBS beat you to that one both chronologically and numerically. (He wants to pay a fee of minus infinity)

seriously tho...

the average outcome of this game would be:

.5*2 = 1
+.25*4 = 1
+.125*8 = 1... etc.

i'm pretty sure that the most rational person would pay any amount of money to play this game because the weighted mean average of all results is infinity.

DING DING DING

Yes, that's the weirdness I was refering to. Instinctively, this game would be worth prolly less than 10 bucks, but the estimated payoff is infinity.

Of course BBS is right, time becomes a factor. In order to get infinity dollars we'd have to stay there for infinity coin tosses, which seems infeasible.
Calculating the outcome by devaluing the result based on how long it would take to flip X coins would be interesting.

e.g. if 1,000,000 people each paid $1000 to play this game, they would probably end up with more money than they started with (as a group)

that is, they would probably have over $1,000*1,000,000 = 1 billion dollars combined at the end of the games.

maybe i'll write a program to test it

Go for it.
You might be able to do the math directly, but I'm too lazy atm.

Now, here's the weirder part

One might try to take the BBS approach and say "bah! humbug! devaluation and the other time related costs make this moot."

However, devaluation really isn't the limiting factor of this scenario. We can do the coin flips on a computer, let's be generous and say a coin toss takes 1 ms to simulate. This means in 10 seconds we could simulate 10,000 coin tosses. 10 seconds is a period short enough that all that fancy stuff BBS posted can be reduced away and yet the estimated payoff for 10,000 coin tosses is still $10,000.

So, apparently, a rational person should be willing to pay $9,000 for the above game and have an expected profit of $1,000. But this seems ridiculous to our common sense.

I dunno, but the above is pretty weird for me.

[Haggis_McMutton]

i just thought of a modified version of the problem:

suppose that, for each failed coin toss, you had to split your winnings with someone. for example, if you flipped tails twice, and then flipped heads the third time, you would split your winnings with two other people.

how much money would a rational person pay to play this game, assuming that they are only seeking to maximize their own winnings?

So the new expected value would be:
Sum from x=1 to x=infinity ( (1/2^x) * (2^x / x) ) =
Sum from x=1 to x=infinity (1/x)
Which is the harmonic series and still sums up to infinity.

So you still have infinite payoff. Infinity is weird.

[BigBallinStalin]

i just thought of a modified version of the problem:

suppose that, for each failed coin toss, you had to split your winnings with someone. for example, if you flipped tails twice, and then flipped heads the third time, you would split your winnings with two other people.

how much money would a rational person pay to play this game, assuming that they are only seeking to maximize their own winnings?

So the new expected value would be:
Sum from x=1 to x=infinity ( (1/2^x) * (2^x / x) ) =
Sum from x=1 to x=infinity (1/x)
Which is the harmonic series and still sums up to infinity.

So you still have infinite payoff. Infinity is weird.

You're weird.

I have a box that gives you infinite cash money at the push of a button. How much are you willing to pay for it?


Which bidders would value the capital (i.e. money) itself, and which bidders would value the consequences (e.g. causing/avoiding hyper inflation)?

[Haggis_McMutton]

You're weird.

I have a box that gives you infinite cash money at the push of a button. How much are you willing to pay for it?


Which bidders would value the capital (i.e. money) itself, and which bidders would value the consequences (e.g. causing/avoiding hyper inflation)?

I'd pay you infinity dollars assuming you'd lend me infinity dollars(which you should do since you'd know I would soon be able to pay your infinity dollars back at any rate of interest with my infinity dollars)

As long as only you and me had infinity dollars I would be relatively confident that we'd both be able to keep our spending at a level that wouldn't cause hyperinflation.

:p

Btw. the thing in the OP is the St. Petersburg paradox

[Army of GOD]

I don't understand the point of the thought experiment. I would pay AS LITTLE as possible (like john said, 1 cent).

I don't feel like doing the math, but the upper limit as to how much I would pay to do this game would probably be 5ish dollars.

[BigBallinStalin]

You're weird.

I have a box that gives you infinite cash money at the push of a button. How much are you willing to pay for it?


Which bidders would value the capital (i.e. money) itself, and which bidders would value the consequences (e.g. causing/avoiding hyper inflation)?

I'd pay you infinity dollars assuming you'd lend me infinity dollars(which you should do since you'd know I would soon be able to pay your infinity dollars back at any rate of interest with my infinity dollars)

As long as only you and me had infinity dollars I would be relatively confident that we'd both be able to keep our spending at a level that wouldn't cause hyperinflation.

:p

Btw. the thing in the OP is the St. Petersburg paradox

Oh, that's from the 1700s. Those smelly people don't even know what money is!

[john9blue]

this reminds me of (what i will call) the ninja problem...

There are 5 rational ninjas, A, B, C, D and E. They find 100 silver coins. They must decide how to distribute them.

The ninjas have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The ninja world's rules of distribution are thus: that the most senior ninja should propose a distribution of coins. The ninjas, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority or a tie vote, it happens. If not, the proposer is killed, and the next most senior ninja makes a new proposal to begin the system again.

Ninjas base their decisions on three factors. First of all, each ninja wants to survive. Second, given survival, each ninja wants to maximize the number of silver coins he receives. Third, each ninja would prefer to kill another ninja, if all other results would otherwise be equal. The ninjas do not trust each other, and will neither make nor honor any promises between ninjas apart from the main proposal.

what distribution should ninja A propose?

[Army of GOD]

Ninjas would never do that. They would just assassinate each other and whoever comes out as the victor gets it all.

[Haggis_McMutton]

this reminds me of (what i will call) the ninja problem...

There are 5 rational ninjas, A, B, C, D and E. They find 100 silver coins. They must decide how to distribute them.

The ninjas have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The ninja world's rules of distribution are thus: that the most senior ninja should propose a distribution of coins. The ninjas, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority or a tie vote, it happens. If not, the proposer is killed, and the next most senior ninja makes a new proposal to begin the system again.

Ninjas base their decisions on three factors. First of all, each ninja wants to survive. Second, given survival, each ninja wants to maximize the number of silver coins he receives. Third, each ninja would prefer to kill another ninja, if all other results would otherwise be equal. The ninjas do not trust each other, and will neither make nor honor any promises between ninjas apart from the main proposal.

what distribution should ninja A propose?

show

[Haggis_McMutton]

I don't understand the point of the thought experiment. I would pay AS LITTLE as possible (like john said, 1 cent).

I don't feel like doing the math, but the upper limit as to how much I would pay to do this game would probably be 5ish dollars.

Yeah, the upper limit is the question. guess I phrased it poorly.

Also, The expected value you'll win from the contest is infinity, so you should be willing to pay any finite sum of money to play. If you don't it must mean you suck at math.

[john9blue]

Also, The expected value you'll win from the contest is infinity, so you should be willing to pay any finite sum of money to play. If you don't it must mean you suck at math.

harsh...

[Haggis_McMutton]

Also, The expected value you'll win from the contest is infinity, so you should be willing to pay any finite sum of money to play. If you don't it must mean you suck at math.

harsh...

shh, I'm trying to bait AoG into doing some actual calculations.

[Army of GOD]

I don't understand the point of the thought experiment. I would pay AS LITTLE as possible (like john said, 1 cent).

I don't feel like doing the math, but the upper limit as to how much I would pay to do this game would probably be 5ish dollars.

Yeah, the upper limit is the question. guess I phrased it poorly.

Also, The expected value you'll win from the contest is infinity, so you should be willing to pay any finite sum of money to play. If you don't it must mean you suck at math.

Is it "pay once and play an infinite amount of times" or pay per game?

[Haggis_McMutton]

Is it "pay once and play an infinite amount of times" or pay per game?

per game.

[rdsrds2120]

seriously tho...

the average outcome of this game would be:

.5*2 = 1
+.25*4 = 1
+.125*8 = 1... etc.

i'm pretty sure that the most rational person would pay any amount of money to play this game because the weighted mean average of all results is infinity.

e.g. if 1,000,000 people each paid $1000 to play this game, they would probably end up with more money than they started with (as a group)

that is, they would probably have over $1,000*1,000,000 = 1 billion dollars combined at the end of the games.

maybe i'll write a program to test it

That's true on a flip by flip basis, but if I'm understanding the problem correctly, then it's inaccurate.

T = $2
TT = $2 + $4 = $6
TTT = $2 + $4 + $8 = $16

etc.

So, our distribution doesn't converge at any limit and goes to infinite still, so I guess this was just a weird moment for a stats reminder (this plot is growing faster than just 1+1+1+1+1+1....it's growing at 2/2 + 6/4 + 16/8 + 32/16 + 64/32 + 192/64....)

BMO

[fadedpsychosis]

after a lot of math i have come to the conclusion...





that we're all a bunch of cheapskates

[fadedpsychosis]

ok, so I read the st. petersburg thing... one thing the statistics forget is REALITY. for EACH throw, yes there's a chance of infinite wealth... but you also have an infinitely small chance of making infinite wealth. as I said earlier, it's exponential gain vs exponential decline in chance... just because an infinite series adds up to 1, as BBS said it would take infinite time... lets take the number the page used (going back to the original unlimited number set here): $25. in order to at least break even, you would have to have 5 heads before you got tails (giving you 32). so what are the odds of that happening? a little over 3%. that's a 3% chance of coming out a little better than breaking even on one throw.

[rdsrds2120]

ok, so I read the st. petersburg thing... one thing the statistics forget is REALITY. for EACH throw, yes there's a chance of infinite wealth... but you also have an infinitely small chance of making infinite wealth. as I said earlier, it's exponential gain vs exponential decline in chance... just because an infinite series adds up to 1, as BBS said it would take infinite time... lets take the number the page used (going back to the original unlimited number set here): $25. in order to at least break even, you would have to have 5 heads before you got tails (giving you 32). so what are the odds of that happening? a little over 3%. that's a 3% chance of coming out a little better than breaking even on one throw.

For cases like these, there's usually a set alpha value to prevent this sort of thing from happening. However, this is not a normal distribution, so it doesn't work.

A similar value to an alpha value in this exact scenario would be a cap of coin tosses. I'd easily pay 3 dollars to play it until I incurred a profit, but the bigger thing to notice here is that no one will ever design a game like this because it causes them to suffer a greater loss than what is gained, so there's no incentive for this game to ever exist.

BMO5

[john9blue]

the problem is that you guys are primarily thinking about the probability of coming out ahead, instead of considering the amount of money that you could come out ahead by.

[Army of GOD]

If I cared I'd try to write a program but, alas, this is me not caring.

[Timminz]

Both of you should stop being so lazy and just write the ridiculously simple program. Run it a million times and report back with the total winnings.

[fadedpsychosis]

the problem is that you guys are primarily thinking about the probability of coming out ahead, instead of considering the amount of money that you could come out ahead by.

once again, reality. in any game that I pay money for, I have a limited amount of starting money. doesn't do me any good to have spent all my money in a game that I could have made a lot of money but didn't... which is the primary reason I very rarely gamble at all

[Army of GOD]

Both of you should stop being so lazy and just write the ridiculously simple program. Run it a million times and report back with the total winnings.

DONT TELL ME HOW TO LIVE MY LIFE