How much would you pay for this game?

[blackaces13]

As long as the $100 won't break you or make you miserable for a sustained period of time then $100.01 should be fine. You're throwing away a profitable (albeit very marginally) situation if you pass it up and there's really no reason to ever do that unless the price of losing will dramatically affect you.

Its interesting to note that everyone playing BJ in a casino or craps at a high roller table is basically taking this same proposition but instead accepting only $99 if they win. So the fact that everyone here would only say a positive value for x is clearly not the case for anyone who has ever played craps or BJ at a $100 minimum table. Well, unless they count cards I suppose.

[me454555]

Since the payoff is independent of how much money you invest, I guess the real question is what is the maximum amount of money you pay for this game where you can still expect a positive return on your money.

[me454555]

The answer to this question is based solely on your bankroll. You can pay up to ((2^n)*n)/((2^n)-1) $ per play provided you have the bankroll to play this game ((2^n)-1) times.

This is because when you play this game 255 times you can expect the coin to land on tails 7 times in a row before it lands on heads once in the 255 times you play. This leads to a payout of $256.

So the most I would pay for this game depends on how many times I am able to play it. For me to pay $5/play, I would have to be able to play it at least 31 times ((2^5)-1). When the price is increased to $10/play I'd have to play 1023 ((2^10)-1) times to expect to make money.

I guess the relation to poker is that even though this is a +EV game, if you don't have a big enough bankroll, you may not be able to play it enough to get into that +EV. If you only have a bankroll to play 63 times at $10/play there is a good chance you will loose money b/c the highest payout you can reasonably expect is $64. To play the game in the example above and expect to make money, you would need a bankroll of $10230.

[pzhon]

[ QUOTE ]
(Petersburg Paradox)
You get $2^N if it comes up heads on the N'th flip.

What do you pay to play this game?

Why do I ask? Again, I'm not sure about my point, but it revolves around bankroll requirements to play poker.

[/ QUOTE ]

It's a good point that this is a bankroll issue.

Here are some observations:

[img]/images/graemlins/spade.gif[/img]The median result of n repetitions grows not like a multiple of n, but like a multiple of n log n. This is true of other fixed percentiles, too.

[img]/images/graemlins/diamond.gif[/img]The Kelly Criterion is to maximize the expected logarithm of your bankroll. If your starting bankroll is $10,000, the Kelly Criterion recommends walking away if the cost of playing is more than $14.23 (if I calculate correctly). This is not sensitive to whether you actually get paid after 30 consecutive tails or not. If your bankroll is $1,000,000, then you should be willing to pay $20.87.

[img]/images/graemlins/club.gif[/img]Suppose your payoff were (-3)^n, i.e., you might lose 3 dollars, or gain 9, or lose 27. It is not clear whether the game favors you or not.

[BigBiceps]

If I were only playing the game once I would pay $4.

If I were bidding to be the one to get to play the game as many times as I wanted, I would bid $16. I picked $16, by taking the assumption that a 1 in 100,000 event would never happen, so 2^n=100,000 gives a price of $16 to make it worth it.

For the question on the flop in hold-em, I would probably want $110 for my trouble to win $100. ($10 extra, to play the game once.) --10%

Of course as the price($ risked) goes up, I would want a higher amount for example $140,000 if I were risking $100,000. --40%

[Punker]

You people are not gamblers. My x=0.

[jwvdcw]

[ QUOTE ]
As long as the $100 won't break you or make you miserable for a sustained period of time then $100.01 should be fine. You're throwing away a profitable (albeit very marginally) situation if you pass it up and there's really no reason to ever do that unless the price of losing will dramatically affect you.

Its interesting to note that everyone playing BJ in a casino or craps at a high roller table is basically taking this same proposition but instead accepting only $99 if they win. So the fact that everyone here would only say a positive value for x is clearly not the case for anyone who has ever played craps or BJ at a $100 minimum table. Well, unless they count cards I suppose.

[/ QUOTE ]

I think the free drinks and fun factor of gambling have something to do with that, though.