>>You pay me a certain amount of money at the beginning. I flip a coin. If "tail" appears, the game ends - you get nothing in exchange for you initial bet. If "head" appears, I pay you $1, and the game continues. If "head" appears for the second time,I pay you $2. The third time, $4, the fourth time $8, etc. - but as soon as "tail" appears, the game is over.
>>
>>How much would you be willing to pay to play this game?
>>
>>Hilmar.
>
>I think I hadn't made it clear that the payments are additive. For instance, if "head" appears thrice, you get $1 + $2 + $4 = $7.
>
>Well, since nobody responded in many days anyway, I will give an analysis. The probability of getting "head" the first time is 1/2; making a value of 1/2 * $0.50. For the second "head", the probability is 1/4, the total value is 1/4 * $2 = $0.50. For the third head, the total value is 1/8 * $4 = $0.50.
>
>The total value of the game, in dollars, is 1/2 + 1/2 + 1/2 ... that is, infinite.
>
>Of course, playing the game a single time, you will get a finite amount of money. It is only the average that is infinite. Strange, huh?
>
>Hilmar.
I'm not sure the analysis is totally correct, but agree that the possible winnings is theorically infinite, but gets there less rapidly - my analysis is

50% chance losing everything
25% chance winning $1
12.5% chance winning $3
6.25% chance winning $7
...

Betting $1 would give me a 50% chance of winning at least $1, so in the long run I'm likely to profit. 50% of the time I lose, 25% of the time break even, 25% of the time profit by at least $2, so the odds are in my favour.

If I bet $3, I would only have a 25% chance of winning at least $3, 50% of the time I would lose $3, 25% of the time I would lose $2, 12.5% of the time I would break even & only 12.5% of the time would I be in profit. At this level the odds are very much in your favour.

Looking at it from that point of view, I would ask you - what is the minimum bet you would accept from a person before you agreed to play your game.