A game of chance

[phk]

In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed  repeatedly until a tail first appears, ending the game.

The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends.

Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc.

T - $1
H - $2
HHT - $4
HHHT -$8
HHHHT - $16

In short, you win 2^(k−1)  dollars if the coin is tossed k times until the first tail appears.

What would be a fair price to pay for entering the game?

[Verily]

"Fair", according to the rules of rationality? An arbitrarily large amount of money. Come now, you can't assume that no one knows the St. Petersburg Game. And the so-called "solutions" to the St Petersburg Game are frankly awful and fail at logic (or else make assumptions that take us outside the realm of strict rationality, which might be practical but do nothing but suggest that pure rational choice is poorly suited for real-world decisions).

[Franzl]

Well let's just say this:

p (T) = 1/2                     1/2 * $1 = $0.50
p (HT)= 1/4                    1/4 * $2 = $0.50
p (HHT) = 1/8                 1/8 * $4 = $0.50
p (HHHT) = 1/16             1/16 * $8 = $0.50

and etc.

So in theory, the amount you win is infinitely high. The propability for each individual event keeps getting smaller...but it'll never reach 0.

That said...as Verily already stated, it's absurd. There's a 50% chance you only get $1.

You would have to play the game an infinite number of times for the theory to work. So there is no "fair" price really.

[True Federalist (진정한 연방 주의자)]

V(G) = value of the game = ½V(T)+½V(H)
V(T) = value of tails = 1
V(H) = value of heads = 2V(G)

so

V(G) = 1 + V(G), hence V(G) must be ∞ or ±∞ depending on whether you prefer one infinity or two.

[ag]

"Fair", according to the rules of rationality? An arbitrarily large amount of money.

"Rationality", as understood by economists, says exactly nothing about this. You may be rational and not be willing to pay more even a penny for this though if you like money, you should be willing to pay at least $1 (but even that's above and beyond rationality - this is just first order stochastic dominance argument). The expected monetary value of this lottery is, of course, infinity, but this has nothing - exactly, nothing - to do w/ rationality. Most rational people, in my experience, would not pay more than $10 for this. Again, as economists understand it, there is nothing "irrational" about that.

Just in case: all rationality means (for economists) is that an individual is maximizing a complete and transitive preference relation over the set of lotteries. A violation of rationality would be, after paying 10 dollars for this lottery, he exchanges 10 dollars for 75 apples and then exchanges the 75 apples for the initial lottery. There is nothing, really nothing, about maximizing expected monetary values, in the definition of rationality.

[ag]

What would be a fair price to pay for entering the game?

For me? 5 bucks. May be 10. If you mean by "fair", what is the expected monetary payoff, it is, of course, infinity. But that has nothing whatsoever to do w/ the value of this lottery to a sentient being