"St. Petersburg paradox"-like paradox which feels even less intuitive

Question

The original St. Petersburg paradox presents a game whose EV is infinite but no reasonable person would pay large amounts of money to play.

Here's a new version: Alice and Bob have a weighted coin which lands heads 80% of the time. They play a game as follows: A "round" is played by each player putting in an equal amount of money into the pot, and flipping the coin. Alice takes the money in the pot if it's heads, and Bob takes it if it's tails. However, Bob always gets to decide what the wager is, and a round is only played if both players are happy with the wager.

Bob declares that he will start the wager at \$1, and every time he loses, he will double the wager. If he ever wins, he will refuse to play any longer, walking away with \$1.

How many games should Alice want to play before walking away?

The EV-based answer is infinity, but obviously if Alice keeps playing until Bob wins, she will lose money. This situation seems more real than the original paradox (although it still relies on a player having an infinite bankroll), and (at least to me) is much more tempting to play than the original paradox.

My question is: how do the "resolutions" of the original paradox transfer over to this formulation, and is there a reasonable framework in which Alice can ask herself the question of "how many times should I play?"

Answer 1

There is a Kelly criterion type-solution: Alice should play when her current wealth including winnings so far is more than $1.0780950776679$ times the value of the next bet, and walk away otherwise.

That number is the solution (real and greater than $1$) to $x=(x+1)^{0.8}(x-1)^{0.2}$. It can be extended to other probabilities between $0.5$ and $1$.

If you know her starting wealth, you can calculate the maximum number of rounds she is prepared to play. For example if her starting wealth was $2$, then you could use this table

Wealth  Next Bet    Ratio
  2       1           2
  3       2           1.5
  5       4           1.25
  9       8           1.125
 17      16           1.0625

to say that she should walk away after winning four rounds with a gain of $15$. It would give her a probability of $0.4096$ of multiplying her starting wealth by a factor of $8.5$ and a probability of $0.5904$ of halving her starting wealth, which does not seem unreasonable for her. Bob might even be able to afford to play this version of the game.