There is a betting game, which seems to have "positive sum" situations, and I'd like to find out the optimal strategy.
Multiple players place bets against the bank, for and bets that fit winning criteria earn players points of the game. There are two sets of criteria: The first: bet 1 point with 0.475 chance of doubling it or 0.525 – losing it. This is obviously a negative sum game. The second part is the "Jackpot": you can add a fixed amount: 2pt, to your prior bet, to have the bet included in roll for the jackpot. The chance for winning the jackpot is 0.0001.
It's all about how big the jackpot is.
All points you lose, be it to the 0.525 chance of loss in the first part, or the 0.999 chance of loss in the second part, are accumulating in the jackpot. Additionally, if the amount in the jackpot is less than 10,000, the house covers the difference between the amount and 10,000 (you can't win less than 10,000 if you hit the 0.0001 chance). When a winning bet for the jackpot is placed, the jackpot is emptied, and the process of collecting begins anew.
When always playing the first part as bet 1, with house edge of 5%, and jackpot bet cost of 2, the expected value for a bet is (-2.05 + (jackpot × 0.0001)) which means it becomes a positive sum game when jackpot value exceeds 20,500. When that happens I should keep placing bets until I win the jackpot.
But I don't quite grasp my chances for jackpot in the two lower ranges: 0–10,000 and 10,000–20,500. If it was a simple case of a fixed-amount jackpot, that would be plainly a negative sum game, but considering all my bets increase the jackpot value, I'm not so sure.
Also, does the fact that the jackpot is shared between all players (their losses add to it, one takes it all) change the optimal strategy?
