# Describing a betting game involving multiple steps

I am trying to figure out how to describe a betting game that involves multiple steps where at each step the probability of winning decreases but the payout increases. The player must start at step 1, and only wins a prize at step 4, 5, 6, or 7. Each step costs a set amount of money to play. The player cannot go backwards until step 4, 5, and 6 where the player returns to step 1 if he losses.

In addition, the player has the option to pay for increase chances of success for a set cost. At any step he can chose single step bonus (increase chance to move up one step), basic double step bonus (allow for possibility to move 2 steps), advanced double step bonus (increase chance to move up with possibility to move 2 steps).

How can I mathematically describe this problem so that I can determine:

1. How much should each step cost to play if I want a positive expected value?
2. How much should the payouts be for steps 4, 5, 6, and 7?
3. How much should the player pay for increased odds of winning at each step?

If nothing else, I need to know the type of problem this is so I can figure it out on my own.

• What happens if the player loses at steps 2 or 3? Do they remain on their current step with the chance to replay it? Also, by default, do you advance a step when you win? (It seems like this is implied but you never stated it explicitly.) Are there steps beyond 7? If not, what happens if you win at step 7? (E.g. is the game over? Does the game continue and you get to keep playing step 7 until you lose?) Also, when you say "positive expected value", that assumes what strategy? Because you're allowed to pay for benefits, there isn't a single way to approach the game. Jan 9, 2018 at 20:36
• So if the player loses before step 4 they stay at their current step. You advance a step (or two if you pay for the extra chance) when you win. You can cash out any time after step 3, but there is no more advancement after step 7. Basically I just need to understand how to even describe this game in math terms. Jan 10, 2018 at 7:34

You can use MDP (Markov Decision Process), but as is, it won't give you the answers you want.

BUT, as far as I see it, the expected value of a step does not depend on the lower steps. I think that that can help you tremendously.

• If you'd like me to try and give a more specific model then ask and I'll try to elaborate Nov 7, 2018 at 13:07
• Thanks. I've actually put this project on the back burner, but maybe others might like an example. Appreciate the response either way! Nov 28, 2018 at 19:26