i was thinking about the Monty Hall problem lately and thought about a different game that is a variation of the original:
There are $100$ doors. behind one of them there's a "$1$", behind all the rest there's a "$0$". you begin the game with $100$ points, each of which awards you at the end of the game $\$1,000$.
You choose a door. the game goes in rounds. each round, the host reduce $1$ point and opens one door with a "$0$" behind it. you then have three options:
stay with the door you previously chose.
switch a door
end the game
once you decide to end the game, you open the last door you chose. if it is a "$1$" you get $\$1,000$ for every point you have remaining ($100$-the amount of rounds it took you to end the game). if it's a "$0$" you get nothing. your goal is obviously to get as mouch money as posible. the hosts goal is for you to get as little points as posible. my questions for you are:
what is the winning strategy (if any) for the player?
what is the winning strategy (if any) for the host?
will and how will those strategies change if instead of 100 it was N doors and points
how will the strategies change if upon opening a door with "$0$" you will get $x$ money instead of nothing?