I have an interesting problem that I'm trying to resolve for a game idea but have no idea how to find a solution. The scenario is you're a player offered one of N prizes that have a weighted bias chance of being offered, for example:
| Prize | Weight | % Offer chance |
|---|---|---|
| £1 | 500 | %50 |
| £2 | 400 | %40 |
| £10 | 50 | %5 |
| £50 | 40 | %4 |
| £150 | 10 | %1 |
| 1000 | %100 |
The player can either choose to take the reward or reject it and take a chance on getting a better offer (at which point the previous offer is discarded from the table and can't be offered again). The game will make a limited number of offers to the player and the player has a limited number of awards they can choose to take. Given this scenario I'm attempting to work out the optimal value at which the player should accept any given offer to try and maximise the amount they can win.
I've done some basic calculations and I think I can work out the average value that would be offered to the player by summing up their "weighted worth":
| Prize | Weight | % Offer chance | Weighted Worth |
|---|---|---|---|
| £1 | 500 | %50 | 50p |
| £2 | 400 | %40 | 80p |
| £10 | 50 | %5 | 50p |
| £50 | 40 | %4 | £2 |
| £150 | 10 | %1 | £1.50 |
| 1000 | %100 | £5.30 |
My first solution was that the player should simply accept any offer given that is above the average value offered but I'm unsure if I've calculated that correctly or whether that solution is even correct. Any pointers in the right direction and/or which areas of probability I should study up on to get the correct solution would be greatly appreciated.
Thanks in advance.