What is an optimal payoff strategy with an unknown set of rewards?Life is like a box of chocolates. Inside each one of these candies is a cash prize. At a party you win the door prize, a generous millionaire allows you to choose any candy from a box and examine the reward amount inside the first candy you choose. The first candy that you select contains a prize of $ 100.00. You don't know if this is the largest prize or the smallest.
The millionaire encourages you to continue. You can give up the cash of $ 100.00 for another opportunity to select a candy. You can quit and keep the cash amount at any time, or you can give up the prize amount hoping for a bigger prize in the next candy you select. You can continue to the last piece of candy in the box of 50 candies if you wish.
What is the optimal strategy to determine when you should take the cash prize and quit selecting candy assuming you want to optimize your payout.
Can this be solved mathematically without making an assumption about the cash reward payout distribution?
LegFuJohnson
Math means numbers.
How can I even pretend to solve the equation without any numbers?
Of course you need to make some assumptions about the prizes. You have to make an assumption just to decide to continue once, rather than take the $ 100, don't you?
pdq
You posted this in the "Gambling" section. I actually think there may be a logical/mathematical solution, but I think you should re-post this question in the mathematics section to see what they come up with.
I love puzzles like this. Here are just some random thoughts about this, merely based on hunches:
* I believe you'll have to at least go through 25% (12 or 13) candies before you can consider stopping. This way you will at least get some sort of range.
* I also believe you MUST make some assumptions to logically solve this puzzle. One reasonable assumption should be a cap on the top amount. You said this is a generous "millionaire", so you should assume that the top cash amount certainly shouldn't be more than a million dollars, and likely less than that. I believe that once you've gone through 25% of the candies as I have suggested that you'll need to make a logical assumption about the range of potential dollar amounts.
That's all I've got so far. Post this in the mathematics section and see what they come up with. My only other guess is that you need to go through 33.3% of the candies. (About 17 of them.) That would still leave you with twice as many candies still to come. I believe there can be a logical and/or mathematical "optimal strategy" to this puzzle. I just don't know what it is.
Vernon Howell
You need to make some assumptions about the potential payoff to proceed.
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