In this game, you have an infinite deck of cards. Each time you draw a card it's value is a uniformly distributed integer on the interval [0,C]. The game lasts for R rounds. You start the game by drawing a card and adding its value to your running total. At each round you have two choices: 1) draw another card from the deck and add its value to your total 2) add the value of the highest card previously drawn to your total What strategy, based on the constraints R and C, gives you the optimal total at the end of the R rounds?

(In reply to Long analysis of a simple case by Steve Herman)

You are on the right track, however there is in fact a closed form equation for the decision threshold.  Also, when I pointed out the dependence on R in this equation, it is because I count the rounds x, from the beginning and thus use R-x for the rounds remaining.  If instead, as you are, you count x as the rounds remaining then R is not needed in the equation.  There is a clever trick which allows you to solve this using 2 summations to form a quadratic inequality based on current maximum card m.  If the solution is not found in the next day or so I will post a hint in the form of the theorem which allows one to easily solve problems like this.

Posted by Daniel on 2010-09-08 20:32:53