Suppose there is a single player card game in which there is a deck of $n$ cards. A card is either red, green or blank. Initially, the deck consists of $r_0$ red, $g_0$ green and $b_0$ blank cards such that $r_0+g_0+b_0 = n$. All cards initially are put into a draw pile. If variable numbers are difficult, we may assume $n=20,r_0=9,g_0=1,b_0=10$.
Each turn, the player draws cards from his draw pile until the number of drawn red cards is equal to the number of drawn green cards plus 3 (or until there is nothing left to draw). For instance, he may draw
BGRBRRBBR. Each drawn card, regardless of its color, gains 1 point. The goal of the game is to score many points.
After each turn, the drawn cards are put to the side into a secondary pile. Whenever the draw pile is exhausted but the secondary pile is not empty, the secondary pile gets shuffled and put into the draw pile, such that drawing may continue. In this fashion, the deck gets recycled deterministically. (This is also how it works in the card game Dominion.)
Just before putting the drawn cards in the secondary pile, the player may, from the cards he had drawn, either pick a red card and color it blank, or a blank card and color it green. This means the cards slowly improve over time, and it may eventually be possible to draw the whole deck in single turn if enough green (or few enough red) cards exist.
The game ends after $n$ turns. The only choice the player has is his preference in coloring reds vs. blanks. I wonder about three questions:
- Which strategy should he follow to maximize the score?
- Do different strategies offer different variance?
- Do they depend on the initial color distribution?