Redraw a card rank for three dollars

Question

You draw a card and are paid its rank in dollars, where Ace counts as $1$ , the numbers count as themselves, and Jack, Queen, King count as $11, 12, 13$ . Each rank is equally likely. You may pay $3 to redraw (discarding the current card), as often as you like, keeping whatever you finally stop on.

Find the optimal stopping rule and the fair value of the game.

Accepted Answer

Let V be the value of a fresh draw. A redraw costs \3 and returns you to a fresh draw worth V, so it is worth V - 3. You keep a rank x when x \ge V - 3. Guess you keep 5 through 13 and redraw 1, 2, 3, 4, then verify. The kept ranks sum to 5+6+\cdots+13 = 81, and four ranks trigger a redraw: V = \frac{1}{13}\Big[81 + 4(V - 3)\Big] = \frac{1}{13}\big[81 + 4V - 12\big] = \frac{4V + 69}{13}. Solving 13V = 4V + 69 gives 9V = 69, so V = \frac{69}{9} = \frac{23}{3} \approx 7.67. The threshold is V - 3 = 	frac{23}{3} - 3 = 	frac{14}{3} \approx 4.67, so you keep any rank 5 or higher and redraw 1, 2,…

Redraw a card rank for three dollars

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