You sit down with a well mixed deck containing A cards marked "+" and B cards marked "—". You may draw cards from this deck as long as you want, i.e., you can stop playing at any point. Each time you draw a + card you are given $1 and each time you draw a — card you have to pay $1. Cards are not replaced after having been drawn.

What would be a fair amount to pay for the right to play (i.e., what is the expected payoff) and under what circumstance should a player cease drawing?

(In reply to re(2): What's the catch? - it's not that simple by Charlie)

I like the following way of presenting your positive expected value (and therefore fair amount to pay to play) with only 2 plus against 3 minus. Below are the 10 sequences in which the five cards could be located within the deck, and the strategy is to quit when one is either ahead or even:

---++  lose 1  (stopped when out of cards)
--+-+  lose 1  (stopped when out of cards)
--++-  even--0 (stopped when even after 4 cards)
-+--+  even--0 (stopped when even after 2 cards)
-+-+-  even--0 (stopped when even after 2 cards)
-++--  even--0 (stopped when even after 2 cards)
+---+  gain 1 (stopped when ahead at 1st card)
+--+-  gain 1 (stopped when ahead at 1st card)
+-+--  gain 1 (stopped when ahead at 1st card)
++---  gain 1 (stopped when ahead at 1st card)

So the overall question is still unresolved in the general case. And even when one starts with more +'s than -'s, it still could be advantageous to continue playing even when the pluses are outnumbered.

Posted by Charlie on 2008-03-11 16:09:16