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): When to play and when to cease drawing. by ed bottemiller)

"Charlie and Leming appear obdurate in what I take to be a confused reinterpretation of the puzzle as posed.  They seem to be intent on providing tables of odds which would only be relevant if one were forced to play and continue playing (which clearly is not the case).  They seem to focus on only the first phrase of the task: "what would be a fair amount to pay for the right to play," more or less ignoring the second: "under what circumstances should a player cease drawing".  They correctly state that the fair amount is A - B, and surely see that this is negative if B is greater than A -- but provide no rationale for starting or continuing when B is greater."

You are probably misunderstanding the table below. It does not imply being forced to play or to continue. If the initial A and B result in a value of zero do not play. If, during play, the pluses and minuses reach a point where you get to a zero, do not play. For example, if you start out A=2, B=4, don't play.  Likewise, if during play it gets to a point where there are 2 pluses and 4 minuses, stop playing. So we have addressed both issues: when to accept the start of play, and when to stop.

I initially thought that the fair amount to pay to play would be A - B, but Leming pointed out how this was wrong. I reiterated his objection in the form of the following equally likely orderings of the cards when there are 2 pluses and 3 minuses:

---++  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)

In each case above, for every remainder of the deck while playing, the remaining pluses vs. minuses could show in the table below why play was continued or terminated. For example, in -+xxx, play is stopped after the remainder of the deck has become 1 plus and 2 minuses.  It was started initially because the amounts were 2 pluses and 3 minuses, which gives an expected win of 0.20, making that 0.20 the fair value to play. You can see from the above payout list, with four orderings of the cards resulting in gains of 1, and only 2 orderings resulting in losses of 1, and there being three equally likely orderings of the cards, that the expectation is (4-2)/10 = 0.20. So even though B is greater, in this instance it pays (0.20 expected) to play.

 b: 0      1    2     3     4     5     6     7     8     9     10    11    12 
  1.00  0.50  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  2.00  1.33  0.67  0.20  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  3.00  2.25  1.50  0.85  0.34  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  4.00  3.20  2.40  1.66  1.00  0.44  0.07  0.00  0.00  0.00  0.00  0.00  0.00
  5.00  4.17  3.33  2.54  1.79  1.12  0.55  0.15  0.00  0.00  0.00  0.00  0.00
  6.00  5.14  4.29  3.45  2.66  1.91  1.23  0.66  0.23  0.00  0.00  0.00  0.00
  7.00  6.13  5.25  4.39  3.56  2.76  2.01  1.34  0.75  0.30  0.00  0.00  0.00
  8.00  7.11  6.22  5.35  4.49  3.66  2.86  2.11  1.43  0.84  0.36  0.05  0.00
  9.00  8.10  7.20  6.31  5.43  4.58  3.75  2.95  2.21  1.52  0.92  0.43  0.10
 10.00  9.09  8.18  7.28  6.39  5.52  4.66  3.83  3.04  2.30  1.61  1.00  0.50
 11.00 10.08  9.17  8.26  7.35  6.46  5.59  4.74  3.91  3.12  2.38  1.69  1.08
 12.00 11.08 10.15  9.24  8.32  7.42  6.54  5.66  4.81  3.99  3.20  2.46  1.77

Posted by Charlie on 2008-03-12 11:11:42