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 When to play and when to cease drawing. by Leming)

I see, and agree with your methodology. I've extended your table, and it shows that as A increases there's even more that B can exceed A and still have a positive expectation, as in A=9 and B=12, where the expectation is still 0.10.

In the table, when b is zero, the expectation is a, so the identification of the a values is left out, as you can see that from the b=0 column.

 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

DEFDBL A-Z

CLS

DIM prev(30), new(30)
FOR i = 1 TO 12: prev(i) = 0: NEXT

FOR a = 1 TO 12
FOR b = 0 TO 12
 IF b = 0 THEN
   new(b) = a
 ELSE
   new(b) = (a / (a + b)) * (prev(b) + 1) + (b / (a + b)) * (new(b - 1) - 1)
 END IF
 IF new(b) < 0 THEN new(b) = 0
 PRINT USING "###.##"; new(b);
NEXT
PRINT
FOR i = 0 TO 12: prev(i) = new(i): NEXT
NEXT

 

Edited on March 11, 2008, 11:18 pm

Posted by Charlie on 2008-03-11 23:15:11