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 Table of Fair Amount by Leming)

How does one get an expected value of 0.67 (or 2/3, presumably) if the strategy is to quit when one is even or ahead, until the cards run out.  For the 6 possible sequence of cards I get:

--++ 0
-+-+ 0
-++- 0
+--+ 1
+-+- 1
++-- !

For a net win of 3 for every 6 games played making each one worh 1/2.

Or are you following a strategy different from quitting when one gets even or ahead?

What I get for the cases where A equals B or A is one less than B are:

 A  B   total won  plays  expected value     
 2  2         3        6 0.50000000000
 2  3         2       10 0.20000000000
 3  3        10       20 0.50000000000
 3  4        10       35 0.28571428571
 4  4        35       70 0.50000000000
 4  5        42      126 0.33333333333
 5  5       126      252 0.50000000000
 5  6       168      462 0.36363636364
 6  6       462      924 0.50000000000
 6  7       660     1716 0.38461538462
 7  7      1716     3432 0.50000000000
 7  8      2574     6435 0.40000000000
 8  8      6435    12870 0.50000000000
 8  9     10010    24310 0.41176470588
 9  9     24310    48620 0.50000000000
 9 10     38896    92378 0.42105263158
10 10     92378   184756 0.50000000000
10 11    151164   352716 0.42857142857

I note that the A=2, B=3 matches your previous result of 0.2, but not your new values.

DECLARE SUB permute (a$)
DEFDBL A-Z
FOR a = 2 TO 10
FOR b = a TO a + 1
  deckCt = 0
  deck$ = STRING$(a, "+") + STRING$(b, "-")
  h$ = deck$
  totWin = 0
  DO
   winnings = 0
   FOR i = 1 TO LEN(deck$)
     IF MID$(deck$, i, 1) = "+" THEN
      winnings = winnings + 1
     ELSE
      winnings = winnings - 1
     END IF
     IF winnings >= 0 THEN EXIT FOR
   NEXT
   totWin = totWin + winnings
   deckCt = deckCt + 1
   permute deck$
  LOOP UNTIL deck$ = h$
  PRINT USING "## ## "; a; b;
  PRINT USING " ########"; totWin; deckCt;
  PRINT USING " #.###########"; totWin / deckCt

NEXT
NEXT

Posted by Charlie on 2008-03-11 16:56:09