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Explain Jacoby paradox

Posted By: Daniel Murphy
Date: Sunday, 9 December 2012, at 3:42 a.m.

In Response To: Explain Jacoby paradox (Tom Keith)

"It just happens that in Position 1 the value of owning the cube outweighs the small equity increase Blue gets by doubling, but not in Position 2."

	Column A	Column B	Column C	Column D
	Blue's wins immediately, or	White on roll wins in both positions (or redoubles Blue out) whether or not White has the cube	White on roll without the cube loses if his checker is on the 5 point, but wins if his checker is on the 4 point (with the cube, White redoubles Blue out in both positions)	White on roll without the cube loses in both positions (with the cube, White redoubles Blue out in both positions).
Frequency =	19/36	17/36 * 31/36	17/36 * 3/36	17/36 * 2/36
=	684/1296	527/1296	51/1296	34/1296
White's checker is on the 5 point. -(1054+306+204)= -1564, more than 1368, so Blue should not redouble.
ND	+684/1296 * 2 = 1368/1296	-527/1296 * 2 = 1054/1296	+51/1296 * 2 =+102/1296	+34/1296 * 2 = +68/1296
DT	+684/1296 * 4 = 2736/1296	-527/1296 * 4 =-2108/1296	-51/1296 * 4 =-204/1296	-34/1296 * 4 = -136/1296
Gain or loss	1368/1296	-1054/1296	-306/1296	-204/1296
White's checker is on the 4 point. -(1054+102+204) = -1360, less than 1368, so Blue should redouble.
ND	+ 684/1296 * 2 = 1368/1296	-527/1296 * 2 = 1054/1296	-51/1296 * 2 =-102/1296	+34/1296 * 2 = +68/1296
DT	+ 684/1296 * 4 = 2736/1296	-527/1296 * 4 =-2108/1296	-51/1296 * 4 =-204/1296	-34/1296 * 4 = -136/1296
Gain or loss	1368/1296	-1054/1296	-102/1296	-204/1296
It's good to double before a winning sequence (Column A). It's bad to double before a losing sequence (Column B if White's checker is on the 5 point, but Column B and Column C if White's checker is on the 4 point). It's especially bad to double and lose when you would have won if you had not doubled (both Column C and Column D if White's checker is on the 5 point, but only Column D if White's checker is on the 4 point). Holding the cube, and therefore winning a 2-cube instead of losing a 4-cube, is worth (306+204)/1296 = 510/1296 points to Blue if White's checker is on the 5 point, but only 204/1296 points to Blue if White's checker is on the 4 point.

If White's checker is on the 5 point, Blue's double loses
1368/1296	-	1054/1296	-	(306/1296 + 204/1296)	=	1368/1296	-	1054/1296	-	510/1296 points.
If White's checker is on the 4 point, Blue's double gains
1368/1296	-	(1054/1296 + 102/1296)	-	204/1296	=	1368/1296	-	1156/1296	-	204/1296 points.

Messages In This Thread

• Explain Jacoby paradox
  leobueno -- Saturday, 8 December 2012, at 11:50 a.m.
  • Explain Jacoby paradox
    David Startin -- Saturday, 8 December 2012, at 2:36 p.m.
  • Explain Jacoby paradox
    Casper van der Tak -- Saturday, 8 December 2012, at 3:21 p.m.
    • Explain Jacoby paradox
      Chuck Bower -- Sunday, 9 December 2012, at 2:28 a.m.
  • The Jacobs Paradox is far more interesting
    Phil Simborg -- Saturday, 8 December 2012, at 5:34 p.m.
    • The Jacobs Paradox is far more interesting
      Strato -- Saturday, 8 December 2012, at 7:43 p.m.
    • The Jacobs Paradox is far more interesting
      Jake Jacobs -- Sunday, 9 December 2012, at 3:52 a.m.
      • Jacobs's version of the Jacobs Paradox is far more interesting
        Timothy Chow -- Sunday, 9 December 2012, at 4:34 a.m.
      • The Jacobs Paradox is far more interesting
        Phil Simborg -- Sunday, 9 December 2012, at 10:23 p.m.
  • Explain Jacoby paradox
    Tom Keith -- Saturday, 8 December 2012, at 8:05 p.m.
    • Explain Jacoby paradox
      Daniel Murphy -- Sunday, 9 December 2012, at 3:42 a.m.
      • Explain Jacoby paradox
        christian munk-christensen -- Sunday, 9 December 2012, at 11:35 a.m.
        • Explain Jacoby paradox
          Daniel Murphy -- Sunday, 9 December 2012, at 3:59 p.m.
  • Explain Jacoby paradox
    Ray Kershaw -- Sunday, 9 December 2012, at 9:49 p.m.
  • Explain Jacoby paradox
    ah_clem -- Monday, 10 December 2012, at 3:06 p.m.