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A comment about the takepoint formula

Posted By: Timothy Chow
Date: Wednesday, 14 April 2010, at 2:18 a.m.

In Response To: World's incorrect 14-away/8-away pass: Analysis (Timothy Chow)

I wrote:

Mathematically, you can take if

wins + (effective gammon wins)*(gammon price) – (effective gammon losses)*(gammon price)

equals or exceeds your raw takepoint. (Here wins include redouble/drop, and I'm assuming backgammons and re-recubes can be ignored.)

So far I've just been asserting this fact without justifying it. It occurred to me that it might help if I say a word or two about why the above formula is true.

Suppose that there is no cube. Then I assume people are comfortable with the claim that you can take if

wins + (gammon wins)*(price of gammon win) – (gammon losses)*(price of gammon loss)

exceeds your raw takepoint (ignoring backgammons). Now, let's introduce the cube. Assume that the cube gets turned at most once more, and that gammons after the cube is turned are negligible. Then the effect of the cube is just to give you a new way to win 1 point (redouble/drop), a new way to win 2 points (redouble/take/win single), and a new way to lose 2 points (redouble/take/lose single). As far as the math is concerned, it doesn't care how you won or lost 2 points (whether the old-fashioned way with a gammon or the new way with the cube); all it cares is how often you win or lose 2 points. So to take the cube into account, just lump all your 1-point and 2-point wins/losses together, and use the exact same formula.

One can derive the above formula more formally by manipulating equations, but I don't think that would be illuminating; the above verbal description is the real explanation of what's going on, in my opinion.

Messages In This Thread

• World's incorrect 14-away/6-away pass: Analysis
  Timothy Chow -- Tuesday, 13 April 2010, at 11:03 p.m.
  • World's incorrect 14-away/6-away pass: Analysis
    Matt Cohn-Geier -- Tuesday, 13 April 2010, at 11:19 p.m.
    • World's incorrect 14-away/8-away pass: Analysis
      Timothy Chow -- Wednesday, 14 April 2010, at 12:17 a.m.
      • World's incorrect 14-away/8-away pass: Analysis
        Matt Cohn-Geier -- Wednesday, 14 April 2010, at 12:26 a.m.
        • World's incorrect 14-away/8-away pass: Analysis
          Timothy Chow -- Wednesday, 14 April 2010, at 12:50 a.m.
          • A comment about the takepoint formula
            Timothy Chow -- Wednesday, 14 April 2010, at 2:18 a.m.
            • A comment about the takepoint formula
              Matt Cohn-Geier -- Wednesday, 14 April 2010, at 9:18 a.m.
              • A comment about the takepoint formula
                Timothy Chow -- Wednesday, 14 April 2010, at 3:04 p.m.
                • A comment about the takepoint formula
                  Matt Cohn-Geier -- Thursday, 15 April 2010, at 2:00 p.m.
                  • A comment about the takepoint formula
                    Timothy Chow -- Thursday, 15 April 2010, at 3:07 p.m.
                    • A comment about the takepoint formula
                      Phil Simborg -- Saturday, 17 April 2010, at 7:30 a.m.
                      • A comment about the takepoint formula
                        Daniel Murphy -- Saturday, 17 April 2010, at 8:27 a.m.
                      • A comment about the takepoint formula
                        Timothy Chow -- Sunday, 18 April 2010, at 4:47 a.m.
    • World's incorrect 14-away/6-away pass: Analysis
      Daniel Murphy -- Wednesday, 14 April 2010, at 1:08 a.m.
      • World's incorrect 14-away/6-away pass: Analysis
        Matt Cohn-Geier -- Wednesday, 14 April 2010, at 1:22 a.m.
  • World's incorrect 14-away/8-away pass: Some recubes
    Timothy Chow -- Sunday, 2 May 2010, at 11:57 p.m.
    • World's incorrect 14-away/8-away pass: Some recubes
      Daniel Murphy -- Monday, 3 May 2010, at 12:51 a.m.