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Ughh, scratch the last sentence

Posted By: Fabrice Liardet
Date: Friday, 15 October 2010, at 12:26 p.m.

In Response To: Ughh, scratch the last sentence (Maik Stiebler)

So how did you analyse the match to seven case originally? Simple, straight-forward analysis of the painfully long binomial expression for the total winning chance depending on 13 variables or is there something more elegant?

No, it was even worse than that, I computed the -2,-1 case and the way it went convinced me that it should hold for any of the induction steps. Then I thought I would appeal to the ingenious BGOnline posters to find an elegant proof, and I was not disappointed. Maybe I should have mentioned that I didn't have any rollout ?

Bob's argument is elegant, but for me the way from "every game is equally important" to "equal time allotting is correct" is still a bit foggy, although I am nearly convinced by "the more one thinks, the less the added value of an extra second".

I think I can prove it from the last assumption. That assumption means that ER is decreasing and that ER' is increasing, right ?

Then consider the possibility of allotting times t₁ and t₂ to two equally important games, with t₁ < t₂. Then ask yourself whether you should equalize the times a little by replacing t₁ by t₁+h and t₂ by t₂-h, for a very small h.

Your error rate will then be ER(t₁+h) + ER(t₂-h) = ER(t₁) + ER(t₂) + h * (ER'(t₁) - ER'(t₂)). Since ER' is strictly increasing, the multiplier of h is negative, so your will have decreased your error rate.

This proves that no solution with t₁ different from t₂ can be optimal.

Messages In This Thread

• The Time Management Problem
  Fabrice Liardet -- Wednesday, 13 October 2010, at 11:37 a.m.
  • The Time Management Problem
    Casper van der Tak -- Wednesday, 13 October 2010, at 12:03 p.m.
    • The Time Management Problem
      Fabrice Liardet -- Wednesday, 13 October 2010, at 12:53 p.m.
      • The Time Management Problem
        Casper van der Tak -- Wednesday, 13 October 2010, at 2:09 p.m.
  • The Time Management Problem
    Bob Koca -- Wednesday, 13 October 2010, at 1:44 p.m.
    • The Time Management Problem
      Bob Koca -- Wednesday, 13 October 2010, at 4:21 p.m.
      • The Time Management Problem
        Rich Munitz -- Wednesday, 13 October 2010, at 4:35 p.m.
        • The Time Management Problem
          Bob Koca -- Wednesday, 13 October 2010, at 4:59 p.m.
        • The Time Management Problem
          Bob Koca -- Wednesday, 13 October 2010, at 5:00 p.m.
        • The Time Management Problem
          Bob Koca -- Wednesday, 13 October 2010, at 5:06 p.m.
          • The Time Management Problem
            Rich Munitz -- Wednesday, 13 October 2010, at 6:49 p.m.
            • The Time Management Problem
              Bob Koca -- Wednesday, 13 October 2010, at 9:11 p.m.
      • The Time Management Problem
        Fabrice Liardet -- Thursday, 14 October 2010, at 7:53 p.m.
  • The Time Management Problem
    Gregg Cattanach -- Wednesday, 13 October 2010, at 3:18 p.m.
  • The Time Management Problem
    eXtreme Gammon -- Wednesday, 13 October 2010, at 8:40 p.m.
    • The Time Management Problem
      Fabrice Liardet -- Thursday, 14 October 2010, at 8:00 p.m.
      • The Time Management Problem
        eXtreme Gammon -- Friday, 15 October 2010, at 2:08 a.m.
        • The Time Management Problem
          Maik Stiebler -- Friday, 15 October 2010, at 7:53 a.m.
          • Ughh, scratch the last sentence
            Maik Stiebler -- Friday, 15 October 2010, at 8:25 a.m.
            • Ughh, scratch the last sentence
              Casper van der Tak -- Friday, 15 October 2010, at 9:11 a.m.
              • Ughh, scratch the last sentence
                Fabrice Liardet -- Friday, 15 October 2010, at 9:29 a.m.
                • Ughh, scratch the last sentence
                  Casper van der Tak -- Friday, 15 October 2010, at 9:36 a.m.
                • Ughh, scratch the last sentence
                  Maik Stiebler -- Friday, 15 October 2010, at 10:59 a.m.
                  • Ughh, scratch the last sentence
                    Fabrice Liardet -- Friday, 15 October 2010, at 12:26 p.m.
                    • Ughh, scratch the last sentence
                      Maik Stiebler -- Friday, 15 October 2010, at 5:23 p.m.
                      • Ughh, scratch the last sentence
                        Fabrice Liardet -- Sunday, 17 October 2010, at 6:17 p.m.
                • Ughh, scratch the last sentence
                  eXtreme Gammon -- Friday, 15 October 2010, at 12:40 p.m.
                  • Ughh, scratch the last sentence
                    Fabrice Liardet -- Friday, 15 October 2010, at 2:35 p.m.