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Followup queston

Posted By: Maik Stiebler
Date: Friday, 21 December 2018, at 8:44 a.m.

In Response To: Followup queston (Bob Koca)

Let the match equity at M away, N away be x (or in a shorter notation q(M,N)=x). Let q(M-1,N-1)=x+s (think of s as match shortening value) and q(M-1,N)=x+p (think of p as point value). Generally, q(M,N)=(q(M-1,N)+q(M,N-1))/2. Then it follows that q(M,N-1)=x-p [1], q(M-2,N)=x+2p-s [2], q(M,N-2)=x-2p-s [3].

Thus, for the first player the risk of taking at (M,N) is q(M,N-1)-q(M,N-2)=p+s, while risk+gain is q(M-2,N)-q(M,N-2)=4p. TP=risk/(risk+gain)=(p+s)/(4p). Now, if M=N, s is obviously 0, so TP=25%. Intuitively, match shortening should benefit the leader [4], thus s > 0 and TP > 25% if M < N, while s < 0 and TP < 25% if M > N.

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[1]:

q(M,N)=(q(M-1,N)+q(M,N-1))/2

x=(x+p+q(M,N-1))/2

q(M,N-1)=2*x-x-p

[2]:

q(M-1,N)=(q(M-2,N)+q(M-1,N-1))/2

x+p=(q(M-2,N)+x+s)/2

q(M-2,N)=2*(x+p)-x-s

[3]:

q(M,N-1)=(q(M,N-2)+q(M-1,N-1))/2

x-p=(q(M,N-2)+x+s)/2

q(M,N-2)=2*(x-p)-x-s

[4]:

I think that for a proof it suffices to show that for the trailer at N away, M away it is more likely to reach 2away, 1 away than 1 away, 2 away later in the match.

Messages In This Thread

• Cube Action when rest of match is cubeless and gammonless
  Bob Koca -- Wednesday, 19 December 2018, at 6:46 p.m.
  • Cube Action when rest of match is cubeless and gammonless
    Colin Owen -- Wednesday, 19 December 2018, at 11:01 p.m.
    • Followup queston
      Bob Koca -- Thursday, 20 December 2018, at 2:19 a.m.
      • Followup queston
        Colin Owen -- Thursday, 20 December 2018, at 4:45 a.m.
        • Followup queston
          Bob Koca -- Thursday, 20 December 2018, at 2:44 p.m.
      • Followup queston
        Maik Stiebler -- Friday, 21 December 2018, at 8:44 a.m.
  • Cube Action when rest of match is cubeless and gammonless
    Florin Popa -- Wednesday, 19 December 2018, at 11:43 p.m.
    • Cube Action when rest of match is cubeless and gammonless
      Bob Koca -- Thursday, 20 December 2018, at 3:09 a.m.
      • Cube Action when rest of match is cubeless and gammonless
        Florin Popa -- Thursday, 20 December 2018, at 9:18 a.m.
  • Cube Action when rest of match is cubeless and gammonless
    Dmitriy Obukhov -- Thursday, 20 December 2018, at 2:25 a.m.
    • Cube Action when rest of match is cubeless and gammonless
      Bob Koca -- Thursday, 20 December 2018, at 3:04 a.m.
      • Cube Action when rest of match is cubeless and gammonless
        Dmitriy Obukhov -- Thursday, 20 December 2018, at 5:23 a.m.
        • Correction
          Dmitriy Obukhov -- Thursday, 20 December 2018, at 5:36 a.m.
          • Correction
            Bob Koca -- Thursday, 20 December 2018, at 2:54 p.m.
  • Cube Action when rest of match is cubeless and gammonless
    David Rockwell -- Thursday, 20 December 2018, at 2:22 p.m.
    • Cube Action when rest of match is cubeless and gammonless
      Bob Koca -- Thursday, 20 December 2018, at 2:40 p.m.