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Counterexample by Lamford?

Posted By: Timothy Chow
Date: Tuesday, 16 September 2014, at 8:59 p.m.

In Response To: Losing your market at 2a2a (Timothy Chow)

AP kindly sent me a scan of a few pages from Lamford, in which he seems to claim that it's possible for both players to play optimally yet for one player to lose market. I say "seems to claim" because Lamford uses the word "correct" at one point and I'm not sure if he means "correct against an optimal player" or "correct against an imperfect player." But anyway, here is the possible example. I will state explicitly the assumptions being made along the way. The score is 2a2a.

1. White wins the opening roll with 21 and plays 24/23 13/11. (Assumption: 24/23 13/11 is the optimal way to play an opening 21 at DMP.)

2. Now Black is on roll. (Assumption: Black has an optional double.) Black does not double.

3. Black rolls 52, played 24/22 13/8. (Assumption: This is the optimal way to play 52 in response to an opening 24/23 13/11 at 2a2a when the cube is still centered.)

4. Now White is on roll. (Now comes the key assumption: Lamford says that not doubling is "correct" for White. I'm interpreting this to mean that it's an optional double for White even against a perfect player. But it's possible that this isn't what Lamford means, since it is prefaced by some informal general discussion about imperfect human players.) White does not double.

5. White rolls 55, played 8/3*(2) 6/1*(2). (Assumption: Again, this is perfect checker play for White.)

6. Black fans. (Assumption: White has lost market—in fact he is too good.)

If all six assumptions are correct then this provides a surprising answer to the original question in my thread. But are they all correct? Assumption 4, the way I've interpreted it, naturally looks the most suspect, but I don't see a way to rule it out on purely logical grounds (i.e., without detailed analysis of the specific position). I think it has to be the case that not only is White too good in Step 6, but he is better off not doubling even if he knew Black would take a double. That surprises me, but I don't see why it's impossible.

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