Match Play at 2-away/2-away

 Proof of doubling with market losers

 From: Walter Trice Address: wagt@worldnet.att.net Date: 18 July 2001 Subject: Re: reading 2 away 2 away in the archive Forum: rec.games.backgammon Google: i1757.42256\$C81.3562015@bgtnsc04-news.ops.worldnet.att.net

```> > Against a perfect opponent, double at the first sign of a market loser.
>
> I've heard that put forward, but without what I would consider a proof.
> Does "the first sign of a market loser" include a 1295 to 1 shot?  If
> not, where do you draw the line, and how do you prove that that's where
> the line should be drawn?
>
> If there is a simple rule, I don't think there's a simple proof of the
> rule!

I believe there is a simple proof of a slightly more general proposition:
against an opponent who handles the cube perfectly, the strategy of
[doubling if there is any chance of losing your market] is optimal.

Proof in two stages:

First, if your opponent handles the cube perfectly, then your chance of
winning the 2 point match is no greater than your chance of winning a 1
point match. The reason is that your opponent can double at his first
opportunity, making your match winning probability equal to your chance of
winning a 1 point match.

Second, by doubling before you lose your market you ensure that your match
winning probability is *at least* equal to your chance of winning a 1 point
match.

QED

If there's a flaw in this argument, I'd certainly be interested in knowing
what it is :-)

-- Walter Trice
```

 Douglas Zare  writes: ```Actually, I'm not sure I agree with the more general proposition. I'd like to see a definition of "perfect cube handling" and "any chance of losing your market" first. Should you double if there is a chance that your opponent's checker misplay will cause you to lose your market, but only because you are known to play badly while leading Crawford 2-away? ```

 Walter Trice  writes: ```Def'n of perfect cube handling: always makes the double and take decisions that maximize match winning chance. Any chance of losing your market: yes, that would include losing your market as a result of opponent's bad checker play. Yes, even if you could only lose your market because you would play badly in the Crawford game. > One implicit assumption is that there is no market-losing sequence of two > initial rolls. That's not a priori obvious, but no reasonable plays seem > to come close. What about 3-1 8/5 6/5 followed by 4-4 6/2(4)? I'd pass. Yes, that is an implicit assumption. Whether the assumption is met in all practical circumstances is a side issue. I'm still interested in how to get around the simple argument that by adopting a strategy of doubling before market loss Player B can limit A's chance to X, while by adopting the same strategy A can ensure that his chance is X, X being his chance of winning a 1 point match. It seems to me that this works even with pathological assumptions about checker play, relative skill, and match equities at the C/2-away scores. --walter trice ```

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### Match Play at 2-away/2-away

Basic strategy  (Darse Billings, Feb 1995)
Counterexample?  (Jim Williams+, Mar 1998)
Do you need an advantage to cube?  (Keene Marin+, Feb 2006)
Double immediately?  (Chuck Bower, Oct 1998)
Ever too good to double?  (Kit Woolsey, July 1995)
Minimum game winning chances to double  (Walter Trice, Mar 1999)
Practical strategy  (Walter Trice, July 1995)
Practical strategy  (Albert Steg+, Feb 1995)
Proof for doubling immediately  (Robert Koca+, May 1994)
Proof of doubling with market losers  (Walter Trice+, July 2001)
Sample game  (Ron Karr, Dec 1996)

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