## Saturday, March 10, 2012

### Match equity table: Crawford games

Crawford games happen in matches the first time a player gets within one game of winning the match. In the Crawford game the opponent cannot use the doubling die.

To compute match equity for these games, we start with the post-Crawford match equities and backward induct.

The Crawford game match equity table (for player 1-away and opponent n-away) I calculated is:

1-away
2-away
3-away
4-away
5-away
6-away
7-away
8-away
9-away
10-away
11-away
12-away
1-away
0
0.363
0.511
0.636
0.692
0.781
0.821
0.870
0.893
0.923
0.937
0.954

Here's how I calculated it:

Call Mc(n) the Crawford game match equity. By symmetry, Mc(1) must be 0.

The doubling cube cannot be used in the Crawford game, so the more general match equity calculation is fairly straightforward. First, you decide the match equity conditional on different game end states:

Any player win: match equity = +1 (probability 50%)
Single player loss: match equity = Mp(n-2) (probability = 50% - gammon probability)
Single player gammon: match equity = Mp(n-4) (probability = gammon probability)

where Mp(n) = post-Crawford match equity for player 1-away and opponent n-away.

For terminological convenience, Mp(n) = -1 for n<=0. We ignore backgammons for this calculation.

Then the weighted Crawford match equity must be

Mc(n) = 1/2 + ( 1/2 - Pg ) Mp(n-2) + Pg Mp(n-4)

where Pg = probability of a gammon loss in a game as seen from before the first die throw, so symmetric (that is, the same as a the probability of a gammon win). Using 2-ply Player 3.2 I estimated this before as 13.735%.

One difference to note in the table above vs the table here (an example of a modern match equity table): for 1-away 3-away the match equity is greater than 0.5. That's because mine includes the proper post-Crawford match equity for 1-away 2-away, rather than approximating that as zero.

Other differences for the Crawford game match equities are smaller than that equity error of 0.011; mostly around 0.008 or less. So ignoring the non-zero value of Mp(2) adds a relatively significant error.

The biggest source of error when computing the Crawford match equities is on 1-away, 2-away (that is, Mc(2)), and is choosing an accurate gammon probability. This reference chooses a 20% chance that a cubeless game ends in a gammon, which is too low and gives too high a Mc(2) by as much as 0.04 points.