Holland Rule
Andrew Bokelman
whichever way you look at it.
I agree. Whenever a match gets to post Crawford the whole cube thing seems
silly to me.
One variation I've come up with is as follows:
1. A match consists of a decided-upon number of games.
2. When that number of games is played, whoever has more points wins.
3. If there is a tie, one more game is played to break the tie.
But I guess that this can have its problems too. For example, if it is
near the end of the match the trailer can start doing automatic doubles. But
for some reason the above method strikes me as better.
Of course, matches could be played with no cube, no gammon, and no
backgammon. This would level the playing field quite a bit. But it just
doesn't seem like as much fun.
BobSMan
>One variation I've come up with is as follows:
>
>1. A match consists of a decided-upon number of games.
>2. When that number of games is played, whoever has more points wins.
>3. If there is a tie, one more game is played to break the tie.
>
>But I guess that this can have its problems too. For example, if it is
>near the end of the match the trailer can start doing automatic doubles. But
>for some reason the above method strikes me as better.
But remember, if the trailer starts doubling early there will now be a
price to pay. Once a player needs more than three points per game remaining,
the match is over. If he doubles, the double will be refused, and winning an
undoubled backgammon in every game will not be enough to save him. If a player
begins making early doubles before this point (in an effort to avoid this
situation) the price will be an increased probability of reaching this point
even earlier.
In fact, I've thought of this type of match myself, and it has always
seemed a good solution to the problem, IMO. A player who is behind will never
be able to double automatically in an effort to catch up, because of the
probable benefit to the opponent. And once a player becomes sufficiently far
behind, any attempt to use the cube at all will result in the loss of the
match. The structure of the match itself removes the need for any artificial
controls on the cube.
Bob Sisselman
EdmondT
>
>>One variation I've come up with is as follows:
>>
>>1. A match consists of a decided-upon number of games.
>>2. When that number of games is played, whoever has more points wins.
>>3. If there is a tie, one more game
is played to break the tie.
>>
If you think about it, the problem with the Crawford rule is not the rule, but
the match set up. If you play a match to a certain number of points, when one
player gets to within one point of the win, the losing player loses nothing by
doubling, since losing by two is no worse than losing by one.
Its this defect in the match that Crawford tries to address, albeit
imperfectly. But the problem is not Crawford, its the match.
Note if you were playing for money, there'd be no reason to do this.
Tom Keith
>
> Andrew Bokelman writes:
>
> >One variation I've come up with is as follows:
> >
> > 1. A match consists of a decided-upon number of games.
> > 2. When that number of games is played, whoever has more points wins.
> > 3. If there is a tie, one more game is played to break the tie.
> >
> > near the end of the match the trailer can start doing automatic doubles. But
> > for some reason the above method strikes me as better.
>
> But remember, if the trailer starts doubling early there will now be a
> price to pay. Once a player needs more than three points per game remaining,
> the match is over. If he doubles, the double will be refused, and winning an
> undoubled backgammon in every game will not be enough to save him. If a player
> begins making early doubles before this point (in an effort to avoid this
> situation) the price will be an increased probability of reaching this point
> even earlier.
> In fact, I've thought of this type of match myself, and it has always
> seemed a good solution to the problem, IMO. A player who is behind will never
> be able to double automatically in an effort to catch up, because of the
> probable benefit to the opponent. And once a player becomes sufficiently far
> behind, any attempt to use the cube at all will result in the loss of the
> match. The structure of the match itself removes the need for any artificial
> controls on the cube.
>
> Bob Sisselman
Gary Wong did a nice critique of this type of match a week or so ago.
Here are a few observations about what happens if you run matches this
way:
1. Near the end of the match, the match equities are not very evenly
distributed -- especially at 1-game-to-go, where there is no real
difference between being ahead by one point, being tied, or being
behind by one point. Besides appearing somewhat unfair, this
skews the take points in the 2-to-go game.
2. Some match scores leave the trailer with only a very remote chance
of turning things around. For example, when you are behind by 4
with 2 games to go, you must win two consecutive gammons and then
a tie-breaker to win the match -- that's less than a one percent
chance. Playing with so little at stake would not be very
enjoyable for many players. (Behind by 6 with 3 games to go is
even worse.)
3. At certain match scores, such as down-3 with 2-to-go, you've lost
your market even before the game begins. Your initial strategy
must be to play for a gammon, and then only if things go badly
will you double. (This isn't bad, necessarily -- just different
than normal match play.)
Some match equities:
Games left: 0 1 2 3 4 5 6 7 8 9 10
Lead 10: 1.000 1.000 1.000 1.000 1.000 1.000 .999 .986 .953 .939 .921
Lead 9: 1.000 1.000 1.000 1.000 1.000 1.000 .990 .954 .932 .912 .893
Lead 8: 1.000 1.000 1.000 1.000 1.000 .993 .956 .922 .902 .880 .863
Lead 7: 1.000 1.000 1.000 1.000 .997 .960 .910 .890 .864 .845 .827
Lead 6: 1.000 1.000 1.000 .999 .969 .894 .874 .845 .824 .805 .789
Lead 5: 1.000 1.000 1.000 .980 .876 .855 .821 .798 .778 .761 .747
Lead 4: 1.000 1.000 .992 .877 .830 .792 .765 .745 .727 .714 .701
Lead 3: 1.000 1.000 .891 .793 .756 .724 .705 .687 .674 .663 .653
Lead 2: 1.000 .938 .695 .718 .669 .658 .636 .628 .618 .610 .603
Lead 1: 1.000 .500 .678 .584 .598 .573 .572 .564 .559 .555 .552
Lead 0: .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500
Lead -1: .000 .500 .322 .416 .402 .427 .428 .436 .441 .445 .448
Lead -2: .000 .062 .305 .282 .331 .342 .364 .372 .382 .390 .397
Lead -3: .000 .000 .109 .207 .244 .276 .295 .313 .326 .337 .347
Lead -4: .000 .000 .008 .123 .170 .208 .235 .255 .273 .286 .299
Lead -5: .000 .000 .000 .020 .124 .145 .179 .202 .222 .239 .253
Lead -6: .000 .000 .000 .001 .031 .106 .126 .155 .176 .195 .211
Lead -7: .000 .000 .000 .000 .003 .040 .090 .110 .136 .155 .173
Lead -8: .000 .000 .000 .000 .000 .007 .044 .078 .098 .120 .137
Lead -9: .000 .000 .000 .000 .000 .000 .010 .046 .068 .088 .107
Lead -10: .000 .000 .000 .000 .000 .000 .001 .014 .047 .061 .079
(Assumptions: 25% gammon rate, 0% backgammon rate, efficient doubles,
free drop ignored.)
Tom Keith
BobSMan
>Gary Wong did a nice critique of this type of match a week or so ago.
>Here are a few observations about what happens if you run matches this
>way:
>
>
<snip>
>2. Some match scores leave the trailer with only a very remote chance
> of turning things around. For example, when you are behind by 4
> with 2 games to go, you must win two consecutive gammons and then
> a tie-breaker to win the match -- that's less than a one percent
> chance. Playing with so little at stake would not be very
> enjoyable for many players. (Behind by 6 with 3 games to go is
> even worse.)
I did consider this. It does not strike me as being particularly
important. Similar conditions exist in ordinary matches all the time. For
example, a player may lead 6-1 in a seven point match. The trailer must first
win the Crawford game, followed by three single games (two games, if there is a
gammon). Would this be much better? Of course, in both cases, the match ends
as soon as the trailer loses. There's no point in playing on once a player is
mathematically eliminated.
>3. At certain match scores, such as down-3 with 2-to-go, you've lost
> your market even before the game begins. Your initial strategy
> must be to play for a gammon, and then only if things go badly
> will you double. (This isn't bad, necessarily -- just different
> than normal match play.)
And I thought of this too. It doesn't bother me at all. It's not unfair
to either player. In fact, it's one of the things I most like about this type
of match. The value of the cube to the trailing player is reduced rather than
increased. Large differences in the score become a more significant handicap
than in a "normal" match. Fine with me.
And, backing up a bit:
>1. Near the end of the match, the match equities are not very evenly
> distributed -- especially at 1-game-to-go, where there is no real
> difference between being ahead by one point, being tied, or being
> behind by one point. Besides appearing somewhat unfair, this
> skews the take points in the 2-to-go game.
This is not really unfair. As I see it, it's not much different from
being 2-away/1-away in a post Crawford game. As for skewing the take points in
the previous game...well, why not? Similar thing occur in ordinary matches
when both players are within three points of winning. In both cases, it just
means that knowing the correct equities will affect the correctness of your
decisions.
It is true that a one point lead in the last game is roughly equivalent to
the value of a free drop in ordinary match play. So what's the difference?
The primary advantage of this type of match, IMO, is that it does not
require any artificial rules (Crawford, Holland, etc.) And it does not seem to
have any significant disadvantages. That's all I meant to suggest originally,
and nothing that's been said so far changes my opinion.
Bob Sisselman
Andrew Bokelman
>>>But remember, if the trailer starts doubling early there will now be a
price to pay. Once a player needs more than three points per game remaining,
the match is over. If he doubles, the double will be refused, and winning an
undoubled backgammon in every game will not be enough to save him. If a
player begins making early doubles before this point (in an effort to avoid
this situation) the price will be an increased probability of reaching this
point even earlier.
I hadn't thought it through this far, but I see what you mean. And now that
I've thought about it more I see a possible reverse problem. It removes the
power of the cube from the trailer even in cases of nonautomatic
doubles because all the leader has to do it drop -- placing himself
one game closer to winning. Of course, this only works if the games/points
ratio is correct. And in this position the leader would not be turning the
cube either, so it would be head to head backgammon.
..Andrew