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I wrote:
: Leader's drop point at 3-away, 5-away is quite sensitive
: to the match equity you assume for 3-away, 4-away.
: The higher the value of Trailer's equity at 3-away, 4-away,
: the more reluctant Leader will be to give up one point when
: the score is 3-away, 5-away.
:
: Kit Woolsey's table lists Trailer's 3-away, 4-away match
: equity at 41%. But there are other tables (e.g., Norman
: Zadeh's) which place the value for this score as high as 43%.
: Each one point increase in the assumed value of Trailer's
: equity at 3-away, 4-away produces a two or three point rise
: in Leader's drop point at 3-away, 5-away.
:
: So, as much as anything else, the position of Leader's drop
: point at 3-away, 5-away depends on which match equity table
: you use.
Kit Woolsey wrote:
> Good point, Tom. Let's see what this means if we use Zadeh's 43%. I
> don't know what his figure is for 1 away, 5 away Crawford, so let's
> assume it is the same as mine (85% for the leader). Now, let's look at
> the leader's drop point, assuming no gammons:
>
> Leader passes: He is ahead 4 away, 3 away for 57% equity.
> Leader takes and wins: He is ahead 1 away, 5 away for 85% equity.
> Leader takes and loses: He is even, for 50% equity.
>
> Thus he would be risking 7% to gain 28%, so he would be getting 4 to 1
> odds on his take. Therefore, he could take with 20% winning chances,
> ignoring possible recube vig.
>
> This just doesn't feel right, does it? All our intuition tells us that
> the leader should be more cautious in taking the cube than he should be
> in money (except possibly where the double puts him out exactly), but
> these figures say he can take positions which are clear money passes.
> To me, this indicates that Zadeh's 43% figure has to be off.
Maybe not. There is an "odd-even" effect on match scores that can
produce surprising results. Since points in backgammon often come
in twos and fours, the opponent's match-winning chances take a big
step forward when he goes from 5-away to 4-away. At 4-away he can
win the match with one doubled gammon or two 2-point games.
Because a one-point loss is so expensive when the opponent is 5-away,
and a two-point loss not that much worse, the leader is more willing
than usual to try turning the game around rather than give up exactly
one point.
> When I constructed my match equity table, one of the things I did was to
> introduce smoothing effects so anomalies such as the one above would be
> avoided. Thus I believe my table will lead to more practical results in
> one's match equity calculations.
Here's an interesting question: Which is better, leading a match at
at 3-away/4-away or leading at 4-away/5-away?
Tom
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