 Forwarded message 
From: William Arlinghaus <warlingha@ltu.edu>
Date: Sun, Apr 3, 2011 at 8:39 PM
Subject: The Role of 1.5 in Masterpoint Formulas
To: Bill Arlinghaus < warlingha@ltu.edu>,
"BOD#25(Rich DeMartino)" < rademr@optonline.net>,
"BOD#3(Joan Levy Gerard)" < joanandron@worldnet.att.net>,
"BOD#4(craig robinson)" < 4spot@comcast.net>,
Claire Jones < stats@accesscomm.ca>,
Robert Heller < hellerb@mindspring.com>,
The Subecks < stansubeck@prodigy.net>
to the masterpoint committee:
In the grand old days of yesteryear, nobody worried about such things
as 12board sessions, 18board sessions, etc. Everybody ran at
least 24 boards, and most ran more if they could. How diefferent
life is today, when people want short morning sessions, afternoon
sessions that finish well before dinner, 12board internet sessions,
and even dinner bell knockouts with 12board sesssions.
Nonetheless, I believe history has something to teach us. In
1993, the masterpoint gurus seemed to be solidly behind the fact that
doubling the number of sessions multiplied the award by 1.5. So in that
year, two sessions were worth 1.5 times one session, 4 sessions were
worth 1.5 times two sessions (or 2.25 times one session), and six
sessions were worth 1.5 times three sessions. This was reflected
in the S factors of that time. Unfortunately, the S factor was
2.0 for three sessions when it should have been 1.90
Mathematical note: to raise 1.5 to
the correct power, for N sessions one should raise 1.5 to the power 2
would have to be raised to to get N. For instance, for N=4, one
should raise 2 to the second power to get 4, so one should square 1.5
to get 2.25, the proper value for S for 4 sessions. Of course,
that number that 2 should be raise to to get N is the logarithm base 2
of N. 1.5 raised to the logarithm base 2 of 3 gives the 1.90
referred to above.
Please feel free to ignore the
mathematical note, if you desire. In any case, someone (I think
in 1998) noticed that the S factors were 1,1.5, and 2 for 1,2,3
sessions, so just made them 2.5,3,3.5 for 4,5,6 sessions. This
created the inconsistencies present in the current formulas. For
instance, a 4session event is now worth 5/3 of a 2session event, a
6session event is now worth 1.75 times a 3sessions event. This
may lend some credence to complaints that KOs pay too much, since they
are 10/9 of what they would have been using S=2.25
Another place we still recognize the 1.5 number is that knockout
matches of 48 boards pay 1.5 the value of 24board knockout
matches.(see p. 4 of current regulations.
This suggests that 12board sessions, being half the length of 24board
sessions, should pay 2/3 of 24board sessions (or, viewing it the other
way, 24board sessions should pay 1.5 times 12board sessions.
To me, the way to solve this problem would be to make the formula for
1st overall = B x R x S x M x P x L (remember we are eliminating T),
where L is 1.5 to the power log2(#bds/24). Thus L=1 for 24board
sessions, L=2/3 for 12board sessions. Other values of L are in
the second sheet of the attached spreadsheet. (The first sheet
contains S factors).
In fact, I think our goal is to replace B x M by a strength of field
factor, which I call FS. Since B X M is almost always between 0
and 2 (it takes more than 400 tables to make B>2), I think FS should
be a number between 0 and 2.
There is a discussion of the masterpoint formula in the third sheet of
the attached spreadsheet.
The knockout formula was written in 2007 to relect values for 4session
regional knockouts. I have translated this formula so that it
fits into the general masterpoint formula. When R=14 and S=2.25,
the FS factor generated produces exactly 9/10 of the present awards (I
chose 9/10 to reflect the change from S=2.5 to S=2.25. This is in
the fourth sheet of the spreadsheet.
The various changes in S make 3session knockouts worth more than they
were. For this reason, the awards for brackets of sizes less than
16 have to be changed, so the awards are higher than for a 3session
knockout. This is accomplished by using a noninteger value for
the number of sessions in a knockout, again based on logarithms base
2. These are illustrated in the fifth sheet of the spreadsheet.
The sixth and 8th sheets of the spreadsheet contain examples. The
7th sheet deals with an anomaly called the D factor, which was
artificially created to compute match awards for swiss teams. The
new values for L allow some of that artificiality to be removed.
As I hope you can tell, I have been working for some time to create a
cohesive whole on which to base our masterpoint structure for the
future. Like the treatment of club games as tournaments, this
seems like a process whose time has come, particularly in light of the
development of a more modern ACBLscore. I know I have given you
much to digest. I hope you will read it carefully, particularly
for consistency.
One of my reasons for trying to do so much is my desire for
consistency. I believe some of the discussions about internet
games have not taken into account this consistency. I hope this
will put the entire picture on a mathematical basis devoid of emotionl
Thanks for your attention to this long message and even larger
spreadsheet.
Bill Arlinghaus

William C. Arlinghaus, Ph.D.
Professor
Emeritus, Mathematics and Computer Science
ACBL Board of Directors,
District 12
