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Groups search result 303 for group:rec.running author:Terry author:R. author:McConnell

Search Result 303
From: Terry R. McConnell (
Subject: Cross Country (XC) Scoring
Newsgroups: rec.running
View complete thread (4 articles)
Date: 1999/11/22

In an article on the recent National Masters XC race I neglected to mention
one important and unusual feature of the race: scoring was done by totaling
the times of the scoring members on each team. Lowest total time wins. (There
were varying numbers of "scoring" members in the various age divisions. For
example, the first 5 finishers' times were totaled in the Men's 40-49 
division, but only the first 3 finishers' times were used in the Men's 70-79

Why is this worthy of note? The normal method used to score XC races is
to total the *places* of finishers rather than their times. One effect of
the usual method is deemphasize the contributions of very fast runners (and the
failings of very slow ones.) The score is more reflective of team than
individual preformance, and seems more suitable to XC -- it being, after all,
the quintessential team sport.  I wonder why USATF has chosen the unusual
total time method? I'm sure they must have a reason, but it escapes me.

While I have the floor, I can't help rambling on a bit more about the usual
XC scoring, as it exhibits some interesting features from a mathematical
point of view. Consider, e.g, a 3-way meet involving teams we will call
(unimaginativly) A, B, and C. In addition to the usual 3-way competition,
it is not unusual for such meets to function as dual meets between each
of the 3 possible pairs of teams. It may seem counterintuitive, but it is
possible for team A to beat team B, team B to beat team C, and team C to
beat team A! This holds in the following example; moreover, the margin of
C's victory over A is more decisive than in any other pairing.

Example. Suppose for simplicity that 3 runners on each team score and
that the runners finish as shown:

Place:	1   2   3   4   5   6   7   8   9
Runner: A   B   C   B   C   C   A   A   B

( Only the team affiliation of a runner is recorded. )

In the 3-way meet, Team C wins with 14 points, team B is second with 15 points,
and team A is third with 16 points.

Now consider the A vs B dual meet. Runners for team C are irrelevant for this
meet, and so are ignored. We have

Place:  1  2  3  4  5  6
Runner: A  B  B  A  A  B,

and team A beats team B 10 points to 11. Similarly, it is easy to check that
team B beats team C 10 points to 11, and team C beats team A 9 points to 12.   

This phenomenon is a well known and serious issue in the mathematics of 
voting. The interested reader may wish to consult the following monograph
for further discussion:  D. Saari, Basic Geometry of Voting, Springer-
Verlag, 1995. 

Added, 11/25/2010: Recently the question arose whether it is possible to 
have a tie score in a cross country meet between two teams. In the most common 
scenario, where 5 runners on each team (and no others) count toward the score, 
the answer is certainly "no". The sum of all ten places is 55, and 55 is an odd
number. It is also common, however, to have one or more additional 
runners on each team who are allowed to influence the score indirectly, by 
"displacement." While such a runner's own place is not added to the total of
his team's, his presence may add to the opposing team's total by 
increasing the finish place of some of its scoring 5. A common setup is to 
allow up to 2 displacing runners. In this case, with 5 scorers, it is 
possible to have a tie. In the following example, each column gives the
finish places of up to seven runners from one team, while only the lowest 5 
places count in the totals:

29	29

Terry R. McConnell   Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150                   

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