# Team scoring

## Track

The Olympics does not have a winning team. Gold, silver, and bronze medals are awarded in each event, and the number of gold, silver and bronze medals each country wins can be counted, but how can that information be combined to get a winning team? Should only the gold medals count? How many silver medals equal one gold medal? One possiblity is to score 3 points for a gold medal, 2 points for a silver medal, and 1 point for a bronze medal. Dual track meets score 5, 3, and 1 point for first, second, and third; does this give a different result? Consider a track meet between Amherst and Bowdoin where the order of finish in seven events are: AAB, BBA, ABB, ABB, ABB, BAB, ABB. If 5-3-1 scoring is used, Amherst has 32 points and Bowdoin has 31 points, hence Amherst wins. But if 3-2-1 scoring is used, Amherst has 20 points and Bowdoin has 22 points, hence Bowdoin wins. The results of these scoring systems are different, a first place is worth more than a second and a third together with 5-3-1 scoring, but is equal to a second and a third together with 3-2-1 scoring

Track meets with more than two teams are sometimes scored 6-4-3-2-1 for first, second, third, fourth, and fifth place. Consider a track meet between Brown, Dartmouth, Harvard, and Yale with four events which finished in the order DHYBB, BDHYB, HYBDD, YDHBB. 6-4-3-2-1 scoring gives Dartmouth 17 points, Brown and Harvard 16 points, and Yale 15 points. But if the results are used to score a dual meet between Brown and Dartmouth, the Harvard and Yale athletes are ignored providing the following finish orders: DBB, BDB, BDD, DBB. Brown wins 19 to 17 with 5-3-1 scoring, and 13-11 with 3-2-1 scoring. Hence the large meet scoring alters the relative ranking of Brown and Dartmouth compared to dual meet scoring.

Exercise: Track teams are often characterized as having strength (meaning athletes who are among the best in their event), and length (meaning several good athletes, allthough they may not be the best).
Does 5-3-1 versus 3-2-1 scoring favor strength or length? How?
Does multi-team scoring versus dual meet scoring in a large meet favor strength or length? How?
Does 6-4-3-2-1 versus 5-4-3-2-1 scoring favor strength or length? How?

## Cross country [We shall not discuss this]

In cross country, there is one race in which each team enters at least five runners, with the score determine by the sum of the positins of the first five runners on each team (the low score wins). Scoring is oftendescribed as five to place and two to displace, which means only the first seven runners on each team are counted in the finish order. For example, if the finish order of a race between Xavier and Yeshiva were XYYXXXYYYYYYXX, Xavier would have 1+4+5+6+12=28 points (although Xavier's fifth runner finished 13, only his four teammates and seven of Yeshiva's runners count as being ahead of him) and Yeshiva would have 2+3+7+8+9=29 points. The same scoing method (five to place, two to displace, with the low score winning) is used in larger meets.

## Regatta [We shall not discuss this]

A regatta often entails several races with every boat competing in each race, from which an overall winner is to be chosen. One method is to sum the positions each boat finished in for all the races (low score wins), and alternative is to use the same method, but score 0 instead of 1 for winning a race in order to provide a bonus for coming in first.

## Elections

Elections can be viewed as track meets where every candidate is a team and every voter is an event. Each voter ranks the candidates, so that if there are three candidates, each voter has a first, second, and third preference. For example, consider an election with three candidates Ambrose, Beulah, and Constantine; and five voters Raphael, Samantha, Theodosius, Ursula, and Victor. If the preferential orders of the voters are R:ABC, S:BAC, T:ACB, U:CBA. V:CAB; then 3-2-1 scoring gives Ambrose 11 points, Beulah 9 points, and Constantine 10 points; and Ambrose wins the election. This is an example of the Borda count method of determining an election. More generally, if there are n candidates, the Borda count method assigns n points for every fist ranking, n-1 points for every secocond ranking, . . . , and 1 point for every n-th ranking.

Competencies: If in a track meet between St. Ambrose and St. Bonaventure the order of finish in five events were ABB, ABA, BAA, AAB, BBB what would the scores of the two teams be, and who would win the meet? (use 5-3-1 scoring)
If the results of a cross country meet between Santa Ana and Santa Barbara were BAABBAAABAAABBBABB, what would be the scores of the two teams, and who would win?
In a three boat regatta between Archibald, Barthowlemew, and Chauncey, the order of finish for three 5 races were ABC, BCA, CAB, CBA, ACB. Who is the winner under 1-2-3 scoring? Who is the winner under 0-2-3 scoring? (include all the scores)

Reflection:Relay races essentially entail adding up the times (rather than relative positions) of the team members. Does this favor a team with strength or length? Why?
Does the fact that the same scoring system is used in large meets in cross country as in dual meets mean that the order from large meet scoring would be the same as if the relative positions of two teams were used for dual meet scoring? Why?

Challenge:

July 2009

campbell@math.uni.edu