# The Eligible Competition Scoring System

Of the many arcane aspects of ice skate competitions, perhaps the most difficult for the average spectator to understand is the scoring system used in eligible competitions. It is not uncommon at competitions to have the audience loudly boo marks that in reality are quite good, and cheer marks that in fact are quite bad. There are two reasons for this. First, the scoring system is not that easy to understand; and second, spectators are not provided all the information they need to understand the marks as they are posted. This month we will describe the ordinal scoring system and how it works. Next month we will discuss whether the method is "good" or "fair", and compare it to other possible scoring methods.

Except for figures, all eligible ice skating events are scored using two marks on a 6.0 scale, specified to one decimal place. A mark of 0.0 means "not skated" and a mark of 6.0 means "perfect and flawless". The lowest mark a skater can receive if they appear on the ice is a 0.1, even if they spend all their time picking their nose. The two marks go by various names in the different events and parts of events, but for now let's just call them the first mark and the second mark.

In eligible ice skating competitions the marks are only an intermediate step in determining the order of finish. What ultimately determines the order of finish is the order of finish each judge comes up with for each skater. The numbers that specify the order of finish determined by the individual judges are called the "ordinals". Unfortunately, ordinals are not posted along with the marks as an event takes place, so unless you have been writing down all the scores, or have an amazing memory, it may not be obvious from the marks alone whether the judges are agreeing with each other or not, and what place they have given each skater. Ordinals are posted following an event, and TV frequently will convert the marks to ordinals for the TV audience, but for the people in the arena interpreting the marks is somewhat of an ordeal. When looking at the marks, one must also remember that although the judges roughly attempt to judge on an "absolute" scale, this is only approximately so. One judge may be giving marks somewhat higher than average while another may be giving marks lower than average, yet both judges may be placing the skaters in the exact same order. As an extreme example, one judge could judge an event using only marks between 5.1 and 6.0 and another using marks between 0.1 and 1.0, with both judges coming up with the same order of finish. Thus, a low mark does not necessarily mean a low ordinal, or a high mark a high ordinal.

So how is it then, that results are determined from the marks?

First the two marks from each judge are added together and the total is used to place the skaters in order of finish from first through last for each judge. These numbers are the ordinals from each judge. If two skaters have the same total mark the tie is broken using the first mark in the short programs, and the second mark in the free skating programs and all parts of the dance event. If two skaters have the same first and second marks they are tied by the judge who (foolishly) did such a thing.

So far this should be pretty easy to follow. Using the marks you first figure out the order of finish for the skaters as decided by each judge. The hard part is understanding how the different orders of finish from all the judges are combined together to determine the final result. This is done using something called the majority principle.

The majority principle is a method of determining the majority opinion of the panel from the individual judge's ordinals. For a panel of nine judges, a skater has a majority for a given place when 5 or more judges give the skater that place or a higher place. To have a majority of firsts, five or more judges must place a skater first. To have a majority of seconds, five or more judges must place a skater first or second. To have a majority of thirds, five or more judges must place a skater first, second, or third. To have a majority of fourths, five or more judges must place a skater first, second, third, or fourth; and so on down to last place.

Consider now a skater with the following ordinals. This skater has two firsts (J1 and J9, not a majority), four seconds (J1, J2, J6, and J9, not a majority), and seven thirds (J1, J2, J5, J6, J7, J8, and J9, a majority).

```Judges Ordinals
J1 J2 J3 J4 J5 J6 J7 J8 J9
1  2  4  4  3  2  3  3  1

```

Here are some more examples.

```Judges Ordinals                 Majority

J1 J2 J3 J4 J5 J6 J7 J8 J9
1  2  1  1  3  2  2  1  1         5/1  (read as five firsts)

J1 J2 J3 J4 J5 J6 J7 J8 J9
4  3  5  6  3  4  4  5  3         6/4  (read as six fourths)

J1 J2 J3 J4 J5 J6 J7 J8 J9
7  7  7  8  7  7  5  9  7         7/7  (read as seven sevenths)

```

Only one skater can have a majority of firsts, but more than one skater might end up with a majority of another place. For example, in the following there is no skater with a majority of firsts and two skaters have majorities of seconds.

```
J1 J2 J3 J4 J5 J6 J7 J8 J9
1  2  1  1  3  2  2  2  1         8/2  (read as eight seconds)

2  1  2  3  4  1  1  1  2         7/2  (read as seven seconds)

4  3  3  4  1  4  3  3  3         6/3  (read as six thirds)

3  4  4  2  2  3  4  4  4         9/4  (read as nine fourths)

```

Skaters are first placed in order by their majorities. A majority of firsts places above a majority of seconds, seconds above thirds, and so on. If two or more skaters have a majority of a given place, the skater with a greater majority places above a skater with a lesser majority. In the above example the top two skaters have majorities of seconds, and since eight of a place is better than seven, the skater with eight seconds places ahead of the skater with seven seconds.

Two skaters could also end up with the same majority. In the following example both skaters have six thirds, and are tied according to their majorities.

```
J1 J2 J3 J4 J5 J6 J7 J8 J9
2  1  2  3  4  5  3  4  2         6/3

5  3  3  5  1  4  2  2  3         6/3

```

To break this tie we first look at the total of the ordinals from the judges that make up each skater's majority, called the "total ordinals of the majority". For the first skater, the six judges who have that skater in third place or above gave ordinals that total up to 13. For the second skater, the six judges who have that skater in third place or above gave ordinals that total up to 14. In this example, then, the skater with total ordinals of the majority of 13 beats a skater with 14.

Now suppose that judge-two in the above example had put the first skater in second place, then we would have the following ordinals.

```
J1 J2 J3 J4 J5 J6 J7 J8 J9
2  2  2  3  4  5  3  4  2         6/3

5  3  3  5  1  4  2  2  3         6/3

```

In this case both skaters have identical total ordinals of the majority of 14. To break a tie in total ordinals of the majority, all of the ordinals are totaled, called the "total ordinals". In this example, the ordinals for the first skater add up to 27, and for the second skater 28. Based on total ordinals the first skater places ahead of the second. If judge-one or judge-four, however, had put the second skater in fourth place, both skaters would have the same majority (six thirds), the same total ordinals of the majority (14), and the same total ordinals (27). In that case the two skaters would then be tied.

### Summary of procedure for placing the skaters

• Total the marks and determine the ordinals for each skater from each judge. Break ties using the first or second mark depending on the program being skated.
• Determine the majority for each skater from their individual ordinals.
• Place the skaters in order from best majority to worst majority.
• If any skaters have the same majority, break the tie using the total ordinals of the majority.
• If any skaters have the same majority and total ordinals of the majority, break the tie using total ordinals.
• If any skaters have the same majority, total ordinals of the majority, and total ordinals, those skaters are tied.

In most eligible competitions, events consist of two or more programs. To determine the result for the entire event the results from the individual parts of the event must be combined. The different parts of an event generally have different assigned weights. The greater the weight, the more important that part of the event. To determine the results for an event, the results from each part of the event are multiplied by the weight for that part and added together. This sum is called the "total factored place" for the event. The lower the total factored place, the better the final place for a skater; i.e, the skater with the lowest factored place wins. The short programs have weights of 0.5 and the free skating programs have weights of 1. A skater who is first in the short program and second in the long program has a total factored place of 1 X 0.5 + 2 X 1.0 = 2.5 If two skaters end up with the same totaled factored place, the results from the free skating (or free dance) breaks the tie.

Next month we will look at the pros and cons of this system and discuss proposals to change it.