ISU officials think the new One-by-One (OBO) scoring system will prevent last-minute "flip-flops" in the medal standings at major international figure skating competitions. But this is not the case, as the following example of the differences between the old and new scoring systems proves.

This example is drawn from the actual marks and skate order of six competitors in the men's short program at the 1998 Olympic Games. In the actual competition, these were the competitors who placed 11th through 16th in the field of 29 skaters.

Here are their marks, listed in their skate order:

Michael Shmerkin

4.7	5.2	5.1	4.7	4.9	4.9	4.9	4.7	5.1
5.0	5.4	5.4	5.0	5.2	5.3	5.2	5.2	5.2

Michael Weiss

4.6	5.1	4.9	4.8	4.8	5.0	4.9	4.6	5.0
5.0	5.6	5.5	5.3	5.3	5.4	5.4	5.3	5.3

Szabolcs Vidrai

5.0	5.0	5.2	5.1	5.1	5.2	5.1	5.2	5.2
4.8	4.9	5.1	5.0	5.1	5.1	5.1	4.9	5.0

Yamato Tamura

4.9	5.0	4.8	5.0	5.0	5.0	5.0	5.0	5.1
5.2	5.2	5.3	5.0	5.2	5.0	5.1	4.8	5.3

Igor Pashkevitch

4.6	4.8	4.9	4.5	4.9	4.9	4.7	4.5	4.8
5.6	5.4	5.7	5.1	5.5	5.6	5.3	5.2	5.4

Viacheslav Zagorodniuk

4.7	5.0	4.7	4.8	4.7	4.7	4.4	4.7	4.7
5.5	5.8	5.5	5.5	5.3	5.6	5.3	5.1	5.5

The old scoring system was based on ordinal rankings on a judge-by-judge basis. So, on the scoring sheet, the first step is to sort the skaters in each judge's column and number them from best to worst accordingly. Then, looking across each skater's row on the scoring sheet, the best placement for which the skater has a majority of ordinals is determined, and this is used to produce the overall placements of the skaters. The tie-breakers are the number of judges in the majority and their total ordinals.

Here is how the results unfold after going through this procedure to compute the incremental results for each skater in our example. Each row in the results table shows the ordinals from the 9 judges, followed by the majority size and place as well as the tie-breaker when needed.

After two skaters:

1. Michael Shmerkin	1	2	1	2	1	2	2	1	1	-- 5/1
2. Michael Weiss	2	1	2	1	2	1	1	2	2	-- 9/2

After three skaters

1. Szabolcs Vidrai	1	3	3	1	1	2	2	1	3	-- 6/2 (8)
2. Michael Weiss	3	1	2	2	3	1	1	3	2	-- 6/2 (9)
3. Michael Shmerkin	2	2	1	3	2	3	3	2	1	-- 6/2 (10)

After four skaters

1. Szabolcs Vidrai	2	4	3	1	1	2	2	1	4	-- 6/2
2. Michael Weiss	4	1	2	2	4	1	1	3	3	-- 5/2
3. Michael Shmerkin	3	2	1	4	3	3	4	2	2	-- 7/3
4. Yamato Tamura	1	3	4	3	2	4	3	4	1	-- 6/3

After five skaters

1. Michael Weiss	5	1	3	2	5	2	1	3	3	-- 7/3
2. Szabolcs Vidrai	3	5	4	1	2	3	2	1	4	-- 6/3 (12)
3. Yamato Tamura	2	3	5	3	3	5	3	4	1	-- 6/3 (15)
4. Michael Shmerkin	4	2	2	4	4	4	4	2	2	-- 9/4
5. Igor Pashkevitch	1	4	1	5	1	1	5	5	5	-- 5/4

After six skaters

1. Michael Weiss	6	2	3	3	5	2	1	3	3	-- 7/3
2. Szabolcs Vidrai	4	6	4	2	2	3	2	1	4	-- 5/3
3. Yamato Tamura	3	4	6	4	3	6	3	4	1	-- 7/4
4. Michael Shmerkin	5	3	2	5	4	5	4	2	2	-- 6/4
5. Igor Pashkevitch	2	5	1	6	1	1	5	6	5	-- 7/5
6. Viacheslav Zagorodniuk	1	1	5	1	6	4	6	5	6	-- 6/5

Notice how, although Shmerkin was initially ahead of Weiss and later Tamura, these placements were reversed as more competitors were marked. This is the kind of "flip-flop" that the new scoring system was supposed to eliminate.

Under the new scoring system, the scoring is computed using a square table the size of the number of competitors -- in this case a 6-by-6 table since there are 6 skaters. Each competitor is compared pairwise or "one-by-one" to all the others to fill in this table.

IN EACH COMPARISON, WE note how many "votes" each skater received from the judges, and count a "win" for the skater who got the majority of votes. Then, looking across each skater's row on the scoring sheet, we count the total number of "wins", which is used to rank the skaters overall, and the total number of "votes", which is used as the tie-breaker.

OBO does not change in any way the rules the judges use to assign marks to a skater's performance; it only affects how the marks are combined to produce the overall competition result. So, it is possible to use the same sample competition with the same marks and skate order to illustrate what the results would have been under OBO. In the interests of space, only the final completed version of the 6-by-6 table is shown here, and all of the separate worksheets for comparing the marks from each judge for each pair of skaters in the table have been omitted. The entries in this table show the ratio of "wins" to the number of "votes" in each pairwise comparison.

Skater	Shmerkin	Weiss	Vidrai	Tamura	Paskevitch	Zagorodniuk
Shmerkin	--	1/5	0/3	0/4	1/5	1/5
Weiss	0/4	--	1/5	1/6	1/5	1/6
Vidrai	1/6	0/4	--	1/6	0/4	1/6
Tamura	1/5	0/3	0/3	--	1/5	0/4
Pashkevitch	0/4	0/4	1/5	0/4	--	1/5
Zagorodniuk	0/4	0/3	0/3	1/5	0/4	--

This, of course, would only represent a small subset of the competitors involved in the actual event. Scoring the complete competition with 29 competitors would require a 29-by-29 table with 406 total pairings -- each of which requires an additional worksheet to compare marks from the 9 judges.

By contrast, the old ordinal system would require only a single 29-by-9 worksheet. The sheer number of computations makes OBO much more cumbersome for large competitions than the old system.

Now, here's a look at the incremental standings under OBO:

After two skaters

Skater	Wins	Votes
1. Shmerkin	1	5
2. Weiss	0	4

After three skaters

Skater	Wins	Votes
1. Vidrai	1	10
2. Weiss	1	9
3. Shmerkin	1	8

After four skaters

Skater	Wins	Votes
1. Vidrai	2	16
2. Weiss	2	15
3. Shmerkin	1	12
4. Tamura	1	11

After five skaters

Skater	Wins	Votes
1. Weiss	3	20
2. Vidrai	2	20
3. Shmerkin	2	17
4. Tamura	2	16
5. Pashkevitch	1	17

After six skaters

Skater	Wins	Votes
1. Weiss	4	26
2. Vidrai	3	26
3. Shmerkin	3	22
4. Pashkevitch	2	22
5. Tamura	2	20
6. Zagorodniuk	1	19

What happened to cause a flip-flop in the standings? Not only was there the same situation as under the old system -- with Shmerkin dropping behind Weiss and Vidrai after initially holding the lead -- but an additional flip-flop was introduced under OBO when Zagorodniuk's marks caused Pashkevitch and Tamura to swap places!

Looking closely at the table recording the one-by-one comparison results, the problem is that the "who-beats-who" relationships between skaters form complicated circular patterns. There is no possible way to rank this group of six skaters without going against at least one of the "wins". And, depending on the skate order, going against a "win" can cause the relative placements of the two skaters concerned to flip-flop.