Olympic medal table - epilogue (more maths)

This is my last word (for now) on the Olympic medal table problem. One or two people have rightly pointed out to me that the choice of three medals is somewhat arbitrary, and that really we should be counting all the placings when trying to rank countries. A country whose athletes came fourth in everything would surely merit some recognition, after all!

One problem with increasing the number of "medals" is that differences in the sizes of events begin to become more apparent. Coming tenth out of 100 participants is clearly more of an achievement than coming tenth out of ten! But setting that problem aside, what happens to my medal weighting system, which is based on an average of plausible weighting systems?

With three medals, one way of writing the relative scores that helps us see a pattern is as follows:

Gold   = 1 + 1/2 + 1/3 = 11/6
Silver =     1/2 + 1/3 =  5/6
Bronze =           1/3 =  2/6

and in my original article I multiplied these all by 6 to get my whole-number points system 11, 5 and 2.

So if we had four medals, it would be

Gold   = 1 + 1/2 + 1/3 + 1/4 = 25/12
Silver =     1/2 + 1/3 + 1/4 = 13/12
Bronze =           1/3 + 1/4 =  7/12
Iron   =                 1/4 =  3/12

so 25, 13, 7 and 3 points.

And with ten medals (I'll give up trying to rank metals by value!), leaving out the calculations, we end up with this rather unwieldy set of scores: 7381, 4861, 3601, 2761, 2131, 1627, 1207, 847, 532, 252.

It is tempting to think that the relative scores might tend to some kind of pattern as we keep adding medals.  We could even ask ourselves what would happen if we had an infinite number of medals!  The resulting "universal" scoring system could then be applied to real-world events by just using the top few scores from the list.

Unfortunately we get into problems with an infinite number of medals.  The gold score would be

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

also written as

Mathematics is sometimes kind to us and allows us to calculate these infinite sums, but sadly this one doesn't work - there is no limit as you add more terms to the sum.  You have to stop somewhere, like I did at 10 in the example above.  Some insight into the finite sum can be gained by replacing the discrete sum by a continuous sum known as the integral, which in this case happens to have an easy solution.  The right hand expression above can be replaced by

So having chosen our value of n, the number of medals, we have an approximation to the gold medal score (I'll call it S(1) meaning score for 1st place) which is

Then if we go back to our patterns to work out the scores for the other medals, we get

or, cutting out the working:

This gives a reasonable approximation given the arbitrary nature of all the original assumptions.  It could probably be improved by tweaking the approximation of the discrete sum by the continuous integral (I just placed each term of the sum in the middle of one unit's worth of the integral).  Oh, so it's probably not the last word after all...

Bonfire Night and Halloween

A hearty greeting of "Happy Guy Fawkes Day" from a dear American friend has prompted this post. That's not a greeting we Brits tend to use, but it's well-meant and perfectly understandable given the American custom of saying "Happy Halloween"; again, that is not something you often hear in the UK. Before anyone concludes that we're a miserable bunch, we do say "Happy Christmas", "Happy Easter", "Happy New Year" and "Happy Birthday"!

The Wikipedia article about Guy Fawkes Day, or as we more commonly call it, Bonfire Night, says "...another old celebration, Halloween, has lately increased in popularity, and according to some writers, may threaten the continued observance of 5 November." Halloween (sometimes spelt Hallowe'en to give us more of a clue that it is the eve of All Hallows' or All Saints' Day) is often seen as an American import that has grown massively in popularity in recent years. In my childhood hardly anyone carved pumpkins, held Halloween parties or did trick-or-treat, but now nearly all children will do those things. One thing that intrigues me given my evangelical Christian formative years is that Halloween seems to be an acceptable celebration in the much more strongly evangelical US, whereas in many of our churches it was (and still is to some extent) a no-no seen as having heavy pagan connotations, to be replaced if possible by a church-based festival with nary a witch or ghoul in sight. Perhaps there is as just much variety of opinion in America, and I should be wary of seeing attitudes there as quite so monolithic.

Anyway, going back to the Wikipedia comment, a few years ago I too thought that the much more commercial Halloween might displace our traditional English Bonfire Night. Thankfully, this does not seem to have happened. There is some mixing of the two, but by and large they now occupy different spaces. Halloween is for family and friends, though with quite a strong commercial input. Bonfire Night is much more village and town-based, with huge bonfires and spectacular firework displays. If Halloween has displaced anything, it might be the family fireworks "display" in which Dad (yes, usually Dad) got everyone to stand back while he lit a couple of rockets and a soggy Catherine wheel which always failed to rotate properly on its nail on the shed. This is a good thing; ten quid spent on tickets for a public display is much better value than the same spent on fireworks for the back garden, as well as being a lot safer. As for Halloween, I have no problem with the witches and scary stuff, but I do find it hard to see trick-or-treating in a favourable light, being as it is the demanding of money with menaces. So my vote goes for Bonfire Night - burning the effigy of a papist... oh, maybe I'm being a bit inconsistent here!

Mike's Olympic medal table, end of 3 Aug 2012

	Country	G	S	B	Score
1	United States	21	10	12	305
2	China	20	13	9	303
3	United Kingdom	8	6	8	134
4	France	8	5	6	125
5	Korea	9	2	5	119
6	Germany	5	9	6	112
7	Russia	3	12	8	109
8	Japan	2	8	11	84
9	Italy	4	5	3	75
10	Australia	1	9	4	64
11	DPR Korea	4	0	1	46
12	Kazakhstan	4	0	0	44
13	New Zealand	3	0	3	39
14	South Africa	3	1	0	38
15	Romania	1	4	2	35
16	Cuba	2	2	1	34
17	Netherlands	2	1	3	33
18	Hungary	2	1	2	31
19	Ukraine	2	0	4	30
20	Poland	2	1	1	29
21	Brazil	1	1	4	24
22	Belarus	1	1	2	20
23	Canada	0	2	5	20
24	Mexico	0	3	1	17
25	Slovenia	1	0	2	15
26	Colombia	0	2	1	12
27	Spain	0	2	1	12
28	Ethiopia	1	0	0	11
29	Georgia	1	0	0	11
30	Lithuania	1	0	0	11
31	Venezuela	1	0	0	11
32	Czech Republic	0	2	0	10
33	Sweden	0	2	0	10
34	Denmark	0	1	2	9
35	Belgium	0	1	1	7
36	Indonesia	0	1	1	7
37	India	0	1	1	7
38	Kenya	0	1	1	7
39	Mongolia	0	1	1	7
40	Norway	0	1	1	7
41	Slovakia	0	0	3	6
42	Croatia	0	1	0	5
43	Egypt	0	1	0	5
44	Thailand	0	1	0	5
45	Chinese Taipei	0	1	0	5
46	Azerbaijan	0	0	1	2
47	Greece	0	0	1	2
48	Hong Kong, China	0	0	1	2
49	Iran	0	0	1	2
50	Moldova	0	0	1	2
51	Qatar	0	0	1	2
52	Singapore	0	0	1	2
53	Serbia	0	0	1	2
54	Uzbekistan	0	0	1	2

Mike's Olympic medal table noon 3 Aug 2012

		G	S	B	Score
1	China	18	11	5	263
2	United States	18	9	10	263
3	Great Britain & N. Ireland	6	6	5	106
4	Germany	5	8	5	105
5	France	6	5	6	103
6	South Korea	7	2	5	97
7	Russian Federation	3	6	8	79
8	Japan	2	6	11	74
9	Italy	4	5	2	73
10	Australia	1	8	4	59
11	North Korea	4	0	1	46
12	Kazakhstan	3	0	0	33
13	South Africa	3	0	0	33
14	Netherlands	2	1	3	33
15	Hungary	2	1	2	31
16	Ukraine	2	0	4	30
17	Romania	1	3	2	30
18	New Zealand	2	0	2	26
19	Brazil	1	1	2	20
20	Canada	0	2	5	20
21	Belarus	1	1	1	18
22	Mexico	0	3	1	17
23	Slovenia	1	0	2	15
24	Cuba	0	2	1	12
25	Colombia	0	2	1	12
26	Georgia	1	0	0	11
27	Lithuania	1	0	0	11
28	Venezuela	1	0	0	11
29	Sweden	0	2	0	10
30	Denmark	0	1	2	9
31	Indonesia	0	1	1	7
32	Spain	0	1	1	7
33	Mongolia	0	1	1	7
34	Norway	0	1	1	7
35	Poland	0	1	1	7
36	Belgium	0	1	1	7
37	Slovakia	0	0	3	6
38	Thailand	0	1	0	5
39	Chinese Taipei	0	1	0	5
40	Croatia	0	1	0	5
41	Egypt	0	1	0	5
42	Czech Republic	0	1	0	5
43	Greece	0	0	1	2
44	Qatar	0	0	1	2
45	India	0	0	1	2
46	Singapore	0	0	1	2
47	Serbia	0	0	1	2
48	Uzbekistan	0	0	1	2
49	Moldova	0	0	1	2
50	Azerbaijan	0	0	1	2