Goal difference vs Head-to-Head - who benefits

This is a question that has bothered me on and off for a while now. Personally, I prefer the goal difference comparison - if only because you can get all the information you need from a simple table of standings (and don't have to go to a list of all the results to see who won).

Still, that's not a really good reason for preferring one method of 'ranking' over another. So, time for some meaningless numbers.

I've got this spreadsheet thing that can randomly generate a set of results for international football matches - with the likelihood of a result based on FIFA ranks and the typical relationship between past international results and FIFA ranks (now, there's a long story). If you run a lot of random trials (say, 50,000) for a group, and compare who would finish where based on GD and HTH rankings, you can get an idea of what the effects are.

So, I've done a number of different scenarios, based on a 4-team group playing a full 6 match schedule. To make it look vaguely realistic, the groups are of 4 African sides, using May 2007 FIFA ranks.

In each case, I've put in a table showing:

• Team
• FIFA rank
• % times finishing 1st under each method
• % times finishing 1st or 2nd under each method

Which gives a handle on what we might actually care about as a result of the group.

TRIAL ONE - A mixed group - no hopeless teams, but a range of abilities.

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Cameroon	968	13.1	1.34	1.36	73.7%	72.6%	93.1%	92.7%
Guinea	593	9.3	2.28	2.29	17.2%	17.8%	62.8%	62.4%
Togo	466	7.6	2.72	2.72	8.2%	8.6%	36.9%	37.2%
Liberia	224	4.0	3.65	3.63	0.9%	1.0%	7.2%	7.7%

TRIAL TWO - One good team - three no hopers.

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Cameroon	968	16.8	1.01	1.01	98.9%	98.7%	99.9%	99.9%
Guinea-Bissau	18	7.5	2.58	2.59	0.8%	0.9%	56.6%	56.0%
São Tomé e Príncipe	0	5.1	3.21	3.20	0.1%	0.2%	21.8%	22.2%
Djibouti	0	5.1	3.20	3.20	0.2%	0.2%	21.6%	21.9%

TRIAL THREE - Three evenly ranked sides - one no-hoper

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Egypt	694	11.0	2.00	2.01	33.8%	33.8%	66.7%	66.8%
Senegal	692	11.0	2.00	2.00	33.7%	33.7%	67.0%	66.8%
Mali	685	10.9	2.03	2.03	32.4%	32.5%	66.0%	66.0%
Somalia	13	1.5	3.97	3.96	0.0%	0.0%	0.3%	0.4%

TRIAL FOUR - One top side - three others around the same

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Cameroon	968	10.8	1.76	1.77	53.1%	52.3%	78.5%	77.9%
Egypt	694	7.7	2.72	2.72	16.2%	16.5%	41.3%	41.5%
Senegal	692	7.6	2.75	2.74	15.6%	15.9%	40.6%	40.9%
Mali	685	7.5	2.77	2.76	15.1%	15.3%	39.7%	39.8%

TRIAL FIVE - Two good - two bad

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Cameroon	968	14.2	1.42	1.42	59.2%	59.0%	99.3%	99.1%
Côte d'Ivoire	850	13.4	1.61	1.61	40.7%	40.9%	98.8%	98.5%
Guinea-Bissau	18	3.7	3.46	3.46	0.1%	0.1%	1.1%	1.3%
Somalia	13	3.4	3.52	3.52	0.0%	0.1%	0.8%	1.0%

TRIAL SIX - One good - two middling - one poor

Team	Rank	Ave Pts	Ave Pos GD	Ave Pos HTH	%1st GD	%1st HTH	%1-2 GD	%1-2 HTH
Cameroon	968	14.1	1.26	1.27	79.0%	77.9%	95.2%	94.9%
Togo	466	9.3	2.40	2.39	10.5%	11.1%	52.4%	52.5%
Zambia	462	9.3	2.41	2.40	10.5%	11.0%	51.8%	51.9%
Somalia	13	1.8	3.94	3.93	0.1%	0.1%	0.7%	0.8%

Pretty much overall, GD gives an advantage to higher ranked sides. Except where ranks are very close (so randomness comes into play), the GD ranking is more likely to see the best ranked side finish higher - or the best two sides finish 1-2.

Of course, there's millions of things that make this unrealistic, but the hope is (I suppose) that by looking at an average of 50,000 trials these are sort of smoothed away. However, I think the results are clear, and match what I might have expected, and match what happens in reality - namely that lower ranked teams tend to get an advantage from using HTH as the tie-breaker.

So, if you prefer the 'best' teams to make the finals, you're probably going to want GD used to rank, if you prefer 'surprises' (Angola ahead of Nigeria for example) then HTH is more your thing.