The maths of "The Group of Death"

Tonight Spain play Croatia in the UEFA Euro 2012 football tournament.  At this stage teams are playing in groups of four - each team plays each of the other three (six matches) - with the top two in each group progressing to the next round.  The results of the first four matches mean that if Spain and Croatia draw tonight and both teams score at least twice, then Italy are knocked out whether or not they win their match against Ireland.

While sporting integrity (a strange concept currently being invoked in discussion about the future of Rangers in the Scottish Premier League) will ensure that Spain and Croatia both play to win, history suggests that if the score is 2-2 with five minutes remaining it would be irrational either side were to take risks in attempting to score a late winner.   A similar situation arose in 1982 when West Germany played Austria in the World Cup. A one-goal victory for the Germans would ensure that both sides progressed.  West Germany scored an early goal and thereafter, according to Wikipedia, "Onlookers noted that both teams played as though they were content with the result", the 1-0 outcome seeing both teams through at the expense of Algeria.  There have been other examples in international and club matches.  Robert Axelrod writes about an English league match in his book about the Prisoners' Dilemma and the mathematics of altruism, "The Evolution of Co-operation".


In fact there is a lot of (relatively simple) mathematics in sporting league tables.  Quite possibly my interest in mathematics began around my eighth birthday in 1965 when Kilmarnock had to beat Hearts 2-0 in the final match to win the league on goal average (the ratio of goals for to goals against), a much more complicated tie-break mechanism than the goal difference used now.


The current rules for deciding ties in mini-leagues in tournaments like Euro 2012 is that, rather than look at goal difference over all matches, instead we look at results over the matches involving the teams who have tied.  So if A and B finish with the same number of points (3 for each win and one for each draw), and A won the match between A and B, then A finish above B.  This avoids the situation which eliminated Scotland in the 1974 World Cup, when all three of Scotland, Yugoslavia and Brazil beat Zaire while the matches between these three were drawn.  Scotland played the weak Zaire side first which meant that Brazil knew exactly how many goals they had to score against Zaire to finish above Scotland.


While the newer rules work well in some circumstances, they still give rise to situations like tonight's Spain-Croatia match.  


As a boy I loved mathematical puzzles based around sporting league tables - typically one was given an incomplete table and had to work out the results of every match.  These appealed to my twin loves of sport and logic.  I noticed however that on the final edition of Dara O'Briain's School of Hard Sums, when such a problem was presented, it didn't enthuse the students on the programme: perhaps these problems no longer have wide appeal.  Nevertheless I still enjoy the mathematics of league tables.


For example, (1) in mini-leaguers of four teams as in Euro 2012, can a team win two matches out of three and still not qualify for the next round?  (2) Can a team win one and lose two, and still not qualify?  (3)What is the smallest number of points with which a team can qualify?


Answers: (1) Yes if three teams win two matches and the fourth loses all three - this would have happened if Denmark had beaten Germany in the "Group of Death" last night, since Germany would have been out despite winning their first two matchers.  (2) Yes, if one team wins three matches and the other three each beat one of the others - this would have happened if Netherlands had beaten Portugal last night. (3) A team can go through with one defeat and two draws, if one team win all three matches and the other three matches are all drawn.


One point which was obvious when league tables were displayed during the TV coverage of last night's matches is that, under today;s tie-breaking rules, traditional league tables (recording numbers of matches, wins, draws,losses, goals for and against, and points) do not contain enough information to show which teams will qualify.  If both last night's matches had been drawn, Denmark and Portugal would each have had four points. Portugal would have finished second, and qualified, because they beat Denmark in the match between these two teams.  But one couldn't have deduced that from thr group table.  Exactly the same numbers could have arisen in the group table with Denmark beating Portugal and drawing with Netherlands, and Portugal beating Netherlands.  


So whereas once a group table gave all the information you needed to decide the order of the teams, this is no longer the case.  How could we create an improved group table which actually included all the information we need?