As a follow-up to yesterday's post, here's more on the maths of football tournament mini-groups.
In last night's UEFA Euro 2012 matches, Croatia played Spain and Italy played Ireland. Because Ireland had lost both previous matches and the others had been drawn, it was likely that the tie-breaking rules would come into effect and the score in the Croatia-Spain match would be crucial. Assuming Italy beat Ireland, the winners of Croatia-Spain would go through and the losers would be out. Croatia and Spain would both qualify if they drew 2-2, Spain and Italy if the result was 0-0, while if Spain and Croatia drew 1-1 then Italy would have to beat Ireland by two goals and score three times to deny Croatia.
So we're in the last couple of minutes of both matches, Spain and Croatia is goalless and Italy lead Ireland 1-0. At this point Spain and Italy are on course to qualify. Croatia need a goal, but the interesting thing is that conceding a goal doesn't damage Croatia. In fact Spain score just before the end, but Croatia need to score just one goal to go through, and the Spanish goal makes no difference to them. At this point there is no likely situation in which a 1-0 win for Croatia is better for them than a 1-1 draw, unless Italy were to score two more goals in injury time. (In fact Italy scored one more to win 2-0 so perhaps that wasn't an impossible scenario.)
Indeed, perhaps Croatia are more likely to score if Spain have scored, since Spain are likely to be less worried about conceding an equaliser which won't put them out.
Spain's goal did change the order of the top two in the group, but, coming when it did, it made no difference at all to Croatia's chances.
I am reminded of the Scottish Premier Division 1990/91. Rangers and Aberdeen are fighting it out with two matches left. In these days it is two points for a win (not three) and one for a draw, and if two teams finish level then the deciding factor is goal difference, with goals scored then being decisive if both teams have the same goal difference. The last match of the season is Rangers against Aberdeen. With two matches left, Rangers are two points above Aberdeen. Aberdeen are winning their penultimate match but, in injury time, Rangers are trailing Motherwell 1-0. If it finishes like that, the teams will go into their last match level on points. Rangers will have scored 60 and conceded 21, Aberdeen have scored 62 and conceded 25. Rangers' goal difference is better and they need only draw the last match against Aberdeen to win the league.
So what do Rangers do? They throw caution to the wind in seeking an equaliser. Had they got an equaliser they would have gone into the last match a point above Aberdeen, and needed a draw to win the league. So scoring an equaliser would not have benefited them in the slightest: they need exactly the same from the last match as if they lose 1-0 to Motherwell. But in fact the reckless attack allowed Motherwell to score twice more in injury time. Rangers lost 3-0, their goal difference is now the same as Aberdeen's, they have scored fewer goals, and now they need to win the final match if they are to win the championship. The reckless pursuit of an equaliser which could make no difference has actually made the league title much harder to win.
(Rangers beat Aberdeen 2-0 so this mathematical madness didn't actually cost them the championship.)