## Tuesday, 9 May 2017

### A Champions League Mathematical curiosity

(Apologies for any reader who doesn't share my interest in football.  You don't have to read on.)

When I'm fed up marking, like so many other middle-aged men, I turn to the computer game Football Manager.  And a disaster for my team last night made me aware of a curiosity regarding the tie-break rules in the Champions League.

This is the situation.  It's the last minute of the last match of the group stage and my team, Arsenal, are losing 2-1 at home to Olympiakos (we haven;t had much luck and the  red card early in the first half for our midfielder Victor Wamyama has cost us).  But it's OK - Bayern Munich (who are losing in Lyon) will win the group and we will come second, qualifying for the knock-out stages, while Olympiakos will be out.

But what's this?  Bayern have scored a last-minute equaliser against Lyon!  They were going to win the group anyway, so presumably we still finish in that all-important second place.

But no - Olympiakos are now above us: we are down to third and we are out!  A goal in the other game, which doesn't change the position of the teams in that game, has caused Arsenal and Olympiakos to swap positions, with disastrous results.

It's all to do with tie-break rules.  Without Bayern's late goal, all four teams would have had eight points,  The tie-break rule looks at the scores in matches involving all the teams which have tied on points.  In this case Bayern (who scored six at home to Olympiakos) had the best goal difference, with Arsenal (who had played well until the last match, despite early red cards ion three matches) second best, and so they finish first and second.  But Bayern's equaliser means they have nine points, and Arsenal and Olympiakos are tied for second place: and since the two teams drew in Greece and Olympiakos won in London, the Greeks are above Arsenal.  That last-minute goal in the other match really has moved Olympiakos above Arsenal.  (If you want to check for yourself, I give all the scores below.)

This consequence of the mathematics of the tie-break rules is something I was vaguely aware of, though I didn't realise the peril my team were in until about ten game-minutes before Bayern's goal.  But it is slightly counter-intuitive that a goal in match B can cause the teams in match A to change positions.  (In fact in my game, before the goal Lyon had been in third place above Olympiakos, so conceding the goal caused them to drop to fourth, but a slightly different set of results could have meant that a game in match B could leave the two teams in that match in the same position while inverting the order of the other two teams.)

Here, for anyone who cares, are the complete results:
Game 1 - Arsenal (2) 2 Bayern (0) 0; Lyon (0) 1 Olympiakos (0) 0
Game 2 - Bayern (0) 0 Lyon (0) 0; Olympiakos (0) 1 Arsenal (0) 1
Game 3 - Lyon (0) 1 Arsenal (1) 3; Olympiakos (0) 1 Bayern (0) 1
Game 4 - Arsenal (0) 0 Lyon (0) 0; Bayern (3) 6 Olympiakos (0) 0
Game 5 - Bayern (0) 1 Arsenal (0) 0; Olympiakos (0) 1 Lyon (0) 0
Game 6 - Arsenal (0) 1 Olympiakos (0) 2; Lyon (3) 3 Bayern (0) 3