Any Ideas?

Why UEFA is turning to ABBA

How to make penalty shootouts fairer.


The match has ended in a draw, so it's all down to penalties.  What's the best thing you can do to improve your chances of winning?  Always aim for the corner?  Save your best striker until the end?

Neither of the above, apparently.  

What you need to do is call correctly when the referee tosses the coin.  Then choose to take the first penalty.  According to a widely quoted analysis in 2010 by Ignacio Palacios-Huerta at the London School of Economics, teams that win the toss and elect to be the first to take a penalty win around 60% of all penalty shootouts.

This won't surprise most footballers or fans, the vast majority of whom already think that it's an advantage to take the first penalty.

So what can be done about it? At the moment, the teams (A and B) take it in turns to take five penalties each, and whichever team is ahead at the end is the winner.  If the two teams are still level after this stage, it goes to sudden death and the first team to score when the other misses is the winner.

Team B always shoots second, and going second is usually the pressure kick: it's the kick that will either save you from being knocked out, or  win the game for your side.  Either way, it's this added mental pressure that is believed to cause the second team to miss more often.

The new system being trialled by UEFA ensures that the team that shoots first alternates.  The first pair of penalties is A then B, the second is B then A, and then AB again.  This is nicknamed the ABBA system because it goes:

ABBA ABBA.... until there's a winner. And the wiinner takes it all.  (Sorry.)

The ABBA order seems a lot fairer.  In the five pairs of penalties, A will kick first three times and B will do so twice, so A only has a slight advantage.  And if it's level after five, B gets to kick first in the opening round of sudden death, which means both teams have had first kick three times. 

ABBA ABBA is similar to the ordering in a tennis tie break.  In tennis, player A serves once and then each player serves twice in turn, so it goes:  A BB AA BB AA.  The same pattern.  The difference is that in tennis, one player has to be ahead by two points in order to win.

There is one small snag with ABBA.  There is never a time when B has had the advantage of first penalty more times than A.  (A, on the other hand, has had more first kicks than B after every odd-numbered round).

Is there a fairer system than ABBA?  Yes.

There is a pattern called the Thue-Morse sequence that turns out to be the optimal pattern for fair-sharing in many situations where two parties are taking turns - including penalties.  The Thue Morse pattern starts like this:

ABBA BAAB BAAB ABBA BAAB...

Here, in the first five penalties, B goes first three times and A twice, so now B has the slight advantage.  The difference between this and ABBA, however, is that as you go deep into sudden death, this overall advantage switches between A and B.  There will be times when A has had more first kicks than B, and times where B has had more first kicks than A.

And when it comes to fairness, this matters. As recently as 2014, Liverpool beat Middlesborough 14-13 in a penalty shootout. (In the end both teams took 15 penalties. Surprise surprise, Liverpool went first.)  With ABBA ABBA there would have been eight moments when Liverpool had had first kick advantage over Middlesborough, but none when Middlesborough had had first kick advantage more often than Liverpool.  With ABBA BAAB, on the other hand, after 15 penalties both teams would have had this advantage four times.

But Thue Morse is harder to get your head around.  And you have to wonder if the relative complexity of the ABBA BAAB pattern justifies the very subtle benefits that it brings.  

ABBA is much fairer than ABAB, but ABBA-BAAB is only marginally fairer than ABBA.

So in choosing ABBA, EUFA has gone for best combination of fairness and simplicity. And in football, simplicity, well....that's the name of the game. (Sorry.)

 

Post Script.

There is another football situation in which Thue Morse is almost certainly the fairest system.  It’s the classic school[boy] lunchtime challenge of picking two teams to play in the playground.  Two captains are appointed, and they then take it in turns to pick the best remaining player.  However this gives captain A an advantage, because the first player picked is better than the second, the third is better than the fourth, and so on.  ABBA would be an improvement.  However, footballing ability does not decay linearly as you go through the squad - no player is worth zero, not even the last pick. More likely, ability decays exponentially towards zero,  When value decreases in this non-linear way, ABBA BAAB BAAB ABBA… is the fairest way to share resources.  So next time you're picking teams in the playground and want fairness, choose Thue Morse. Or you could end up with a case of Gimme Gimme Gimme my ball back. (Enough! Ed.)

 

I would like to thank the maths department at King Edwards School in Birmingham for first alerting me to this story.