The penalty shoot-out (PSO) is monumentally unfair. The last time the World Cup was held in the U.S., for example, the tournament ended in the style of a tragic Italian opera. Roberto Baggio, Italy’s star striker who had scored five goals on the way to the final, stepped up to take the fifth penalty in the PSO. If Baggio missed, the Cup went to Brazil. If he scored Italy remained alive.
Baggio had an 85% career success rate at taking penalties, but with the outcome of the world’s most important trophy riding on his kick, he ran up and kicked the ball five yards over the bar. He later said that this tragedy lived with him for many years since that day in 1994.
The PSO is a way of settling tied games in single-elimination soccer championships, typically after 90 minutes of regulation play and 30 minutes of extra time. Until now they have worked like this: each team takes turns to shoot a penalty, with the team going first being decided by a coin toss. The highest score after five penalties wins. (In that 1994 championship match, Italy had missed three and Brazil scored three, Brazil won the shoot out 3-2 having taken only four penalties.) If still level after five penalties each team takes turns until one team is ahead (each team having made an equal number of attempts).
The system was adopted by FIFA, soccer’s international governing body, in 1970 and since then four World Cup finals have been decided by this method — two for the men (1994 and 2006) and two for the women (1999, when the U.S. defeated China, and 2011, when Japan out-shot the U.S). Dozens of other ties have been decided in the same way in the World Cup and many other competitions. The Germans are notoriously good at them: At the World Cup and European Championships they have participated in six shoot-outs since 1976 and lost only once (in 1976). Their conversion rate is a striking 85%; by contrast England and the Netherlands are the laughing stocks of the PSO, having lost six of seven, and five of six, respectively, with woeful conversion rates of 65% and 63%.
The team shooting first wins 60% of the time, and the team winning the toss almost always chooses to go first.
We know that the PSO is unfair thanks to the work of economist Ignacio Palacios-Huerta of the London School of Economics. The unfairness resides with the order in which the penalties are taken. After collecting data for hundreds of PSOs, Palacios-Huerta found that the team shooting first wins 60% of the time, and the team winning the toss almost always chooses to go first. In other words, simply by winning the toss, you give yourself an enormous edge that may well override merit, skill or preparation. What is the source of this advantage? It is probably psychological pressure that accounts for the discrepancy: If you shoot first and miss, you can still be bailed out either by your opponent’s or your goalie’s performance, but if you shoot second, the entire game rests on your shot and yours alone, the previous 120 minutes be damned.
Palacios-Huerta has done more than simply demonstrate the unfairness. He identified the ideally fair system and then persuaded notoriously conservative soccer authorities to experiment with a fairer system. Economists struggle to influence economic policy these days, so persuading a sports governing body to change its rules is nothing short of miraculous.
First the theory: The sequence in traditional PSOs is the “ABAB” system — team A alternates with team B. The problem is that if there is any advantage in the order (going first or second), then the sequence perpetuates this advantage endlessly. It follows, therefore, that any advantage in the ordering from the first two attempts (AB) can be nullified simply by reversing the order (BA).
So now the sequence is ABBA. If there is any advantage to one side in the sequence ABBA, then this can be nullified by reversing the sequence for next four attempts (BAAB). Fairness then is ensured by reversing the current sequence in the following sequence.
This sequence itself has a name — the Prouhet-Thue-Morse sequence, named for the two mathematicians and an inventor who independently discovered it (this Morse is same one who invented Morse code). Palacios Huerta realized that implementing this ideal sequence might seem a little complicated, so he advocated a close approximation, the “ABBA” system. Tennis fans will note that this is essentially the sequence used to settle tie-breaks in that sport. It’s not quite as fair as the ideal, but is much closer than ABAB.
Palacios-Huerta had one advantage in his efforts to persuade the authorities to change — he is a board member of the celebrated Basque soccer club, Athletic Bilbao. In 2017 FIFA announced that it would experiment with the system and in May 2017 Norway’s under-17 women were the first team to lose an international match under the ABBA system (to Germany, of course). Since then some national federations have adopted ABBA for domestic cup competitions.
FIFA remains tight-lipped about whether the system will be extended to the World Cup. Unsurprisingly, many soccer fans, long noted for their traditionalism, have complained about the newfangled procedure. But if your team loses the toss in a PSO this World Cup under the old ABAB system, don’t say you weren’t warned.
Stefan Szymanski is the Stephen J. Galetti Collegiate Professor of Sport Management at the University of Michigan’s School of Kinesiology. He is the author, with Simon Kuper, of Soccernomics: Why England Loses; Why Germany, Spain, and France Win; and Why One Day Japan, Iraq, and the United States Will Become Kings of the World’s Most Popular Sport (Nation Books)