Analysing World Cup penalty shoot-outs

Game theory and Penalties

Penalty shoot-outs have been an ever-present feature in at least one match in the World Cup finals since France failed to overcome the then West Germany in the 1982 semi-final in Spain. As well as providing an enthralling, if potentially painful spectacle, the penalty kick, in general, has also been a fertile area for both video analysis and as a real-life experiment in game theory.

Penalty takers invariably have a favoured, natural side to which they would prefer to shoot-outs. The swing of a right-footed player’s foot naturally sends the ball to the keeper’s right-hand side, usually more powerfully. In comparison, an attempt that is directed to the opposite corner relies more on placement of the shot and disguise.

Game theory interweaves decision-making rationality with psychological bias. If a taker predominately favours his natural side, the keeper can also regularly dive to this side to increase his chances of making a save, but the taker knows this so must act accordingly.  A degree of gambling is often part of a successful penalty save, because if a keeper waited until the ball had been struck, his reaction time would invariably leave him with insufficient time to reach a decently struck penalty.

Infrequent penalty takers in shoot-outs may prefer to trust more to the natural power of their strong side

Therefore, a taker should randomise his shoot-out decisions, rather than risk becoming predictable by continually choosing his natural side, even though he is likely to strike such an attempt with more power and accuracy. When both keeper and taker chose the same corner of the goal, the success rate for penalty kicks falls to below 70%. So it is vital to keep the goalkeeper guessing.

This aspect of penalty analysis has gradually gained a foothold in the modern game. Many regular penalty takers do vary their shoot-out tendencies in line with best practice, although randomised patterns are sometimes confused with regular change and therefore become predictable.

Infrequent penalty takers in shoot-outs may prefer to trust more to the natural power of their strong side, but more audacious practitioners, dating back to Panenka in the 1976 European Nations Cup and more recently Pirlo (Euro 2012 vs. England), have introduced a third alternative by chipping the ball in a gentle arc centrally to the area recently vacated by a gambling keeper.

Is it better to take the first penalty?

If game theory adds an additional layer of complexity to the penalty shoot-out, a more gambling-related headline statistic relates to the apparent advantage enjoyed by the team taking the first kick. 

Fifa will not use the ABBA system in penalty shoot-outs-outs in the 2018 World Cup despite research showing it is fairer to both teams.

Popularised in Soccernomics, a study by Ignacio Palacios-Huerta of 129 pre-2003 penalty shoot-outs resulted in just over 60% wins for the side taking the first kick of the sequence. This timescale was used because post this date, the side winning the toss could choose whether to kick first or second, but before 2003 they were compelled to. In the earlier dataset the order was decided by the coin toss and was therefore truly random.

Understandably, the 60% headline figure has become associated with penalty shoot-outs played out nowadays. An individual penalty from a front line taker is likely to be successful just under 80% of the time. So more often than not, the side going second will be playing from behind and it is easy to rationalise that the build-up of pressure increasingly erodes confidence and performance of the team kicking second.

A 60% success rate over 129 trials is possible if both sides had a 50% chance of winning the shoot-out, but it is unlikely and such figures are not considered statistically significant. So the evidence for a large first shooter advantage initially appears compelling but it may not be reflected in the odds available about each side once the coin toss and choice has been made.

However, there are objections. Firstly, on average, the side taking the first penalty may have an advantage, but it might not be as large as the 60% widely quoted. If teams had a marginally elevated overall 54% chance of winning when kicking first, they would be more likely to record a 60:40 split over 129 matches compared to a scrupulously fair contest. So the 60% rate may have arisen partly by chance in 129 iterations of an only slightly unfair contest.

Also, various alternative studies by Kocher, Lenz and Sutter, ranging from 262 to 470 shoot-outs from the same timeframe have failed to repeat the 60% figure. Instead they have recorded success rates for first shooters of around 53%, statistically consistent with the possibility of a fair contest. In World Cups this figure is around 58%.

First penalty advantage at World Cups

*Includes Spain vs. Russia and Poland vs. Croatia shootouts from the 2018 World Cup

More recent studies comprising some results from latter-day competitions, such as Prozone’s analysis of Euro and World Cups from 1998 until the present, do again appear to confirm the original premise, citing winning success rates of 75%.

However, there appears little reason to choose 1998 as the starting point and sample size is small. If sample sizes are increased to include all shoot-outs in these finals competitions since the practice was introduced, the success rate drops to 54%. These are levels more consistent with random variation in a predominately fair contest, rather than overwhelming psychological pressures being the cause.

The penalty shoot-out has been the preferred method of breaking a prolonged stalemate in many competitions worldwide. World and European international tournaments, Asian, African and Copa America competitions, as well as international and domestic club tournaments, have used this method. shoot-out results are relatively easy to find, but shoot-out order often requires match reports or YouTube to gather this evidence.

On average, the side taking the first penalty may have an advantage, but it might not be as large as the 60% widely quoted

Nevertheless, an afternoon of Google searching can easily yield 100+ shoot-outs from such domestic English competitions as the divisional playoffs and the cup competitions under their various guises and it is far from unusual to produce three figure subsets where it is the second shoot-out team which emerges with superior winning rates. In the UEFA European Championship, for example, the first team to take a penalty have lost 11 of the 18 shoot-outs-outs on record.

The 60% figure is almost bound to crop up if any of the latter stages of the World Cup go to a penalty shoot-out. But it would be unwise to assume an edge of this strength for the team electing or required to shoot-outs first. There is ample contradictory evidence, both latterly and from the same timescale as the original study.

Are some teams worse at penalties?

If, as the data suggests, penalty shoot-outs are effectively random, then England’s tag as penalty ‘bottlers’ is nothing more than an extension of the gamblers’ fallacy. For instance, If they have a 50/50 chance of winning, three defeats is no more abnormal than landing on black three times at the roulette table. With that in mind, if England go to penalties in the World Cup, bettors should seriously consider this logic before making their shoot-out predictions. However, do the odds tell a different story?

Interestingly, Pinnacle’s odds imply that teams do not, in fact, have an exact 50% chance of winning a penalty shoot-out. If the odds reflect reality then, in the event of a penalty shoot-out, the quality of each team needs to be taken into account.

The odds on a team winning a shoot-out are closely matched to the moneyline odds for each round of 16 tie at this year’s tournament:

Percentage chance of winning in the event of a penalty shoot-out (implied by odds)

                                                                    

*Money Line favourites in bold

As is shown here, prior to kick off the best teams are rated as more likely to advance in the event of a penalty shoot-out. This could be due to them potentially possessing marginally better quality penalty takers than weaker sides as well as superior goalkeepers.

The odds also ignore previous shoot-out results. Despite their poor record, England are favoured to advance in the event of a penalty shoot-out against Colombia. This makes sense since England’s previous penalty woes probably do not apply to their current squad, none of whom have taken a penalty in a World Cup shoot-out.

Despite the odds suggesting the stronger team has a greater chance of winning the shoot-out, the advantage enjoyed by the better side in all of these cases could be less than that of the team taking the first penalty.

Given all the factors in play, despite studies to the contrary, a penalty shoot-out should probably be treated as the near fair coin toss it was originally designed to be.

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