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With knockout style tournaments like the World Cup looms the threat of the dreaded penalty shootout. This article examines World Cup penalty shoot-outs, the game theory involved and examines whether or not it favours a team to go first or second.

Game theory and Penalties

Penalty shootouts 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. 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.
Therefore, a taker should randomise his shooting 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 shooting tendencies in line with best practice, although randomised patterns are sometimes confused with regular change and therefore become predictable.
Infrequent penalty takers in shootouts 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 penalties first or second?

If game theory adds an additional layer of complexity to the penalty shootout, a more gambling related headline statistic relates to the apparent advantage enjoyed by the team taking the first kick.
Popularised in Soccernomics, a study by Ignacio Palacios-Huerta of 129 pre 2003 penalty shootouts 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. Therefore, 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 shootouts 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 shootout, but it is unlikely and such figures are considered statistically significant. So the evidence for a large first shooter advantage initially appears compelling and 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 shootouts 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.
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 shootouts 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 shootout 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. Shootout results are relatively easy to find, but shooting order often requires use of match reports or YouTube to gather this evidence.
Nevertheless, an afternoon of Google searching can easily yield 100+ shootouts 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 shooting team which emerges with superior winning rates.
The 60% figure is almost bound to crop up if any of the latter stages of the World Cup go to a penalty shootout. But it would be unwise to assume an edge of this strength for the team electing or required to shoot first. There is ample contradictory evidence, both latterly and from the same timescale as the original study.
If as the data suggests penalty shootouts are effectively random, 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 shootout predictions.
Slicing and dicing data can be used to endorse often contradictory proposals and headline claims often lack the nuance of the extensive studies on which they are based. Despite studies to the contrary, a penalty shootout, if one arises at Brazil 2014, should probably be treated as the near fair coin toss it was originally designed to be.

[MoMoneyMoVaughn]

Excellent stuff try.