When the Data Lies: Finding Optimal Penalty Kicking Strategies with Game Theory

Penalties are among the most decisive and stressful moments in football. A single kick, taken only by the goalkeeper, can decide the outcome of an entire game or championship. From a data science perspective, they offer something even more interesting: a uniquely controlled environment for learning to make decisions under strategic uncertainty.

Unlike open play, penalties feature a fixed distance, one kicker, one goalkeeper, and a limited set of clearly defined actions. This simplicity makes them an ideal setting for understanding how data and techniques interact.

Suppose we want to answer a seemingly simple question:

Where should a kicker shoot to increase the chances of scoring goals?

At first glance, looking at historical data seems sufficient to answer this question. However, as we will see, relying only on raw statistics can lead to misleading conclusions. If the results depend on the cooperation of the strategies, the correct decisions cannot be considered by averages alone.

At the end of this article, we will see why the most successful penalty kick strategy is not the one suggested by the raw data, how game theory explains this apparent paradox, and how the same reasoning applies to many real-world problems involving competition and strategic behavior.

The Trap of Raw Conversion Rates

Imagine accessing a dataset containing many historical observations of penalty kicks. The first natural value we can think of is to measure the hit rate associated with each shooting position.

Suppose we find that penalties directed at the center are converted more often than those directed at the sides. The conclusion may seem obvious: kickers should always aim for center.

The hidden thought behind this assumption is that the goalkeeper's behavior remains unchanged. In fact, however, penalties are not independent decisions. It is a strategic interaction where both players get to know each other continuously.

If the kickers suddenly started pointing in the middle more often, the goalkeepers would react quickly by staying in the middle more often.
So the historical success rate of average shots reflects past strategic behavior rather than the intrinsic dimensions of that choice.

Therefore, the problem is not about identifying the best action to isolate, but about finding a balance where no player can improve their result by changing their strategy. In game theory, this balance is known as a Nash equilibrium.

Legalizing Penalties as a Zero-Sum Game

Penalty kicks can naturally be modeled as a two-player zero-sum game. Both the kicker and the goalkeeper need to choose a path at the same time. To keep things simple, let's assume they have just three options:

• Left (L)
• Center (C)
• Right (R)

In their choice, kickers aim to increase their chances of scoring goals, while goalkeepers aim to reduce them.

If $P$ means the chances of scoring goals, then the kicker's advantage $P$while the keeper's fee i $-P$. The payoff, however, is not fixed, as it depends on the joint choices of both players. We can represent the charge as a matrix:

$P= begin{bmatrix} P_{LL} & P_{LC} & P_{LR}\ P_{CL} & P_{CC} & P_{CR}\ P_{RL} & P_{RC} & P_{RR}\\end{bmatrix})$,

…where each element $P_{ij}$ represents the probability of scoring goals if the kicker chooses a direction $i$ and the goalkeeper chooses the way $j$.

Later we will estimate these probabilities from past data, but first let's make some sense of the problem using a simplified model.

Toy Model

To describe a simple but reasonable model of the payoff matrix, we assume that:

• If the kicker and goalkeeper choose different paths, the result is always a goal ($P_{ij}=1$ for $ine j$).
• If both choose the center, the shot is always saved by the goalkeeper ($P_{CC}=0$).
• If both choose the same side, a goal is scored $60$ of times ($P_{LL}=P_{RR}=0.6$).

This yields the following payoff matrix:

$P= begin{bmatrix} 0.6 & 1 & 1\ 1 & 0 & 1\ 1 & 1 & 0.6\\end{bmatrix}$.

Equilibrium techniques

How can we find the correct strategies for a kicker knowing the payoff matrix?

It is easy to understand that having a consistent strategy, i.e. making the same choices, cannot be right. If the kicker is always facing the same direction, the goalkeeper can quickly use this prediction. Likewise, a goalkeeper who always dives in the same way can be easy to beat.

In order to in order to reach equilibrium and remain unexploited, players must make their own random choiceswhich in game theory is called having a mixed strategy.

A mixed strategy is defined by a vector, whose elements are the probabilities of making a particular choice. Let's show the kicker's mixed strategy as

$p = (p_L, p_C, p_R)$,

and a mixed goalkeeper strategy as

$q = (q_L, q_C, q_R)$.

Equality is achieved when neither player can improve their outcome by changing their strategy unilaterally. In this context, it means that kickers should randomize their shots in a way that makes goalkeepers not concerned about diving left, right, or staying in the middle. If one direction provided the highest expected save rate, goalkeepers would use it, forcing kickers to adjust.

Using the payoff matrix defined earlier, we can calculate the scoring probabilities for all possible goalkeeper choices:

• if the goalkeeper dives to the left, the expected scoring probabilities are:

$V_L = 0.6 p_L + p_C +p_R$

• if you have a goalkeeper who stays in the middle:

$V_C = p_L +p_R$

• if the goalkeeper dives well:

$V_R = p_L + p_C + 0.6 p_R$

For a kicker strategy to be a balancing strategy, we need to find $p_L$, $p_C$, $p_R$ so that the goalkeeper's chance of conceding a goal does not change with their choice, which means we need that.

$V_L = V_C = V_R$,

which, and the general nature of the strategy

$p_L+p_C+p_R=1$,

gives a linear system of three equations. By solving this system, we find that the kicker's equilibrium strategy is

$p^* simeq (0.417, 0.166, 0.417)$.

It's interesting that even medium shots are much easier to maintain when expected, shooting in the middle about $16.6$ of times makes all options equally successful. Medium shots work well because they are rare.

Now that we are armed with knowledge of game theory and Nash equilibria, we can finally turn to real-world data and test whether professional players behave well.

Learning from Real-World data

We analyze an open dataset (CC0 license) that contains 103 penalties from the 2016-2017 English Premier League season. For each penalty, the data set records the direction of the shot, the direction chosen by the goalkeeper, and the final result.

By examining the data, we find that the total number of penalty goals is approx $77.7$and that medium shot seems to be very effective. In particular, we find the following values ​​for different shooting directions:

• On the left: $78.7$;
• Institution: $88.2$;
• Right: $71.2$.

To find the right strategies, however, we need to reconstruct the payoff matrix, which requires estimating nine conversion rates – one for each possible combination of kicker and goalkeeper options.

However, with only 103 observations in our dataset, certain combinations are rarely encountered. As a result, estimating these probabilities directly with raw calculations will introduce significant noise.

Since there is no strong reason to believe that the left and right sides of the goal are significantly different, we can improve the robustness of our model by putting balance between the two sides and combining equal conditions.

This effectively reduces the number of parameters to estimate, thereby reducing the variance of our probability estimates and increasing the robustness of the resulting payoff matrix.

Under this assumption, the empirical payoff matrix becomes:

$Psimeq begin{bmatrix} 0.6 & 1 & 0.86\ 0.94 & 0 & 0.94\ 0.86 & 1 & 0.6\end{bmatrix}$.

We can see that the estimated payoff matrix is ​​exactly the same as the toy model we described earlier, the main difference is that kickers can actually miss a goal even if the goalkeeper chooses the wrong way.

Solving equity strategieswe get:

$begin{aligned} p^* &simeq (0.39, 0.22, 0.39) \ q^* &simeq (0.415, 0.17, 0.415) end{aligned}$.

Are Players Really Good?

Comparing measurement techniques with observed behavior reveals an interesting pattern.

The kickers are very well behavedalthough they aim at the center less than they should ($16.5$ of times instead of 22%).

On the other hand, goalkeepers deviate too much from their proper techniques, only the center is left $6$ of times instead of correct $17$.

This explains why average images seem to be unusually successful in historical data. Their high conversion rate does not reflect intrinsic superiority, but rather a systematic dysfunction of goalkeeper behavior.

If both the goalie and goalkeeper follow their balancing techniques correctly, mid-range shots can be hit almost $77.8$ of time, which is close to the global average.

Beyond Football: A Data Science Perspective

Although penalty kicks provide an intuitive example, the same phenomenon is seen in many real-world data science applications.

Online pricing systems, financial markets, recommendation algorithms, and cybersecurity all involve agents that adapt to each other's behavior. In such cases, Historical data shows strategic fit rather than passive results. A pricing strategy that seems right on past data may no longer work if competitors react. Similarly, fraud detection systems change user behavior as soon as they are implemented.

In competitive environments, learning from data requires modeling interactions, not just correlation.

Conclusions

Penalty kicks represent a broader study of data-driven decision-making.

Historical estimates do not always reveal the right decisions. If results arise from strategic cooperation, the observed data reflect the balance between competing agents rather than the intrinsic quality of individual actions.

Understanding the mechanism that generates the data is therefore important. Without modeling strategic behavior, descriptive statistics can easily be considered objective guidelines.

The real challenge for data scientists is not just to analyze what happened, but to understand why rational agents do it in the first place.