Are Players Chasing the Best Outcome or Just Playing It Safe?
Investigating the Influence of Behavioral Considerations on Penalty Kick Location with Game-Theory and Regression.
The following summary critically reviews the research paper titled "Do Behavioral Considerations Cloud Penalty-Kick Location Optimization in Professional Soccer: Game Theory & Empirical Testing using Polynomial Regression and ML Gradient Boosting" by Alivia Uribe, Shane Sanders, Justin Ehrlich, James J. Reade and Carl Singleton. All data, figures, and analysis presented here are drawn from their original work; I do not claim any authorship or ownership of the content. This summary has been written to provide a concise and technically informed synthesis of the paper’s findings, methodologies, and implications, while maintaining fidelity to the authors’ intellectual contributions.
1. Introduction
Penalty kicks (PKs) in football occur roughly once every four matches and carry outsized importance in determining match outcomes due to the sport’s low-scoring nature. In a dataset of nearly 295,000 matches, common scorelines such as 1-0 (17.9%) and 1-1 (11%) underscore the frequency with which a single goal can decide a game [2]. PKs account for approximately 6.8% to 7.6% of total goals in recent English Premier League seasons, highlighting their strategic significance [3,4]. The paper investigates whether professional players choose PK shot locations solely to maximize expected conversion rates or if other behavioral motivations, like avoiding visibly embarrassing misses, influence their decisions.
Analogous behavior has been documented in other sports. Despite the underhand free throw's statistical benefits, NBA players like Shaquille O’Neal and Andre Drummond have refused to adopt it due to its perceived optics, what O’Neal called a “shot for sissies”, even though their career free-throw percentages (52.7% and 48.2%, respectively) significantly underperform the league average of 78.4% [7,8]. These choices have led to defensive strategies like “Hack-a-Shaq,” which exploit such players' shooting inefficiencies. Similar aesthetic pressures affect corner three-point shots, where players tend to miss short (front rim) rather than commit socially awkward errors like hitting the backboard's side [9].
These examples suggest a broader behavioral pattern: athletes may prefer choices that maintain the appearance of technical competence, even at the expense of performance. The present study hypothesizes that this phenomenon also applies to football PKs, players may intentionally avoid riskier but potentially more rewarding shot placements if those placements carry a higher risk of visibly missing the target. The study examines this hypothesis by blending game theory, statistical modeling, and empirical data to assess how shot allocation decisions reflect a trade-off between optimizing scoring chances and managing reputational optics.
2. Methods
This section combines theoretical modeling and empirical analysis to examine penalty kick (PK) shot-location decisions. The authors first construct two theoretical frameworks: a classical game-theoretic model and a statistical game-of-chance. These are complemented by an empirical analysis of 536 PKs from the 2015–2020 UEFA Champions and Europa Leagues.
The classical game-theoretic model is structured as a two-player, constant-sum game between the penalty taker (T) and goalkeeper (G), with simplified strategies: shoot center or shoot corner. The model assumes that each player maximizes their respective payoffs (conversion for T and save (non-conversion) for G). Best-response strategies are analyzed to identify potential Nash equilibria.
The empirical component maps the goal into eight equal partitions ({Top Left, Top Middle Left, Top Middle Right, Top Right, Bottom Right, Bottom Middle Right, Bottom Middle Left, Bottom Left}) as in Almeida et al. (2016) [10]. Off-target shots are assigned to the nearest partition, assuming shooters intend to target that area. Partition-level shot volume and conversion rate are measured to identify patterns in spatial shot allocation.
Statistical estimation is conducted using linear, polynomial, and Lasso-regularized regression models to explain shot-volume as a function of both optimization variables (expected conversion rate) and behavioral variables (expected off-target rate, conditional on a miss). These models test whether players prioritize scoring efficiency alone or also factor in reputational risk when choosing shot locations.
3. Results
3.1 Classical Game-Theoretic Model: PK-Attempt as a Classical Game between PK-Taker and Goalkeeper
The authors formulate a discrete, constant-sum two-player game between a penalty taker (T) and goalkeeper (G), with binary strategy sets: shoot center or corner (for T), and stay center or dive corner (for G). Payoffs are based on empirical conversion rates, with T’s payoff being the probability of scoring and G’s the complement.
The normal-form matrix reveals no pure strategy Nash equilibrium (NE), as there is no strategy profile where both players simultaneously play their best responses. Instead, a mixed-strategy Nash equilibrium (msNE) is derived. Solving the indifference conditions yields equilibrium strategies:
pT∗=1/4 (T shoots center 25% of the time),
qG∗=1/4 (G stays center 75% of the time),
with expected payoffs πT∗=3/4 and πG∗=1/4.
These results reflect strategic interdependence: T mixes to avoid being predictable, while G optimally counters. Crucially, the msNE predicts zero correlation between partition-level shot volume and conversion rate; since mixed strategies equalize expected outcomes across choices, rendering spatial allocation theoretically uninformative about efficiency. This benchmark sets the stage for empirical testing of behavioral deviations.
3.2 Alternative Theoretical Model: PK-Attempt as a Mixed (Continuous-Discrete) Statistical Game of Chance between PK-Taker and Nature
This section reformulates the PK scenario as a statistical game of chance between the taker (T) and “Nature” (C), where C determines match-specific conditions affecting expected conversion rates for center and corner shots. Each PK attempt is treated as a realization from two independent normal distributions: πT,i(SCe)∼N(0.8,0.05) for center shots and πT,i(SCo)∼N(0.75,0.05) for corner shots.
T, assumed to have rational expectations, chooses the shot with the higher expected payoff on each trial. Simulating 1 billion such trials yields a corner shot selection rate of 76%, closely matching the mixed strategy NE derived earlier.
Critically, the observed conversion rates (conditioned on being optimal and thus chosen) are 81.4% for corners and 79.6% for center shots. These values are higher than their underlying distributions’ means due to selection bias: only optimal decisions are observed. This leads to a positive correlation between location frequency and conversion rate.
Bootstrapped resampling over 100 simulated samples confirms a strong positive correlation (ρ=0.729). Thus, in contrast to the classical game, the statistical model predicts a significant positive volume-performance relationship.
This contrast between models yields a joint null hypothesis (H_0): if either game governs behavior, we should observe non-negative correlations between shot volume and conversion rate across all location subsets.
Empirical analysis reveals that players disproportionately avoid high-conversion partitions when these carry a higher miss risk, violating predictions from both game-theoretic models. Spatial heatmaps of conversion rate and shot volume show inverted patterns: top partitions, despite higher conversion rates, receive fewer shots, while safer bottom partitions attract more attempts.
Top-partition shots have a 12.88% miss rate versus 2.48% for bottom ones, while save rates flip: 5.30% (top) vs. 23.51% (bottom). This suggests players avoid high-risk, high-reward locations, likely due to reputational concerns tied to missing the goal entirely.
Partition-level correlation analysis between shot volume and conversion rate reveals strongly negative correlations: -0.749 for corner partitions and -0.997 for middle ones. Bootstrap resampling confirms these are statistically significant, violating both classical and statistical game predictions which require non-negative correlations.
This mismatch suggests that behavioral considerations (particularly the optics of missing) bias decision-making away from expected value maximization.
Two regression models (one linear, one polynomial) estimate how expected conversion rate (optimization) and expected off-target rate (behavioral optics) influence shot location. Both variables are statistically significant in both models. As expected conversion rate increases, shot volume rises; as expected off-target rate increases, volume falls. The polynomial model (Model 2) confirms a non-linear, concave relationship, suggesting diminishing returns and risk aversion.
Model 2 explains 86.8% of variation in shot volume, supporting intentional shot allocation. A Lasso regression confirms robustness to multicollinearity, yielding similarly signed and significant coefficients: 1.154 (conversion rate), −0.387 (off-target rate).
Regression estimates suggest players balance optimization and behavioral concerns. A derived indifference equation from Model 1 shows that players would “trade” a 1% increase in conversion rate for a 3.12% increase in on-target rate. This indicates that players moderately value appearance-based outcomes alongside goal-maximization.
4. Conclusion
Theoretical models predict that optimal penalty-kick (PK) strategies should yield a non-negative correlation between shot volume and conversion rate across goal partitions. However, empirical analysis of 536 UEFA PKs reveals strong negative correlations within top-vs-bottom goal partitions, inconsistent with classical or statistical equilibria. Regression and Lasso models show that both optimization (conversion rate) and behavioral concerns (miss optics) significantly influence shot location. Players avoid high-miss-risk areas, even at the cost of lower scoring potential. Indifference curve analysis indicates that while optimization utility dominates (3.12x more important), behavioral utility still meaningfully shapes decisions. Players are thus best described as hybrid agents; “part rational optimizers, part appearance-driven”.
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References
Uribe, A., Sanders, S., Ehrlich, J., Reade, J. & Singleton, C. (2025). Do Behavioral Considerations Cloud Penalty-Kick Location Optimization in Professional Soccer: Game Theory & Empirical Testing using Polynomial Regression and ML Gradient Boosting. https://cdn.prod.website-files.com/5f1af76ed86d6771ad48324b/67920109fe4182e7da894011_Do%20Behavioral%20Considerations%20Cloud%20Penalty-Kick%20Location%20Optimization%20in%20Professional%20Soccer.pdf
To keep this article concise, please refer to the original paper for the full list of references.