Penalty kicks: an adversarial risk analysis (ARA) perspective

Samuel Luxenberg; Refik Soyer; Sudip Bose

doi:10.1515/jqas-2024-0022

Abstract

Penalty kicks are critical to game outcomes in soccer. The typical quantitative strategy is as follows. First, model such an interaction as a two or three strategy game between the kicker and the goalkeeper. Second, find the mixed-strategy Nash equilibrium (MSNE) to determine the “optimal” probabilities of each player choosing to kick or dive to either side or to the center of the goal. While this is the usual path to a solution, it is also fraught with many assumptions due to the nature of penalty kick data as well as due to implicit assumptions within the game theory model. In this paper, we introduce an alternative set of strategies, known as adversarial risk analysis (ARA), to determine optimal decisions for the penalty kick game. ARA, which is grounded in the principles of Bayesian decision analysis, allows the decision maker to avoid many of the assumptions and limitations encountered when using the game theory approach. By examining 2018–2019 Major League Soccer (MLS) penalty kicks on an individual basis, we show that our most accurate ARA model predicts the correct goalkeeper decision 65 % of the time, while the game-theoretic model predicts correctly 51 % of the time, a statistically significant difference.

Keywords: Bayesian decision analysis; game theory; penalty kicks; soccer analytics; sports analytics

Corresponding author: Samuel Luxenberg, Department of Decision Sciences, The George Washington University, Washington, DC, USA, E-mail: samluxenberg@gwu.edu

Acknowledgement

The authors would like to thank Professor David Rios Insua of ICMAT for his insightful comments on an earlier version of this manuscript.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: None declared.
Data availability: The authors confirm that the data supporting the findings of this study are available within the article or its supplementary materials.
Software availability: All analyses in this study were conducted using R (4.2.2), and custom scripts were written. The authors confirm that the code supporting the findings of this study is available in the article’s supplementary materials.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/jqas-2024-0022).

Received: 2024-02-04

Accepted: 2024-12-26

Published Online: 2025-01-27

Published in Print: 2025-06-26

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