When does a soccer team really face a crisis - and in which situations is the dismissal of a coach nothing but an over-hasty reaction? In a recently published study, researchers at Karlsruhe Institute of Technology (KIT) describe the development of a system that objectively measures crises in professional soccer. Based on three crisis indicators derived from real-world soccer characteristics, it is possible to accurately capture the performance level of a team and detect critical phases at an early stage. The results are published in the German Journal of Exercise and Sport Research (DOI: 10.1007/s12662-025-01078-1 ).
In their study, the KIT researchers show how mathematical models can be used to map emotional and mental processes in sports. Mathematics is used here as a medium to track down team dynamics. "We want to understand when and why a team enters a state of crisis and how such phases can be predicted by looking at objective data," says Professor Darko Jekauc, sports psychologist at KIT's Institute of Sports and Sports Science (IFSS), who headed the study. "Our indices translate expectations and actual results into mathematical key indicators that reflect psychological processes such as change of mood, confidence, and performance dynamics."
After analyzing the matches played in the 2023/2024 Bundesliga season, the research team came up with three key indicators: "Relative Position" (RP) quantifies how a team falls short of the expected place in the standings, taking its market value and last season's performance into account. "Linear Rate of Change" (LRC) shows long-term overperformance or underperformance trends. "Exponential Rate of Change" (ERC) captures recent performance variations and provides insight into the psychological momentum of a team. In particular, the ERC value proved to be a valuable early-stage indicator: In nearly all cases, this value dropped significantly before coaches were dismissed. "A sudden decline in ERC reflects a loss of energy and collective belief. This exactly signals the tipping point in dynamics, turning a mere feeling of uncertainty into a true crisis," Jekauc says.
Golden Ratio as a Means of Crisis Prevention
The crisis indicator ERC is based on phi, a value denoting the so-called Golden Ratio. This classic concept is applied in mathematics, art, and architecture as well as natural sciences - for example, when describing black hole dynamics or when modeling the demographic development. Now, the study is drawing attention to parallel phenomena in psychological processes.
The new crisis indicators can be applied in a wide variety of fields. A club's management can use them to monitor performance trends over the season and react to imminent downturns at an early stage. They facilitate data-driven decisions on training, team leadership, and communication issues. Sports psychologists can use them as a tool to promote mental strength and team resilience in a targeted manner and to provide scientific support during a crisis. "It is not our aim to take the emotion out of soccer," says Jekauc. "However, we want to show that even passion, pressure, and confidence levels can be mapped as mathematical patterns. If we understand these patterns, we cannot only anticipate, but also prevent crises."
Original publication
D. Jekauc, J. Jekauc, C. Rausch: Performance Crises in Professional Soccer: Developing and Validating Crisis Indices for Team Performance Assessment. German Journal of Exercise and Sport Research, Springer Verlag, 2025. DOI: 10.1007/s12662-025-01078-1 .
More information (in German only)
In close partnership with society, KIT develops solutions for urgent challenges - from climate change, energy transition and sustainable use of natural resources to artificial intelligence, sovereignty and an aging population. As The University in the Helmholtz Association, KIT unites scientific excellence from insight to application-driven research under one roof - and is thus in a unique position to drive this transformation. As a University of Excellence, KIT offers its more than 10,000 employees and 22,800 students outstanding opportunities to shape a sustainable and resilient future. KIT - Science for Impact.
