Despite growing interest in quantifying and modeling the scoring dynamics within professional sports games, relative little is known about what patterns or principles, if any, cut across different sports. Using a comprehensive data set of scoring events in nearly a dozen consecutive seasons of college and professional (American) football, professional hockey, and professional basketball, we identify several common patterns in scoring dynamics. Across these sports, scoring tempo-when scoring events occur-closely follows a common Poisson process, with a sport-specific rate. Similarly, scoring balance-how often a team wins an event-follows a common Bernoulli process, with a parameter that effectively varies with the size of the lead. Combining these processes within a generative model of gameplay, we find they both reproduce the observed dynamics in all four sports and accurately predict game outcomes. These results demonstrate common dynamical patterns underlying within-game scoring dynamics across professional team sports, and suggest specific mechanisms for driving them. We close with a brief discussion of the implications of our results for several popular hypotheses about sports dynamics. Professional team sports like American football, soccer, hockey, basketball, etc. provide a rich and relatively well-controlled domain by which to study fundamental questions about the dynamics of competition. In these sports, most environmental irregularities are eliminated, players are highly trained, and rules are enforced consistently. These features produce a level playing field on which competition outcomes are determined largely by a combination of skill and luck (ideally more the former than the latter). Modern sports in particular produce large quantities of detailed data describing not only competition outcomes and team characteristics, but also the individual events within a competition, e.g., scoring events, referee calls, timeouts, ball possessions, court positions, etc. The availability of such data has enabled many quantitative analyses of individual sports [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] . Relatively little work, however, has asked what patterns or principles, if any, cut across different sports, or whether there are fundamental processes governing some dynamical aspects of all such competitions. These questions are the focus of this study, and our results shed light on several other phenomena, including the roles of skill and luck in determining outcomes, and the extent to which events early in the game influence events later in the game. Game theory provides an attractive quantitative framework for understanding the principles and dynamics of competition [13] . Given a set of payoffs for different actions, formal game theory can identify the optimal strategy or probability distribution over actions against an intelligent adversary. In simple decision spaces, like penalty shots in soccer [14] or serve-and-return play in tennis [15] , professional athletes appear to behave as * sears.merritt@colorado.edu † aaron.clauset@colorado.edu game theory predicts (although some do not [16] ). However, most professional team sports exhibit large and complex decision spaces, with many possible actions of uncertain payoffs, and execution is carried out by an imperfectly coordinated team. Game theory provides less guidance within such complex games, and the resulting dynamics are often better described using tools from dynamical systems [17, 18] . Using such an approach, we investigate the withingame scoring dynamics of four team sports, college and professional (American) football, professional hockey, and professional basketball. Our primary goals are (i) to quantify and identify the common empirical patterns in scoring dynamics of these sports, and (ii) to understand the competitive processes that produce these patterns. We do not consider non-stationary effects across games, e.g., evolving team rosters or skill sets, playing field variables, etc. Instead, we focus explicitly on the sequence of scoring events within games. For each sport, we study three measurable quantities: scoring event tempo, balance, and predictability. We take an inferential approach to investigating their cross-sport patterns and present a generative model of competition dynamics that can be fitted directly to scoring event data within games. We apply this model to a comprehensive data set of 1,279,901 scoring events across 9 or 10 years of consecutive seasons in our four team sports. There are many claims in both the academic literature and the popular press about scoring dynamics within sports, and sports are often used as exemplars of decision making and dynamics in complex competitive environments [16, [19] [20] [21] . Our results on common patterns in scoring dynamics and the processes that generate them serve to clarify, and in several cases directly contradict, many of these claims, and provide a systematic perspective on the general phenomenon.
Work in REF models within-game scoring dynamics to predict game outcomes.
Scoring dynamics across professional team sports: tempo, balance and predictability
EPJ Data Science 3, 4 (2014)
Computer Science Mathematics Physics