# Sports Dynamics

Which team wins the points is coin flip a Bernoulli process that depends on the relative skill difference of the teams on the field. A strong focus on short-term maximization of scoring opportunities, while blocking the other team from the same. There is no evidence of strategic planning across plays, as in games like chess or Go.

Teams largely react to events as they occur. Every moment is equally easy or difficult.

But, teams try harder at the end of a period. Pro basketball, where lead sizes spreads tend to shrink back to zero. Pro basketball is the only sport where the spread tends to shrink.

In football and hockey, the spread tends to grow over time. Does being behind help you win, as argued by [ 21 ]? Being behind helps you lose. Being ahead and being lucky helps you win. We combine these insights within a generative model of gameplay and demonstrate that it accurately reproduces the observed evolution of lead-sizes over the course of games in all four sports, and also makes highly accurate predictions of game outcomes, when only the first few scoring events have occurred.

Cursory comparisons suggest that this model achieves accuracy comparable to or better than several commercial odds-makers, despite this model knowing nothing about teams, players, or strategies, and instead relying exclusively on the observed tempo and balance patterns in scoring events. We first introduce the limiting case of an ideal competition , which provides a useful tool by which to identify and quantify interesting deviations within real data, and to generate hypotheses as to what underlying processes might produce them.

Although we describe this model in terms of two teams accumulating points, it can in principle be generalized to other forms of competition. Furthermore, each side is perfectly skilled, i. This is an unrealistic assumption, as real competitors are imperfectly skilled, and possess both imperfect information and incomplete strategic knowledge of the game. However, increased skill generally implies improved performance on these characteristics, and the limiting case would be perfect skill.

Finally, each side exhibits a slightly imperfect ability to execute any particular chosen strategy, which captures the fact that no side can control all variables on the field. In other words, two perfectly skilled teams competing on a level playing field will produce scoring events by chance alone, e.

An ideal competition thus eliminates all of the environmental, player, and strategic heterogeneities that normally distinguish and limit a team. Real competitions will deviate from this ideal because they possess various non-ideal features. The type and size of such deviations are evidence for competitive mechanisms that drive the scoring dynamics away from the ideal. Summary of data for each sport, including total number of seasons, teams, competitions, and scoring events.

Playing fields are flat, featureless surfaces. Gameplay is divided into three or four scoring periods within a maximum of 48 or 60 minutes not including potential overtime. The team with the greatest score at the end of this time is declared the winner. Past work on the timing of scoring events has largely focused on hockey, soccer and basketball [ 4 , 6 , 10 ], with little work examining football or in contrasting patterns across sports. However, these studies show strong evidence that game tempo is well approximated by a homogenous Poisson process, in which scoring events occur at each moment in time independently with some small and roughly constant probability.

Analyzing the timing of scoring events across all four of our sports, we find that the Poisson process is a remarkably good model of game tempo, yielding predictions that are in good or excellent agreement with a variety of statistical measures of gameplay. However, we do find some evidence for modest non-Poissonian patterns in tempo, some of which are common to all four sports.

Tempo summary statistics for each sport, along with simple derived values for the expected number of events per game and seconds between events. Parenthetical values indicate standard uncertainty in the final digit. Scoring events per game. Time between scoring events. Empirical distribution of time between consecutive scoring events, shown as the complementary cdf, along with the estimated distribution from the Poisson model dashed.

Insets show the correlation function for inter-event times. If C n is positive, short intervals tend to be followed by other short intervals or, large intervals by large intervals , while a negative value implies alternation, with short intervals followed by long, or vice versa. However, in CFB, NFL and NHL games, we find a slight negative correlation for very small values of n , suggesting a slight tendency for short intervals to be closely followed by longer ones, and vice versa.

Our results above provide strong support for a common Poisson-like process for modeling game tempo across all four sports. We also find some evidence for mild non-Poissonian processes, which we now investigate by directly examining the scoring rate as a function of clock time. Within each sport, we tabulate the fraction of games in which a scoring event associated with any number of points occurred in the t th second of gameplay.

Game tempo. Empirical probability of scoring events as a function of game time, for each sport, along with the mean within-sport probability dashed line. Each distinct game period, demarcated by vertical lines, shows a common three-phase pattern in tempo. Thus, without regard to other aspects of the game, it must take some time for players to move out of these initial positions and to establish scoring opportunities. This would reduce the probability of scoring relative to the game average by limiting access to certain player-ball configurations that require time to set up.

These behaviors would also reduce the probability of scoring by encouraging risk averse behavior in establishing and taking scoring opportunities. We find evidence for both mechanisms in our data. Both CFB and NFL games exhibit short and modest-sized dips in scoring rates in periods 2 and 4, reflecting the fact that player and ball positions are not reset when the preceding quarters end, but rather gameplay in the new quarter resumes from its previous configuration.

In contrast, CFB and NFL periods 1 and 3 show significant drops in scoring rates, and both of these quarters begin with a kickoff from fixed positions on the field. Similarly, NBA and NHL games exhibit strong but short-duration dips in scoring rate at the beginning of each of their periods, reflecting the fact that each quarter begins with a tossup or face-off, in which players are located in fixed positions on the court or rink.

In contrast, NHL games exhibit a prolonged warmup period, lasting well past the end of the first period. This produces a flat, stable or stationary pattern in the probability of scoring events. A stable scoring rate pattern appears in every period in NFL, CFB and NBA games, with slight increases observed in periods 1 and 2 in football, and in periods in basketball. NHL games exhibit stable scoring rates in the second half of period 2 and throughout period 3. Within a given game, but across scoring periods, scoring rates are remarkably similar, suggesting little or no variation in overall strategies across the periods of gameplay.

The end of a scoring period often requires players to reset their positions, and any effort spent establishing an advantageous player configuration is lost unless that play produces a scoring event.

This impending loss-of-position will tend to encourage more risky actions, which serve to dramatically increase the scoring rate just before the period ends. The increase in scoring rate should be largest in the final period, when no additional scoring opportunities lay in the future. In some sports, teams may effectively slow the rate by which time progresses through game clock management e. This effectively compresses more actions than normal into a short period of time, which may also increase the rate, without necessarily adding more risk. We find evidence mainly for the loss-of-position mechanism, but the rules of these games suggest that clock management likely also plays a role.

## Scoring dynamics across professional team sports: tempo, balance and predictability

In football, this increase is greatest at the end of period 2, rather than period 4. This likely creates a mild incentive to initiate some play before the period ends which is allowed to finish, even if the game clock runs out. NHL games exhibit no discernible end-phase pattern in their intermediate periods 1 and 2 , but show an enormous end-game effect, with the scoring rate growing to more than three times its game mean.

Regardless of the particular mechanism, the end-phase pattern is ubiquitous. In general, we find a common set of modest non-Poissonian deviations in game tempo across all four sports, although the vast majority of tempo dynamics continue to agree with a simple Poisson model. Perfectly balanced games, however, do not always result in a tie. Comparing the simulated distribution against the empirical distribution of c provides a measure of the true imbalance among teams, while controlling for the stochastic effects of events within games.

Across all four sports, we find significant deviations in this fraction relative to perfect balance. Within a game, scoring balance exhibits unexpected patterns.

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In contrast, NFL, CFB and NHL games exhibit the opposite effect, in which the probability of winning the next scoring event appears to increase with the size of the lead - a pattern consistent with a heterogeneous distribution of team skill. The fraction of all events in the game that were won by a randomly selected team provides a simple measure of the overall balance of a particular game in a sport.

Let r and b index the two teams and let E r E b denote the total number of events won by team r in its game with b. Game balance. Modes at 1 and 0 indicate a non-trivial probability of one team winning or losing every event, which is more common when only a few events occur.

This is likely a result of the broader range of skill differences among teams at the college level, as compared to the professionals. Like CFB and NFL, NHL games also exhibit substantially more blowouts and fewer ties than expected, which is consistent with a heterogeneous distribution of team skills.

Surprisingly, however, NBA games exhibit less variance in the final relative lead size than we expect for perfectly balanced games, a pattern we will revisit in the following section. Although many non-Bernoulli processes may occur within professional team sports, here we examine only one: whether the size of a lead L , the difference in team scores or point totals, provides information about the probability of a team winning the next event. Across all four of our sports, we tabulated the fraction of times the leading team won the next scoring event, given it held a lead of size L.

Lead-size dynamics. This model is identical to the popular Bradley-Terry model of win-loss records of teams [ 29 ], except here we apply it to each scoring event within a game. The function flattens out at large L assuming the value representing the largest skill difference possible among the league teams. NBA games, however, present a puzzle, because no distribution of skill differences can produce a negative correlation under this latent-skill model.

Coasting could occur for psychological reasons, in which losing teams play harder, and leading teams less hard, as suggested by [ 21 ]. Again, however, the absence of this pattern in other sports suggest that the mechanism is not psychological. For instance, when a team is in the lead, they often substitute out their stronger and more offensive players, e. When a team is down by an amount that likely varies across teams, these players are put back on the court.

If both teams pursue such strategies, the effective ratio c will vary inversely with lead size such that the leading team becomes effectively weaker compared to the non-leading team. The previous insights identify several basic patterns in scoring tempo and balance across sports.

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However, we still lack a clear understanding of the degree to which any of these patterns is necessary to produce realistic scoring dynamics. Here, we investigate this question by combining the identified patterns within a generative model of scoring over time, and test which combinations produce realistic dynamics in lead sizes. In particular, we consider two models of tempo and two models of balance. For each of the four pairs of tempo and balance models for each sport, we generate via Monte Carlo a large number of games and measure the resulting variation in lead size as a function of the game clock, which we then compare to the empirical pattern.

Our two scoring tempo models are as follows. Our two balance models are as follows. The four combinations of tempo and balance models thus cover our empirical findings for patterns in the scoring dynamics of these sports. The simpler models called Bernoulli represent dynamics with no memory, in which each event is an iid random variable, albeit drawn from a data-driven distribution.

The more complicated models called Markov represent dynamics with some memory, allowing past events to influence the ongoing gameplay dynamics. In particular, these are first-order Markov models, in which only the events of the most recent past state influence the outcome of the random variable at the current state.

Modeling lead-size dynamics. Comparison of empirical lead-size variation as a function of clock time with those produced by Bernoulli B or Markov M tempo or balance models, for each sport. That being said, some small deviations remain. For instance, the Markov model slightly overestimates the lead-size variation in the first half, and slightly underestimates it in the second half of CFB games. In NFL games, it provides a slight overestimate in first half, but then converges on the empirical pattern in the second half. NBA games exhibit a similar pattern to CFB games, but the crossover point occurs at the end of period 3, rather than at period 2.

These modest deviations suggest the presence of still other non-ideal processes governing the scoring dynamics, particularly in NHL games. Instead, the pattern provides a compact and efficient summary of scoring dynamics conditioned on unobserved characteristics like team skill. Our model generates competition between two featureless teams, and the Markov model provides a data-driven mechanism by which some pairs of teams may behave as if they have small or large differences in latent skill.

It remains an interesting direction for future work to investigate precisely how player and team characteristics determine team skill, and how team skill impacts scoring dynamics. In this section, we study the predictability of game outcome using the Markov model for scoring balance, and compare its accuracy to the simple heuristic of guessing the winner to be the team currently in the lead at time t.

The probability that team r is the predicted winner depends on the probability distribution over lead sizes at time T. In this way, we capture the information contained in the magnitude of the current lead, which is lost when we simply predict that the current leader will win, regardless of lead size.

Instead of evaluating the model at some arbitrarily selected time, we investigate how outcome predictability evolves over time. When the number of cumulative events is small, game outcomes should be relatively unpredictable, and as the clock runs down, predictability should increase. To provide a reference point for the quality of these results, we also measure the AUC over time for a simple heuristic of predicting the winner as the team in the lead after the event. Predicting game outcome from dynamics. This occurs in part because small leads are less informative than large leads for guessing the winner, and the heuristic does not distinguish between these.

Although there is increasing interest in quantitative analysis and modeling in sports [ 31 â€” 35 ], many questions remain about what patterns or principles, if any, cut across different sports, what basic dynamical processes provide good models of within-game events, and the degree to which the outcomes of games may be predicted from within-game events alone.

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The comprehensive database of scoring events we use here to investigate such questions is unusual for both its scope every league game over seasons , its breadth covering four sports , and its depth timing and attribution information on every point in every game. As such, it offers a number of new opportunities to study competition in general, and sports in particular. These variations, however, follow a common three-phase pattern, in which a relatively constant rate is depressed at the beginning of a scoring period, and increases dramatically in the final few seconds of the period.

CFB games are much less balanced than NFL games, suggesting that the transition from college to professional tends to reduce the team skill differences that generate lopsided scoring. This reduction in variance is likely related both to only the stronger college-level players successfully moving up into the professional teams, and in the way the NFL Draft tends to distribute the stronger of these new players to the weaker teams.

Furthermore, we find that all three of these sports exhibit a pattern in which lead sizes tend to increase over time. As with overall scoring balance, the size of this effect in CFB games is much larger about 2. That is, NFL teams are generally closer in team skill than CFB teams, and this produces gameplay that is much less predictable. Both of these patterns are consistent with a kind of Bradley-Terry-type model in which each scoring event is a contest between the teams. Ken Pendleton discuss group dynamics with guest Dr. Mark Eys, co-author of Group Dynamics in Sport. Successful group performance is the result of many complex factors, but Dr.

## Statistics > Applications

Mark Eys of Wilfrid Laurier University specializes in teasing out which factors matter, or at least which can make the difference between winning and losing in a close matchup. So what makes a cohesive team? These factors can be related to leadership, role acceptance, environment, and personality. Two of the most studied are social cohesion, a sense of belonging, and task cohesion, how united a group is around completing a particular task.

To improve both types of cohesion, Eys recommends teams pay special attention to goal setting and not just individual goals. Set performance and process goals that are very specific, that athletes can buy into during practice, that coaches are involved, and that you monitor over time. I think it keeps the team directed. There could also be such a thing as too much cohesion.