|Ivars Peterson's MathTrek|
May 20, 1996
This sort of time management is seen most clearly in games such as football and basketball, in which teams alternate between offense and defense. On offense, each team gets a chance to score with every possession of the ball (or puck). How long such an effort takes, whether successful or not, has a bearing on the game's outcome. One oft-used tactic to manage time during a game is known as ball control. In effect, a team attempts to maintain possession of the ball for a much greater proportion of the entire game than its opponent. Its goal is not only to score but also to take as long as possible doing so.
Indeed, ball control has become common practice in a number of sports. In basketball, for example, coaches often urge their teams to slow down a game at certain times or to control its tempo. In professional football, "time of possession" has become a statistic of major importance in game summaries, alongside the number of first downs and yards gained.
Is ball control an effective strategy? The answer is no, say Harold and Daniel Sackrowitz. In an article in the latest issue of Chance, they argue that "a team using ball control may reduce the number of possessions and points scored by its opponent, but it will lose more often than if it did not use ball control."
Harold Sackrowitz is a statistics professor at Rutgers University, and his son Daniel is a software engineer at AT&T. Daniel once worked briefly as a statistics editor at the now-defunct National Sports Daily, and Harold used to coach Daniel's Little League team.
The conclusion that Harold and Daniel Sackrowitz arrive at based on their analysis of time-limited games may come as a shock to coaches, players, commentators, and fans who are sold on the notion of ball control. The dominant rationale underlying this deeply rooted faith really stems from the popular, compelling truism that "the other team can't score if it doesn't have the ball."
How valid is this truism when applied to a ball-control strategy? The Sackrowitz team developed a mathematical model for evaluating the efficacy of three styles of play: unconstrained (a team's normal mix of offensive maneuvers and actions), time-consuming (using additional time), and hurry-up (using less time than normal).
They modeled a typical time-limited game by a multidimensional Markov chain -- a sequence of random vectors, each of which can be written as a string of numbers. The first three numbers, or coordinates, may indicate which team has possession, the current point difference between the teams, and the amount of time remaining. The remaining numbers of a given vector could represent the values of other variables deemed important for the particular game being considered, such as the number of time-outs remaining. Such a vector describes the state of the game at the beginning of each possession.
Assumptions about the probability of scoring and the amount of time used for a style of play when in a particular state (described by a vector) determines the probability distribution for the next state of the game (described by a new vector). The rules of a game determine the initial state.
To evaluate specific strategies and offer an optimal approach, the Sackrowitzes applied dynamic programming methods, in which they worked backwards from the end of a game to determine scoring probabilities with one unit of time remaining, two units of time remaining, and so on.
Such a computational model allowed the researchers to check game outcomes over a broad range of possible situations involving weak and strong teams adopting different styles of play at different times. A consistent pattern emerged in all the cases studied: An unconstrained strategy is preferable to either the time-consuming or the hurry-up strategies for both teams, even when one team is demonstrably weaker than the other.
"The results also force us to the realization that, despite what one feels emotionally, a proficient ball-control offense reduces the number of possessions for both teams," the authors note. "Thus, if anything, one might guess that the better team would decrease its probability of winning by using ball control, particularly if it had reduced its probability of scoring."
The article by Harold and Daniel Sackrowitz contains much that ought to tickle the minds of sports fans settling in to watch the NBA and NHL playoffs or awaiting the start of the football season. Conventional wisdom, even when it comes from esteemed commentators, doesn't necessarily reflect the true state of a game.
In a 1993 New York Times article about football coach Bill Parsells, the reporter stated, "His masterpiece was the 1991 Super Bowl, in which his Giants defused the powerful and innovative offense of the Buffalo Bills through the simple expedient of denying Buffalo the use of the football."
The Sackrowitzes have a different view. "Even in the supposed ultimate endorsement of ball control, the 1991 Super Bowl, the Bills had ten possessions (but punted six times)," they remark. "In that game, a great defense helped to create the illusion that ball control is effective."
Copyright © 1996 by Ivars Peterson.
Harold Sackrowitz can be reached at email@example.com.
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