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Game Theory & Time in Sports

Source:  http://www.sportshistory.us/uncertainty.html

In most sports, the winner is decided when the clock comes to an end.  Football has four 15-minute quarters, basketball has four 12-minute quarters, and hockey has three 20-minute periods.  Based on how the teams perform up to the final minutes, their game-end strategies are developed.  If a football team is up by seven with a minute to go in a game, they will use a prevent defense to stop the opposing team from completing any long plays, while giving up shorter plays more easily.  If the football team is on offense in this situation, they will kneel to waste the clock, as the payoff is high.  If a football team kneeled at a game’s start, the payoff would be horrendous.  Similarly, in basketball, when a team is down by 10 points with two minutes left, they will try to foul the other team to conserve time and hope the other team misses the foul shots.  This would be considered a bad strategy until the final minutes of the game, but game theory suggests that the payoff of this strategy increases under end-of-game circumstances.

Contrarily, many sports do not use clocks, and thus the strategy is more consistent throughout the game.  For example, in baseball, there is always nine innings and the defending team will always try to get the offensive team out, and the offensive team will always try to get a hit.  The payoff matrices stay the same throughout the game for both teams, so their strategies do not change.  Likewise, soccer strategies basically remain the same throughout the entire game.  However, soccer is a timed game, but the players do not know when the game will end because the referee adds an unknown amount of stoppage time to the end of the game.

The article suggests that sports could take an approach similar to soccer, where the average length of games stays the same, but they end on a normally distributed time about the mean.  For example, in football, the game would have a mean time of 60 minutes, but the game would end on a random, normally distributed time with a standard deviation of 5 minutes.   This would reduce the effect of game theory on sports, as team’s strategies would remain more consistent throughout games because they do not know when the game will end.  The payoff matrices wouldn’t change depending on the amount of time left.  There would be no fouling at the end of basketball games because the payoff would be as low as it is throughout the rest of game.   There would be less prevent defenses in football, as they would not want to give up the short gains without knowing the game is ending soon.   Nevertheless, sports would be considerably more boring without the buzzer-beater shots and hail mary’s of set-time sports games.

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