Suspense vs. Purity: A Game Theory Perspective on Timed Sports

There’s almost nothing more exciting in the sports world than a buzzer-beating jump shot, a Hail Mary pass, or a last second wrist shot.  We all know that when time is running out in a close game, we may be in for something very special.  Knowing that the end of the game is near can certainly be entertaining, but is it pure?  That is, is a contest hurt by its participants’ ability to change their behavior when they know the end is near?

Game theorists have known for a long time that players will take advantage of the knowledge that a game is about to end.  Axelrod’s iterated prisoner’s dilemma study demonstrated this.  In 1979, Professor Robert Axelrod designed a contest to determine which strategy (a combination of choices to “cooperate” or “defect”) was most successful in a prisoner’s dilemma game over a number of repetitions.  He asked people to send in their strategies, and then he pitted them against each other over 200 iterations. 

His contest had several interesting results, but there is one in particular that has implications for clocked sports.  What Axelrod found was that strategies would take advantage of the fact that the game was designed to last exactly 200 turns.  Players would defect on the 200th turn, knowing that there was no 201st turn in which an opponent could punish them for defection.  Knowing that defection was the dominant strategy on the 200th turn, they would defect on the 199th, the 198th, and so on, which made the contest essentially meaningless.  To fix this problem, Axelrod again solicited strategies, but this time he ran the contest under uncertainty concerning the duration of the game.  Rather than play 200 iterations, he ran every iteration with a certain (very small) probability that it would be the last.  Axelrod’s adjustment brought some needed integrity to the contest, but it did so by taking away foreknowledge of the end of a game, which we like to have.    

In games whose length is determined by a clock, like basketball, football, and hockey, both the players and the fans gain some excitement from knowing when the contest will end.  But in order to give sports fans the suspense they crave, we allow sports teams to do some pretty bizarre things.  In football, a team with a tentative lead will often resort to running the ball late in the game in order to keep the clock running and avoid the risk of an interception.  For teams with a larger lead, a “prevent” defense places defenders further downfield, making the big play less likely while forfeiting smaller gains.  Also in play when time is about to run out is the “squib kick,” a kick that bounces along the ground and, though it surrenders a greater average field position than higher kicks, is associated with fewer long returns because it does not allow blockers to assemble and is often fielded by players other than the kick returner.  The most extreme example of “gaming” a football contest is allowing the other team to score late in the ballgame.  Some teams have given up a safety so as not to give up blocked kicks, and others have even allowed a touchdown they feel is inevitable so that they will have enough time to score afterwards. 

Basketball is also no stranger to impure behavior.  Any basketball fan has seen his team try to erase a late deficit by fouling, stopping the clock and putting the opposing team’s poorest shooters at the line.  The Four Corners offense, a stall technique used by a team with a lead, has been eliminated by the shot clock, but teams in the lead will still run more time off the shot clock late in the game than they normally would.  On the final defensive possession, a team with a two point lead may foul rather than give up a three-point attempt.  And in hockey, teams that are behind late in the game will pull their goalies because the increased risk of giving up a goal matters less when compared to the opportunity to score with an extra offensive skater. 

These approaches to the knowledge that the game is about to end can be viewed as overly opportunistic, but there is something we can do about them.  Timed sporting events can have their consistency restored if we play them under uncertainty, keeping the participants in the dark about when the game will end.  Some sports already do this, with varied results.  In most soccer matches, stoppage time is added to the end of the match in an amount known only to the referees.  In boxing, the scores are kept from the participants, which means that a boxer may be less likely to go for a knockout late in the match because he does not know whether the scoring dictates such a strategy. 

A purely Axelrodian approach would be to ensure the purity of sports contests by applying some sort of formula that all sports could use to bring an end to contests.  For instance, an NBA game lasts 48 minutes.  In an NBA contest played under uncertainty, the game would end at a time chosen randomly from a normal distribution centered on 48 minutes with a standard deviation of 4 minutes.  This time would be known only to an official who would end the game at the prescribed time.  If this rule were applied to NBA games, most games would end around 48 minutes, but about 16% would end in less than 44 minutes and 16% would last longer than 52 minutes.  With this kind of uncertainty, players would not worry as much about fouling intentionally or running out the clock because they would probably be hurting their cause more than helping it.  This system ensures a purer result—players will be more concerned with playing the game than with managing the clock. 

Would anyone want to live in this kind of world?  Players might find it refreshing just to play the game without worrying about managing the situation, though I suspect that having a clock conveys a level of certainty that would be missed.  Coaches would probably have the most difficult time, as they make their money by telling players what to do down the stretch.  And fans would probably be a mixed bag.  Some would enjoy watching a full contest without having to see bogus sports inventions like the prevent defense and the Four Corners offense, but others would probably be upset if they paid for a ticket that ended up giving them 43 minutes of a 48-minute game.  We would all miss the sports moments created by a ticking clock.  On the whole, we like experiencing the desperation and surprise of NC State winning the 1983 Final Four, even if it means we have to endure watching the same Wolfpack team foul mercilessly to get out of the first round (they erased a six-point overtime deficit in 1:10 against Pepperdine).  So it looks like we’re safe from an “unspoiled” world in which players react to the game and not to the clock.  After all, game theory is just that-theory, and games in the real world have to satisfy a public that craves a little more excitement.