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Julian D. A. Wiseman & Drew C. Phillips
Publication history: sent to the NHL Commissioner in early December 2006, and then here. Usual disclaimer and copyright terms apply.
Drew Phillips took Julian Wiseman to see an ice hockey game. Though great entertainment, the scoring system incentivized dull play near the end of regulation time. So this proposal to repair the problem was sent in early December 2006 to Mr. Gary Bettman, the NHL Commissioner.
Dear Mr. Bettman,
Recently, the two of us, friends and work colleagues, watched the Atlanta Thrashers beat the New York Rangers 5-4 in overtime. One of us, a diehard Ranger fan and season ticket holder, was riveted by the two+ minutes of overtime. The other, an Englishman and mathematician, very much enjoyed his first NHL game, but was quite intrigued by the comparison between the end of regulation play and the start of overtime. The Englishman liked ice hockey: fast-paced, physical, with clear and simple rules (even the ‘offside’ rule is easy). The league scoring rule, however, seems completely wrong, and creates a perverse incentive.
Approaching the end of regulation, both the Rangers and Thrashers had a strong incentive to get the game to overtime, thereby securing one point each. In some sense, they both knew that cautious play would result in sharing three points rather than two. So they played cautiously. The current scoring system encourages conservative play at the end of the game — at what ought to be the time for the most thrilling action. That is a bad scoring system.
After a beer and a description of that night in May 1994, when the Rangers beat the Devils in triple overtime of Game 7 of the Eastern Conference Finals, we concluded the league scoring system should be as follows:
|Result in regular time:||2||-||0|
|Result in overtime:||1||-||0|
|Result from penalty shootout:||½||-||0|
This would be far more brutal. As the end of regulation approached, it wouldn’t be enough to win eventually. Each team would need to win now!
Let’s reason more numerically, and focus on the strategy of one team. Assume that in the last two minutes of regulation, that team can play ‘defensively’ or ‘aggressively’. If playing defensively, the probability of a regulation win:draw:loss is 20:60:20. If playing aggressively, that team is more likely to score, but so is their opponent (perhaps even a bit more likely — call it 40:10:50). Also assume that if the game goes into overtime it’s 50:50. Under each strategy, as the end of regulation approaches, that team’s expected score is the probability of an immediate win times its score, plus the probability of an overtime win times its score, plus the probability of an overtime loss times its score.
First let’s apply this formula to the current scoring system. Playing defensively has an expected score of 2×20% + 2×60%×50% + 1×60%×50% = 1.3, and playing aggressively has an expected score of 2×40% + 2×10%×50% + 1×10%×50% = 0.95. Accordingly, the optimal strategy near the end of regulation is currently defensive, cautious, dull play.
What about the proposed new scoring system? For simplicity assume that if the game goes to overtime, it will be resolved in overtime. Then the expected score from defensive play, using the same formula as above is 0.7, and from aggressive play is higher at 0.85. So the optimal strategy is aggressive, attacking play. Similar reasoning applies near the end of overtime: the imminent halving of the winner’s score creates an incentive to win quickly, even if that entails a larger risk of losing.
The only caveat to this proposal is that not all NHL games are awarded 2 points. But the NHL is not wedded to the number two as it was pre the 2005-06 season: most games currently get 2, but some get 3.
So, Mr. Bettman, do you want bigger television audiences? Make the end of regulation a climax, not a whimper. (And a bonus question asked by the diehard Ranger fan: even if you don’t like our proposal, please may we drive the Zambonis at the All-Star game in Dallas?)
Drew C. Phillips & Julian D. A. Wiseman.
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