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Why do hockey players support helmet rules, even though they choose not to wear helmets when there is no rule?

In the 1950s and 1960s, hockey players were often badly injured because they were not wearing helmets. Everyone was aware of the danger, but players persisted in not wearing helmets. An article in Newsweek (1969) that followed a serious head injury of a player stated: "Players will not adopt helmets by individual choice for several reasons. Chicago star Bobby Hull cites the simplest factor: 'Vanity.' But many players honestly believe that helmets will cut their efficiency and put them at a disadvantage, and others fear the ridicule of opponents. The use of helmets will spread only through fear caused by injuries like Green's or through a rule making them mandatory.... One player summed up the feelings of many: 'It's foolish not to wear a helmet. But I don't because the other guys don't. I know that's silly, but most of the players feel the same way. If the league made us do it, though, we'd all wear them and nobody would mind.' "

If the players know that they could be injured from playing without a helmet (indeed a player died in the 1960s from a head injury) and would vote to make them mandatory, why wouldn't they wear them voluntarily?

Consider two hockey teams: the Penguins or the Thrashers. Players on each team can choose to wear helmets or not wear helmets. Not wearing a helmet increases the odds of winning by reducing overheating, making it slightly easier to see and hear, and by intimidating opposing players (or equivalently, not wearing a helmet makes one look weak and scared, which are not desirable characteristics for hockey players). However, not wearing a helmet increases the odds of getting hurt. Players on both teams value the higher odds of winning more than they value the extra safety. Why will they choose not to wear helmets, even though the odds of winning when players on both sides wear helmets are no different from the odds when no one wears helmets?

How can we construct a game matrix for this situation? The players cannot calculate specific payoffs for each possible outcome, but they do know the ranking of the outcomes: 1 is the most desirable, 2 is the second-best, 3 is the third-best, and 4 is the least desirable outcome.

The dominant strategy for each team is to go without helmets, even though this combination of choices is worse for both teams than the alternative in which each team wears helmets: a Prisoner's Dilemma.

So players don't have an incentive to wear helmets and for different reasons, the team owners did not have an incentive to create a mandatory helmet rule (they were worried that every player would look the same and that it would be more difficult to market players based on their individual looks). It wasn't until the start of the 1979-1980 season that the NHL passed a rule decreeing that anyone who came into the NHL from that point on had to wear a helmet. By the early 1990s there were only a few players left who did not use a helmet, with the last one retiring after the 1996-97 season.

 
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