September 19, 2006

Diving Preferences

I was going to throw this on Tom Benjamin's NHL Weblog, but it turns out I’m a spammer, so I decided to expand and present it a little better and put this longer post on my webpage.

The first diving infraction would result in a warning letter being sent to the player. A second infraction would be accompanied by a $1,000 fine. A third infraction would result in a telephone hearing with the league and a possible one-game suspension. The length of the suspension would double for any subsequent violation. ~ NHL new Diving Rules

Last I checked $1,000 is 0.2% of the league min (should be at least 2% of salary - yah $135,000 for Luongo!). Also a one game suspension is much more expensive to Naslund than let's say Goren (although he may never come back...).

However this is a math website, what would it be without some sort of equations and numbers?

If you'd like to control diving in the current rules you need to first consider the value of a dive:

Value of dive = p_power_play * 0.2 Goals/power_play * Value/Goal

If p_power_play is close to 0.5 you're looking at 0.1 Goals per dive... To prevent dives one needs to have costs in excess of these benefits. A marginal goal is worth around $200,0001 so a dive would be worth about $10,000. There are likely some "opinion" costs (other players hate you). Of course this isn't a fixed cost: a marginal goal is worth more when the game is tied in the end of third period. It's doubtful you could prevent diving with any tools; due to the value can approach $1,000,000 in the end of the game in an average game and almost unlimited to make the playoffs (bounded by $40,000,000?)

The other side is the cost:

Actual costs = p_caught * costs

Currently p_caught sits around 0.05, (IMHO) as this keeps the false positives down (only penalize the obvious). At this rate the cost to the player and team would have to be around $200,000 (a marginal goal), raising p_caught to 0.25 would lower this to $40,000 (penalty shot works here...). Since Actual costs need to exceed benefits in order to prevent them the equation we want here is:

Costs > Value of a dive/p_caught.

This really makes the job quite simple: decrease the value of a dive, increase probability of being caught or increase costs associated with diving.

Of course diving isn’t quite that simple the other has to do with the subject quality of diving. That is of course is false positives (player doesn't dive and gets called) and false negatives (player dives doesn't get caught). Even if you could figure out a fool proof set of rules you're going to have these problems - note: referees are afraid of these (if management disagrees with you then you could be in the dog house, mostly for the false positives). So when considering rules you want to make sure they’re not too severe so that referees are comfortable making the false positive periodically.

For example when a referee sees a player fall there’s a potential dive, he considers the situation in order to determine if it’s a dive. In general statistics a false positive is much more serious than a false negative, it is much more serious for the NHL to hand out serious punishment (one game suspension) to a player who didn’t dive and as such a false positives becomes much more embarrassing. “There is no [way] that will simultaneously make both [false positives] and [false negatives] [lower]” (Devore - Quote simplified for non-math people). Reducing the false positive rate means increasing the false negatives. I suspect the NHL uses something 99-99.5% confidence to determine dives. This means a significant number of false negatives (we all know this already). However in terms of entertainment and respect for the game false negatives are much more costly, as they are making the sport a joke.

In order to fix the above problem the NHL has to make false positives much less costly. As such a referee who calls a dive that isn’t a dive doesn’t get criticized and lambasted for it. A called dive results in fines and other punishment outside of the sport that garners a lot of attention. A dive should be controlled within the confines of the given game. There are many ways to do this.

So what are the solutions?

  1. Simplest: fines to team and player. This is as explained above and calculated as $10,000/p_caught, or likely around $200,000 (equivalent to 1 marginal goal). These rules could lower p_caught further making these rules something that’s never used and just a waste of the paper it’s written on.
  2. Of course don’t call any penalties that include a dive (don’t call either player). This would reduce penalties called in total; p_caught would have to be very high. False positives wouldn’t be that bad (neither player was called, no one is actually penalized) as the NHL already misses a lot of calls.
  3. The opposite of the above is to penalize both players (current standard) this also requires a high p_caught value. The NHL throws fines and other such penalties lowering p_caught and as such this has been ineffective.
  4. Of course the best way to control diving is that the value of a dive is nothing. Most fans realize this and referees would likely like this as well, but in practical terms it’s very hard to see infractions in hockey, but a falling player is quite noticeable. Also the NHL seems to have a general understanding that the punishment should be a function of the results and not the action itself (take note of Bertuzzi and other suspensions and actions).
  5. Penalty shot for each dive. As a penalty shot is worth ¼ of a goal this would be worth about $50,000, a p_caught rate of 20% would be needed. False positives would not be significantly costly in terms of the game ($10,000/$1,000,000 = 1%). But it would equal the benefits associated with diving.

There are many other things that can be done obviously within the above context; the goal is the same, control the costs associated with false positives. In general it’s better to give the costs associated with false positives to a team rather than a player as they will average out over the course of a season. The second goal is to make sure that players can see the costs and that these costs will exceed the benefits. However, the NHL has made it more costly for false positives and they still don’t have rules that exceed the benefits.

So will the new rules make a difference, absolutely not $1,000 is nothing to a hockey player, and missing a game isn’t even worth as much as dive except to the team's best players (who the rule “doesn’t apply to”). And as such the benefits still exceed the costs so players will likely continue to dive at the same rate and the NHL won’t know what to do about it.

So enjoy a season filled diving. There's nothing new here. Also, enjoy a season filled with a ton of garbage calls as a result of this diving and expect the NHL to do nothing about it. The NHL wants more scoring, to get that you need more power plays and diving produces more power plays. The NHL has no incentive to get rid of diving. They just want to make it appear like they’re trying to control them so the fans are happy.