A Mathematical Model to Explain
the Success of the Las Vegas Golden Knights
I think most hockey fans, and sports fans in general, are
aware of the surprising success of the Las Vegas Golden Knights, in their first
year in the National Hockey League. For
an expansion team to finish as high as they did, and to then contend in the Stanley
Cup final seems like a remarkable feat.
Indeed, some fans find it so remarkable as to be incredible, and
therefore suspect.
To recap, Las Vegas finished their 2017-18 expansion season with
109 points, winning their division. This
put them in 5th place overall, in a 31 team league, of which 16
teams make the playoffs. They then won
the quarter finals, semi-finals, and conference finals. These are feats unprecedented in professional
hockey history for an expansion team, and rarely equaled in other professional sports.
So, how can we account for this amazing achievement? After doing some fairly simple Monte Carlo mathematical modelling,
based on the expansion draft rules and assumptions about underlying "hockey value", I believe the solution lies in the
intersection of four factors:
- The draft rules were actually quite generous to
Las Vegas.
- The presence of superstars is not as definitive
for success as is commonly believed.
- Management is not as at evaluating talent and
predicting player performance as commonly believed.
- As the Bible says, time and chance happeneth to them all.
The model, described below, is relatively simple, but by
varying model parameters, it can come up with scenarios that show Las Vegas’s final
standings to be unexpectedly good, but not impossibly so.
The Expansion Draft Rules
The draft rules allowed each team to protect 7 forwards, 3 defense men and 1 goalie. Alternatively,
they could protect 8 skaters and a goalie.
My model assumes that teams will use the first alternative, when
protecting their preferred players.
Furthermore, Las Vegas was to choose one player from each team.
The probability of keeping unprotected players was included
in the model, with the constraint that there would be a total expectation of
each team losing one player’s hockey value.
For simplicity, I made the probability of any one of the unprotected
players being drafted the same (17%), except for the second goalie (7%). That way, Las Vegas will draft its allotment
of one skater from each team, and 2 goalies overall.
The Model
The Concept of “Hockey Value”
My model focused on the usual lineup dressed for games – 4
forward lines, 3 defense pairings, and 2 goalies. I simulated 30 teams, each protecting the
players on their top lines.
I use a concept I call “hockey value” to measure the gains
that Las Vegas makes in the draft, and the corresponding losses to the other
teams. Again, this is meant to be a
general concept of what value a particular position contributes to a team. The highest valued position is given 100
points, with other positions receiving correspondingly lower point values. The total hockey value of a team's players is
then summed, and the results are ranked to determine the overall
competitiveness of the team.
Note that the non-Las Vegas teams are not identified, nor
strictly speaking is Las Vegas. The
model is a general one, seeking to understand the likely outcomes of a
simulated set of drafts, such as the one that actually produced the Las Vegas
Golden Knights, under the rules for that draft.
Note that this use of the term “hockey value” is not related
to the financial value of the various franchises, or the teams’ payrolls. Instead, it is a relative measure of hockey
playing ability and the importance of the various positions on a hypothetical
hockey team. By varying the underlying hockey value parameters, one can test different assumptions and see which ones give a reasonable probability of a Las Vegas type expansion draft team to end up high in the standings.
To reflect the fact that teams that finish higher in the
standings must have more hockey value than those lower down, the total hockey
value of each team was diminished by 2.5%, for each rank lower in the standings. That gave the hockey value of the 30th
team at about 65% of the leading team’s value.
That conforms quite well with the difference between first and last in
goals for (67%), goals against (69%), and average player salary (64%), all
metrics that common sense tells us should be reasonable proxies for each team’s
“hockey value”. The accompanying graph
shows how those metrics compare.
Computing the Hockey Value of Each Team
The hockey value of a given team is calculated by
multiplying the value of each position by the overall value of the team (given
by its position in the standings), minus the value lost in the draft to Las
Vegas. Those values are then multiplied by the team's overall relative hockey value, as determined by its position in the standings.
Las Vegas’s total hockey value is
given by adding up the hockey value of the players that it gains in the draft
(one per team). Las Vegas’s position in
this hockey value scale can then be computed, and used as a proxy for how well
it would do in reality, should actual hockey correspond fairly well to the
assumptions of the models being tested.
Given these results, we can see which model results in some
reasonable prospect for a team put together under these draft rules to have an
expansion year as highly placed as Las Vegas actually did.
Four scenarios were modelled, with different assumptions:
- A moderate model (1), where there is
differentiation between the lines and defense pairings, in terms of hockey
value, but it is not extreme. It represents
a moderate view of how a hockey team actually works – for example, the first
line players are more important than their second line counterparts, but a top
ranked team will probably have a second line that is also very good. Similarly for the defense pairings.
- A second moderate model (2), which is
intermediate between the moderate model 1 and the egalitarian model. It places a bit more importance on the second
line and below than the first model does.
- An extremely egalitarian model (3), where each
player on the 20 man roster is actually as important as the rest. This model is highly unlikely to represent
reality, but is there as a “logical extreme” test.
- An extreme superstar model (4), where the top
line’s value was much higher than the second line’s value, which was in turn
much higher than the third line, and so on. This model probably represents the average
person’s view of how hockey teams work – in other words, very superstar driven.
Here is a typical result of a run, using Model 2 and a 10% randomization factor (explained below). The results for Las Vegas are at the bottom of this model run result, and indicate that it would have a predicted #5 finish in the standings.
Using these models gives the following results, if a randomization factor isn't included:
- Model 1 – the expansion team is predicted to
come in 10th.
- Model 2 – the expansion team is predicted to
come in 6th.
- Model 3 – the expansion team is predicted to
come in 2nd.
- Model 4 – the expansion team is predicted to
come in 24th .
It seems clear that Model 3, the Egalitarian model is far
too optimistic to be likely.
Essentially, it says that if every position is as important as every
other position, and which line the player is on doesn’t matter, the expansion
team will get an excellent team, as it will receive a top player from all the
top teams.
Model 4, the superstar model is very pessimistic about an
expansion team’s chances. It says that
the expansion team will only be able to draft relatively inferior players from
each club, and will therefore have no real hope.
Models 1 and 2 seem to be promising, though in both cases
the actual Las Vegas expansion team did better than what these models
predicted.
Adding Some Randomization, and Running Monte Carlo Models
This elaboration of the model is meant to be somewhat more
realistic, adding some random chance to the draft, as is always evident in real
life. In this case, the hockey values of
the various positions within the model were allowed to vary, within specified
limits. These limits were 10%, 25%, 50%
and 100%.
For instance, Model 1 says that a third line forward is
worth 80 points to a team, compared to the 100 points that a first line forward
is worth. With 25% randomization, that
third line forward could actually be somewhere between 60 and 100 points, while
the first line forward would be somewhere between 75 and 125 points. Basically, this says that it is hard to
predict just what value a player will actually add to a team in any given year. He might be injured, and thus be less
valuable than expected. Or, he might
have a banner year, maturing earlier than expected, and have a great year.
Each of these four models was run 100 times, for each of the
possible randomizations. The results are
as follows:
- The key finding here is that an expansion team
drafted under these rules, can sometimes come in within the top 4 places in the
standings, and can quite frequently can come in between 5th and 8th
(about 25% of the time).
- Model 2
- Similarly to model 1, this model finds that the
expansion club has a good chance of finishing within the top 4, and is quite
likely to finish within the top 8.
- ·
Model 3
- This model strikes on as very unrealistic – it has the
expansion team finishing in the top 4 most of the time.
- ·
Model 4
- Model 4 results are interesting. Even with very high levels of randomization,
the superstar model rarely allows the expansion team to finish in the playoffs,
let alone in the top 5.
I should note that as the randomization goes up, the range
of predicted finishes grows larger. This
makes sense, as the randomization is basically adding uncertainty to the
models, thus allowing a greater range of predictions (larger confidence
intervals).
Conclusion
So, this simple modelling leads us to the (tentative) conclusion
that the relatively high placing of Las Vegas this year can be understood,
without resorting to theories of owner collusion or sports miracles. Though the Las Vegas result is somewhat
unlikely, such a result is probably within the 10% to 20% range. We should think of it as being similar to getting 3 or 4 heads in a row, in a game of coin flipping - unexpected, but not all that unlikely.
In my opinion, the draft rules were probably crafted to
ensure that Las Vegas had a very good chance of making the playoffs, as that
would be good for hockey in the U.S. southwest.
However, NHL and individual team management probably erred on the side
of making the expansion team’s chances a little too good. Thus, Las Vegas ends up 5th overall, and contending for the Stanley Cup. That might be because owners and league officials intuitively thought
that the underlying hockey model is more star-driven than it really is.
If this is the case (hockey is less superstar driven than commonly thought), there is hope for
teams that can’t pay the big bucks. So, dream on, Oiler fans (and fans of similar franchises).
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Here's the part where I try to persuade blog readers to buy one of our Dodecahedron Books stories from Amazon. Bronco Billy has some hockey references and A Dark Horse is about gambling.
On the Road with Bronco Billy
Sit back and go on a ten day trucking trip in a
big rig, through western North America, from Alberta to Texas, and back again. Explore the countryside, learn some trucking
lingo, and observe the shifting cultural norms across this great continent.
A Dark Horse
Just what might a
gambler give up, to go on the winning streak of his life? Even he can't know
for sure. Christopher Marlowe's Doctor Faustus legend is given a Damon Runyon
spin, in this short story.
