Every NHL player affects their team's results in a number of different ways. Comparing the impact of two different players at the same skill is fairly straightforward, but it is much trickier to compare the value of two different skills. For example, Alexander Ovechkin is possessed of a very remarkable finishing talent, turning the shots that he and his teammates generate for him into many more goals than an ordinary shooter would. On the other hand, he's also well-known to be weak defensively, causing his opponents to take more shots than a league-average skater would. Which of these two effects is stronger? Once this question is asked, of course, we would like to have a comprehensive way of valuing all of the different ways that a player can help or harm their team, all on the same scale.

The guiding principle behind all of my player evaluation models is that they should estimate a player's **intrinsic**
ability, that is, isolated from any effects outside of their control. In practice, this means accounting for teammates,
opponents, game state, where relevant, coaching systems, and, vitally, minutes played, since playing time is determined
by (somewhat constrained) coaching choices, and is not in any player's ability to affect, no matter how strong or weak
they may be.

The elements of player ability that I include here are:

- Impact on 5v5 shot rates, for and against, as measured by my 5v5 shot rate model.
- Impact on power-play shot rates (for) and on penalty-kill shot rates (against), as measured by my special teams shot rate model.
- Impact on goal scoring, most obviously by finishing talent but also through setting talent; both of these are measured via my xG model.
- Impact on team penalties drawn and taken, as measured by my penalty model.

While I am reasonably happy with each of these models, they were all developed separately and each one has its own units: the impact is calculated per 5v5 hour, or per power-play hour, or per penalty-kill hour, or per shot, or per all-situation hour, as appropriate. In other to synthesise a total impact we must bring all of these skill estimates together in a way that is independent of context, so that all players are treated the same.

Naturally, the units of our result will be goals, but these goals are not any particular goals that have been scored,
for which credit is being divvied up; they are instead notional goals that a player **would** cause to be scored, in contrived
circumstances, rather than their past circumstances or any particular future circumstances. For both of these reasons,
I call the resulting unit "synthetic goals", or sG for short. Furthermore, a given skill may impact goals scored by a player's
team, or by their opponents, or in some cases, both. Sometimes it will be convenient to distinguish offensive sG from
defensive sG, sometimes it will be convenient to combine them.

Since we cannot use a player's past icetime, we must choose a "reference basket" of icetime, which I have chosen to be:

- One thousand minutes of 5v5 time.
- One hundred minutes of power-play time.
- One hundred minutes of penalty-kill time.

While sG is an "all-in-one" stat, that reductively simplifies a player's manifold abilities into a single number, it is
**not** designed to be similar to a WAR/GAR-style stat. It is different in two important ways: first, sG is not
descriptive of the past; like all of the constituent model measurements that go into sG, it is an estimate of ability *at
a given moment in time*. In this sense sG is somewhat closer to the "predictive" side of the (slightly over-accentuated)
descriptive-predictive "dichotomy". Second, sG is not a counting stat; it is instead a rate stat. You should think of it like
you would think of a speedometer reading on your car, where instead of distance per hour the units are goals per reference
basket of icetime.

The conversion for 5v5 shot rate impact is the simplest. For example, at time of writing, Sidney Crosby's isolated impact on 5v5 offence
is 0.31 xG per hour (that is, per 5v5 hour). Thus, in a thousand 5v5 minutes, this amounts to 5.17 xG; leaving aside for
the moment that some of these shots will be taken by Crosby himself, assuming the shots are taken by league average shooters,
with league average setters, and against league average goaltenders, thee 5.17 xG should become 5.17 goals. Thus, this skill
is worth 5.17 sG. His defensive impact is currently more modest, just -0.05 xG per hour, so his defensive ability over a thousand
minutes is -0.83 xG, that is, Crosby's imagined opponents can be expected to score 0.83 **fewer** goals than they would have
scored had Crosby been replaced with a league-average player at this skill. These two impacts are felt in two different areas
(goals scored and goals allowed) but they can be gathered together to give a total of almost exactly 6.00 sG.

Continuing to use Crosby-as-of-the-northern-summer-of-2023 as our running example, my estimate of his impact on his team's power-play shot rates as 0.42 xG per hour, that is, per power-play hour. Again leaving aside the fact that many of these shots that Crosby generates for his team will be taken by him, we expect that over one hundred minutes of power-play time in average conditions this will amount to 0.70 sG. Crosby's estimated impact on penalty-kill shot rates against is -0.05 xG per hour, that is, per penalty-kill hour, so we expect this will amount to -0.08 goals over the course of 100 short-handed minutes. Combining the two impacts together, we have an impact of 0.62 sG.

Crosby's estimate is close to zero in part because he is not observed killing penalties very much. In his most recent season (22-23), he played 322 minutes on the power-play and just 4 short-handed. This eminently wise coaching decision is deliberately excluded from consideration here, because we are trying to measure Crosby himself, not the synergy between the player and their coach's decisions, be they wise or unwise.

Impact on shot **quality** is somewhat more complicated to value than impact on shot quantity. First, we compute a
distribution of all shots taken by all NHL teams over a broad swath of time, specifically all shots taken in the 2018-2023
regular seasons. For all of these shots, I can compute the probability of the shot being blocked, being missed if not blocked,
and being scored if not missed; that is, each shot can be assigned to a point in a three-dimensional configuration space.
By discretizing this space, we can make a tractable distribution of what kinds of shots are taken in NHL games.
The setting and shooting estimates for each player that I produce from my xG model have units of
logits, that is, impact on goal probability when expressed on the logit scale. For every element in our distribution of shots,
we can compute the change in goal odds that would result from Crosby taking on a specific role in that shot. For instance,
suppose a shot is described by conditional sucess probabilities \( (p_b,p_m,p_g) \), that is, a probability \(p_b\) of being
unblocked, a probability \(p_m\) of being on net rather than missed, given that it is unblocked, and a probability \(p_g\) of
being a goal given that the shot is on net; and suppose that Crosby's shooter talent estimate
is to have logit-scale impacts of \(c_b\), \(c_m\), and \(c_g,\) at each stage. Then Crosby acting as shooter shifts those probabilities to
\( (p_b',p_m',p_g') = (l(l^{-1}(p_b)+c_b), l(l^{-1}(p_m) + c_m), l(l^{-1}(p_g) + c_g) ) \) where \(l\) is the logistic function that maps
probabilities from the logit scale to the unit interval. Then the impact on this particular shot can be computed as
\( p_b'p_m'p_g' - p_bp_mp_g \). Then, by forming the weighted sum of this using the weights in the distribution of shots we computed
earlier, we can compute the impact of Crosby's shooting on an "average shot". Finally, we can multiply this per-shot impact by
0.94, the all-situations average number of shots taken per minute in 2018-2023, and then multiply the result by 1200, the total
number of minutes in our reference basket of minutes. In the future, we could consider how likely specific players are to *be*
the shooter of a given shot, but for now let us assume that forwards take two thirds of the shots and defenders one third, in line
with recent tendencies. Then multiplying Crosby's shooter impact by 2/9 (since there are usually three forwards on the ice), we
obtain a final value of +2.6 sG through shooting talent. A similar calculation for setting, where we assume that 80% of shots have
setters, and 7/9 of those setters are forwards, gives a setting impact of +3.5 sG.

The chance of a 5v5 goal being scored in a given two-minute stretch in the last several years is 8.5%, at 5v4 the chance rises to 24.0%, and at 4v5 it falls to 3.3%. Thus drawing a penalty has an offensive value of 24.0% - 8.5% = +0.155 goals for and a defensive value of 3.3% - 8.5% = -0.052 goals against, for a net benefit of 0.207 goals; similarly, taking a penalty has a net detriment of 0.207 goals. The outputs of my penalty model are impacts on team penalties drawn and taken, per thousand all-situations minutes; multiplying these estimates by 0.207 and then by 1200/1000, to stretch the impact to our 1200 minutes, gives the estimates of sG for drawing and taking penalties. For Crosby, whose presence on the ice is associated with the Penguins both taking and drawing slightly more penalties than average, the sG values are +0.7 (drawing) and -0.3 (taking).

For the moment I've kept all of the individual contributions to sG strictly separated—so, for instance, a player who creates offence and who shoots well gets credit for both of those things, but in fact such a player has an even larger positive impact on their team's scoring because some of the extra shots that they create will be taken by them. I've decided not to include these "synergy" effects at the moment, for reasons of complexity, but I might add them in the future.

As long as skills are reasonably independent of one another, they can be added to a player's sG. I have one model in preparation that I mean to add, measuring a player's tendency to gain or lose territory in their shifts. In principle any number of things could be added as we learn more.

Similarly, sG values could be computed for goaltenders and for coaches, which I mean to do over the course of the coming months.