Naturally Including A Linear Usage-Efficiency Tradeoff

[v-zero]

Since the usage-efficiency tradeoff is popular in box score metrics I thought this was worth posting. Mostly this seems to be done by adding in a squared term, however if you do the algebra then you find that this implies a non-linear tradeoff; but the evidence we have seen suggests that the tradeoff is more-or-less linear, as such the formulation involving a squared term is not ideal, but there is a better way.

Forget points, we should cast our scoring rating in terms of efficiency and usage if we want to accomodate a linear tradeoff.

Let EFF = PTS/USG, USG = FGA + 0.44*FTA + TOV (or however you prefer really).

Then consider:

Rating = EFF*k0 + USG*k1

Where k0 and k1 are the coefficients (linear weights).

(We can ignore all other terms of a possible box score metric in order to illustrate this in isolation).

Now consider increasing the value of USG by one, calling it USG_n, then:

USG_n = USG + 1, and let us call the new efficiency at this usage EFF_n, and the new rating Rating_n, then we have:

Rating_n = EFF_n*k0 + USG_n*k1 = EFF_n*k0 + (USG + 1)*k1

Now, the tradeoff implies that the original Rating, and Rating_n should be equal, so we can equate the two and solve for EFF_n:

EFF*k0 + USG*k1 = EFF_n*k0 + USG*k1 + k1,
--> EFF_n*k0 = EFF*k0 - k1,
--> EFF_n = EFF - k1/k0

So, that says, increasing your usage by one should decrease your efficiency on average by k1/k0, and more generally varying your usage by N will on average change your efficiency by -N*k1/k0, which is clearly a linear tradeoff.

So, rather than PTS, USG and USG^2 we can use EFF and USG to better reflect the findings we have on the tradeoff.

Whilst I do not use this tradeoff myself (experimented with it, found it too general for what I'm trying to do) I have 'run the numbers', and I found over a long period of box score data that coefficients of k0 = 5, k1 = 0.076 were more or less the best fit, which leaves us with k1/k0 = 0.0152, or roughly 1.5 pts per 100 possessions, more or less in line with other findings.

One thing to keep in mind is that this won't work unless the normalize usage to a certain number of possessions. So if a player plays for 30 possessions then whilst his USG will vary linearly with this his efficiency will not, so USG per 100 pos is how I formulated it (so in this case you'd do USG*100/30 to get the USG per 100 pos, and then calculate the rating per 100, then pro-rate back to 30 if desired).

[Crow]

I must have missed this earlier. You provide an analysis that is both rigorous and concise.

[nbacouchside]

This was pretty great. It'd be pretty cool to add this Alternate Win Score and see how it works out as a very simple box score based predictor. In fact, maybe I will start doing that.