## Defining the cost of a player

I'm looking for a bit of philosophical and mathematical input.

I'm in the early stages of a project attempting evaluate players' contributions to their teams relative to the costs of having them on the roster. The simplest explanation is that I'm trying to take the crude observations we make about whether players are "worth" their cap hits, and test them against the true costs of various types of player contributions around the league. I want to base estimations of the costs of missing pieces on real data, rather than relying on highly subjective selections of "cap comparables," or the doomed prospect of trying to evaluate the cost of a certain type of player by glancing at the cap hits of players who happen to come to our minds as matching the profiles we seek.

I'll explain the methods and goals of my analysis in more detail as I progress. For today, I want to focus on how I'm defining "cost," and seek suggestions on how to improve that definition.

The currency of NHL roster management is cap space, but simply looking at a player's AAV is not sufficient for determining the true cost of having him on the roster, as not every player costs his team the full amount of his AAV over the course of a season. Since cap hits accrue daily, and many players who will contribute data under consideration will not spend every day of the season on the NHL roster, we can only "charge" them for the days they do if we want an accurate evaluation of the costs of the things they produce. This is easy enough to accomplish by defining a player's cost as his number of days spent on the NHL roster multiplied by his daily cap hit.

What I'm less sure of is how to treat players who spend time on Long Term Injured Reserve. Since teams are allowed to exceed the salary cap to replace LTIR players if they must, cap hits of LTIR players do not obstruct their teams' abilities to add production from other players as much as those of healthy players. This would suggest that counting a player's cap hit for days he spends on LTIR, producing zero of the "goods" we want to evaluate the costs of will distort our prices. At the same time, not counting that hit would ignore that the team's ability to add production from other players is limited by the cap dollars already allocated to the player on LTIR, and that they're typically not, in practice, able to replace the full value of the injured player. Ignoring these practical limitations would also create a distortion.

The intuitive solution appears to be to apply an LTIR discount, acknowledging that an injured player costs his team cap dollars, but that they are able to get production from some of the cap dollars earmarked for him. But what should the discount be? I'm stumped. Going on a case by case basis, and discounting the actual cap dollars spent on replacement players over the course of a particular LTIR listing ignores a number of organizational structure variables affecting a team's ability to bring in replacements at the particular time a player is injured, so all LTIR listings should probably be discounted at a common rate. So what's the rate? The league-wide average rate of replaced cap dollars per LTIR dollar seems appealing, but does that effectively deal with the noise inherent in each LTIR replacement event? Is there a different approach to discounting LTIR time I'm overlooking? Is there a capology-based case that it should be discounted totally, or not at all?

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