calculating the diminishing returns of team quality

[ampersand5]

I imagine this has already been answered, or has an easy enough solution that I'm just missing right now.

When we do projections, we calculate team quality by adding up the value of each individual player X minutes played. We then simulate games based on the quality of each team.

If a team can only win/lose a maximum of 82 games, then there must be a cutoff where it doesn't matter how good/bad a team is, as long as they are past the cutoff (ie a team with a cumulative RAPM of +100 will win 82 games, just as a team with a cumulative RAPM of +200 will win 82 games).

This means that the ability a player has to impact a team is not linear. What I want to know is what is the curve. Practically speaking, at what team quality is it the easiest to add wins to? Furthermore, what is the rate of change?

To conceptualize this issue, if a player with an RAPM of 4 is going to join a team, which team would get the most additional wins - a team with a cumulative RAPM of -10, -7.5, -5,-2.5, 0, 2.5, 5, 7.5 or 10?

(I assume the answer is a team with a cumulative rapm of 0/500% record. What I really want to know is how quickly the value a player can add changes)

[Crow]

It may be -2.5 along with or over 0 because of impact on road wins. Overcoming home court advantage?

Should be fairly easily for someone with a bit of time to calculate.

[xkonk]

I've always thought the limit on games that could be won (the max of 82) should be taken into account, as you say, but I think that practically speaking it doesn't matter. Evan Z has a post (that I bring up just because I remembered it; he isn't the originator) showing the correspondence between wins and point differential: https://thecity2.wordpress.com/2010/12/ ... valuation/ . That's just one year in the graph, but the equation should cover the vast majority of teams. A team with a point differential of 12 is expected to win 71 or 72 games, for example, which is only a little high for the three teams (https://en.wikipedia.org/wiki/List_of_N ... percentage) that have hit that mark (the '96 Bulls hit it pretty much on the nose). The 9-73 76ers team had a margin of -12, which predicts about 10 wins. Long story short, for the point differentials and wins that actually occur, a linear model seems to be close enough and maybe only breaks down at the extremes.

From that point of view, a player with a given RAPM (or whatever measure that maps onto point differential/wins) is going to add the same amount to any team assuming he plays the same number of minutes and replaces the same minutes/levels of players from any team he would be added to. But I'm sure there are lots of threads in here you could read that discuss diminishing returns of rebounds, usage, etc., which means it isn't quite that straightforward.

[Crow]

The values on Evan's chart from -3 to 0 all outperformed the slope line. The values just beyond -3 all underperformed. I think the threshold for winning a decent % of roads is a relevant and key factor as I suggested above.

[ampersand5]

It may be -2.5 along with or over 0 because of impact on road wins. Overcoming home court advantage?

Should be fairly easily for someone with a bit of time to calculate.

I don't get how roadwins should have anything to do with anything seeing how a team plays just as many games at home as they do on the road...?

[ampersand5]

I've always thought the limit on games that could be won (the max of 82) should be taken into account, as you say, but I think that practically speaking it doesn't matter. Evan Z has a post (that I bring up just because I remembered it; he isn't the originator) showing the correspondence between wins and point differential: https://thecity2.wordpress.com/2010/12/ ... valuation/ . That's just one year in the graph, but the equation should cover the vast majority of teams. A team with a point differential of 12 is expected to win 71 or 72 games, for example, which is only a little high for the three teams (https://en.wikipedia.org/wiki/List_of_N ... percentage) that have hit that mark (the '96 Bulls hit it pretty much on the nose). The 9-73 76ers team had a margin of -12, which predicts about 10 wins. Long story short, for the point differentials and wins that actually occur, a linear model seems to be close enough and maybe only breaks down at the extremes.

From that point of view, a player with a given RAPM (or whatever measure that maps onto point differential/wins) is going to add the same amount to any team assuming he plays the same number of minutes and replaces the same minutes/levels of players from any team he would be added to. But I'm sure there are lots of threads in here you could read that discuss diminishing returns of rebounds, usage, etc., which means it isn't quite that straightforward.

Thanks. This is a very interesting find. It would then appear as if teams are too close in skill for there to be any diminishing returns of this nature (which definitely exist in a conceptual world).

[Nate]

Lets supposs that the score difference at the end of a game is the difference between team strengths plus some 'normally distributed' random variable. IIRC the standard deviation in that distribution is around 12.5 points to best match scoring data. By the time that things get really non-linear, the number of expected losses per season is quite small, so sample size issues tend to be bigger than the non-linearity.

[Crow]

Road wins is probably a non-linear function, especially for the road team between -3 and 0 on team strength across a distribution of opponents. The extra points in this range affects win% most dramatically. I may not be explaining it well but I continue to maintain it is a relevant factor. It may be true at home too especially for teams in that -3 to 0 team strength range. Perhaps road vs home isn't that different but clearly the impact of home court advantage is going to affect the win curve more in that range more than elsewhere.

[xkonk]

The values on Evan's chart from -3 to 0 all outperformed the slope line. The values just beyond -3 all underperformed. I think the threshold for winning a decent % of roads is a relevant and key factor as I suggested above.

That's just the graph for one season. I grabbed a much larger data set (the team data link in http://wagesofwins.com/2012/09/20/break ... -produced/), covering '78 to '12. If you make the same graph with the linear prediction (essentially the same as what Evan got, 40.2+2.62*point dif) and a loess smoothed line to look for wiggles, the two lines are virtually identical from about -9 to +9 point differential. If someone tells me how to post jpegs or the like, I can post the figure.