last season, the Moon suggested using probabilistic methods to determine the worth of the Charlotte Bobcat's pick. I suggested a refined strategy using Markov Chains as a sounder methodology in determining the value. As we are now the proud owners of the Sacramento King's protected 1st rounder, I think that it would be good in forthcoming discussions if we maintained a consensus on what this pick will actually get us.
The King's pick is top-12 protected this season, and top-10 protected for the 2014-2015, 2015-2016 and 2016-2017 seasons.
The underlying idea is to use the past performances of NBA teams to provide probabilities for teams improving and declining. I have split the teams into seven categories:
A : 0 -23 Wins (picks 1-3 on average)
B: 24-30 Wins (Picks 4-7)
C: 31-34 Wins (Picks 8-10)
D: 35-40 Wins (Picks 11-14)
E: 41 - 45 Wins (Picks 15-20)
F: 46-53 Wins (Picks 21-25)
G: 54 - 82 Wins (Picks 26 - 30)
That is, the third worst team in the NBA has 23 wins on average (this is post-expansion) and so on. There are a few hidden assumptions here that I will discuss in the section on the possible flaws in the calculation, namely the fact that the correlation between the number of wins and placement in the draft is variable by season (and this isn't even including the lottery system).
The probabilities that we use in our calculations are the transition probabilities. To illustrate what this is by way of example, consider those teams in group B (incidentally, Sacramento is projected to finish the season in group B). Of the 64 teams in group B in the post-expansion NBA, we have the following:
18.75% ended the next season in group A.
28.125% ended the next season in group B.
14.0625% ended the next season in group C.
14.0625% ended the next season in group D.
12.5% ended the next season in group E.
9.375% ended the next season in group F.
3.125% ended the next season in group G (hello '99 Spurs).
Thus, using this model, if we agree to assume that the Kings will end this season in group B, then we will get their 2015 first rounder with probability 39.0625%. That is the sum of the probabilities that they land in groups D,E,F or G.
We now consider the question of whether or not the Bulls get the King's pick at all. Essentially, we will compute the probability that the King's do not break 35 wins in one of the next three seasons (2015, 2016, 2017 where they are top-10 protected) assuming that they are in group B this season (2014).
Thus, we are computing the probability that the King's stay in group A-C over the next 3 seasons. Consider the following 3 x 3 matrix:
The entry in the A column and C row is the probability that a team transitions from group A to group C in one season (12.9%).
This is where it gets tricky so feel free to skip over this if you are of questionable intelligence (that means you Sacramento fans!).
We may multiply these entries in order to gain the probabilities of the King's placement over multiple years. That is, assuming that the Kings are in group B this year, what is the probability that they are in group C in 2015, group A in 2016 and group B in 2017 (we shall encode this event by the string BCAB)? This is computed through multiplying the entries of the transition matrix above, leading to the following computation:
Probability (BCAB) = .140625 x .18181818 x .2580645161 = .00659824
(Probability that team transitions from B to C) x (Probability that team transitions from C to A) x (Probability that team transitions from A to B)
Thus, there is a 0.65% chance that this specific outcome occurs. In order to compute the total probability that the Kings never break 35 wins, it is merely a question of cubing the above matrix and reading off the 2nd column. Here is the computation:
In decimal format, this means the following:
The King's will end the 2017 season in group A, never having left groups A-C, with probability 10.27%.
The King's will end the 2017 season in group B, never having left groups A-C, with probability 9.76%
The King's will end the 2017 season in group C, never having left groups A-C, with probability 5.47%
Thus, the King's will keep their draft pick with probability 25.49%.
THE BULLS WILL GET THEIR DRAFT PICK WITH PROBABILITY 74.51%.
Essentially, we have a 3/4 chance of getting their draft pick. Not bad if you ask me.
Possible flaws in the methodology:
Overall, I think that the Markov Chain is the best way to predict these type of events. That said, I would have been more thorough in my approach if i were being paid for this and I was not on my lunch break.
First, the use of only post-expansion NBA data leads to possible aberrations due to sample size. I should probably have used post-1979 data but this would require some creative bookkeeping and some additional assumptions and I am on a limited time allotment.
Second, I assumed that the number of wins correlates roughly with draft position. This could have been avoided if, instead of using the total number of wins as my data, I simply used the teams placement in the standings. Once again, if this were a professional job, I would have done so, but I have bills to pay suckas.
Third, I did not factor in the possibility that the King's end a season in group D but we do not gain their pick due to lottery considerations (hello 1.7%). This should have a marginal, but non-trivial affect.
Lastly, I did a lot of scratch arithmetic and am not very good with spreadsheets so there might be small errors in there. I do not think there is anything major (or any error that could become major through its use in a larger calculation), but I would not use this calculation if I were, say, Gar Forman, but it is fine for you and me.
I used the 3 x 3 matrix in order to compute the probability that the Kings never leave groups A-C. I could use the expanded 7 x 7 matrix in order to compute the probabilities associated to the placement of the pick as well as the year that it will be received. Thus, if there is popular demand, we can use this method to compute the probability that the Bulls will receive the 11-14 overall pick in 2015 and maintain a chart with all of these possible outcomes.
For now, we can simply say that with probability 3/4, the Kings will not suck at some point in the next 3 seasons. Isn't that comforting, Sacramento fans?