**“The New York Knicks, with their first pick, select Patrick Ewing of Georgetown,”** called out David Stern in the 1985 NBA draft. A draft that has since been heavily debated for three-plus decades as being the epitome of the NBA draft being “rigged” or fixed. Discussion about bent corners of envelopes, all the way to large leaps for cities to draft their hometown superstar, such as the Chicago Bulls did with Derrick Rose in the 2008 NBA Draft; the first time an eighth pick or higher had leap-frogged the competition to obtain the first pick of the draft. Similarly, the arguments that the Lakers, Bulls, and Celtics being favored every year of the draft while teams like the Kings, Pacers, and Clippers are screwed over every year… The NBA Draft has been wit controversy for a long time running.

So here, we’d like to take a look at the actual distribution of selections over the years and ask ourselves: Is the NBA draft really fixed? Can we actually show real evidence that certain teams are favored? Or will we find that there is absence of favoritism, suggesting that the NBA Draft Lottery is exactly what it is… random.

So let’s walk through the data and find out!

## NBA Draft: 1994 – 2017

Despite the NBA Draft Lottery starting in 1985 with the infamous Patrick Ewing selection; the current format of the NBA draft has only been around since 1994. In 1985 and 1986, every non-playoff team had a single card thrown into a hopper; of which the entire set of lottery teams were selected in entirety. Each team had an equal probability of selection to avoid tanking in the previous season.

In 1987, the Lottery changed to be of the same format, but only the top three seeds were selected. The remaining teams filled in the rest of the lottery picks by reverse order of win totals.

In 1990, a weighted lottery system started with only 66 random numbers. The worst team in the league obtained 11 of the 66 numbers; second worst obtained 10; and so on.

Starting in 1994, the number of random numbers changed from 66 to 1000. The odds for each team changed drastically as well. The worst team in the league went from having an 11/66 chance to a 250/1000 chance of obtaining the first pick in the draft. Similarly, the worst team moved from 1/66 chance to 5/1000 chance; dropping to roughly a third of its original probability of obtaining the first draft pick.

The current format then has the following probability distribution for each round, assuming no ties between any teams.

## Exceptions: Changes to the Lottery Odds

If multiple teams have the same record, the lottery odds are averaged such that each team has effectively the same probability. In cases of odd probabilities, teams resort to a coin flip to determine order and the extra probability point. As an example, for the 2017 season, the New York Knicks and the Minnesota Timberwolves had identical records, therefore taking 0.043 and 0.063 spots to average out to 0.053 probability each of obtaining the first pick. The Minnesota Timberwolves then won the coin toss, giving them one spot ahead of the Knicks if neither team were selected into the top three.

Other exceptions to the draft included the expansion teams of Toronto, Vancouver/Memphis, and Charlotte. When the Raptors and Grizzlies entered into the league in 1995, the 6th and 7th spots were given to the Grizzlies and Raptors, respectively. For the following drafts in 1996, 1997, and 1998, both the Raptors and Grizzlies were not allowed to obtain the first pick in the draft. They could, however, end up with second or third in the draft; a feat repeatedly performed by the Grizzlies.

In the 2004 draft, the Charlotte Bobcats were barred from obtaining a top-3 pick due to their expansion status. Unlike the Raptors and Grizzlies, the Bobcats were able to vie for the top pick in their second season.

## With The First Pick of the Draft…

There have been 24 seasons of the NBA Draft Lottery under its current format. For each season, we take the probabilities and add across each of the 24 seasons. We can do this since there can only be one first pick in each draft; treating each year as a single draw from an independent Multinomial distribution. Let’s look at the distribution of each lottery outcome for each year.

Year | First | Second | Third |

1994 | 4 | 1 | 2 |

1995 | 5 | 1 | 4 |

1996 | 2 | 3 | 1 |

1997 | 3 | 5 | 2 |

1998 | 3 | 5 | 1 |

1999 | 3 | 1 | 13 |

2000 | 7 | 5 | 4 |

2001 | 3 | 8 | 5 |

2002 | 5 | 2 | 1 |

2003 | 1 | 6 | 2 |

2004 | 1 | 5 | 2 |

2005 | 6 | 1 | 4 |

2006 | 5 | 2 | 3 |

2007 | 7 | 5 | 4 |

2008 | 9 | 1 | 3 |

2009 | 3 | 6 | 4 |

2010 | 5 | 6 | 1 |

2011 | 8 | 1 | 6 |

2012 | 4 | 1 | 2 |

2013 | 3 | 1 | 8 |

2014 | 9 | 1 | 2 |

2015 | 1 | 4 | 3 |

2016 | 1 | 2 | 3 |

2017 | 1 | 3 | 8 |

We see that it took 10 years before the team with the best chance won the actual draft lottery! Over the 24 years, the top team has only won 5 times. Now, despite saying that the worst team has 250/1000 (25% odds) chance of winning the top pick; this does not mean we should expect six (24 times .25) worst teams to obtain the pick. By the way… 5 isn’t bad even if we did expect six of these results. This is due in part to the 2002 and 2003 drafts, where the worst team in the league tied with yet another worst team in the league. This dropped the probability for the worst team from being 250/1000 to 225/1000. So we see an expected number of worst teams obtaining the first pick to be 22*.250 + 2*.225 = 5.95. So we actually expect less than 6 times for the worst team to obtain the top pick. Holy cow… this actually came true!

We should not expect numbers to match up identically. For instance, the second worst team is expected to have 4.74 top picks. This time, the numbers don’t magically align and we have seen only one… **ONE!** second worst team obtain the top pick in the draft over the 24 years. That lucky team? The 1996 Toronto Raptors. Effectively all of the second worst teams’ first picks were lost out to the third worst team, who has nabbed the top spot 6 times and the ninth worst team, who has picked up first pick twice.

Lottery Position | Expected # First Picks | Actual # First Picks |

1 | 5.95 | 5 |

2 | 4.742 | 1 |

3 | 3.775 | 6 |

4 | 2.805 | 2 |

5 | 2.04 | 4 |

6 | 1.521 | 1 |

7 | 1.125 | 2 |

8 | 0.721 | 1 |

9 | 0.467 | 2 |

10 | 0.307 | 0 |

11 | 0.192 | 0 |

12 | 0.156 | 0 |

13 | 0.129 | 0 |

14 | 0.07 | 0 |

So as we see the expected number of first picks compared to the actual number of first picks based on their lottery odds over the 24 years, we find that there really isn’t anything disturbing. However, we are merely looking at numbers and guessing if they show any “fixing.” So let’s look at the tests.

## The Usual (wrong) Test:

So the basic test to use is a Chi-Square goodness-of-fit test. While we can compute the mathematics necessary, the result is a little misleading. This is due to the fact that the statistical test assumes that there is a large sample. The general rule of thumb is that every cell has an expected count of five. Go back and look at the above table. All but the worst team has an expected count of 5. In fact, we would require another 1,700 years of drafts to be able to use the Chi-Square Goodness-of-Fit test. But ehhh… let’s see what it says anyways.

Since there are a total of 14 teams across all 24 years (note that the years teams 12, 13, and 14 don;t exist, they are simply **zero**), we have a 13-degree-of-freedom Chi-Square test. The associated test statistic looks at the squared-difference of the observed and expected value and divides this by the expected value. This means if the distribution is correct **(no fixing in the Draft lottery), **then the top value should be **very** close to zero. In this case, the statistic is 13.383439. Is this close enough to zero?

Since this is a 13-degree-of-freedom test, we look to see what the quantiles of the 13-df Chis Square distribution gives us. At 90 percent accuracy (this means we throw away 10 good results at random in order to get an answer) we have 19.8119. This means anything below 19.8119 is close enough to zero. **This suggests that the NBA Draft Lottery is not “fixed” for selecting the first pick in the draft. **

OK, let’s get a sharper bound. Suppose we have 60 percent accuracy. Then we obtain the value of 13.6355. This means that anything below 13.6355 is close enough to zero. At this point, we again find that the **NBA Draft Lottery is not “fixed” for selecting the first pick in the draft. **

If we look for the cut-off where we can finally suggest that the NBA Draft Lottery is fixed, we must select 58.14% accuracy. This is almost random-coin flip country. Meaning that the NBA Draft is **definitely not rigged. **But wait…

## But What About the “Right” Test?

Now, the argument can be made that the goodness-of-fit test is not correct. Instead, we can look into building an **exact test**; or also called a **permutation test**. The underlying problem is to find all instance of where a test statistic is larger than the test statistic observed using the actual data. By adding up these probabilities, we find the exact p-value associated with the 24 years’ worth of data.

Above, we argued that we have a constrained Multinomial distribution where the worst team has up to 24 possible attempts to obtain the top pick in the draft. Similarly, the best team in the lottery has only 14 attempts; as there have only been 14 teams in the lottery since 2004.

Thus there are (11^2)*(12^3)*(13^6)*(14^13) ~ 8×10^26 different combinations. That’s a lot to crawl through. So we wrote a MATLAB script that “quickly” iterates through all the combinations. By quickly, we mean 53 hours on a single processor. Below is a MATLAB snapshot of processing all the possible outcomes of obtaining the first pick over the 24 year period. The exact p-value for the permutation test is **.1841. This suggests that the NBA Draft Lottery is random. **

In fact, if we attempted to carry this out for all three lottery selections for each of the 24 years, we will only have 24 samples against 364 possible outcomes; compared to the 14 outcomes for obtaining the first pick. By nature of small numbers and the fact that the top three picks are primarily between spots 1 and 5 (11 of the 24 years); it is safe to say the resulting permutation test will show that the NBA Draft Lottery is random.

We can take this one step further below and visualize the distribution of obtaining the first pick (green line) against the theoretical values (red line). We see that the data slowly converges towards the truth; despite being only 24 data points.

## OK, So the NBA is Hiding Their “Rigging” for Specific Teams in the Noise… Right?

To test this, let’s take a look at the distribution of every team’s movement within the draft. To this end, we look at when teams move up in the draft, move down in the draft, and stay put. Since the draft order is ordered 1 through 14, for every spot a team moves up, that many spots must be moved down by other teams.

For example, the 1999 draft saw the Chicago Bulls move up two spots to number one and the Charlotte Hornets (New Orleans Pelicans) move up 10 spots to the number three. This means that 12 spots need to drop in the draft. The Los Angeles Clippers drop two spots from second to fourth; while 10 other teams all drop a single spot.

Now, we note each time a team drops in the draft, stays put, and moves up in the draft. In this case, we obtain these results:

Here we note that over the 24 years, it is only possible for 72 teams to move up in the draft. More specifically, every year there is effectively a 75% chance for a team other than the worst team in the draft to obtain the first pick in the draft. So we should effectively see at least 18 teams move up in the draft. Continuing on in this manner, we should see a 59.7% chance of a team moving up into the second spot. This gives us another expected 14 teams to move up. Similarly, there is a 49.5% chance of moving up to the third spot in the draft; for another 11 teams to move. In total, we should expect 43 spots moved up over the 24 years. And in fact, we only have **40**. This is easily within noise of a standard draft.

Taking a look at the distribution of all teams, there are a few apparent things…

### 1. Since 1994, the San Antonio Spurs have been in the Lottery ONCE.

That year was 1997. That year they landed Tim Duncan. The Spurs have been fortunate for staying relevant in the league over this 24 year period, thanks to shrewd moves, international inclusion, and player development. In that one year, the Spurs moved up in the draft from number 3 to number 1 to get Duncan. The probability of a move up was over 21%; so this happening is no surprise. Just very fortunate.

### 2. There are only five teams in the “Green”

These teams are the Houston Rockets, Los Angeles Lakers, Philadelphia 76ers, and Portland Trailblazers, and San Antonio Spurs. “Green” teams mean that they have moved up in the lottery more than moved down in the lottery. The Rockets, Lakers, Trailblazers, and Spurs are only up one. The Lakers, specifically, just went into the “Green” thanks to their move from 3rd to 2nd in this year’s draft lottery. How did these teams move into the “Green”?

**Houston** moved up from 5th to 1st in 2002 with an 8.9% chance. **Los Angeles** moved up from 4th to 2nd in 2015 with a 15.9% chance and from 3rd to 2nd with a 20.8% chance. **Portland **moved up from 4th to 3rd in 2005 with a 17.3% chance and from 7th to 1st with a 4.3% chance in what goes down as one of the biggest draft busts in NBA history (Greg Oden).

The **Philadelphia 76ers** are the only team to have a moved up two times more than move down in the NBA Draft Lottery. They have moved up **FOUR** times with moves from 4th to 3rd in 1995, 2nd to 1st in 1996, 5th to second in 1997, and 6th to 2nd in 2010. That said, the 76ers have had horrible luck of late with missing out on multiple picks in the Process due to protected picks and a recent move from 2nd to 4th in the 2017 NBA Draft.

### 3. There are 12 teams that have never moved up… But that’s relatively expected…

Eighteen teams have moved up over the 24 years, the most recent being the Sacramento Kings in 2017. Still, however, twelve teams have been on the short end of the stick; eleven have moved down. This is actually expected as we look at the teams who move down. For instance, the first seed has a 75% chance of moving down. Similarly, the second seed has a 61.3% chance of moving down while the third seed has a 53.1% chance. This means if teams are consistently in the top 3; they should expect to be in the “Red” over time.

Clear example: **Golden State Warriors**. They have been in the lottery for 16 of the previous 24 drafts. Despite moving up only once (5th to 1st in 1995 to grab **Joe Smith…**) they have fallen **FIVE **times. These drops were one spot (from 4th in 1998, 9th in 1999), two spots (from 1st in 2002, 4th in 2010), and three spots (from 2nd in 2001). The lone crazy exception was in 1999 when the **Charlotte Hornets** jumped from 13th to 3rd to grab Baron Davis. Otherwise, the Warriors typically held firm with their 7th through 14th pick. This is exactly the case for **Indiana**.

The Pacers have never moved up nor moved down in the draft with six “stays” at their expected position. This is effectively due in part to their draft positions: 10th (2010), 11th (2007, 2008, 2015), 12th (1997), 13th (2009).

In fact, here is the history of how each seed fared:

### 4. Three teams have NEVER moved down in the Draft.

These three teams are the Houston Rockets, Indiana Pacers, and San Antonio Spurs. We’ve already explained the Pacers and Spurs. These are obvious. The Houston Rockets, however, are a mix of the Spurs and Pacers. They moved up like the Spurs because they were in the sweet spot of 3rd through 6th. In fact, they were the second ever 5th to 1st jump when they grabbed Yao Ming in 2002. The other 7 times? Here’s their positions: 8th (2006), 9th (2001), 13th (2000, 2003), and 14th (2010, 2011, and 2012).

There’s no surprises here.

### 5. If there’s any team with a gripe… the Minnesota Timberwolves have it…

The Timberwolves have been in the current format of the draft lottery a record 17 times over the past 24 years. Over these 17 times, the Wolves have dropped in the draft eight times while holding still nine times. Never have they improved.

Here are all positions when the Timberwolves dropped: 1st (2011), 2nd (2010), 3rd (1994, 1995), 5th (1999, 2009), 6th (2007, 2017)

Similarly here are the positions when the Timberwolves stayed: 1st (2015), 3rd (2007), 5th (1996, 2016), 6th (2006), 9th (2013), 10th (2012), 13th (2014), 14th (2005)

We see that this is fairly consistent with the above distribution of upward, staying, and downward movement in the draft. However… we will test this…

## Testing Team By Team Basis

Since the sample sizes are relatively small (there’s only one value for San Antonio!!!), we adhere to a nonparametric test called the Mann-Whitney test. We will then test the position of each team against the entire league. Here, we assume that the draft movement is independent of the entire league’s movement; which is not entirely true as in 1999 the Charlotte Hornets movement forced the Timberwolves to fall one spot in the draft. However, the violation of this assumption will not cause egregious errors in the results; provided the small sample teams are not affected by drastically. They effectively are not thanks to the Spurs taking the top spot in their lone lottery since 1994.

We then take a look at the p-values for these small samples and state anything small is considered “significantly different than random.” To insinuate any fraudulent activity, we assume that at least one bad sample must occur over the 24 years and therefore we look for any evidence below the 1/24 = .04167 range.

For each of the thirty teams, we obtain the following p-values:

Here, we find that there are two teams with significant values: Minnesota and Philadelphia. We previously walked through Minnesota’s movement. This test triggers on three particular values, **all drops**, coming from their one seed drops from 5th (twice), and 6th. However, if we look at the directional test, this evidence does not support a negative bias against the Timberwolves. Therefore, due to the effect of the test, we find the Timberwolves to be the **spurious case**.

Similarly for the Philadelphia 76ers, recall we briefly discussed their foray into being the team who has gained the most from the draft lottery. Their movement upwards came from the 2nd, 4th, 5th, and 6th. The only team to grab the top spot from the second seed. Their two drops were from 2nd (2015) and 4th (2017). The latter bailing them out thanks to a previous trade with Sacramento. This again follows the distribution above and we find that the 76ers are, like the Timberwolves the **spurious case**.

## Conclusions

Breaking down the exact permutation tests, we found that the NBA Draft Lottery is indeed random. Furthermore, in a team-by-team analysis, we find that the Minnesota Timberwolves and Philadelphia 76ers are significantly different than the rest of the league; however, the evidence does not support that they are being biased against in a particular direction; indicating that they are spurious cases.

It is safe to say that the NBA Draft Lottery is **not fixed**.

Was going to attempt this analysis on my own after some loose bar talk with my old man, but I would not have been as thorough as you! Well constructed and thought-out, thanks for posting.

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