NBA “3-2-1 Lottery”
On April 28, ESPN reported new details regarding a proposed reform of the NBA Draft lottery. The proposal, referred to as the “3–2–1 lottery,” modifies both the number of participating teams and the allocation of lottery odds.
The "3-2-1 lottery" proposal, named to represent the number of lottery balls per team, would expand the lottery from 14 to 16 teams. Teams that do not qualify for the playoffs or play-in tournament but stay out of the relegation zone (spots four through 10) would receive three lottery balls each. Teams with a bottom-three record -- the relegation area -- would have only two lottery balls but have a floor of the No. 12 pick, and the rest of the 13 lottery teams could fall as far as the No. 16 pick.
The Nos. 9 and 10 play-in seeds in each conference receive two lottery balls each, and the losers of the 7-8 play-in games receive one lottery ball each.
This proposal has been criticized for being punitive towards the team with three worst records (“the Bottom Three”) who are given only two lottery balls each. However, the crux of biscuit is in how the Top-12 guarantee is implemented in practice. At the time of writing, no procedural details have been publicly specified.
The guarantee can be implemented in (at least) two ways: using a Hard Boundary method or by using an Accept/Reject approach.
In the following we will see that the implementation of the guarantee actually determines the entire structure of the probability table describing the pick probabilities.
Hard Boundary Method
In the Hard Boundary method, the teams are selected one at a time until nine picks have been determined. After the first nine selections, any remaining Bottom Three teams are assigned picks no lower than No. 12 using a random tie-breaking mechanism. For example, if two of the Bottom Three are still without a pick when ten picks have been drawn, the two teams flip a coin to determine who gets No. 11 pick and who is pushed to No. 12 pick.
The results of a Monte Carlo simulation of the Hard Boundary method (with 1,000,000 simulation replications) are presented below.
Monte Carlo simulation results for the NBA 3-2-1 Lottery with Hard Boundary.
The simulation indicates that the Hard Boundary implementation substantially disadvantages the Bottom Three teams. Because they have only two lottery balls in the drawing, each Bottom Three team has approximately a 5.4% probability of receiving the first overall pick, compared to 8.1% for teams seeded 4–10. Moreover, there is roughly a 25% probability that a Bottom Three team is pushed all the way back to the pick No. 12.
Accept/Reject Approach
In the Accept/Reject approach, the lottery balls are drawn to first determine the entire draft order. Then, the draft order is checked to see if any of the Bottom Three have fallen below No. 12 pick. If this is the case, the entire draft order is rejected and all picks are determined once again. This is repeated until an acceptable draft order is found.
The results of a Monte Carlo simulation of the Accept/Reject approach (10,000,000 replications, out of which 3,470,244 were accepted) are presented below.
Monte Carlo simulation results for the NBA 3-2-1 Lottery with Accept/Reject Approach.
The first observation from the Monte Carlo simulation is that 65% of the randomly generated draft orders were rejected because they did not fulfill the Top 12 guarantee for the Bottom Three. Compared to the Hard Boundary method, these were the situations were at least one of the Bottom Three teams was pushed beyond the boundary line.
The second observation of the results is that this method is far less punitive compared to the first alternative. The Bottom Three teams have approximately the same odds for getting top picks as the teams seeded in the 4-10 range. In other words, the Bottom Three teams only having two lottery balls is almost completely compensated by the Top 12 guarantee.
Notably, the expected draft position for Bottom Three teams is 6.8, compared to 7.7 for teams seeded 4–10, indicating a slight advantage instead of a penalty.
Conclusion
The analysis demonstrates that the impact of the proposed “3–2–1 lottery” depends critically on the implementation of the Top-12 guarantee. When considering potential tanking boundary near the Bottom Three of the standings, the same nominal rule can produce either a strongly punitive or nearly neutral outcome for the teams landing in the Bottom Three.
In particular:
The Hard Boundary method introduces a significant downward bias and punishes teams for falling into the Bottom Three.
The Accept/Reject approach largely offsets the reduced number of lottery balls.
Consequently, any evaluation of the proposal remains incomplete without explicit procedural details. One should hold judgment before the implementation mechanism is specified, as it effectively determines the resulting probability structure.
