My preferred solution is to use the current system, but reverse the percent chance of winning.
Give the best non-playoff team a 25% chance of picking first overall, decreasing down to the 0.5% for the 30th place team.
Basically, it would break down this way:
17th: 25% 1st pick, 75% 17th pick
18th: 18.8% 1st pick, 56.2% 16th pick, 25% 17th pick
19th: 14.2% 1st pick, 42% 15th pick, 43.8% 16th pick
20th: 10.7% 1st pick, 31.3% 14th pick, 54.5% 15th pick
...
28th: 1.1% 1st pick, 1.3% 3rd pick, 97.6% 4th pick
29th: 0.8% 1st pick, 0.5% 2nd pick, 98.7% 3rd pick
30th: 0.5% 1st pick, 99.5% 2nd pick
Basically, the worse you are, the lower your odds of drafting first overall, and the greater your odds of being bumped down one position. This gives teams a reason to be the best they can, while also still allowing the truly awful teams the chance to improve, as dead last is still guaranteed the 2nd overall pick.
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