I think there was a short experiment in the 1950s where the regular draft started with the second pick, and the first pick was a pure lottery among all the teams. Let me see if I can find out what happened. I only read it about once a long time ago, and have never heard anything more about it.
I'm not really sure what a lottery solves on first glance. Without a lottery, you might tank to get a higher pick. With a lottery, you might tank for a chance at a higher pick. The strategy is the same, except the lottery adds a little more risk. But I'm not sure that would really change the minds of teams who would be inclined to tank.
Maybe a better approach would be to move your draft spot up by one if you win on the final week of the season. That would prevent weak teams from tanking and strong teams from sitting starters. You'd have a little bit of math to adjust for the fact that half the teams will win, but that's easy to solve.
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