The push back on that is, if the starting lineup would be hampered with Monk starting, then why isn't the closing lineup? They can still do the staggering with him there, in a different way.
If you have four clear top offensive players, the hockey line shift plays them all together (only one ball) and then puts out five complementary (non-primary scoring threat) players together. In six minute intervals, the number of top players on the court might look like this:
4 4 0 4 half 4 4 0 4
That's 12 minutes of junk ball and 36 minutes where your top players are competing for shots (giving our top players 36 minutes each, which is...optimistic).
In the staggered rotation scheme (again 36 minutes per top player) those minutes can look like, for example:
3 3 2 3 half 3 3 3 4
That's the same amount of minutes for the top four players. Now you've got none of those dead minutes. The idea is that because the ball has to be shared the following approximate relationships hold:
4 > 3
4 >>> 2
4 >>>>>> 1
4 >>>>>>>>>>>>> 0
So the staggered rotation avoids the horrible "0" and "1" lineups and minimizes "2" lineups to get a lot of "3" lineups. This is at the expense of "4" lineups, but "4" lineups aren't much better than "3" lineups - you'd rather have [3 3] than [4 2]. You save your "4" lineups for the end, when it counts. So you sacrifice it at the beginning. Sure, you could start Monk to have a "4" lineup on the floor to start, but then you'd have to run more minutes of "2" lineups, which you don't want to do.
Obviously these are toy numbers, but I hope they illustrate the point.
Barnes hasn't been part of the closing lineup, it makes sense to move him to the 2nd unit where he'll get more shots
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