(1)The reason for the deprecation is explained in the docstring:

parameter ``reorder`` has been deprecated in version 0.20 and will
be removed in 0.22. It's introduced for roc_auc_score (not for
general use) and is no longer used there. What's more, the result
from auc will be significantly influenced if x is sorted
unexpectedly due to slight floating point error (See issue #9786).
Future (and default) behavior is equivalent to ``reorder=False``.

(2)We get different result with reorder=False because we're actually looking at different points:

 x = [-2.0, 0.0, 0.0, 0.0, 1.0]
 y1 = [2.0, 0.0, 0.5, 1.0, 1.0]
# auc(x, y1) = 3
# looking at point (-2, 2), (0, 0), (0, 1), (1, 1)
# (2 + 0) * 2 / 2 + (1 + 1) * 1 / 2 = 3 
y2 = [2.0, 1.0, 0.0, 0.5, 1.0]
# auc(x, y2) = 3.75
# looking at point (-2, 2), (0, 1), (0, 0.5), (1, 1)
# (2 + 1) * 2 / 2 + (0.5 + 1) * 1 / 2 = 3.75

(3) I agree with @lesteve that we are checking the behaviour of the deprecated behaviour, so ignoring the warning might be fine. Will submit a PR now.

Uh oh!

regression AUC reorder? #11572

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