Repository: pandas-dev/pandas

Issue:
(Assuming we don't want to disallow 1-level MultiIndexes), stop identifying MultiIndex by nlevels == 1
#### Code Sample, a copy-pastable example if possible

```python
In [2]: mi = pd.MultiIndex.from_product([[0, 1]])

In [3]: mi.set_names(('first',), level=0)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-3-1f4ad6c7f0bf> in <module>()
----> 1 mi.set_names(('first',), level=0)

/home/nobackup/repo/pandas/pandas/core/indexes/base.py in set_names(self, names, level, inplace)
   1386 
   1387         if level is not None and self.nlevels == 1:
-> 1388             raise ValueError('Level must be None for non-MultiIndex')
   1389 
   1390         if level is not None and not is_list_like(level) and is_list_like(

ValueError: Level must be None for non-MultiIndex

```

#### Problem description

The case of ``MultiIndex`` with ``self.nlevels == 1`` is currently perfectly valid (despite e.g. ``MultiIndex.droplevel`` returning a normal ``Index`` when only one level is left). Unless we want to change this, we should always identify a ``MultiIndex`` with ``isinstance``, not by looking at ``self.nlevels``.

#### Expected Output

```python
In [4]: mi.set_names(('first',))
Out[4]: 
MultiIndex(levels=[[0, 1]],
           labels=[[0, 1]],
           names=['first'])
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit: 5cf4957a2cc8429170742dd1e6435ddb8c323811
python: 3.5.3.final.0
python-bits: 64
OS: Linux
OS-release: 4.9.0-6-amd64
machine: x86_64
processor: 
byteorder: little
LC_ALL: None
LANG: it_IT.UTF-8
LOCALE: it_IT.UTF-8

pandas: 0.24.0.dev0+14.g5cf4957a2.dirty
pytest: 3.5.0
pip: 9.0.1
setuptools: 39.0.1
Cython: 0.25.2
numpy: 1.14.3
scipy: 0.19.0
pyarrow: None
xarray: None
IPython: 6.2.1
sphinx: 1.5.6
patsy: 0.5.0
dateutil: 2.7.3
pytz: 2018.4
blosc: None
bottleneck: 1.2.0dev
tables: 3.3.0
numexpr: 2.6.1
feather: 0.3.1
matplotlib: 2.0.0
openpyxl: 2.3.0
xlrd: 1.0.0
xlwt: 1.3.0
xlsxwriter: 0.9.6
lxml: 4.1.1
bs4: 4.5.3
html5lib: 0.999999999
sqlalchemy: 1.0.15
pymysql: None
psycopg2: None
jinja2: 2.10
s3fs: None
fastparquet: None
pandas_gbq: None
pandas_datareader: 0.2.1


</details>



Discussion:

Resolving patch:
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -91,7 +91,8 @@ def cmp_method(self, other):
         if needs_i8_conversion(self) and needs_i8_conversion(other):
             return self._evaluate_compare(other, op)
 
-        if is_object_dtype(self) and self.nlevels == 1:
+        from .multi import MultiIndex
+        if is_object_dtype(self) and not isinstance(self, MultiIndex):
             # don't pass MultiIndex
             with np.errstate(all='ignore'):
                 result = ops._comp_method_OBJECT_ARRAY(op, self.values, other)


Test patch:
diff --git a/pandas/tests/indexes/test_multi.py b/pandas/tests/indexes/test_multi.py
--- a/pandas/tests/indexes/test_multi.py
+++ b/pandas/tests/indexes/test_multi.py
@@ -3307,3 +3307,20 @@ def test_duplicate_multiindex_labels(self):
         with pytest.raises(ValueError):
             ind.set_levels([['A', 'B', 'A', 'A', 'B'], [2, 1, 3, -2, 5]],
                            inplace=True)
+
+    def test_multiindex_compare(self):
+        # GH 21149
+        # Ensure comparison operations for MultiIndex with nlevels == 1
+        # behave consistently with those for MultiIndex with nlevels > 1
+
+        midx = pd.MultiIndex.from_product([[0, 1]])
+
+        # Equality self-test: MultiIndex object vs self
+        expected = pd.Series([True, True])
+        result = pd.Series(midx == midx)
+        tm.assert_series_equal(result, expected)
+
+        # Greater than comparison: MultiIndex object vs self
+        expected = pd.Series([False, False])
+        result = pd.Series(midx > midx)
+        tm.assert_series_equal(result, expected)