It seems that the default version of Python is 2.7 in the 2015/03 release of Amazon Linux, which made Ansible via yum a little addictive. Amazon Linux AMI 2015.03 Release Notes
$ sudo yum install ansible
Dependent packages include python 2.6 series.
==================================================================================================================================================================================================
Package architecture version repository capacity
==================================================================================================================================================================================================
During installation:
 ansible                                                          noarch                            1.9.0.1-2.el6                                   epel                                    1.7 M
Dependency related installation:
 python-crypto2.6                                                 x86_64                            2.6.1-2.el6                                     epel                                    513 k
 python-keyczar                                                   noarch                            0.71c-1.el6                                     epel                                    219 k
 python26                                                         x86_64                            2.6.9-1.80.amzn1                                amzn-updates                            5.7 M
 python26-PyYAML                                                  x86_64                            3.10-3.10.amzn1                                 amzn-main                               186 k
 python26-babel                                                   noarch                            0.9.4-5.1.8.amzn1                               amzn-main                               1.8 M
 python26-backports                                               x86_64                            1.0-3.14.amzn1                                  amzn-main                               5.2 k
 python26-backports-ssl_match_hostname                            noarch                            3.4.0.2-1.12.amzn1                              amzn-main                                12 k
 python26-crypto                                                  x86_64                            2.6.1-1.10.amzn1                                amzn-main                               697 k
 python26-ecdsa                                                   noarch                            0.11-3.3.amzn1                                  amzn-main                                77 k
 python26-httplib2                                                noarch                            0.7.7-1.5.amzn1                                 amzn-main                                81 k
 python26-jinja2                                                  noarch                            2.7.2-2.15.amzn1                                amzn-main                               899 k
 python26-libs                                                    x86_64                            2.6.9-1.80.amzn1                                amzn-updates                            692 k
 python26-markupsafe                                              x86_64                            0.11-4.6.amzn1                                  amzn-main                                27 k
 python26-paramiko                                                noarch                            1.15.1-1.5.amzn1                                amzn-main                               1.3 M
 python26-pyasn1                                                  noarch                            0.1.7-2.7.amzn1                                 amzn-main                               174 k
 python26-setuptools                                              noarch                            12.2-1.30.amzn1                                 amzn-updates                            582 k
 python26-simplejson                                              x86_64                            3.6.5-1.12.amzn1                                amzn-main                               210 k
 python26-six                                                     noarch                            1.8.0-1.23.amzn1                                amzn-main                                31 k
Transaction summary
==================================================================================================================================================================================================
Installation 1 package(+18 dependency packages)
$ ansible localhost --connection=local -m ping
Traceback (most recent call last):
  File "/usr/bin/ansible", line 36, in <module>
    from ansible.runner import Runner
ImportError: No module named ansible.runner
It doesn't work.
$ python --version
Python 2.7.9
The Python version is 2.7 series.
Try changing the Python version to 2.6 series.
$ sudo alternatives --config python
There are 2 programs'python'To provide.
Select command
-----------------------------------------------
*+ 1           /usr/bin/python2.7
   2           /usr/bin/python2.6
Press Enter to select the current[+]Or enter the selection number:2
$ python --version
Python 2.6.9
This works for the time being.
$ ansible localhost --connection=local -m ping
 [WARNING]: The version of gmp you have installed has a known issue regarding
timing vulnerabilities when used with pycrypto. If possible, you should update
it (i.e. yum update gmp).
localhost | success >> {
    "changed": false,
    "ping": "pong"
}
However, this doesn't work for yum.
---
- hosts:
    - localhost
  connection: local
  tasks:
    - name: update all packages
      yum: name=* state=latest
      sudo: yes
$ ansible-playbook test.yml
 [WARNING]: The version of gmp you have installed has a known issue regarding
timing vulnerabilities when used with pycrypto. If possible, you should update
it (i.e. yum update gmp).
PLAY [localhost] **************************************************************
GATHERING FACTS ***************************************************************
ok: [localhost]
TASK: [update all packages] ***************************************************
failed: [localhost] => {"failed": true, "parsed": false}
BECOME-SUCCESS-cowfsuudxofapfzgotceycqgfeksicos
Traceback (most recent call last):
  File "/home/ec2-user/.ansible/tmp/ansible-tmp-1430645872.15-203675717416361/yum", line 27, in <module>
    import yum
ImportError: No module named yum
FATAL: all hosts have already failed -- aborting
PLAY RECAP ********************************************************************
           to retry, use: --limit @/home/ec2-user/test.retry
localhost                  : ok=1    changed=0    unreachable=0    failed=1
I don't think it's common to apply a playbook to localhost in Ansible, so it may not be a problem, but it's a bit unpleasant.
Let's install Python gently with pip using 2.7 series.
$ python --version
Python 2.7.9
$ sudo pip install ansible
...
Successfully installed ansible-1.9.1
This will work. (Warning is out ...)
$ ansible-playbook test.yml
/usr/lib64/python2.7/dist-packages/Crypto/Util/number.py:57: PowmInsecureWarning: Not using mpz_powm_sec.  You should rebuild using libgmp >= 5 to avoid timing attack vulnerability.
  _warn("Not using mpz_powm_sec.  You should rebuild using libgmp >= 5 to avoid timing attack vulnerability.", PowmInsecureWarning)
PLAY [localhost] **************************************************************
GATHERING FACTS ***************************************************************
ok: [localhost]
TASK: [update all packages] ***************************************************
changed: [localhost]
PLAY RECAP ********************************************************************
localhost                  : ok=2    changed=1    unreachable=0    failed=0