@article{pritchard2007exact,
  title={Exact results on manipulability of positional voting rules},
  author={Pritchard, Geoffrey and Wilson, Mark C.},
  journal={Social Choice and Welfare},
  volume={29},
  number={3},
  pages={487-513},
  year={2007},
  publisher={Springer-Verlag},
  keywords={social choice, voting},
  url_Paper={https://link.springer.com/content/pdf/10.1007/s00355-007-0216-5.pdf},
  url_Link={https://link.springer.com/article/10.1007/s00355-007-0216-5},
  abstract={We consider 3-candidate elections under a general scoring rule and
derive precise conditions for a given voting situation to be
strategically manipulable by a given coalition of voters. We present an
algorithm that makes use of these conditions to compute the minimum size
$M$ of a manipulating coalition for a given voting situation. The
algorithm works for any voter preference model - here we present
numerical results for IC and for IAC, for a selection of scoring rules,
and for numbers of voters up to 150. A full description of the
distribution of $M$ is obtained, generalizing all previous work on the
topic. The results obtained show interesting phenomena and suggest
several conjectures. In particular we see that rules {``} between
plurality and Borda{"} behave very differently from those {``} between
Borda and antiplurality{"}.}
}

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