# On stable outcomes of approval, plurality, and negative plurality games

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## Abstract

We prove two results on the generic determinacy of Nash equilibrium in voting games. The first one is for negative plurality games. The second one is for approval games under the condition that the number of candidates is equal to three. These results are combined with the analogous one obtained in De Sinopoli (Games Econ Behav 34:270–286, 2001) for plurality rule to show that, for generic utilities, three of the most well-known scoring rules, plurality, negative plurality and approval, induce finite sets of equilibrium outcomes in their corresponding derived games—at least when the number of candidates is equal to three. This is a necessary requirement for the development of a systematic comparison amongst these three voting rules and a useful aid to compute the stable sets of equilibria Mertens (Math Oper Res 14:575–625, 1989) of the induced voting games. To conclude, we provide some examples of voting environments with three candidates where we carry out this comparison.

## JEL Classification

C72 D72## Notes

### Acknowledgments

A previous version of this paper was circulated under the title “Scoring Rules: A Game-Theoretical Analysis”. We thank the Associate Editor and two anonymous referees for insightful comments that improved the paper. We also thank Claudia Meroni and José Rodrigues-Neto for very useful suggestions. Francesco and Giovanna gratefully acknowledge financial support from the Italian Ministry of Education, PRIN 2010–2011 “New approaches to political economy: positive political theories, empirical evidence and experiments in laboratory”. Carlos thanks financial support from *UNSW ASBRG* and from the Australian Research Council’s Discovery Projects funding scheme DP140102426. The usual disclaimer applies.

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