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

- 169 Downloads
- 1 Citations

## 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.

## References

- Bochnak J, Coste M, Roy M (1998) Real algebraic geometry. Springer, New YorkCrossRefGoogle Scholar
- Brams S, Fishburn P (1978) Approval voting. Am Polit Sci Rev 72:831–847CrossRefGoogle Scholar
- Brams S, Sanver M (2006) Critical strategies under approval voting: who gets ruled in and ruled out. Elect Stud 25(2):287–305CrossRefGoogle Scholar
- Buenrostro L, Dhillon A, Vida P (2013) Scoring rule voting games and dominance solvability. Soc Choice Welf 40(2):329–352CrossRefGoogle Scholar
- De Sinopoli F (2000) Sophisticated voting and equilibrium refinements under plurality rule. Soc Choice Welf 17(4):655–672CrossRefGoogle Scholar
- De Sinopoli F (2001) On the generic finiteness of equilibrium outcomes in plurality games. Games Econ Behav 34(2):270–286CrossRefGoogle Scholar
- De Sinopoli F, Dutta B, Laslier J-F (2006) Approval voting: three examples. Int J Game Theory 35(1):27–38Google Scholar
- De Sinopoli F, Iannantuoni G, Pimienta C (2013) Counterexamples on the superiority of approval vs plurality. J Public Econ TheoryGoogle Scholar
- Debreu G (1970) Economies with a finite set of equilibria. Econometrica 38(3):387–392CrossRefGoogle Scholar
- Dhillon A, Lockwood B (2004) When are plurality rule voting games dominance-solvable? Games Econ Behav 46(1):55–75CrossRefGoogle Scholar
- Farquharson R (1969) Theory of voting. Yale University Press, New HavenGoogle Scholar
- Fishburn P, Brams S (1981) Approval voting, Condorcet’s principle, and runoff elections. Public Choice 36(1):89–114CrossRefGoogle Scholar
- Gershgorin S (1931) Uber die abgrenzung der eigenwerte einer matrix. Izv Akad Nauk SSSR 7:749–754Google Scholar
- Govindan S, McLennan A (2001) On the generic finiteness of equilibrium outcome distributions in game forms. Econometrica 69(2):455–471CrossRefGoogle Scholar
- Govindan S, Wilson R (2001) Direct proofs of generic finiteness of Nash equilibrium outcomes. Econometrica 69(3):765–769CrossRefGoogle Scholar
- Harsanyi J (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2(1):235–250CrossRefGoogle Scholar
- Kalai E, Samet D (1984) Persistent equilibria in strategic games. Int J Game Theory 13(3):129–144CrossRefGoogle Scholar
- Mertens J-F (1989) Stable equilibria—a reformulation, part I: definition and basic properties. Math Oper Res 14(4):575–625CrossRefGoogle Scholar
- Mertens J-F (1992) The small worlds axiom for stable equilibria. Games Econ Behav 4(4):553–564Google Scholar
- Myerson R (2002) Comparison of scoring rules in Poisson voting games. J Econ Theory 103(1):219–251CrossRefGoogle Scholar
- Myerson R, Weber R (1993) A theory of voting equilibria. Am Polit Sci Rev 87(1):102–114CrossRefGoogle Scholar
- Ostrowski A (1955) Note on bounds for some determinants. Duke Math J 22(1):95–102CrossRefGoogle Scholar
- Park I (1997) Generic finiteness of equilibrium outcome distributions for sender-receiver cheap-talk games. J Econ Theory 76(2):431–448CrossRefGoogle Scholar
- Plemmons RJ (1977) M-matrix characterizations. I—nonsingular M-matrices. Linear Algebra Appl 18(2):175–188CrossRefGoogle Scholar
- Price G (1951) Bounds for determinants with dominant principal diagonal. Proc Am Math Soc 2(3):497–502CrossRefGoogle Scholar
- van Damme E (1991) Stability and perfection of Nash equilibria. Springer, BerlinCrossRefGoogle Scholar