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

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.

[1]  E. Damme.  Stability and perfection of Nash equilibria , 1987 .

[2]  Carlos Pimienta,et al.  Counterexamples on the superiority of approval versus plurality , 2014 .

[3]  A. Ostrowski.  Note on bounds for some determinants , 1955 .

[4]  Roger B. Myerson,et al.  Comparison of Scoring Rules in Poisson Voting Games , 2002, J. Econ. Theory.

[5]  Peter C. Fishburn,et al.  Approval voting, Condorcet's principle, and runoff elections , 1981 .

[6]  Ben Lockwood,et al.  When are plurality rule voting games dominance-solvable? , 2004, Games Econ. Behav..

[7]  Francesco De Sinopoli.  Sophisticated voting and equilibrium refinements under plurality rule , 2000 .

[8]  Gérard Debreu.  Economies with a Finite Set of Equilibria , 1970 .

[9]  G. B. Price.  Bounds for determinants with dominant principal diagonal , 1951 .

[10]  Jean-François Laslier,et al.  Approval voting: three examples , 2006, Int. J. Game Theory.

[11]  A. McLennan,et al.  On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms , 2001 .

[12]  Douglas Muzzio,et al.  Approval voting , 1983 .

[13]  Jean-François Mertens.  The Small Worlds Axiom for Stable Equilibria , 1992 .

[14]  R. Weber,et al.  A Theory of Voting Equilibria , 1993, American Political Science Review.

[15]  Robin Farquharson,et al.  Theory of voting , 1969 .

[16]  R. Plemmons.  M-matrix characterizations.I—nonsingular M-matrices , 1977 .

[17]  In-Uck Park.  Generic Finiteness of Equilibrium Outcome Distributions for Sender–Receiver Cheap-Talk Games , 1993 .

[18]  Steven J. Brams,et al.  Critical Strategies Under Approval Voting: Who Gets Ruled In And Ruled Out , 2006 .

[19]  J. Harsanyi.  Oddness of the number of equilibrium points: A new proof , 1973 .

[20]  Lucia Buenrostro,et al.  Scoring rule voting games and dominance solvability , 2013, Soc. Choice Welf..

[21]  E. Kalai,et al.  Persistent equilibria in strategic games , 1984 .

[22]  Jean-François Mertens.  Stable Equilibria - A Reformulation: Part I. Definition and Basic Properties , 1989, Math. Oper. Res..

[23]  Robert Wilson,et al.  Direct proofs of generic finiteness of nash equilibrium outcomes , 2001 .

[24]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[25]  Francesco De Sinopoli,et al.  On the Generic Finiteness of Equilibrium Outcomes in Plurality Games , 2001, Games Econ. Behav..