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Social Choice and Welfare

Erschienen in:

01.04.2015

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

verfasst von: Francesco De Sinopoli, Giovanna Iannantuoni, Carlos Pimienta

Erschienen in: Social Choice and Welfare | Ausgabe 4/2015

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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.

Vorheriger Artikel Extremism drives out moderation

Nächster Artikel Voting power apportionments

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See van Damme (1991) for an excellent review.

Debreu (1970) makes an analogous argument but in the context of pure exchange economies. See also Harsanyi (1973), Park (1997), Govindan and McLennan (2001), Govindan and Wilson (2001).

Given the election rule below, (1,1,1) is equivalent to abstention \((0,0,0)\).

Govindan and McLennan (2001) offer a counterexample that shows that this result does not extend to general games.

A set is semi-algebraic if it is defined by a finite system of polynomial inequalities. A function or a correspondence is semi-algebraic if its graph is a semi-algebraic set. Every set and correspondence defined in this paper is semi-algebraic.

Typically, voting games do not have strictly dominated strategies. Here and throughout the paper, by dominated strategy we mean a weakly dominated strategy.

Every strict equilibrium is an absorbing retract (Kalai and Samet 1984) and every absorbing retract contains a stable set (Mertens 1992, p. 562).

This comes from the fact that, in every undominated strategy profile, no voter casts a negative vote against \(a\). Furthermore, in any undominated strategy profile such that some other candidate also receives zero negative votes some voter has an incentive to deviate.

Note that in either system, no voter votes for candidate \(c\), therefore, the voting game is reduced to a two-candidate contest between candidates \(a\) and \(b\).

Note that we do not need to know how any voter \(i\) distributes probability among elements in \(\fancyscript{A}_i(C)\). This distribution only affects the distribution of probability between ballot profiles with the same set of winning candidates and different number of negative votes for losing candidates.

In this case the system of equations is quite simple. For each voter whose set of pure best responses has two or three elements we only have one equation and one unknown.

Note that we do not need to know how a voter distributes probability between two elements of an equivalence class. This distribution only affects the distribution of probability between ballot profiles with the same set of winning candidates and different number of approval votes for losing candidates.

The strategy \(\sigma '\) is well defined. Candidates \(c_1,\, c_2\) and \(c_3\) (and only them) all win with positive probability under \(\sigma \). Hence, if \(\sigma \) is an equilibrium, every voter approves at least one of them in every pure strategy that is played with positive probability.

Bochnak J, Coste M, Roy M (1998) Real algebraic geometry. Springer, New YorkCrossRef

Brams S, Fishburn P (1978) Approval voting. Am Polit Sci Rev 72:831–847CrossRef

Brams S, Sanver M (2006) Critical strategies under approval voting: who gets ruled in and ruled out. Elect Stud 25(2):287–305CrossRef

Buenrostro L, Dhillon A, Vida P (2013) Scoring rule voting games and dominance solvability. Soc Choice Welf 40(2):329–352CrossRef

De Sinopoli F (2000) Sophisticated voting and equilibrium refinements under plurality rule. Soc Choice Welf 17(4):655–672CrossRef

De Sinopoli F (2001) On the generic finiteness of equilibrium outcomes in plurality games. Games Econ Behav 34(2):270–286CrossRef

De Sinopoli F, Dutta B, Laslier J-F (2006) Approval voting: three examples. Int J Game Theory 35(1):27–38

De Sinopoli F, Iannantuoni G, Pimienta C (2013) Counterexamples on the superiority of approval vs plurality. J Public Econ Theory

Debreu G (1970) Economies with a finite set of equilibria. Econometrica 38(3):387–392CrossRef

Dhillon A, Lockwood B (2004) When are plurality rule voting games dominance-solvable? Games Econ Behav 46(1):55–75CrossRef

Farquharson R (1969) Theory of voting. Yale University Press, New Haven

Fishburn P, Brams S (1981) Approval voting, Condorcet’s principle, and runoff elections. Public Choice 36(1):89–114CrossRef

Gershgorin S (1931) Uber die abgrenzung der eigenwerte einer matrix. Izv Akad Nauk SSSR 7:749–754

Govindan S, McLennan A (2001) On the generic finiteness of equilibrium outcome distributions in game forms. Econometrica 69(2):455–471CrossRef

Govindan S, Wilson R (2001) Direct proofs of generic finiteness of Nash equilibrium outcomes. Econometrica 69(3):765–769CrossRef

Harsanyi J (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2(1):235–250CrossRef

Kalai E, Samet D (1984) Persistent equilibria in strategic games. Int J Game Theory 13(3):129–144CrossRef

Mertens J-F (1989) Stable equilibria—a reformulation, part I: definition and basic properties. Math Oper Res 14(4):575–625CrossRef

Mertens J-F (1992) The small worlds axiom for stable equilibria. Games Econ Behav 4(4):553–564

Myerson R (2002) Comparison of scoring rules in Poisson voting games. J Econ Theory 103(1):219–251CrossRef

Myerson R, Weber R (1993) A theory of voting equilibria. Am Polit Sci Rev 87(1):102–114CrossRef

Ostrowski A (1955) Note on bounds for some determinants. Duke Math J 22(1):95–102CrossRef

Park I (1997) Generic finiteness of equilibrium outcome distributions for sender-receiver cheap-talk games. J Econ Theory 76(2):431–448CrossRef

Plemmons RJ (1977) M-matrix characterizations. I—nonsingular M-matrices. Linear Algebra Appl 18(2):175–188CrossRef

Price G (1951) Bounds for determinants with dominant principal diagonal. Proc Am Math Soc 2(3):497–502CrossRef

van Damme E (1991) Stability and perfection of Nash equilibria. Springer, BerlinCrossRef

Titel: On stable outcomes of approval, plurality, and negative plurality games
verfasst von: Francesco De Sinopoli
Giovanna Iannantuoni
Carlos Pimienta
Publikationsdatum: 01.04.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 4/2015
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0866-z

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