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Social Choice and Welfare

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01-04-2015

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

Authors: Francesco De Sinopoli, Giovanna Iannantuoni, Carlos Pimienta

Published in: Social Choice and Welfare | Issue 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.

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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.

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Title: On stable outcomes of approval, plurality, and negative plurality games
Authors: Francesco De Sinopoli
Giovanna Iannantuoni
Carlos Pimienta
Publication date: 01-04-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0866-z

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