Abstract
Approval voting has attracted considerable attention in voting theory, but it has rarely been investigated in an Arrovian framework of collective preference (”social welfare”) functions and never been connected with Arrow’s impossibility theorem. The article explores these two directions. Assuming that voters have dichotomous preferences, it first characterizes approval voting in terms of its collective preference properties and then shows that these properties become incompatible if the collective preference is also taken to be dichotomous. As approval voting and majority voting happen to share the same collective preference function on the dichotomous domain, the positive result also bears on majority voting, and is seen to extend May’s and Inada’s early findings on this rule. The negative result is a novel and perhaps surprising version of Arrow’s impossibility theorem, because the axiomatic inconsistency here stems from the collective preference range, not the individual preference domain.
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Notes
Fishburn (1978a, b) axiomatized approval voting in terms of a ballot aggregation function, which is an anonymous social choice function taking profiles of individual choices as its arguments. This framework is taken up in Sertel (1988), Alos-Ferrer (2006) and Sato (2014), and it inspires Baigent and Xu (1991) and Xu’s (2010) related variants. The less formalistic, but pioneering paper by Brams and Fishburn (1978) emphasized the strategic properties of approval voting, and this line is recovered by Vorsatz (2007; 2008) in a framework where the social choice function is defined on profiles of individual preferences, as in the Gibbard–Satterthwaite theorem. Some explicit body of game-theoretic analysis now exists (see Laslier and Sanver 2010). All this theoretical work is scant compared with the many practical studies on approval voting (see Brams and Fishburn 2005, and the relevant chapters in Laslier and Sanver 2010).
Approval voting does satisfy a weaker form of \(IIA\), to wit: for all \( (R_{1},\ldots ,\) \(R_{n})\), \((R_{1}^{\prime },\ldots ,R_{n}^{\prime })\in \mathcal {D} ^{n}\) and all \(x,y\in X\), if \(xR_{i}y\approx xR_{i}^{\prime }y\) for all \( i\in N\), then \(xPy\Longrightarrow xR^{\prime }y\). As Campbell and Kelly (2000) demonstrate after Baigent (1987), this \(Weak\) \(IIA\) delivers a variant of Arrow’s impossibility theorem in which \(ND\) is replaced by a no-vetoer condition.
The following example adapts a suggestion made by Bill Zwicker.
Essentially the same paper appears as Ju (2011).
Both Vorsatz (2007) and Sato (2014) actually have more conditions. They allow for a varying set of candidates \(X\) and—following Fishburn (1978a, b)—for a variable finite set of individuals \(N\). This leads them to introduce consistency conditions to connect the social choice functions obtained for these diverse sets.
References
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The authors thank Claude d’Aspremont, Franz Dietrich, Jérôme Lang, Arunava Sen, Bill Zwicker, and two anonymous referees, for useful comments on an earlier version. The second author is grateful to CORE and the Australian National University for their hospitality when he was working on the paper.
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Maniquet, F., Mongin, P. Approval voting and Arrow’s impossibility theorem. Soc Choice Welf 44, 519–532 (2015). https://doi.org/10.1007/s00355-014-0847-2
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DOI: https://doi.org/10.1007/s00355-014-0847-2