Addendum to L. Lauwers and L. Van Liedekerke, “Ultraproducts and aggregation” [J. Math. Econ. 24 (3) (1995) 217–237]

Journal of Mathematical Economics 46 (2010) 277–278

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Addendum

Addendum to L. Lauwers and L. Van Liedekerke, “Ultraproducts and aggregation” [J. Math. Econ. 24 (3) (1995) 217–237] Frederik Herzberg a,∗ , Luc Lauwers b , Luc van Liedekerke c , Emmanuel Senyo Fianu d a b c d

Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld, Universitätsstraße 25, D-33615 Bielefeld, Germany Katholieke Universiteit Leuven, Faculteit Economie en Bedrijfswetenschappen, Onderzoeksgroep Econometrie, Naamsestraat 69, 3000 Leuven, Belgium Katholieke Universiteit Leuven, Universiteit Antwerpen, Centrum voor Economie en Ethiek, Naamsestraat 69, 3000 Leuven, Belgium Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld, Universitätsstraße 25, D-33615 Bielefeld, Germany

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Article history: Received 21 May 2009 Received in revised form 21 June 2009 Accepted 21 June 2009 Available online 26 June 2009

a b s t r a c t This addendum fills a minor gap in the key lemma of a paper by Lauwers and Van Liedekerke [Lauwers, L., Van Liedekerke, L., 1995. J. Math. Econ. 24 (3), 217–237].

JEL classification: D71 Keywords: Arrow aggregation Ultrafilters

Lauwers and Van Liedekerke (1995) argue that model theory provides a fruitful mathematical framework for social choice theory. In particular, they show that Arrovian aggregation functions can be viewed as (restrictions of) ultraproduct constructions. The key step in their proof is a lemma which asserts that the collection D of decisive coalitions forms an ultrafilter [Lauwers and Van Liedekerke, 1995, Lemma 2, p. 227]. Unfortunately, the proof of this lemma is incomplete as it stands: The proof purports to show that D satisfies the ultrafilter properties (F1), (F2), (F4)—which would be enough since (F3) (closedness under supersets) follows logically from the combination of (F1), (F2), (F4) [Lauwers and Van Liedekerke, 1995, p. 221]. However, in the original proof that D satisfies (F2), Lauwers and Van Liedekerke [Lauwers and Van Liedekerke, 1995, p. 228, line 9, “hence”] inadvertently assume property (F3). Let us now establish that D does satisfy (F3). Let C ∈ D and C  ⊆ I with C ⊆ C  . We have to show that C  ∈ D. It was assumed that there exist ,  ∈ Tb = {P(a, b)|a, b ∈ A} such that  ∧ ,  ∧ ¬, and ¬ ∧  are each consistent with the theory T . Hence there exists a profile ω = Ai |i ∈ I as follows: Ai   ∧  Aj   ∧ ¬ Ak  ¬ ∧ 

for all i ∈ C for all j ∈ C  \ C for all k ∈ I \ C  .

DOI of the original article:10.1016/S0304-4068(94)00684-3. ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (F. Herzberg), [email protected] (L. Lauwers), [email protected] (L. van Liedekerke). 0304-4068/$ – see front matter doi:10.1016/j.jmateco.2009.06.001

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F. Herzberg et al. / Journal of Mathematical Economics 46 (2010) 277–278

Note that C(ω,  ∧ ) = C ∈ D. Therefore A(ω)   ∧  due to axiom (A3) and the definition of D. In particular, A(ω)  , whence readily C(ω, ) ∈ D (by definition of D). On the other hand, however, the choice of ω entails C  = C(ω, ). Therefore C  ∈ D. Reference Lauwers, L., Van Liedekerke, L., 1995. Ultraproducts and aggregation. J. Math. Econ. 24 (3), 217–237.