Precisely dictatorial social welfare functions
This note is in response to an error in a previous paper of the author [ibid. 7, 55-75 (1980; Zbl 0427.90005)] pointed out by \textit{H. Sarbadhikari} [unpublished]. Proposition 3.2 asserts a bijection between Arrow Social Welfare Functions (ASWFs) \(\sigma\) and their ultrafilters \(k=k_{\sigma}\) of determining coalitions. Specifically it is asserted that \(\sigma =\sigma_ k\) for any \(\sigma\). However, the map \(\sigma \to k_{\sigma}\) is only surjective in general and an example with \(\sigma \neq \sigma_ k\) may be constructed. In this note we give a natural condition on ASWFs which ensures that \(\sigma =\sigma_ k\) hence ensures bijectivity of \(\sigma \to k_{\sigma}\). We shall call such ASWFs precisely dictatorial. It is only for precisely dictatorial ASWFs that the remainder of the author's previous paper is valid especially section 4 on competitive social welfare functions. The conditions we give here essentially reduce, in the case of finitely many agents to those given by \textit{B. Mirkin} [''Group choice'' (1979; Zbl 0406.92023), chap. 3].
precisely dictatorial social welfare functions
scientific article; zbMATH DE number 3920154