Abstract
In this paper I investigate the properties of social welfare functions defined on domains where the preferences of one agent remain fixed. Such a domain is a degenerate case of those investigated, and proved Arrow consistent, by Sakai and Shimoji (Soc Choice Welf 26(3):435–445, 2006). Thus, they admit functions from them to a social preference that satisfy Arrow’s conditions of Weak Pareto, Independence of Irrelevant Alternatives, and Non-dictatorship. However, I prove that according to any function that satisfies these conditions on such a domain, for any triple of alternatives, if the agent with the fixed preferences does not determine the social preference on any pair of them, then some other agent determines the social preference on the entire triple.
Highlights
Introduction and motivationSocial choice theory in the Arrovian style concerns functions from domains containing n-tuples of orders over some set of alternatives that map to an overall order on those alternatives
The restriction of an ordering R to a pair of alternatives x, y is denoted by R|x,y. Such a function f : D → R satisfies the Arrovian conditions of Independence of Irrelevant Alternatives (IIA), Weak Pareto (WP), and Non-dictatorship (ND) if and only if: IIA For all profiles α, β ∈ D, and for all x, y ∈ X, if α|x,y = β|x,y, ( f (α))|x,y = ( f (β))|x,y
Theorems 1 and 2 demonstrate that domains of profiles constructed by holding the preferences of one agent fixed are Arrow consistent
Summary
Introduction and motivationSocial choice theory in the Arrovian style concerns functions from domains containing n-tuples of orders over some set of alternatives that map to an overall order on those alternatives. I prove here that according to any function on such a domain that satisfies WP, IIA, and ND, for any triple of alternatives, if the agent with the fixed preferences does not determine the social preference on any pair of them, some other agent determines the social preference on the entire triple.
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