Necessary and sufficient conditions for a resolution of the social choice paradox

Topological Complexity of Manifolds of Preferences

“…These manifolds therefore satisfy the necessary and sufficient condition of Chichilnisky and Heal (1983) for the existence of aggregation rules which are continuous, anonymous and respect unanimity. However, the larger space of all preferences given by regular foliations is shown to be topologically complex : it contains a sphere as a retract.…”

“…However, the CW complex S" is not contractible. Therefore, by Theorem 1 of Chichilnisky and Heal (1983),~cannot exist. An alternative proof of this proposition is provided in Chichilnisky (1980) .…”

From Competition and Collusion to Consent-Based Collaboration: A Case Study of Local Democracy

“…The informed consent principle in decision-making, also known as the "consent" principle (Kapp, 1997), essentially says that a decision is taken when there are no remaining "paramount" objections to the proposed decision. The decision principle of informed consent has a strong fundament in the literature on the Pareto criterion and unanimity rule (e.g., Chichilnisky & Heal, 1983;Sen, 1995;Sobel & Holcombe, 2001). Buchanan and Tullock (1962, p. 250) argued that "political theorists have perhaps shrugged off the unanimity requirement too early in their thinking."…”

Topological aggregation of inequality preorders

“…Here we want to tackle the same issue using, instead, the topological approach. This framework has been introduced by Chichilnisky [4] and developped extensively by Chichilnisky and Heal (see, for instance, [5] and [7]). We shall recall briefly the general setting.…”

“…This has an important consequence in our work. Under the additional assumption that the space of admissible preferences allows a parafinite parametrization (in short, the space is a parafinite CW complex), Chichilnisky and Heal [5] have proven that a necessary and sufficient condition for the existence of a continuous, anonymous and unanimous social choice rule, for all m > 3, is that the space of preferences be contractible (we shall not discuss here the implications of this far-reaching result; for an analysis of the implications of the contractibility property, the reader is referred to the papers previously quoted). The space of inequality preorders does not admit a parafinite parametrization, but the sufficient part of Chichilnisky and Heal's result does not depend heavily on this assumption.…”