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# Social Choice Problem

Last updated Dec 2, 2022 Edit Source

1. How to aggregate many individual preference orderings into a single group or social preference ordering; or,
2. How to have rational individuals make rational choices as a group

We can define a Social welfare function (SWF) combining individual preference orderings (over social states) into a social preference ordering (over those same states).

Suggestions

1. Use majority rule to aggregate individual preferences into group preferences.
1. Problem: The voting paradox. Individual preferences may be transitive but the group preference can be cyclic when we do a majority vote. The violates a Klmogorov Axiom about ordering
2. Use maximum total utility to determine group preference
1. Different scales (equivalent vNM scales) yield a different social preference ordering
3. Just use ordinal rankings

Definitions:

1. A group $D \in G$ is decisive with respect to some pair of social states $(a,b)$ iff $a \succ b$ by the whole group $G$ whenever everyone in $D$ prefers $a \succ b$
2. A group is decisive if it is decisive over all pairs of social states