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# Arrow's Impossibility Theorem

Last updated Nov 25, 2022 Edit Source

Related: social choice

Assume there is more than one individual, and there are at least three distinct social states. Then there is no SWF that meets the following four conditions:

1. Non-dictatorial: no individual is decisive
2. Ordering: must produce social preference orderings which are complete, asymmetric and transitive (see also: utility)
3. Pareto condition: If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
4. Independence of Irrelevant Alternatives (IIA): If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged

Unstated: Arrow also requires the unrestricted domain assumption (U)

How can we get around this?

1. Sen: give up liberalism (Pareto)
1. Argues that liberalism + Pareto leads to a contradiction in ordering axioms
2. Nozick’s solution: give up the unrestricted domain assumption (U)
1. Liberalism excludes certain kinds of states (‘private’ alternatives) from social scrutiny in advance.

## # Similarities to Group Membership

Suppose we encode access in terms of some function $A_{i,j}$ where $A_{i,j}$ is true if subject $i$ considers $j$ to be in the group and false otherwise.

A social state $S$ for some collection of individuals $G$ is $i \in G$ such that $A_{j,i} \forall j \in G$.

We suppose there is some function that takes in individual preference orderings (what it thinks the group membership currently looks like) and produces a group preference ordering (what the true group membership is). This is normally called the SWF.

Some properties of said SWF:

• Non-dictatorial: we don’t want a single admin who has power to dictate who is in the group. If this were to happen, we couldn’t remove the admin if they were compromised
• Intention-preserving: If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• Strong requirement is needed. Consider a weaker version: If the majority of voters prefers alternative X over alternative Y, then the group prefers X over Y
• This may not work when we have a Sybil actor that can add new accounts to overwhelm the majority
• …anything else I’m missing?

If we can show that these properties are equivalent to the Arrow Axioms, then there may be no way to come to a singular group where all members agree.