Arrow's Impossibility Theorem

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: Interval Scales)
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?

• Sen says that we should give up liberalism (the Pareto condition). He argues that liberalism + Pareto leads to a contradiction in ordering axioms
• Nozick’s solution is to give up the unrestricted domain assumption (U). Liberalism excludes certain kinds of states (‘private’ alternatives) from social scrutiny in advance.

Similarities to Group Membership

Rough thoughts on how we might prove decentralized access control to be impossible.

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.