Arrow's Impossibility Theorem Notes

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Related Readings

This post is a reading note of the articles listed in the Related Readings section. We first present the Arrows' impossibility theorem:

Arrow's Theorem. Any election process that respects transitivity, independence of irrelevant alternatives, and unanimity is a dictatorship.

This is a quite shocking result, although we need to be careful about the nuance in the interpretation of the theorem. The Arrow's theorem is built on top of the following five assumptions/conditions:

conditions.png

Another interesting and related topic is Condorcet paradox. It says a society that consists of rational individuals may not be rational if it uses majority voting to make decisions. Here rational means there is no circular preference. For example, a rational individual cannot have preference such as A > B > C > A. Irrational may be a strong word; at least with circular preference, the individual becomes indecisive. If the individual is willing to make additional efforts to exchange a less liked item for a preferred item, then it becomes irrational because the efforts will be wasted and it will get the same results.

Notation and Concepts

Proof

The first and third proof in the paper are accessible.

A clarification on profile III in the first proof and the last statement in the penultimate paragraph of the first proof:

By independence of irrelevant alternatives, the social preference over ac must agree with \(n^*\) whenever \(a >_{n^*} c\).

For any profile \(\langle R_1, R_2, ..., R_d, ..., R_n \rangle \), we could construct profile III such that

$$ \forall j \,\,\, R_{j}^{III}(a,c) = R_j(a,c) $$

and by independency of irrelevant alternatives, we have

$$ f( \langle R_1, R_2, ..., R_d, ..., R_n \rangle )(a,c) = f( \langle R_1^{III}, R_2^{III}, ..., R_d^{III}, ..., R_n^{III} \rangle )(a,c) $$

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