Question

What are similarities between Arrow's and McKelvey's Theorem in social choice mechanism?

Answer 1

Answer:

Arrow's Theorem in social choice mechanism:

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.

The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem.

The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values.

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.

• If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).

• There is no "dictator": no single voter possesses the power to always determine the group's preference.

McKelvey–Schofield chaos theorem in social choice mechanism:

The McKelvey–Schofield chaos theorem is a result in social choice theory.

It states that if preferences are defined over a multidimensional policy space, then majority rule is in general unstable: there is no Condorcet winner.

Furthermore, any point in the space can be reached from any other point by a sequence of majority votes.

The theorem can be thought of as showing that Arrow's impossibility theorem holds when preferences are restricted to be concave in $\mathbb {R} ^{n}$ .

The median voter theorem shows that when preferences are restricted to be single-peaked on the real line, Arrow's theorem does not hold, and the median voter's ideal point is a Condorcet winner.

The chaos theorem shows that this good news does not continue in multiple dimensions.