Voting: Arrow's Impossibility Theorem

Remember the different criteria for voting procedures:

(0a) Universality: the voting scheme should be deterministic (no randomness), and all rankings of the voters are accepted by the scheme.
(0b) Non-imposition or citizen sovereignty: Every ranking of the alternatives should be possible as outcome.

(1) Majority criterion: If there is a majority solution (being number one preference for the majority (more than n/2) of voters), then that option should be the winner.

(2) Condorcet criterion: If there is a Condorcet solution, then that option should be the winner.

(3) Independence of irrelevant alternatives: If one alternative is removed, then the voting system should still create the same ordering of the remaining alternatives.

(4) Monotonicity: ("better is better") If one voter promotes on alternative, then the ranking of that alternative in the result should at least not drop.

All voting methods considered obey criteria (0a) and (0b). As we have seen, some of the other criteria are not obeyed by some of the voting methods considered. The following table displays which of our voting methods obeys which of the criteria.

Kenneth Arrow, *1921

is one of the founders of modern neoclassical economical theory. "The" Theorem was from his 1951 Ph.D. thesis. He won the Nobel Prize in Economics in 1972.

Marquis de Condorcet (1743-1794)

He was mainly working in probability theory and economics. He had a leading role in the French revolution, but still died in prison.

Vilfredo Pareto (1848-1923)

was a learned mathematician and engineer, but later working in economics, making contributions in income distribution theory and choice theory.

Plurality Borda Hare Inverse Hare Runoff Pairwise comparison
Majority Criterion yes no yes yes yes yes
Condorcet Criterion no no no .. .. yes
Monotonicity Criterion yes no no .. .. no
Irrelevant Alternatives no no no .. .. no

This could just indicate that we haven't found the perfect voting method---those obeying all the criteria---yet, but unfortunately we cannot hope for such a perfect voting scheme:

Arrow's Theorem (1950): If there are at least three options, then no voting system except dictatorship can obey criteria (0a), (0b), (3), and (4) above.

Dictatorship is a method that just takes the preferences of voter 1 and takes it as the resulting short list, no matter what the votes of the other voters are. Of course such a "voting" method is not acceptable.

Another version of Arrow"s Theorem uses still another criterion:

(5) Pareto-efficiency: If every voter prefers a certain alternative over another, then the result must prefer this alternative over the other too.

Another Version of Arrow's Theorem: If there are at least three options, then no voting system except dictatorship can obey criteria (0a), (3), and (5) above.

This version is actually slightly stronger, since criteria (0b), (3), (4) together imply criterion (5), whereas conditions (0b), (3), (5) together do not imply criterion (4).

There is also a version for elections where just one wins, formulated independently by Gibbard- and Satterthwaite around 1973/75: If there at least two voters and at least three options, and we are just looking for methods to elect one option, then such a method must be

  1. either dictorial, or
  2. such that one (or more) options cannot win, or
  3. it is rule manipulable---suspectible for tactical voting (some voter, knowing all preferences, will have to decide to vote different to his/her preferences to make his/her option the winner.

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