Arrow’s TheoremAlex Tabarrok

Review

Individual RankingsB D

AC C

CA B

DD A

B

Voting System

Election Outcom

eBDAC

The Axioms

1. Universal domain – all individually rational preference orderings are allowed as inputs into the voting system.

2. Completeness and Transitivity – the derived social preference ordering should be complete and transitive.

3. Positive association – suppose that at some point the voting rule outputs X>Y then it should continue to output X>Y when some individuals raise X in their preference orderings.

4. Independent of Irrelevant Alternatives – the social ranking of X and Y should depend only on how individuals rank X and Y (and not on how they rank some “irrelevant” alternative W relative to X and Y).

5. Non-imposition – an outcome is not to be imposed which is independent of voter preferences.

6. Non-dictatorship – the voting rule cannot be based solely on one person’s preferences.

If individual rankings are transitive and unrestricted and the election outcome is transitive and complete

Arrow’s Theorem

Individual Rankings

B DA

C CC

A BD

D AB

Voting System

Election Outcome

BDAC

Transitive and unrestricted

Transitive and complete

Independence of Irrelevant Alternatives

and Unanimity condition

If individual rankings are transitive and unrestrictedIf individual rankings are transitive and unrestricted and the election outcome is transitive and complete then the only voting system which satisfies independence of irrelevant alternatives and the unanimity condition

If individual rankings are transitive and unrestricted and the election outcome is transitive and complete then the only voting system which satisfies independence of irrelevant alternatives and the unanimity condition is a dictatorship.

Arrow’s Theorem

Individual RankingsB D

AC C

CA B

DD A

B

Voting System

Election Outcom

eBDAC

Transitive and unrestricted

Transitive and complete

Independence of Irrelevant Alternatives

and Unanimity condition

Alternative reading: All democratic voting system will fail to satisfy at least one of independence of irrelevant alternatives, the unanimity condition or transitivity of the outcome – thus all voting systems will sometimes result in “paradoxical” outcomes.

Review of Voting Paradoxes

None of the voting systems we look at earlier was dictatorial or imposed so they each must violate at least one and perhaps several of Arrow's other axioms.

Positional vote systems like plurality rule violate the Independence of Irrelevant Alternatives axiom. (Nader was relevant).

Pairwise voting with majority rule violates the Transitivity axiom (i.e. majority rule can create cycles).

Positive Association was violated by runoff procedures.

A Voting System is an Aggregation Mechanism

Individual Rankings

(Inputs)B D

AC C

CA B

DD A

B

Voting System

(Aggregation Mechanism)

Election Outcome

(Global Ranking)BDAC

Escaping Arrow?

Arrow’s theorem says the 6 axioms cannot all be true at the same time. What if we modify or drop one of the axioms.

For a democratic system we don’t want to drop non-imposition or non-dictatorship. So that leaves us with:

1. Universal domain – all individually rational preference orderings are allowed as inputs into the voting system.

2. Completeness and Transitivity – the derived social preference ordering should be complete and transitive.

3. Positive association – suppose that at some point the voting rule outputs X>Y then it should continue to output X>Y when some individuals raise X in their preference orderings.

4. Independent of Irrelevant Alternatives – the social ranking of X and Y should depend only on how individuals rank X and Y (and not on how they rank some “irrelevant” alternative W relative to X and Y).

Giving up Universal Domain

Giving up UD is the same as looking for a voting system which will work well for some but not all distributions of individual preference rankings.

If everyone has identical preferences, for example, then majority rule is a perfectly acceptable voting system (i.e. it will satisfy the remaining axioms).

But a voting system which works well only when everyone has identical preferences is not very useful. We are thus interested in knowing how much homogeneity we need to impose on preference orderings if we want a voting system which satisfies the remaining 5 axioms.

The answer is that quite a lot of homogeneity is required but perhaps not so much to be uninteresting.

Single Peaked Preferences on a Single Dimension

If everyone's preferences are single peaked on the same single dimension then majority rule satisfies the remaining 5 axioms.

Single dimension, e.g. left-right. Single-peaked – each voter has an ideal point and the further away

from the ideal point the lower their utility.

Left Right

Single peaked “left” voterSingle peaked “right” voter

Single peaked “moderate” voterNon single peaked voter

The Median Voter Theorem(Anthony Downs, 1957, An Economic Theory of Democracy)

LessSpending

MoreSpending

R D R R D

Median

Voter

• If every voter’s preferences are single peaked on a single dimension then majority rule with pairwise voting satisfies Arrow’s Theorem. Irrelevant alternatives are irrelevant and there are no cycles because the median voter’s preferences are unbeatable in pairwise voting (i.e. the median voter’s preferences are a Condorcet winner)

Median Voter Theorem

If people are homogeneous enough that everyone fits on a left-right or other single-dimension spectrum then majority rule with pairwise voting works well.

The MVT is also very useful because it implies that the group will behave as if it were an individual wtih rational preferences. Thus, one can make predictions and models of voter behavior assuming the MVT.

We will return to the MVT.

Dropping Completeness

The completeness axiom requires given any question of the form `Is X socially preferred to Y or is Y socially preferred to X or are X and Y socially indifferent?' the voting system must return a definite answer.

But suppose that X is the outcome, "tax Peter to pay Paul," and Y the outcome "tax Paul to pay Peter." A libertarian would argue that the question `Is X socially preferable to Y' has no answer (Rothbard 1956). In an ideal libertarian society the only legitimate exchanges are between individuals who agree to those exchanges. A `voting system' for such a society is nothing more than the market.

The libertarian believes that the only meaning that `X is socially preferred to Y 'can have is `X was arrived at by voluntary exchange from Y'. In the libertarian view, the fact that non-voluntary exchanges cannot be ranked is not a fault of the market as a social choice mechanism it is rather an expression of the fact that there is no social preference ordering between non-voluntary exchanges.

We can satisfy Arrow’s Theorem if we allow that many options cannot be ranked. But is true that the two outcomes Paul kills Peter and Paul taxes Peter one penny cannot be ranked?

Weakening Transitivity

Transitivity requires if X>Y and Y>Z then X>Z and also if X~Y and Y~Z then X~Z

Quasi-transitivity allows X~Y and Y~Z but X>Z. e.g. X is 4 grams of sugar in coffee, Y is 4.5 grams and Z is 5 grams.

Surprisingly, if weaken transitivity of the outcome to quasi-transitivity then all of Arrow’s other axioms can be satisfied but instead of a dictatorship we get an oligarchy.

Interesting but probably not a useful path.

Dropping IIA

Dropping IIA

Dropping IIA

If we drop IIA there are lots of voting systems that satisfy Arrow’s other axioms. The positional voting systems, for example, ask voters to rank their candidates from best to worst and then assign points from to best to worst.

Winner of the election is the candidate who receives the most points.

1

0

0

2

1

0

Plurality Rule

BordaCount

14

9

8

MVPBaseba

ll

Dropping IIA

If we use a positional voting system then “irrelevant” candidates and preferences will matter. The Nader Problem.

Defenders of these systems say that is ok because these systems are measuring relative intensity and that is desirable.

But which is the right system for measuring intensity? Should first place votes get 3 points and second place 2 or should first place votes get 10 points and second place votes 4?

Also are these systems really measuring intensity?

Dropping IIA

Arrow’s Theorem

Bottom Line: No (easy?) escape!Group choice is not like individual choice and never will be.

All democratic voting systems are subject to certain paradoxes and inconsistencies!