Alex Tabarrok
Election
Outcome
Individual
Rankings
B
D
A
C
C
C
A
B
D
A
D
A
B
C
Voting System
B
D
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.
1.
Independence of Irrelevant
Alternatives and Unanimity
condition
Transitive and
unrestricted
Transitive and
complete
Election
Outcome
Individual
Rankings
B
D
A
C
C
C
A
B
D
A
D
A
B
C
Voting System
B
D
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.
Independence of Irrelevant
Alternatives and Unanimity
condition
Transitive and
unrestricted
Transitive and
complete
Election
Outcome
Individual
Rankings
B
D
A
C
C
C
A
B
D
D
A
B
Voting System
B
D
A
C
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.




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.
Individual
Rankings
Voting System
(Inputs)
Election
Outcome
(Global Ranking)
B
D
A
C
C
C
A
B
D
A
D
A
B
C
(Aggregation
Mechanism)
B
D


1.
2.
3.
4.
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 nondictatorship. So that leaves us with:
Universal domain – all individually rational preference orderings are allowed
as inputs into the voting system.
Completeness and Transitivity – the derived social preference ordering should
be complete and transitive.
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.
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 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.




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.
Single peaked “left” voter
Single peaked “moderate” voter
Non single peaked voter
Single peaked “right” voter
Left
Right
• 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)
R
Less
Spending
D R
R D
Median
Voter
More
Spending



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.
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 nonvoluntary 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?





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 quasitransitivity 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.


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.
Plurality
Rule
Borda
Count
MVP
Baseball
1
2
14
0
1
9
0
0
8




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?
 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!