arrow's impossibility theorem
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Arrow's Impossibility
Theorem
Kevin Feasel
December 10, 2006
http://36chambers.wordpress.com/arrow/
The Rules 2 individuals with 3 choices (x, y, z) --> 6
profiles for each.
x > y > z, x > z > y, y > x > z, y > z > x, z > x > y,
z > y > x
36 profiles in all for our two-person example (6 * 6)
The 36 Profiles1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
Explanation Of Symbols Colored Text
Blue – First choice for both
Green – Second choice for both
Red – Third choice for both
Black – Inconsistent choices (e.g., in profile 10,
Z > Y for person A but Y > Z for person B)
Numbers along the left-hand side depict the #1, #2, and #3
choices for person A.
Numbers along the bottom depict the #1, #2, and #3 choices for
person B.
Each profile has a number (e.g., 1 and 10 here).
How to read this: for profile 10, person A (left-hand side) likes
Z > X > Y. Person B (bottom) prefers X > Y > Z.
1
3 Z
2 Y
1 X
1 2 3
>
10
3 Y
2 X
1 Z
1 2 3
>
The Assumptions Completeness – All profiles must be solvable. In
this case, we have 3 choices among 2 people, so
36 total sets of preferences could arise. We must
have a solution for each one of the 36.
The Assumptions Unanimity – If all individuals agree on a single
position, that position will be guaranteed.
In our case, if an option is red, both individuals rank
this option as the #3 choice, so it will end up as #3 in
the preference ranking setup.
The Assumptions Independence of Irrelevant Alternatives – The
relationship between X and Y should be independent of
Z. In general terms, the relationship between two
elements will not change with the addition of another
element.
Example: in profile 25, X > Y for person A (X > Y
> Z) and for person B (X > Z > Y). Adding in option
Z, we assume, does not alter this relationship.
Example: in profile 26, Y > X for person A (Y > Z >
X) and for person B (Z > Y > X). If we removed
option Z, we would still expect Y > X for both.
25
3 Z
2 Y
1 X
1 2 3
>
26
3 X
2 Z
1 Y
1 2 3
>
Proving The Final Rule Non-dictatorship. We want to understand
whether we can come up with a social rule which
follows the rules of completeness, unanimity, and
independence of irrelevant alternatives, and
which is simultaneously non-dictatorial.
“Dictatorial” here means that all 36 profiles will
match one person's profiles exactly. In other
words, all social choices will precisely match the
individual's preferences.
Solving The Problem 6 preference sets are already solved, thanks to
unanimity. In profile 1, for example, all parties
agree that X > Y > Z, so X > Y > Z is the social
rule. These six completed profiles have the “>”
highlighted in yellow.
Solving The Problem1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > > >
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > > > > > > > >
Solving The Problem In addition to this, we know that any profile with
colored text is also solved—both people agree on
where to place this option. So these can all be
filled in as well.
Solving The Problem1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X > X >
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > Z > Z > Y > Y > > > X >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > Y > Y > Z > > > X > X > Y
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> Z > > > Z > Z > Y > X > >
Solving The Problem There are two remaining sets of relationships, as shown
in profile 23.
a > b: in this example, Y > Z for both individuals. Y
is the #2 choice for person A and Z #3, whereas Y is
the #1 choice for person B and Z #2. In a case such
as this, we know that Y > Z in the social preferences
because both people prefer Y to Z.
a ? b: in this example, Z > X for person B, but X > Z
for person A. A's preferences are X > Y > Z and B's
are Y > Z > X. In this type of situation, we do not
yet know if Z or X will be socially preferred because
there is no agreement between individuals.
23
3 Z
2 Y
1 X
1 2 3
>
Solving The Problem In this case, because we know that Y > Z socially,
we can put that down. But we do not know if Y > Z
> X, Y > X > Z, or X > Y > Z, so we will not write
these down just yet, but once things start clearing
up, these rules will be used.
The key here is that Z is above and to the right of
Y, which means that there is agreement that Y > Z.
However, Z is above and to the left of X, meaning
there is conflict. Y is also above and to the left of
X, so there is another conflict, due to the fact that X
> Y > Z for person A, but Y > Z > X for person B.
23
3 Z
2 Y
1 X
1 2 3
> ? ? ?
The Single Choice If we look at profile 7, there is a single choice to be made.
Person A (left-hand column): X > Z > Y
Person B (bottom row): X > Y > Z
We know that X will be the #1 preference, but we have to
decide whether Y > Z or Z > Y here. We can choose either,
but let us pick that Y > Z. In other words, we support
person B's preference here.
This means that the social preference will be X > Y >
Z. In addition, because of the Independence of
Irrelevant Alternatives axiom, any time we see a
conflict between Y and Z similar to the one in profile 7,
we know to choose Y > Z.
Let us fill in the chart with this new information...
7
3 Y
2 Z
1 X
1 2 3
> X Y Z
Results1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X Y Z > Y Z X > Y Z X
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > Z > Z > Y > Y > > > Y X Z >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > Y > Y > Z > > > X > X > Y
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> Z > > > Z > Z > X Y Z > X > >
Explanation The profiles which have been newly solved (old solutions are colored green;
new are yellow) are profiles 7, 8, 9, 10, 17, and 33. 7, 8, 17, and 33 were
solved due to knowing one position already because of unanimity. 9 was
solved because both individuals support Z > X, so we know that Y > Z > X
there. 10 was solved because both individuals support X > Y, so we know that
the result must be X > Y > Z.
In doing this, we also have a new rule: in 33, X is above and to the left of Y,
and X > Y. Because of Independence of Irrelevant Alternatives, we can state
that X > Y whenever X is above and to the left of Y, like in 33. This will allow
us to fill in more profiles, which we will do now.
Step 2 Results1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X Y Z > Y Z X > Y Z X
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > Z > Z > Y > Y > > > Y X Z >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > Y > Y > Z > > > X > X > Y
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Z Y > > X Z Y > Z X Y > X Y Z > X Y Z > Z X Y > Z X Y > X Y Z
Step 2 Results Explanation We have 7 new completed profiles: 28, 30, 31, 32, 34, 35, 36. In addition, we
have two new rules, as determined by profile 28. When X is above and to the
left of Z, X > Z. Also, when Z is above and to the left of Y, Z > Y Let us see
how many new solutions we can find knowing that:
Y above and to the left of Z --> Y > Z
X above and to the left of Y --> X > Y
X above and to the left of Z --> X > Z
Z above and to the left of Y --> Z > Y
Step 3 Results1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X Y Z > Y Z X > Y Z X
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > Z > Z > Y X Z > X Z Y > X Z Y > Y X Z > Y X Z >
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> > Y > Y > Z > > Z X Y > X Z Y > Z Y X > Z Y X
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Z Y > Z Y X > X Z Y > Z X Y > X Y Z > X Y Z > Z X Y > Z X Y > X Y Z
Step 3 Results Explanation Now we have profiles 13, 14, 15, 16, 24, 25, 26,
27, and 29 filled in. And, once more, we have
another rule, determined from 27: when Y is
above and to the left of X, Y > X. This will lead
to yet more results.
Step 4 Results1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X Y Z > Y Z X > Y Z X
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > Z Y X > Y X Z > Y X Z > X Z Y > X Z Y > Y X Z > Y X Z > Y X Z
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> Z Y X > Y > Y > Y Z X > Y Z X > Z X Y > X Z Y > Z Y X > Z Y X
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Z Y > Z Y X > X Z Y > Z X Y > X Y Z > X Y Z > Z X Y > Z X Y > X Y Z
Step 4 Results Explanation Now profiles 11, 12, 18, 19, 22, and 23 are filled
in. And, from profile 22, we can determine that
when Z is above and to the left of X, Z > X. This
results in:
Step 5 Results1 2 3 4 5 6 7 8 9
3 Z 3 Y 3 Z 3 X 3 Y 3 X 3 Y 3 X 3 Y
2 Y 2 Z 2 X 2 Z 2 X 2 Y 2 Z 2 Y 2 X
1 X 1 X 1 Y 1 Y 1 Z 1 Z 1 X 1 Z 1 Z
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > X Z Y > Y X Z > Y Z X > Z X Y > Z Y X > X Y Z > Y Z X > Y Z X
10 11 12 13 14 15 16 17 18
3 Y 3 Y 3 Z 3 X 3 Y 3 X 3 X 3 Y 3 Y
2 X 2 X 2 Y 2 Z 2 X 2 Y 2 Y 2 X 2 Z
1 Z 1 Z 1 X 1 Y 1 Z 1 Z 1 Z 1 Z 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Y Z > Z Y X > Y X Z > Y X Z > X Z Y > X Z Y > Y X Z > Y X Z > Y X Z
19 20 21 22 23 24 25 26 27
3 Y 3 Z 3 Y 3 Y 3 Z 3 Z 3 Z 3 X 3 Z
2 Z 2 X 2 Z 2 Z 2 Y 2 Y 2 Y 2 Z 2 Y
1 X 1 Y 1 X 1 X 1 X 1 X 1 X 1 Y 1 X
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> Z Y X > Y Z X > Z X Y > Y Z X > Y Z X > Z X Y > X Z Y > Z Y X > Z Y X
28 29 30 31 32 33 34 35 36
3 X 3 Z 3 Z 3 X 3 Z 3 X 3 Z 3 X 3 X
2 Z 2 X 2 X 2 Y 2 X 2 Y 2 X 2 Z 2 Z
1 Y 1 Y 1 Y 1 Z 1 Y 1 Z 1 Y 1 Y 1 Y
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
> X Z Y > Z Y X > X Z Y > Z X Y > X Y Z > X Y Z > Z X Y > Z X Y > X Y Z
Step 5 Results Explanation What happened?
In all 36 cases, person B (bottom row) matches up exactly
with the social preferences. In other words, person B is
our dictator.
How did this happen?
We made one decision and used unanimity (in cases
where X, Y, or Z was above and to the right) and
independence of irrelevant alternatives (where X, Y, or Z
was above and to the left) to solve the rest.
What Does This Mean? Any complete social preference order which
obeys both unanimity and independence of
irrelevant alternatives will be dictatorial (as
described on slide 8).
A non-dictatorial social preference ranking would
require dropping one of the three other desirable
conditions. Normally, completeness and the
independence of irrelevant alternatives are
dropped.
Practical Results Cannot develop a rational social preference
function like what we have for individual
preferences
No voting mechanism will simultaneously satisfy
completeness, unanimity, independence of
irrelevant alternatives, and non-dictatorship.
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