Arrow's impossibility theorem
In
social choice theory
,
Arrow’s impossibility theorem
, the
General Possibility Theorem
,
or
Arrow’s paradox
, states that, when voters have three or more distinct alternatives (options),
no
rank order
voting system
can convert the
ranked preferences
of individuals into a community-
wide (complete and transitive) ranking while also meeting a pre-specified set of criteria. These pre-
specified criteria are called
unrestricted domain
,
non-dictatorship
,
Pareto efficiency
,
and
independence of irrelevant alternatives
. The theorem is often cited in discussions of election
theory as it is further interpreted by the
Gibbard–Satterthwaite theorem
.
The theorem is named after economist
Kenneth Arrow
, who demonstrated the theorem in his
doctoral thesis and popularized it in his 1951 book
Social Choice and Individual Values
. The original
paper was titled "A Difficulty in the Concept of Social Welfare".
[1]
In short, the theorem states that no rank-order voting system can be designed that 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.
Voting systems that use
cardinal utility
(which conveys more information than rank orders; see
the
subsection
discussing the cardinal utility approach to overcoming the negative conclusion) are
not covered by the theorem.
[2]
The theorem can also be sidestepped by weakening the notion of
independence. Arrow rejected cardinal utility as a meaningful tool for expressing social welfare,
[3]
and
so focused his theorem on preference rankings.
The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on
preferences) within one unified framework. In that sense, the approach is qualitatively different from
the earlier one in voting theory, in which rules were investigated one by one. One can therefore say
that the contemporary paradigm of social choice theory started from this theorem.
[5]
Statement of the theorem
[
edit
]
The need to aggregate
preferences
occurs in many disciplines: in
welfare economics
, where one
attempts to find an economic outcome which would be acceptable and stable; in
decision theory
,
where a person has to make a rational choice based on several criteria; and most naturally in
voting
systems
, which are mechanisms for extracting a decision from a multitude of voters' preferences.
The framework for Arrow's theorem assumes that we need to extract a preference order on a given
set of options (outcomes). Each individual in the society (or equivalently, each decision criterion)
gives a particular order of preferences on the set of outcomes. We are searching for a
ranked voting
system
, called a
social welfare function
(

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