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# Arrow's Impossibility Theorem:

## Is There a Best Procedure?  Arrow’s Impossibility Theorem

Because plurality rule, the Borda count, singular transferable vote, instant runoff, and approval voting all reduce to majority rule for two candidates and an odd number of voters, there needs to be some other way to evaluate and compare election procedures.  Kennth Arrow’s Impossibility Theorem was born out of this idea.  Can one determine an election procedure to satisfy a collection of reasonable properties?  And, what are the properties?  In Arrow’s language, the properties are called “axioms.”  A description of these axioms are preceded by a brief historical sketch.

Kenneth Arrow

Kenneth Arrow’s Impossibility Theorem was part of his doctoral research in economics at Columbia University where he earned his doctrate in 1951; his research was published the same year as the book, Social Choice and Individual Values.  Because of his work in social choice/voting theory, as well as his work in economics on the general equilibrium theory he received the Nobel Prize in Economics in 1972, sharing the prize with John Hicks.  At 51, Arrow was the youngest person ever to receive the Prize.  Not only has the implication of Arrow’s Impossibility Theorem had great impact on voting theory, but Arrow’s axiomatic approach has had great impact, too.  Here is one formulation of Arrow’s axioms:

• Universality:  Each individual voter ranks the n candidates from 1 to n.  The candidate ranked i is preferred to all candidates ranked j where j > i.  There are no restrictions on which rankings are allowed.
• Transitive Outcomes:  The societal outcome of an election procedure is a ranking possibly with ties of the candidates.  If the society prefers candidate A to candidate B and prefers candidate B to candidate C, then A is ranked above B and B is ranked above C.  Because A is ranked above C, then the society prefers A to C, too.
• Unanimity: If all voters prefer candidate A to candidate B, then the societal ranking should place A above B.
• Independence of Irrelevant Alternatives: The societal ranking of a pair of candidates A and B should only depend on how the voters rank the two candidates.  To rank these two candidates, it is irrelevant to factor in how the voters rank other candidates.  More specifically, suppose that the society ranks A above B and C.  If some voters decide to change their ranking of B and C, then it should not affect the societal ranking of A and B:  A should still be ranked above B.
###### Anecdotal Interpretation of Independence of Irrelevant Alternatives

Consider the shopper that tries on a pair of shoes.  The clerk indicates that the shoes are available in brown and black.  The customer decides to get the shoes in black.  The clerk finds a pair of the shoes in mahogany and lets the customer know.  Armed with new information, the customer decides to get the shoes in brown instead!  Despite having chosen black over brown when mahogany was not available, the option of having the shoes in mahogany (a supposedly irrelevant alternative) caused the shopper to switch to brown (thereby affecting the decision).  The shopper’s decision making fails to adhere to the Independence of Irrelevant Alternatives axiom.

Taken individually, Arrow’s axioms appear to place reasonable requirements upon an election procedure, especially when the possible affect of the independence of irrelevant alternatives is highlighted in the boxed anecdotal description (see above).  To Arrow’s surprise, the only election procedure that satisfies the above axioms isn’t much of an election procedure at all!

Arrow’s Impossibility Theorem  For three or more candidates, the only procedure that satisfies the above four axioms is a dictatorship, in which the outcome of an election always agrees with a specified voter’s preferences.  That voter is referred to as the “dictator.”

The language of Arrow’s theorem can be misunderstood.  It does not mean that democracy is doomed to failure and that only dictators are sound political institutions.  Arrow’s theorem states that there is no procedure to decide among three or more candidates that satisfies the above properties.  If the axioms are an ideal that cannot be met, then other, less stringent, criteria should be used to compare and evaluate election procedures.  Or, as may be the case, the selection of an election procedure is a decision about which of the above axioms is less important to the application at hand.

Don Saari of the University of California, Irvine, provides a movie, “What Causes Voting Paradoxes,” that describes the misinterpretation of Arrow’s theorem at his website.
 Understanding Arrow’s Impossibility Theorem Hodge and Klima (p.79, Mathematics of Voting and Elections: A Hands-On Approach) include the following number-theoretic example to develop intuition about Arrow’s theorem.  They ask the following question: Is it possible for a positive integer to be divisible by 2, 11, and 23 and to be less than 500? Notice how each of the characteristics (from the end of the list to the front) reduces the set of possible positive integers.  The set of all positive integers less than 500 consists of 499 integers, S1 = {1, 2, 3, … , 499}.  The 21 integers in S1 that are multiples of 23 form the set S2: {23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483}. Let S3 be the set of integers in S2 that are evenly divisible by 11.  Then, S3 = {253}.  Because there are no even integers in S3, there is no positive integer less than 500 that is evenly divisible by 2, 11, and 23. Arrow’s theorem is referred to as Arrow’s Impossibility Theorem because the inclusion of a fifth axiom (No Dictatorship) ensures that there is no procedure that satisfies all five axioms.  Hence, it is impossible for a procedure to satisfy all five axioms.  As such, Arrow’s theorem is in the spirit of the above example.

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