ECONOMY

398 a tool kit for voting theory
This approach sounds too good to be true, and that is the case. My cautionary
advice is to avoid putting too much trust in these conclusions as they can be seriously
misleading. The reason is that many “axiomatic” studies in the social sciences have
little, if anything, to do with axiomatics. Rather than “axioms,” they are using what
we mathematicians call “properties” or “hypotheses.” In blunt terms, most of these
conclusions only show which particular properties happen to identify uniquely a
particular decision rule. But “uniquely identifying” and “characterizing” are very
different traits: in particular, most results from this approach fail to characterize what
yougetfromadecisionrule.
To illustrate with a non-technical illustration, consider the three traits (1) Finnish-
American heritage, (2) born in a particular year in the Upper Peninsula of Michigan,
and (3) does research in social choice and the Newtonian N-body problem. While
these traits uniquely identify me, they do not characterize me. Knowing just this,
you do not know “what you are getting.” Similarly the traditional “axiomatic studies”
often identify properties that, by emphasizing special settings, uniquely identify an
election rule. But unless these properties can be used to derive all properties of the
rule, then instead of being “axioms,” they are just special properties the rule happens
to satisfy.
If the axiomatic approach does not accomplish what we want, how do we find
all properties of a voting rule? In addressing this question, I was influenced by the
research of Nurmi, Brams, and Fishburn. In particular, Fishburn (1981) published
an intriguing paradox (that is, a counterintuitive outcome) where the sincere plurality
election outcome is A ≻ B ≻ C ≻ D, butifD drops out, the sincere election
outcome for the same voters—nobody changes preferences!—becomes the reversed
C ≻ B ≻ A. Fishburn’s reversal example is more than an amusing curiosity: it illustrates
a peculiar and unexpected property of the plurality vote.
The ranking “paradoxes” in the literature must be taken seriously because they
identify properties of election rules. Similarly, “paradoxes that cannot occur” identify
a rule’s positive properties. To illustrate with the Borda Count (where 2, 1, 0 points
are assigned, respectively, to a voter’s top-, second-, and bottom-ranked candidate),
this rule does not allow the list
(B ≻ C ≻ A, A ≻ B, A ≻ C, B ≻ C)
to ever happen. This and other lists of Borda rankings that cannot occur define
the important property that the Borda Count never ranks the Condorcet loser (the
candidate that loses to all candidates in pairwise elections) above the Condorcet
winner. Does this property extend to any other positional method? (No.)
To find all ranking properties of election procedures, we could find all possible
lists of election rankings that could ever occur. To illustrate with Fishburn’s example,
rather than just the plurality (A ≻ B ≻ C ≻ D, C ≻ B ≻ A) listing,wewantto
know what happens with all subsets of candidates. Can we have, for instance, a
plurality listing
(A ≻ B ≻ C ≻ D, C ≻ B ≻ A, D ≻ B ≻ A, D ≻ C ≻ A, D ≻ C ≻ B,
A ≻ B, B ≻ C, C ≻ D, D ≻ A, A ≻ C, B ≻ D)