Theoreticians' attempts to find "optimal" voting systems. (Haven't worked well...)
Condorcet ≈1780:
Consider this probability model. There is some true God-given quality-ordering
among the N candidates. Each voter if asked "who is better, candidate A or B?"
gets the wrong answer with some fixed probability p. All voters and all answers
are independent.
Then: Condorcet's "least reversal" voting system elects candidate most-likely to be
"true" winner.
Flaws:
a⇒Condorcet did not understand prob. theory?
e⇒theorem lives in some other world.
#voters
Their Vote
3
A>B>X>C
3
C>A>X>B
3
B>C>X>A
2
A>X>B>C
2
C>X>A>B
2
B>X>C>A
Actually, your answers (in a single rank-ordering vote) depend.
(C.L.Dodgson attempted to fix that flaw...)
All systems that always elect Condorcet winners suffer "no show" paradoxes in which
voter gets a better election result (in her view) by refusing to vote than
by casting an honest vote.
They also always are "inconsistent" in the sense that if W wins in district 1 and
in district 2, that does not imply W wins in combined country.
The least-reversal system can declare X "the unique best" candidate... while
simultaneously declaring X "the unique worst" candidate.
Real human voters use strategy. They are not honest idiots.
D.G.Saari ≈1990:
Among "weighted positional" voting systems,
Borda uniquely has
the "fewest paradoxes."
Also, Borda is the unique WP system satisfying both "fairness" and
"reversal symmetry."
Flaws:
a,b⇒who cares about WP systems?
d⇒theorem lives in some other world.
It can be shown that all WP voting systems can act arbitrarily crazily
in the sense that (a) for any pair (X, Y) of nontrivial WP systems, and any two orderings
(A, B) among C and C-1 candidates respectively, and any k∈{1,2,...,C}
there is an election in which system X outputs societal ordering A,
but after deleting candidate k
and using the same votes, system Y outputs ordering B.
#voters
Their Vote
5
C>B>A
4
B>A>C
2
B>C>A
2
A>C>B
Range voting and Condorcet systems and IRV and approval are not WP systems.
Every WP system elects B and not the Condorcet winner C in pictured situation
(C>B by 7:6, C>A by 7:6).
But Gehrlein, Fishburn, & van Newenhizen
showed that the probability (assuming each vote is random permutation)
that Borda elects a non-Condorcet-Winner (conditioned on a CW existing)
is about 10% if N=3 and #voters→∞.
They showed Borda minimizes this conditional
probability over all WP systems, for each N≥3.
Real human voters use strategy. They are not honest idiots.
W.D.Smith 1999:
Range voting
is
the unique COAF voting system (Compact, One-vote, Additive, Fair)
with "maximum voter expressivity" subject to "avoiding favorite betrayal."
Also has "max-measure strategically accessible vote-sets."
Related theorems, but cleaned up and nicer,
were shown by Marcus Pivato in 2011-2013.
W.D.Smith 2007:
Range features both "immunity to cloning" and
"avoiding favorite betrayal"; but no rank-order system does.
Dhillon & Mertens 1999:
Range voting is the unique voting system satisfying certain Arrow-like axioms.