Today I start three weeks of
blogging “On Troikas”. These 15 chapters are about the logic of voter’s
paradoxes. The topics are:
The Troika
Glitches
More Glitches
Expanding Troikas
Linear Loops
Delta Deduction
Trilemmas
Religious Troikas
Republican Troikas
Kantian Troikas
Causation Troika
Surreality Troikas
Paradox of the Second Best
Paradox of Distribution
Agenda Manipulation and the
Chairman’s Paradox
**********
The Troika
A “troika” is my name for what is
usually called a “voter’s paradox”, and it is at the heart of Kenneth Arrow’s
Impossibility Theorem. These logic knots have a habit of bollixing political
systems. These tiny tangles give politics its notorious perversity.
To simplify presentation of the
Troika, I introduce three fictional characters; none other than the Three
Stooges.
General Moe rules the Scissors
Party with an iron hand. His politics are fascistic; he favors power over logic
over fairness. He would rather be decisive than consistent, and he would rather
be consistent than share power. Naturally he prefers monarchy, most preferably
if the monarch is himself.
Judge Larry is senior
theoretician for the Paper Party. His politics are legalistic: he favors logic
over fairness over power. Naturally he prefers to govern by consensus.
Mayor Curly is lead singer for
the Rock Party. His politics are populistic: he favors fairness over power over
logic. Naturally he prefers to govern by majority rule.
Moe: Fairness < Logic <
Power
Larry: Power < Fairness < Logic
Curly: Logic < Power <
Fairness
Each single Stooge has a
consistent linear ranking of fairness, power, and logic; but when you put them
all together, something has got to go.
Twothirds of the Stooges
(namely, Moe and Larry) put logic above fairness; Larry and Curly put fairness
above power; and Curly and Moe put power above logic.
Moe Larry Curly
Power < Fairness? no yes yes
Logic < Power? yes no yes
Fairness < Logic? yes yes no
This gives us a Condorcet
Election, or “Voter’s Paradox”:
Logic
< <
fairness > power

by 2/3 majority each; yet they all agree that the ranking is
linear!
There are several partial
resolutions to this.
If we appoint a single voter as
tyrant (Moe, say) then we can decide this consistently; but this is not a fair
system.
If we attempt to decide by
consensus (as Larry suggests) then that is fair and consistent; but we decide
nothing, and that is a weak system.
If we have faith in majority rule
(as Curly professes) then we accept the nonlinear order, and the
linearity of the order. This is fair and decisive, but it is inconsistent.
Finally, we can accept the
nonlinear ranking, and accept it as nonlinear; this goes with every 2/3
majority, but reverses a consensus; and that is perverse.
This political knot is an
instance of Arrow’s Theorem, which says that no voting system has all
four of these virtues:
it is fair: it gives all voters
equal power
it is decisive: it decides all
questions posed to it
it is logical: it does not
believe contradictions
it is responsive: it never defies
a voter consensus.
In other words, any government is
at least one of:
cruel ;
weak ; absurd
; perverse.
Moe prefers cruelty, Larry
prefers weakness, and Curly prefers folly; none of them want perversity,
but that of course is what they always get!
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