Arrow’s Trilemma
Consider this troika, about the politics and values of the Stooges. Together they will validate Arrow’s Impossibility Theorem.
General Moe rules the Scissors Party with an iron hand. His politics are fascistic; he favors Lies over Truth over Imagination. 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 Truth over Imagination over Lies. He prefers to govern by consensus.
Mayor Curly is lead singer for the Rock Party. His politics are populistic: he favors Imagination over Lies over Truth. He prefers to govern by majority rule.
Together they form this troika:
Moe: Imagination < Truth < Lies
Larry: Lies < Imagination < Truth
Curly: Truth < Lies < Imagination
Each Stooge has a consistent linear ranking of imagination, lies and truth; but when you put them all together, something has got to go. Two-thirds of the Stooges (namely, Moe and Larry) put truth above imagination; Larry and Curly put imagination above lies; and Curly and Moe put lies above truth. Thus this trilemma:
Lies < Imagination;
Imagination < Truth;
Truth < Lies.
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 non-linear order, and the linearity of the order. This is fair and decisive, but it is inconsistent.
Finally, we can accept the non-linear ranking, and accept it as non-linear; this goes with every 2/3 majority, but reverses a consensus; and that is perverse.
This political knot is an instance of Arrow’s Impossibility 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 and lies, Larry prefers weakness and truth, and Curly prefers absurdity and imagination; none of them want perversity and paradox, but of course that is what they keep getting!
Here is Arrow’s Troika:
Moe: the government should be decisive and logical but unfair;
Larry: the government should be logical and fair but indecisive;
Curly: the government should be fair and decisive but illogical.
They all want responsive government.
This troika supports Arrow’s Trilemma:
A responsive government is fair;
A responsive government is decisive;
A responsive government is logical;
Choose two!
The associated triad is:
From: the government is responsive, decisive and logical
Deduce: The government is unfair.
From: the government is responsive, logical, and fair
Deduce: The government is indecisive.
From: the government is responsive, fair and decisive
Deduce: The government is illogical.
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