On Troikas 1: The Troika

            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

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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.

Two-thirds 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 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 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!