# Voting systems

Mathematics and economics of aggregate group preferences; the will of the people (whatever that might be); social choice theory

September 22, 2014 — April 21, 2022

There’s lots of interesting mathematics around this democracy business, and its shortcomings. Insert disclaimers about the complicated relationship between *is* and *ought*, and *model* and *actuality*. Anyway… I’ll look at that here. Whining about modern democratic failure I’ll leave to capitalism’s end game and practical analysis to psephology.

Arrow-style Impossibility Theorems

Neat summary by Alex Tabarrok:

We know or should always have known that a group doesn’t have preferences anymore than a group smiles. What Arrow showed, however, is that without invoking special cases we can’t even rationalize group choices as if leviathan had preferences. Put differently, the only leviathan that rationalizes group choice has the preferences of a madman.”

See also a snappy analysis on Encyclopaedia of math.

The Gibber-Satterthwaite theorem says, basically, that voting systems are subject to strategic abuse.

C&C Condorcet paradox

C&C Anscombe imposibility

C&C Arrow-Sen

Voting process construction

Public choice theory

To write: short necessary disclaimer of the wrong headedness of the formulation in terms of “optimal” choice for group decisions

Iterated Arrow results, for lots of polls. I think this is the Gibber-Satterthwaite model?

Oligopolistic game theory of the reverse case - what if parties have incentives to offer a shit range of options to the punters; in essence, what if parties systems effectively create cartels? Cost of entry to electoral processes etc.

Voter models: the fusion of statistical mechanics, graph theory, and a semblance of human behavior

David Chaum makes the case (that I think every statistician thinks is obvious) that Random-Sample Elections are “Far lower cost, better quality and more democratic”

To Build a Better Ballot is Nicky Case’s incredible voting-system visualizer.

## 1 Alternatives to/improvements upon voting for deciding things

## 2 References

*Journal of Political Economy*.

*The Calculus of Consent: Logical Foundations of Constitutional Democracy*.

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*.

*Social Choice and Welfare*.

*Journal of Economic Theory*.

*Public Choice*.

^{1}.”

*Synthese*.

*Journal of Statistical Mechanics: Theory and Experiment*.

*Advances in Complex Systems*.

*Ethics*.

*Ethics*.

*Journal of Economic Theory*.

*The American Mathematical Monthly*.