# Voting Systems

### Mathematics and economics of ballots

Usefulness: đź”§
Novelty: đź’ˇ
Uncertainty: đź¤Ş đź¤Ş đź¤Ş
Incompleteness: đźš§ đźš§ đźš§

Thereâ€™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. Anywayâ€¦ Iâ€™ll look at that here. Whining about modern democratic failure Iâ€™ll leave to capitalismâ€™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 doesnâ€™t have preferences anymore than a group smiles. What Arrow showed, however, is that without invoking special cases we canâ€™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.â€ť

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

• The brood of descendants of these theorems

• Voting process construction

• Public choice theory

• To write: short necessary disclaimer of the wrong headedness of the formulation in terms of â€śoptimalâ€ť 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 â€śFar lower cost, better quality and more democraticâ€ť

## Alternatives to voting for deciding things

Sortition - government by random sampling of representatives from the population. What statistician could avoid at least toying with this idea?

David Van Reybrouckâ€™s Against Elections (2014). I should also shout out to futarchy, probably. For more of that, consider mechanism design.

# Refs

Buchanan, James M. 1954. â€śSocial Choice, Democracy, and Free Markets.â€ť Journal of Political Economy 62 (2): 114â€“23. http://www.jstor.org/stable/1825570.

Buchanan, James M, and Gordon Tullock. 1962. The Calculus of Consent: Logical Foundations of Constitutional Democracy. University of Michigan Press.

Chaum, David. n.d. â€śFar Lower Cost, Better Quality and More Democratic,â€ť 17.

Duggan, John, and Thomas Schwartz. 2000. â€śStrategic Manipulability Without Resoluteness or Shared Beliefs: Gibbard-Satterthwaite Generalized.â€ť Social Choice and Welfare 17 (1): 85â€“93. https://doi.org/10.1007/PL00007177.

Geanakoplos, John. 1996. â€śThree Brief Proofs of Arrowâ€™s Impossibility Theorem.â€ť Cowles Foundation Discussion Paper 1123R3. Cowles Foundation for Research in Economics, Yale University. http://ideas.repec.org/p/cwl/cwldpp/1123r3.html.

Masuda, N, and S Redner. 2011. â€śCan Partisan Voting Lead to Truth?â€ť Journal of Statistical Mechanics: Theory and Experiment 2011: â€“02002.

Perony, Nicolas, RenĂ© Pfitzner, Ingo Scholtes, Claudio J Tessone, and Frank Schweitzer. 2013. â€śEnhancing Consensus Under Opinion Bias by Means of Hierarchical Decision Making.â€ť Advances in Complex Systems 16 (06): 1350020. https://doi.org/10.1142/S0219525913500203.

Satterthwaite, Mark Allen. 1975. â€śStrategy-Proofness and Arrowâ€™s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions.â€ť Journal of Economic Theory 10 (2): 187â€“217. https://doi.org/10.1016/0022-0531(75)90050-2.

Taylor, Alan D. 2002. â€śThe Manipulability of Voting Systems.â€ť The American Mathematical Monthly 109 (4): 321â€“37. https://doi.org/10.2307/2695497.