Voting Systems

Mathematics and economics of ballots

Tom Gauld:

Voting system

See also eigenmorality, distributed sensing, the wisdom of crowds, insurgency, mechanism design.

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.

  • 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 “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”

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 Reybrouck’s Against Elections (2014). I should also shout out to futarchy, probably. For more of that, consider mechanism design.

Buchanan, James M. 1954. “Social Choice, Democracy, and Free Markets.” Journal of Political Economy 62 (2): 114–23. 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. “Far Lower Cost, Better Quality and More Democratic,” 17.

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

Geanakoplos, John. 1996. “Three Brief Proofs of Arrow’s Impossibility Theorem.” 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. “Can Partisan Voting Lead to Truth?” Journal of Statistical Mechanics: Theory and Experiment 2011: –02002.

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

Satterthwaite, Mark Allen. 1975. “Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions.” Journal of Economic Theory 10 (2): 187–217. https://doi.org/10.1016/0022-0531(75)90050-2.

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