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

Arrow’s Impossibility Theorem

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.”

The GibberSatterthwaite theorem says, basically, that voting systems are subject to strategic abuse

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.

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
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)
To read
 Buchanan, J. M.(1954). Social Choice, Democracy, and Free Markets. Journal of Political Economy, 62(2), 114–123. Online.
 Duggan, J., & Schwartz, T. (2000). Strategic manipulability without resoluteness or shared beliefs: GibbardSatterthwaite generalized. Social Choice and Welfare, 17(1), 85–93. DOI. Online.
 Satterthwaite, M. A.(1975). Strategyproofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10(2), 187–217. DOI. Online.
 Taylor, A. D.(2002). The Manipulability of Voting Systems. The American Mathematical Monthly, 109(4), 321–337. DOI. Online.