Biased voting and the democratic primary problem

Abstract

Inspired by the recent Democratic National Primary, we consider settings in which the members of a distributed population must balance their individual preferences over candidates with a requirement to quickly achieve collective unity. We formalize such settings as the "Democratic Primary Problem" (DPP) over an undirected graph, whose local structure models the social influences acting on individual voters. After contrasting our model with the extensive literature on diffusion in social networks (in which a force towards collective unity is usually absent), we present the following series of technical results: An impossibility result establishing exponential convergence time for the DPP for a broad class of local stochastic updating rules, which includes natural generalizations of the well-studied "voter model" from the diffusion literature (and which is known to converge in polynomial time in the absence of differing individual preferences). A new simple and local stochastic updating protocol whose convergence time is provably polynomial on any instance of the DPP. This new protocol allows voters to declare themselves "undecided", and has a temporal structure reminiscent of periodic polling or primaries. An extension of the new protocol that we prove is an approximate Nash equilibrium for a game-theoretic version of the DPP. © 2008 Springer Berlin Heidelberg.