The maximum likelihood approach to voting on social networks

One view of voting is that voters have inherently different preferences - de gustibus non est disputandum - and that voting is merely a method for reaching a reasonable compromise solution. An alternative view is that some of the alternatives really are better in an objective sense, and by voting over the alternatives we hope to be more likely to reach the correct outcome. In this latter view, we can see the votes as noisy estimates of the truth. Specifying a probabilistic noise model gives us a natural “optimal” voting rule for determining the outcome based on the votes, namely, the function that takes the votes as input and produces the outcome that maximizes the likelihood of these votes as output. We will first review some of the work on the maximum likelihood approach to voting. Most of this work supposes that, conditional on the correct outcome, votes are independent. In reality, however, voters are clearly influenced by the opinions of those close to them. How should we model the effects of the social network, and what does this imply for the maximum likelihood approach? We will first review an earlier result [1] that states that, under certain assumptions, the social network structure should not affect the voting rule. We then consider a new model under which this is not true, and prove that computing the probability of the votes given the correct outcome is #P-hard under this model. On the other hand, if the goal is to simultaneously also give a point estimate of the hidden variables in the model, then the optimization problem can be solved in polynomial time.