Maximum Likelihood Approach

Abstract: In voting, the joint decision is made based on the agents' preferences. Therefore, in some sense, this means that the agents' preferences are the " causes " of the joint decision. However, there is a different (and almost reversed) point of view: there is a " correct " joint decision, but the agents may have different perceptions (estimates) of what this correct decision is. Thus, the agents' preferences can be viewed as noisy reports on the correct joint decision. Even in this framework, the agents still need to make a joint decision based on their preferences, and it makes sense to choose their best estimate of the correct decision. Given a noise model, one natural approach is to choose the maximum likelihood estimate of the correct decision. The maximum likelihood estimator is a function from profiles to alternatives (more accurately, subsets of alternatives, since there may be ties), and as such is a voting rule (more accurately, a correspondence). This maximum likelihood approach was first studied by Condorcet (1785) for the cases of two and three alternatives. Much later, Young (1995) and Young (1988) showed that for arbitrary numbers of alternatives, the MLE rule derived from Con-dorcet's noise model coincides with Kemeny's rule (Kemeny, 1959). The approach 182