 The distance-based approach can clarify the distinction between arbitration and majority operators. Leximax is an example of arbitration operator. A leximax operator may take d as the Hamming distance and, for each interpretation, the distances between that interpretation and the n bases \(K_i\) form a list. A pre-order over interpretations is defined by taking the lexicographical order between sequences of distances, fixing an order over the set of agents. Finally, \(D_{\textit{leximax}}\) selects the minimum. The intuition is that, unlike a majority operator that selects the option that minimizes the total disagreement (by minimizing the sum of the individual distances, for example), an arbitration operator looks at the distribution of such disagreement and selects the option that is fairer to all individuals, that is, it aims at equally distributing the individual dissatisfaction with the chosen outcome (recall the birthday gift example above). This follows from the definition of the Hamming distance: the larger the Hamming distance, the more disagreement there is between two interpretations (here disagreement simply means that the interpretations assign different truth values to the same formula). Suppose that a belief profile E has three bases. Suppose as well that the distances from the two models of IC (\(\omega\) and \(\omega'\)) are \(D_{\Sigma} (\omega, E) = D_{\Sigma} (\omega', E) = 6\) when we take the sum of the Hamming distances, and \(D_{\textit{leximax}}(\omega, E)= (2,2,2)\) and \(D_{\textit{leximax}}(\omega', E)= (5,1,0)\) when we take the lexicographic order on the distances. In this example, the majority operator cannot distinguish between \(\omega\) and \(\omega'\) (because in both cases the sum is 6), while the arbitration operator will prefer \(\omega\) to \(\omega'\) as \(\omega\) distributes the individual disagreement in a fairer way than \(\omega'\).