 Example 3: Let us go back to our main example. Typically Countries have preferences over the outcome of the decision, e.g., Italy think we should spend between 5 and 10 million euros for the scheme, Germany think we should spend between 1 and 2, Belgium between 4 and 5, Luxembourg, The Netherlands and France exactly 5. This means, for instance, that Italy’s preference relation is such that \(w \succ_{\textrm{Italy}} w'\) whenever \(\textrm{5M} \leq w \leq \textrm{10M}\) and either \(w'&gt;\textrm{10M}\) or \(0\leq w'&lt;\textrm{5M}\). What about all other couples of outcomes? In the simplest case Italy are indifferent between them. So \(w \sim_{\textrm{Italy}} w'\), otherwise. However, we could also assume a more complex preference. For instance, Italy would like to spend as much money as possible within their desired budget. In this case the preference relation is: \(w \succ_{\textrm{Italy}} w'\) whenever \(\textrm{5M} \leq w \leq \textrm{10M}\) and either \(w'&gt;\textrm{10M}\) or \(0\leq w'&lt;\textrm{5M}\), \(w \succ_{\textrm{Italy}} w'\) whenever \(\textrm{5M} \leq w' &lt; w \leq \textrm{10M}\) while \(w \sim_{\textrm{Italy}} w'\), otherwise. Not all outcomes of a vote are going to reach an agreement. We then, for technical purposes, define an auxiliary outcome \(w^{d}\), interpreted as a disagreement outcome. The idea is that this is an outcome of the vote that does not reach any consensus. We assume that any agreement is strictly better for any player than disagreement, i.e., \(w \succ_{{i}} w'\) whenever \(w'=w^{*}\) and \(w\neq w^{*}\), for each \(i\in N\).