Multi-utilitarianism in two-agent quasilinear social choice (Q2572443)

Let \(\{1,2\}\) be a set of agents. A problem is a non-empty subset \(B\) of \(\mathbb{R}^{2}\) that is closed, convex, comprehensive and bounded above. \(\overline{x}(B)\) is by definition \(\max_{x \in B}x_1 + x_2\). \(H(B) = \{x \in \mathbb{R}^2 : x_1 + x_2 = \overline{x}(B)\}\) is the set of efficient points that the agents can achieve by making transfers. Let \(\mathcal{B}\) denote the set of all problems. A rule is a function \(f : \mathcal{B}\to\mathbb{R}^2\) such that for all \(B \in \mathcal{B}\), \(f(B)\) is a solution for \(B\). The main theorem says that a rule satisfies efficiency, translation invariance, transfer monotonicity, continuity and additivity if and only if it is a multi-utilitarian rule. The independence of the axioms is shown in an appendix. A multi-utilitarian rule is of the form \(U^{\nu}(B) = \int_{[0,1]-\{1/2\}} U^\lambda(B)d\nu (\lambda)\), where \(\nu\) is a probability measure on the measurable space \(([0,1] -\{1/2\},\Sigma\)) with \(\Sigma\) the Borel sets restricted to \([0,1]-\{1/2\}\). A rule \(f\) is efficient if for all \(B \in \mathcal{B}\), \(f(B) \in H(B)\); and translation invariant if for all \(B \in \mathcal{B}\) and for all \(x \in \mathbb{R}^2\), \(f(B + x) = f(B) + x\). A problem \(A\) dominates problem \(B\) for agent 1 if \(A\) gives agent 1 `better possibilities' than \(B\) does. Transfer monotonicity means that for \(A, B \in \mathcal{B}\), if \(A\) dominates \(B\) for agent 1, then \(f_1(B) \leq f_1(A)\). \(f\) is continuous if it is continuous in the Hausdorff topology and \(f\) is additive if for all \(A, B \in \mathcal{B}\), \(f(A + B) = f(A) + f(B)\). By adding a condition of recursive invariance the author obtains a characterization of bi-utilitarian rules and by changing transfer monotonicity into strict transfer monotonicity a characterization of full multi-utilitarian rules.