Efficient points in a sum of sets of alternatives
Anas Mifrani
TL;DR
The paper addresses when the efficient-point set of a sum, $\mathscr{E}(A+B)$, coincides with $\mathscr{E}(A)$ under a general relation $R$ on a (not necessarily abelian) group $(G,+)$. It develops a framework with properties (P1)–(P5) and distinguishes finite vs infinite $A$, providing broad sufficient conditions (extending the Yu–Ehrgott result) and a range of counterexamples showing failure under other configurations. Key contributions include precise theorems that guarantee equality in certain isotone/additive settings and that guarantee non-equality when $A$ is finite and $B$ contains elements incomparable with $0_G$, plus a detailed appendix with cycle-based proofs that underpin the finite-case results. The work broadens the scope of efficient-point analysis beyond Euclidean spaces, offers insights for potential algorithmic approaches to computing $\mathscr{E}(A+B)$, and points to extensions to semigroups, lattices, cones, and stochastic/graph-structured decision problems.
Abstract
The concept of efficiency plays a prominent role in the formal solution of decision problems that involve incomparable alternatives. This paper develops necessary and sufficient conditions for the efficient points in a sum of sets of alternatives to be identical to the efficient points in one of the summands. Some of the conditions cover both finite and infinite sets; others are shown to hold only for finite sets. Examples are provided that illustrate these results.