The Equal Loss Choice Function Revisited

Choice theory is a mathematical discipline which studies the problem of choosing a point from a set of points by studying the mathematical properties of maps assigning an outcome to each choice problem in some class of choice problems. A large literature has grown up concerning choice problems in Euclidean spaces. A typical choice problem is then a compact, convex, comprehensive subset of the non-negative orthant of a finite dimensional Euclidean space, containing a strictly positive vector. For such choice problems, Yu (1973) and Freimer and Yu (1976) have introduced a class of solutions obtained by minimizing the distance of the “ideal point”, measured by some norm. The equal loss solution is one such. However neither Yu (1973) nor Freimer and YU (1976), succeeded in characterizing such solutions axiomatically. It was in Chun (1988) that we find a complete axiomatic characterization of the equal loss solution for the first time. A brief glance at the proof of Chun’s theorem, begs the questions, whether there is a simple alternative proof. The purpose of this paper is to provide such a proof, by modifying the technique suggested by Thomson and Lensberg (1989), in their axiomatic characterization of the egalitarian solution. In the later sections of the paper we consider choice problems with variable dimensions and obtain an axiomatic characterization of the equal-loss-choice function using a reduced choice problem property, first invoked in the relevant literature by Peters, Tijs and Zarzuelo (1994). We are thereby able to drop the assumption of Strong Monotonicity with Respect to the Ideal point, which is used in the original characterization.