Lattices,bargaining and group decisions

This essay aims at constructing an abstract mathematical system which, when interpreted, serves to portray group-choices among alternatives that need not be quantifiable. The system in question is a complete distributive lattice, on which a class of non-negative real-valued homomorphisms is defined. Reinforced with appropriate axioms, this class becomes a convex distributive lattice. If this lattice is equipped with a suitable measure, and if the mentioned class of homomorphisms is equipped with a metric, then the class and its convex sets are seen to possess certain characteristic properties. The main result (Theorem 6) follows from a combination of these results and a famous result due to Choquet.The mathematical scheme is then interpreted in the subject-language of choice among alternatives. It is shown, by means of an example, that the system furnishes all the ingredients for describing multi-group choices. Whether or not the same ingredients are also adequate for a behavioural theory of multi-group choices is an issue that will not be gone into. However, the example effectively illustrates how a process of bargaining can be described with the aid of the mathematical scheme.In the second example, a class of bargaining situations is modelled in the symbolism of linear programming with several objective functions combined with unknown weights; the cost vectors in such formulations are identified with homomorphisms, and the main theorem of this essay is applied.