Extension Functions on Power Sets

In Kannai and Peleg(1984) the following problem was posed: Given a positive integer ‘n’, is it possible to define a positive integer valued function on all non-empty subsets of the first n positive integers, so that singletons preserve their original ranking and further the function satisfies two apparently reasonable properties? The same paper shows that for n greater than five, such a function cannot be defined. A large literature spawned out of this work, where modifications of the properties desired by Kannai and Peleg lead to possibility results. Notable among them are the following: Barbera, Barrett and Pattanaik (1984), Barbera and Pattanaik(1984) Fishburn(1984), Heiner and Packard(1984), Holzman (1984), Nitzan and Pattanaik(1984), Pattanaik and Peleg(1984), Bossert (1989). Our own efforts in this direction culminated in Lahiri(1999), where several of the above contributions have been discussed and studies. The above mentioned result lead to the search for a possibility result for n equal to five, resulting in the paper by Bandopadhyay (1988). In this paper we provide another different possibility result for n equal to five. Out method of proof suggests an alternative (: and perhaps simpler) approach to the result established in Bandopadhyay (1988) as well.