The Notion of a Rational Convex Program, and an Algorithm for the Arrow-Debreu Nash Bargaining Game

We introduce the notion of a rational convex program (RCP) and we classify the known RCPs into two classes: quadratic and logarithmic. The importance of rationality is that it opens up the possibility of computing an optimal solution to the program via an algorithm that is either combinatorial or uses an LP-oracle. Next, we define a new Nash bargaining game, called ADNB, which is derived from the linear case of the Arrow-Debreu market model. We show that the convex program for ADNB is a logarithmic RCP, but unlike other known members of this class, it is nontotal. Our main result is a combinatorial, polynomial-time algorithm for ADNB. It turns out that the reason for infeasibility of logarithmic RCPs is quite different from that for LPs and quadratic RCPs. We believe that our ideas for surmounting the new difficulties will be useful for dealing with other nontotal RCPs as well. We give an application of our combinatorial algorithm for ADNB to an important "fair" throughput allocation problem on a wireless channel. Finally, we present a number of interesting questions that the new notion of RCP raises.