Rational Convex Programs, Their Feasibility, and the Arrow-Debreu Nash Bargaining Game

TL;DR

This paper introduces rational convex programs, develops methods for non-total problems like the Arrow-Debreu Nash bargaining game, and applies these algorithms to fair wireless channel allocation.

Contribution

It extends combinatorial algorithms to handle non-total convex programs, specifically addressing the Arrow-Debreu Nash bargaining game and related market models.

Findings

Developed primal-dual algorithms for feasibility and equilibrium computation.

Proved infeasibility in certain parameter settings.

Applied algorithms to wireless throughput fairness problem.

Abstract

Over the last decade, combinatorial algorithms have been obtained for exactly solving several nonlinear convex programs. We first provide a formal context to this activity by introducing the notion of {\em rational convex programs} -- this also enables us to identify a number of questions for further study. So far, such algorithms were obtained for total problems only. Our main contribution is developing the methodology for handling non-total problems, i.e., their associated convex programs may be infeasible for certain settings of the parameters. The specific problem we study pertains to a Nash bargaining game, called ADNB, which is derived from the linear case of the Arrow-Debreu market model. We reduce this game to computing an equilibrium in a new market model called {\em flexible budget market}, and we obtain primal-dual algorithms for determining feasibility, as well as giving a proof of infeasibility and finding an equilibrium. We give an application of our combinatorial algorithm for ADNB to an important "fair" throughput allocation problem on a wireless channel.