New Convex Programs for Fisher's Market Model and its Generalizations

—We give two natural generalizations of Fisher’s market model: In model M1, sellers can declare an upper bound on the money they wish to earn (and take back their unsold good), and in model M2, buyers can declare an upper bound on the amount to utility they wish to derive (and take back the unused part of their money). —We derive convex programs for the linear case of these two models by generalizing a convex program due to Shmyrev and the Eisenberg-Gale program, respectively. —We generalize the Arrow-Hurwicz theorem to the linear case of these two models, hence deriving alternate convex programs. — For the special class of convex programs having convex objective functions and linear constraints, we derive a simple set of rules for constructing the dual program (as simple as obtaining the dual of an LP). Using these rules we show a formal relationship between the two seemingly different convex programs for linear Fisher markets, due to Eisenberg-Gale and Shmyrev; the duals of these are the same, upto a change of variables.