  We study Fisher markets and the problem of maximizing the Nash social welfare
(NSW), and show several closely related new results. In particular, we obtain:
  -- A new integer program for the NSW maximization problem whose fractional
relaxation has a bounded integrality gap. In contrast, the natural integer
program has an unbounded integrality gap.
  -- An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn
showing that the integrality gap of the above relaxation is at most 2. The
approximation factor shown by [7] was $2e^{1/e} \approx 2.89$.
  -- A lower bound of $e^{1/e}\approx 1.44$ on the integrality gap of this
relaxation.
  -- New convex programs for natural generalizations of linear Fisher markets
and proofs that these markets admit rational equilibria.
  These results were obtained by establishing connections between previously
known disparate results, and they help uncover their mathematical
underpinnings. We show a formal connection between the convex programs of
Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent
up to a change of variables. Both programs capture equilibria of linear Fisher
markets. By adding suitable constraints to Shmyrev's program, we obtain a
convex program that captures equilibria of the spending-restricted market model
defined by [7] in the context of the NSW maximization problem. Further, adding
certain integral constraints to this program we get the integer program for the
NSW mentioned above.
  The basic tool we use is convex programming duality. In the special case of
convex programs with linear constraints (but convex objectives), we show a
particularly simple way of obtaining dual programs, putting it almost at par
with linear program duality. This simple way of finding duals has been used
subsequently for many other applications.
