Sadra Yazdanbod

We introduce a new class of combinatorial markets in which agents have covering constraints over resources required and are interested in delay minimization. Our market model is applicable to several settings including scheduling, cloud computing, and communicating over a network. This model is quite different from the traditional models, to the extent that neither do the classical equilibrium existence results seem to apply to it nor do any of the efficient algorithmic techniques developed to compute equilibria seem to apply directly. We give a proof of existence of equilibrium and a polynomial time algorithm for finding one, drawing heavily on techniques from LP duality and submodular minimization. We observe that in our market model, the set of equilibrium prices could be a connected, non-convex set. To the best of our knowledge, this is the first natural example of the phenomenon where the set of solutions could have such complicated structure, yet there is a combinatorial polynomial time algorithm to find one. Finally, we show that our model inherits many of the fairness properties of traditional equilibrium models.

We present the following results pertaining to Fisher's market model: -We give two natural generalizations of Fisher's market model: In model M_1, sellers can declare an upper bound on the money they wish to earn (and take back their unsold good), and in model M_2, 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.

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and the problem of counting the number of non-symmetric NE in a symmetric game is #P-complete. In 2005, Kannan and Theobald defined the "rank of a bimatrix game" represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be computed in rank 1 games in polynomial time. Observe that the rank 0 case is precisely the zero sum case, for which a polynomial time algorithm follows from von Neumann's reduction of such games to linear programming. In 2011, Adsul et. al. obtained an algorithm for rank 1 games; however, it does not solve the case of symmetric rank 1 games. We resolve this problem.