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19
Market Equilibrium via a PrimalDual Algorithm for a Convex Program
"... We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primaldual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by ..."
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Cited by 127 (27 self)
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We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primaldual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it.
The Notion of a Rational Convex Program, and an Algorithm for the ArrowDebreu Nash Bargaining Game
, 2012
"... 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 us ..."
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Cited by 11 (4 self)
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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 LPoracle. Next, from the linear case of the ArrowDebreu market model, we define a new Nash bargaining game, which we call ADNB. 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. Finally, we present a number of interesting questions that the new notion of RCP raises.
A perfect price discrimination market model with production, and a rational convex program for it
 Mathematics of Operations Research
"... Recent results showing PPADcompleteness of the problem of computing an equilibrium for Fisher’s market model under additively separable, piecewiselinear, concave utilities (plc utilities) have dealt a serious blow to the program of obtaining efficient algorithms for computing equilibria in “tradit ..."
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Cited by 10 (5 self)
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Recent results showing PPADcompleteness of the problem of computing an equilibrium for Fisher’s market model under additively separable, piecewiselinear, concave utilities (plc utilities) have dealt a serious blow to the program of obtaining efficient algorithms for computing equilibria in “traditional ” market models and has prompted a search for alternative models that are realistic as well as amenable to efficient computation. In this paper we show that introducing perfect price discrimination into the Fisher model with plc utilities renders its equilibrium polynomial time computable. Moreover, its set of equilibria are captured by a convex program that generalizes the classical EisenbergGale program, and always admits a rational solution. We also give a combinatorial, polynomial time algorithm for computing an equilibrium. Next, we introduce production into our model, and again give a (rational) convex program that captures its equilibria. We use this program to obtain surprisingly simple proofs of both welfare theorems or this model. Finally, we also give an application of our price discrimination market model to online display advertising marketplaces. 1
Strongly polynomial algorithm for a class of minimumcost flow problems with separable convex objectives.
 In Proceedings of the 44th symposium on Theory of Computing, STOC ’12,
, 2012
"... ABSTRACT A wellstudied nonlinear extension of the minimumcost flow problem is to minimize the objective ij∈E Cij(fij) over feasible flows f , where on every arc ij of the network, Cij is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad c ..."
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Cited by 7 (2 self)
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ABSTRACT A wellstudied nonlinear extension of the minimumcost flow problem is to minimize the objective ij∈E Cij(fij) over feasible flows f , where on every arc ij of the network, Cij is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad class of such problems. The key characteristic of this class is that an optimal solution can be computed exactly provided its support. This includes separable convex quadratic objectives and also certain market equilibria problems: Fisher's market with linear and with spending constraint utilities. We thereby give the first strongly polynomial algorithms for separable quadratic minimumcost flows and for Fisher's market with spending constraint utilities, settling open questions posed e.g. in
Nash bargaining via flexible budget markets
, 2009
"... We initiate a study of Nash bargaining games [Nas50] via combinatorial, polynomial time algorithms and we carry this program over to solving nonsymmetric bargaining games of Kalai [Kal77] as well. Since the solution to a Nash bargaining game is also the optimal solution to the corresponding convex p ..."
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Cited by 6 (4 self)
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We initiate a study of Nash bargaining games [Nas50] via combinatorial, polynomial time algorithms and we carry this program over to solving nonsymmetric bargaining games of Kalai [Kal77] as well. Since the solution to a Nash bargaining game is also the optimal solution to the corresponding convex program, this involves obtaining a combinatorial algorithm for the latter. This in turn can be viewed as the problem of finding an equilibrium for a market in a new model, called a flexible budget market. Our main result pertains to a natural Nash bargaining game derived from the linear case of the ArrowDebreu market model. The structural insights gained from the combinatorial nature of our algorithms has led to novel insights into gametheoretic properties of the solution concepts of Nash and nonsymmetric bargaining games, see [CGV + 09].
The complexity of nonmonotone markets
 In Proceedings of ACM STOC
, 2013
"... We introduce the notion of nonmonotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPADhard to compute an approximate ArrowDebreu market equilibrium in markets with linear and nonmonotone utilities. Building on this result, we settle the ..."
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Cited by 5 (0 self)
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We introduce the notion of nonmonotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPADhard to compute an approximate ArrowDebreu market equilibrium in markets with linear and nonmonotone utilities. Building on this result, we settle the longstanding open problem regarding the computation of an approximate ArrowDebreu market equilibrium in markets with CES utilities, by proving that it is PPADcomplete when the Constant Elasticity of Substitution parameter, ρ, is any constant less than −1. Categories and Subject Descriptors F.2 [Analysis of algorithms and problem complexity]
Continuity properties of equilibria in some Fisher and ArrowDebreu market models
 In Proceedings of The 5th Workshop on Internet and Network Economics
, 2009
"... Abstract. Following up on the work of Megiddo and Vazirani [10], who determined continuity properties of equilibrium prices and allocations for perhaps the simplest market model, Fisher’s linear case, we do the same for: – Fisher’s model with piecewiselinear, concave utilities – Fisher’s model with ..."
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Cited by 3 (3 self)
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Abstract. Following up on the work of Megiddo and Vazirani [10], who determined continuity properties of equilibrium prices and allocations for perhaps the simplest market model, Fisher’s linear case, we do the same for: – Fisher’s model with piecewiselinear, concave utilities – Fisher’s model with spending constraint utilities – ArrowDebreu’s model with linear utilities – EisenbergGale markets. 1
Discrete price updates yield fast convergence in ongoing markets with finite warehouses
, 2010
"... This paper shows that in suitable markets, even with outofequilibrium trade allowed, a simple price update rule leads to rapid convergence toward the equilibrium. In particular, this paper considers a Fisher market repeated over an unbounded number of time steps, with the addition of finite sized ..."
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This paper shows that in suitable markets, even with outofequilibrium trade allowed, a simple price update rule leads to rapid convergence toward the equilibrium. In particular, this paper considers a Fisher market repeated over an unbounded number of time steps, with the addition of finite sized warehouses to enable nonequilibrium trade. The main result is that suitable tatonnement style price updates lead to convergence in a significant subset of markets satisfying the Weak Gross Substitutes property. Throughout this process the warehouse are always able to store or meet demand imbalances (the needed capacity depends on the initial imbalances). Our price update rule is robust in a variety of regards: • The updates for each good depend only on information about that good (its current price, its excess demand since its last update) and occur asynchronously from updates to other prices. • The process is resilient to error in the excess demand data. • Likewise, the process is resilient to discreteness, i.e. a limit to divisibility, both of goods and money.