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Spending constraint utilities, with applications to the Adwords market
, 2006
"... The notion of a “market ” has undergone a paradigm shift with the Internet – totally new and highly successful markets have been defined and launched by Internet companies, which already form an important part of today’s economy and are projected to grow considerably in the future. Another major cha ..."
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Cited by 20 (7 self)
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The notion of a “market ” has undergone a paradigm shift with the Internet – totally new and highly successful markets have been defined and launched by Internet companies, which already form an important part of today’s economy and are projected to grow considerably in the future. Another major change is the availability of massive computational power for running these markets in a centralized or distributed manner. In view of these new realities, the study of market equilibria, an important, though essentially nonalgorithmic, theory within mathematical economics, needs to be revived and rejuvenated via an inherently algorithmic approach. Such a theory should not only address traditional market models but also define new models for some of the new markets. We present a new, natural class of utility functions which allow buyers to explicitly provide information on their relative preferences as a function of the amount of money spent on each good. These utility functions offer considerable expressivity, especially in Google’s Adwords market. In addition, they lend themselves to efficient computation, while still possessing some of the nice properties of traditional models. The latter include weak gross substitutability, and uniqueness and continuity of equilibrium prices and utililities.
New Convex Programs and Distributed Algorithms for Fisher Markets with Linear and Spending Constraint Utilities
"... In this paper we shed new light on convex programs and distributed algorithms for Fisher markets with linear and spending constraint utilities. • We give a new convex program for the linear utilities case of Fisher markets. This program easily extends to the case of spending constraint utilities as ..."
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Cited by 7 (1 self)
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In this paper we shed new light on convex programs and distributed algorithms for Fisher markets with linear and spending constraint utilities. • We give a new convex program for the linear utilities case of Fisher markets. This program easily extends to the case of spending constraint utilities as well, thus resolving an open question raised by [Vaz10]. • We show that the gradient descent algorithm with respect to a Bregman divergence converges with rate O(1/t) under a condition that is weaker than having Lipschitz continuous gradient (which is the usual assumption in the optimization literature for obtaining the same rate). • We show that the Proportional Response dynamics recently introduced by Zhang [Zha09] is equivalent to a gradient descent algorithm for solving the new convex program. This insight also gives us better convergence rates, and helps us generalize it to spending constraint utilities. 1
2Player Nash and Nonsymmetric Bargaining Games: Algorithms and Structural Properties
"... The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2player game whose convex program has linear constraints, admits a rational solution and such a solution can be fo ..."
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Cited by 3 (3 self)
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The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are “small”, then its solution can be found in strongly polynomial time. We also give a nonsuccinct linear game whose solution can be found in strongly polynomial time. 1
Fisher Markets and Convex Programs
"... Convex programming duality is usually stated in its most general form, with convex objective functions and convex constraints. (The book by Boyd and Vandenberghe is an excellent reference [2].) At this ..."
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Convex programming duality is usually stated in its most general form, with convex objective functions and convex constraints. (The book by Boyd and Vandenberghe is an excellent reference [2].) At this
Submodularity helps in nash and nonsymmetric bargaining. Available Online: http://www.cc.gatech.edu/grads/g/ gagang/NBprop.pdf
"... Abstract Motivated by the recent work of [Vaz12], we take a fresh look at understanding the quality and robustness of solutions to Nash and nonsymmetric bargaining games by subjecting them to several stress tests. Our tests are quite basic, e.g., we ask whether the solutions are computable in polyn ..."
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Abstract Motivated by the recent work of [Vaz12], we take a fresh look at understanding the quality and robustness of solutions to Nash and nonsymmetric bargaining games by subjecting them to several stress tests. Our tests are quite basic, e.g., we ask whether the solutions are computable in polynomial time, and whether they have certain properties such as efficiency, fairness, desirable response when agents change their disagreement points or play with a subset of the agents. Our main conclusion is that imposing submodularity, a natural economies of scale condition, on Nash and nonsymmetric bargaining games endows them with several desirable properties.
Efficiency, Fairness and Competitiveness in Nash Bargaining Games
"... Abstract. Recently, [8] defined the class of Linear Nash Bargaining Games (LNB) and obtained combinatorial, polynomial time algorithms for several games in this class. [8] also defines two natural subclasses within LNB, UNB and SNB, which contain a number of natural Nash bargaining games. In this pa ..."
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Abstract. Recently, [8] defined the class of Linear Nash Bargaining Games (LNB) and obtained combinatorial, polynomial time algorithms for several games in this class. [8] also defines two natural subclasses within LNB, UNB and SNB, which contain a number of natural Nash bargaining games. In this paper we define three basic game theoretic properties of Nash bargaining games: price of bargaining, fairness and full competitiveness. We show that for each of these properties, a game in UNB has this property iff it is in SNB. 1