Linear programming techniques for algorithms with applications in economics

Abstract

We study algorithms and models for several economics-related problems from the perspective of linear programming.

In network bargaining games, stable and balanced outcomes have been investigated in previous work. However, existence of such outcomes requires that the linear program relaxation of a certain maximum matching problem has integral optimal solution. We propose an alternative model for network bargaining games in which each edge acts as a player, who proposes how to split the weight of the edge among the two incident nodes. We show that the distributed protocol by Kanoria et. al can be modified to be run by the edge players such that the configuration of proposals will converge to a pure Nash Equilibrium, without the linear program integrality gap assumption. Moreover, ambiguous choices can be resolved in a way such that there exists a Nash Equilibrium that will not hurt the social welfare too much.

In the oblivious matching problem, an algorithm aims to find a maximum matching while it can only makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve performance ratio 0:5, which is the expected number of matched nodes to the number of nodes in a maximum matching. We revisit the Ranking algorithm using the linear programming framework, where the constraints of the linear program are given by the structural properties of Ranking. We use continuous linear program relaxation to analyze the limiting behavior as the finite linear program grows. Of particular interest are new duality and complementary slackness characterizations that can handle monotone constraints and mixed evolving and boundary constraints in continuous linear program, which enable us to achieve a theoretical ratio of 0:523 on arbitrary graphs.

The J-choice K-best secretary problem, also known as the (J;K)-secretary problem, is a generalization of the classical secretary problem. An algorithm for the (J;K)-secretary problem is allowed to make J choices and the payoff to be maximized is the expected number of items chosen among the K best items. We use primal-dual continuous linear program techniques to analyze a class of infinite algorithms, which are general enough to capture the asymptotic behavior of the finite model with large number of items. Our techniques allow us to prove that the optimal solution can be achieved by a (J;K)-threshold algorithm, which has a nice \rational description" for the case K = 1.