A Game Theoretic Approach to a Problem in Polymatroid Maximization
Abstract
We consider the problem of maximizing the minimum (weighted) value of all components of a vector over a polymatroid. This is a special case of the lexicographically optimal base problem introduced and solved by Fujishige. We give an alternative formulation of the problem as a zero-sum game between a maximizing player whose mixed strategy set is the base of the polymatroid and a minimizing player whose mixed strategy set is a simplex. We show that this game and three variations of it unify several problems in search, sequential testing and queuing. We give a new, short derivation of optimal strategies for both players and an expression for the value of the game. Furthermore, we give a characterization of the set of optimal strategies for the minimizing player and we consider special cases for which optimal strategies can be found particularly easily.
Cite
@article{arxiv.2104.10236,
title = {A Game Theoretic Approach to a Problem in Polymatroid Maximization},
author = {Lisa Hellerstein and Thomas Lidbetter},
journal= {arXiv preprint arXiv:2104.10236},
year = {2021}
}