A Game Theoretic Approach to a Problem in Polymatroid Maximization

TL;DR

This paper reformulates a polymatroid maximization problem as a zero-sum game, providing new insights, strategies, and unifying various related problems in search, testing, and queuing.

Contribution

It introduces an alternative game-theoretic formulation of the polymatroid maximization problem and derives new, simplified strategies and characterizations for optimal solutions.

Findings

Unified several problems in search, testing, and queuing through the game formulation.

Derived a short, new proof of optimal strategies for both players.

Characterized the set of optimal strategies for the minimizing player.

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.