On seeking efficient Pareto optimal points in multi-player minimum cost flow problems with application to transportation systems

TL;DR

This paper introduces a multi-player extension of the minimum cost flow problem with nonlinear costs and integer flows, proposing algorithms to find Pareto optimal points that are also Nash equilibria, relevant for transportation systems.

Contribution

It develops a novel multi-player model for minimum cost flow problems with nonlinear costs and integer flows, along with algorithms for computing Pareto optimal solutions.

Findings

Algorithms successfully compute Pareto optimal points.

Pareto optimal points correspond to Nash equilibria in the game model.

Applicable to transportation systems with complex flow and cost structures.

Abstract

In this paper, we propose a multi-player extension of the minimum cost flow problem inspired by a transportation problem that arises in modern transportation industry. We associate one player with each arc of a directed network, each trying to minimize its cost function subject to the network flow constraints. In our model, the cost function can be any general nonlinear function, and the flow through each arc is an integer. We present algorithms to compute efficient Pareto optimal point(s), where the maximum possible number of players (but not all) minimize their cost functions simultaneously. The computed Pareto optimal points are Nash equilibriums if the problem is transformed into a finite static game in normal form.