# Variable Demand and Multi-commodity Flow in Markovian Network Equilibrium

## TL;DR

This paper extends Markovian network equilibrium to include variable demand with quitting options and multi-commodity flows with heterogeneous ending times, providing new algorithms and computational analysis.

## Contribution

It introduces two novel extensions to Markovian network equilibrium and develops dynamic programming algorithms with complexity analysis.

## Key findings

- Algorithms outperform Mosek in computational efficiency.
- Extensions handle variable demand and multi-commodity flows.
- Numerical experiments validate the proposed models.

## Abstract

Markovian network equilibrium generalizes the classical Wardrop equilibrium in network games. At a Markovian network equilibrium, each player of the game solves a Markov decision process instead of a shortest path problem. We propose two novel extensions of Markovian network equilibrium by considering 1) variable demand, which offers the players a quitting option, and 2) multi-commodity flow, which allows players to have heterogeneous ending time. We further develop dynamic-programming-based iterative algorithms for the proposed equilibrium problems, together with their arithmetic complexity analysis. Finally, we illustrate our network equilibrium model via a multi-commodity ride-sharing example, and compare the computational efficiency of our algorithms against state-of-the-art optimization software Mosek over extensive numerical experiments.