  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.
