Atomic Splittable Flow Over Time Games

TL;DR

This paper studies atomic splittable flow over time games where players route flow through networks with capacity and transit time constraints, analyzing equilibrium existence and efficiency under different information scenarios.

Contribution

It introduces a formal model for atomic splittable flow over time games and characterizes equilibrium existence and uniqueness depending on information availability.

Findings

No Nash equilibrium exists without network state information.

Dynamic strategies based on congestion data lead to a unique feasible flow.

In parallel networks, Nash equilibria achieve optimal flow with a price of anarchy of 1.

Abstract

In an atomic splittable flow over time game, finitely many players route flow dynamically through a network, in which edges are equipped with transit times, specifying the traversing time, and with capacities, restricting flow rates. Infinitesimally small flow particles controlled by the same player arrive at a constant rate at the player's origin and the player's goal is to maximize the flow volume that arrives at the player's destination within a given time horizon. Hereby, the flow dynamics described by the deterministic queuing model, i.e., flow of different players merges perfectly, but excessive flow has to wait in a queue in front of the bottle-neck. In order to determine Nash equilibria in such games, the main challenge is to consider suitable definitions for the players' strategies, which depend on the level of information the players receive throughout the game. For the most restricted version, in which the players receive no information on the network state at all, we can show that there is no Nash equilibrium in general, not even for networks with only two edges. However, if the current edge congestions are provided over time, the players can adopt their route choices dynamically. We show that a profile of those strategies always lead to a unique feasible flow over time. Hence, those atomic splittable flow over time games are well-defined. For parallel-edge networks Nash equilibria exists and the total flow arriving in time equals the value of a maximum flow over time leading to a price of anarchy of 1.