A Necessary and Sufficient Condition for the Existence of Potential Functions for Heterogeneous Routing Games

TL;DR

This paper establishes necessary and sufficient conditions for the existence of potential functions in heterogeneous routing games, extending theoretical understanding and providing toll strategies to ensure equilibrium, with applications to vehicle platooning.

Contribution

It relaxes previous assumptions for Nash equilibrium existence and characterizes tolls that guarantee potential functions in two-type heterogeneous routing games.

Findings

Derived necessary and sufficient conditions for potential function existence.

Presented toll strategies to ensure potential functions with two user types.

Provided an upper bound for the price of anarchy in two-type affine cost games.

Abstract

We study a heterogeneous routing game in which vehicles might belong to more than one type. The type determines the cost of traveling along an edge as a function of the flow of various types of vehicles over that edge. We relax the assumptions needed for the existence of a Nash equilibrium in this heterogeneous routing game. We extend the available results to present necessary and sufficient conditions for the existence of a potential function. We characterize a set of tolls that guarantee the existence of a potential function when only two types of users are participating in the game. We present an upper bound for the price of anarchy (i.e., the worst-case ratio of the social cost calculated for a Nash equilibrium over the social cost for a socially optimal flow) for the case in which only two types of players are participating in a game with affine edge cost functions. A heterogeneous routing game with vehicle platooning incentives is used as an example throughout the article to clarify the concepts and to validate the results.