Nonatomic Routing Games Research Articles

A note on social learning in non-atomic routing games

This work expands on the existing body of research about Social Learning in Dynamic Non-atomic Routing Games to include infinite capacity multi-commodity instances and capacitated instances with one or more commodities. We define when a network state is identifiable, and show the network conditions sufficient to achieve almost sure learning. In capacitated instances, a sufficient condition to achieve learning is that the network respects a condition called weak capacity conservation, that is independent on the network topology. In infinite capacity instances, a sufficient condition is that the network is series-parallel in each commodity sub-network.

Information Design in Nonatomic Routing Games With Partial Participation: Computation and Properties

We consider a routing game among nonatomic agents where link latency functions are conditional on an uncertain state of the network. The agents have the same prior belief about the state, but only a fixed fraction receive private route recommendations or a common message, which are generated by a known randomization, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">public signaling policy</i> , respectively. The remaining agents choose route according to Bayes Nash flow with respect to the prior. In this article, we develop a computational approach to solve the optimal information design problem, i.e., to minimize expected social latency over all public or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">obedient</i> private signaling policies. For a fixed flow induced by nonparticipating agents, the design of an optimal private signaling policy has been shown to be a generalized problem of moments for polynomial link latency functions, and to admit an atomic solution with a provable upper bound on the number of atoms. This implies that, for polynomial link latency functions, information design can be equivalently cast as a polynomial optimization problem. This, in turn, can be arbitrarily lower bounded by a known hierarchy of semidefinite relaxations. The first level of this hierarchy is shown to be exact for the basic two link case with affine latency functions. We also identify a class of private signaling policies over which the optimal social cost is nonincreasing with an increasing fraction of participating agents for parallel networks. This is in contrast to existing results where the cost of participating agents under a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fixed</i> signaling policy may increase with their increasing fraction.

Social learning in nonatomic routing games

We consider a discrete-time nonatomic routing game with variable demand and uncertain costs. Given a routing network with single origin and destination, the cost function of each edge depends on some uncertain persistent state parameter. At every period, a random traffic demand is routed through the network according to a Wardrop equilibrium. The realized costs are publicly observed and the public Bayesian belief about the state parameter is updated. We say that there is strong learning when beliefs converge to the truth and weak learning when the equilibrium flow converges to the complete-information flow. We characterize the networks for which learning occurs. We prove that these networks have a series-parallel structure and provide a counterexample to show that learning may fail in non-series-parallel networks.

Toll Design for Routing Games With Stochastic Demands

We consider the toll design problem for non-atomic routing games with stochastic demands. Different from standard toll design problems where the demand for each origin-destination pair is a fixed number, the stochastic demand for each origin-destination pair is a random variable following a specified and known distribution. Our goal is to obtain a toll design for the traffic network to minimize a global cost function, that can, e.g., be the price of anarchy. We formulate the problem into a mathematical program with equilibrium constraints. We then develop a two-loop stochastic algorithm to solve this optimization. Our algorithm is guaranteed to converge to a solution under some assumptions. A numerical example is presented to illustrate the efficacy of our method.

The price of anarchy in routing games as a function of the demand

The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.

On weak Pareto optimality of nonatomic routing networks

This paper establishes several sufficient conditions for the weak Pareto optimality of Nash equilibria in nonatomic routing games on single- and multi-commodity networks, where a Nash equilibrium (NE) is weakly Pareto optimal (WPO) if under it no deviation of all players could make everybody better off. The results provide theoretical and technical bases for characterizing graphical structures for nonatomic routing games to admit WPO NEs. We prove that the NE of every nonatomic routing game on a single or multi-commodity network is WPO (regardless of the choices of nonnegative, continuous, nondecreasing latency functions on network links) if the network does not have two links x, y and three paths each of which goes from some origin to some destination such that the intersections of the three paths with $$\{x,y\}$$ are $$\{x\},\{y\}$$ and $$\{x,y\}$$, respectively. This sufficient condition leads to an alternative proof of the recent result that the NE of every 2-commodity nonatomic routing game on any undirected cycle is WPO. We verify a general property satisfied by all feasible 2-commodity routings (not necessarily controlled by selfish players) on undirected cycles, which roughly says that no feasible routing can “dominate” another in some sense. The property is crucial for proving the weak Pareto optimality associated to the building blocks of undirected graphs on which all NEs w.r.t. two commodities are WPO.

How Will the Presence of Autonomous Vehicles Affect the Equilibrium State of Traffic Networks?

It is known that connected and autonomous vehicles are capable of maintaining shorter headway and distances when they form platoons of vehicles. Thus, such technologies can potentially increase the road capacities of traffic networks. Consequently, it is envisioned that their deployment will also increase the overall network mobility. In this paper, we examine the validity of this expected impact, assuming that drivers select their routes selfishly, in traffic networks with mixed vehicle autonomy, that is, traffic networks with both regular and autonomous vehicles. We consider a nonatomic routing game on a network with inelastic (fixed) demands for a set of network origin destination (O/D) pairs, and study how replacing a fraction of regular vehicles by autonomous vehicles will affect the mobility of the network. Using well-known U.S. Bureau of Public Roads traffic delay models, we show that the resulting Wardrop equilibrium is not necessarily unique for networks with mixed autonomy. Then, we state the conditions under which the total network delay at equilibrium is guaranteed to not increase as the fraction of autonomous vehicles increases. However, we show that when these conditions do not hold, counterintuitive behaviors may occur-the total network delay can grow as the fraction of autonomous cars increases. In particular, we prove that for networks with a single O/D pair, if the road degrees of capacity asymmetry (i.e., the ratio between the road capacity when all vehicles are regular and the road capacity when all vehicles are autonomous) are homogeneous, the total delay is: 1) unique and 2) a nonincreasing continuous function of the fraction of autonomous vehicles in the network. We show that for heterogeneous degrees of capacity asymmetry, the total delay is not unique, and it can further grow as the fraction of autonomous vehicles increases. We demonstrate that similar behaviors may be observed in networks with multiple O/D pairs. We further bound such performance degradations due to the introduction of autonomy in general homogeneous networks.

Regrets in Routing Networks

The impact of the recent increase in routing apps usage on road traffic remains uncertain to this day. The article introduces, for the first time, a criterion to evaluate a distance between an observed state of traffic and the user equilibrium of the traffic assignment: the average marginal regret . The average marginal regret provides a quantitative measure of the impact of routing apps on traffic using only link flows, link travel times, and travel demand. In non-atomic routing games (or static traffic assignment models), the average marginal regret is a measure of selfish drivers’ behaviors. Unlike the price of anarchy , the average marginal regret in the routing game can be computed in polynomial time without any knowledge of user equilibria and socially optimal states of traffic. First, this article demonstrates on a small example that the average marginal regret is more appropriate to define proximity between an observed state of traffic and an user equilibrium state of traffic than comparing flows, travel times, or total cost. Then, experiments on two different models of app usage and three networks (including the whole L.A. network with more than 50,000 nodes) demonstrate that the average marginal regret decreases with an increase of app usage. Sensitivity analysis of the equilibrium state with respect to the app usage ratio proves that the average marginal regret monotonically decreases to 0 with an increase of app usage. Finally, using a toy example, the article concludes that app usage could become the new Braess paradox.

Algorithm for Searching an Equilibrium in a Routing Game with Piecewise Constant Cost Functions

We consider nonatomic routing games which are used to model transportation systems with a large number of agents and suggest an algorithm to search for the user equilibrium in such games, which is a generalization of the notion of Nash equilibrium. In general, finding a user equilibrium in routing games is computationally a hard problem. We consider the following subclass of routing games: games with piecewise constant cost functions, and construct an algorithm finding equilibrium in such games. This algorithm is based on the potential function method and the method of piecewise linear (PWL) costs enumeration which finds min-cost flow in a network with PWL cost functions. If each cost function increases, then the complexity of our algorithm is polynomial in the parameters of the network. But if some cost functions have decreasing points, then the complexity is exponential by the number of such points.

Quantum routing games

We discuss the connection between a class of distributed quantum games, with remotely located players, to the counter intuitive Braess’ paradox of traffic flow that is an important design consideration in generic networks where the addition of a zero cost edge decreases the efficiency of the network. A quantization scheme applicable to non-atomic routing games is applied to the canonical example of the network used in Braess’ paradox. The quantum players are modeled by simulating repeated game play. The players are allowed to sample their local payoff function and update their strategies based on a selfish routing condition in order to minimize their own cost, leading to the Wardrop equilibrium flow. The equilibrium flow in the classical network has a higher cost than the optimal flow. If the players have access to quantum resources, we find that the cost at equilibrium can be reduced to the optimal cost, resolving the paradox.

Achieving target equilibria in network routing games without knowing the latency functions

The analysis of network routing games typically assumes precise, detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desired target flow as an equilibrium. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. We give a crisp positive answer to this question. We show that one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes tolls as input and outputs the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and extends to various other settings. We obtain improved query-complexity bounds for series-parallel networks, and single-commodity routing games with linear latency functions. Our techniques provide new insights into network routing games.

Price of Anarchy for Highly Congested Routing Games in Parallel Networks

We consider nonatomic routing games with one source and one destination connected by multiple parallel edges. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we prove that under suitable conditions on the costs the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case, and that these counterexamples already occur in simple networks with only 2 parallel links.

Equilibrium routing under uncertainty

We provide a brief introduction to the basic models used to describe traffic on congested networks, both in urban transport and telecommunications. We discuss traffic equilibrium models, covering atomic and non-atomic routing games, with emphasis on situations where the travel times are subject to random fluctuations. We use convex optimization to present the models in a unified framework that stresses the common underlying structures. As a prototypical example of traffic equilibrium with elastic demands, we discuss some models for routing and congestion control in telecommunications. We also describe a class of stochastic dynamics that model the adaptive behavior of agents and which provides a plausible micro-foundation for the equilibrium. Finally we present some recent ideas on how risk-averse behavior might be incorporated in the equilibrium models.

Topological Uniqueness of the Nash Equilibrium for Selfish Routing with Atomic Users

We consider the problem of selfish routing in a congested network shared by several users, where each user wishes to minimize the cost of its own flow. Users are atomic, in the sense that each has a nonnegligible amount of flow demand, and flows may be split over different routes. The total cost for each user is the sum of its link costs, which, in turn, may depend on the user’s own flow as well as the total flow on that link. Our main interest here is network topologies that ensure uniqueness of the Nash equilibrium for any set of users and link cost functions that satisfy some mild convexity conditions. We characterize the class of two-terminal network topologies for which this uniqueness property holds, and show that it coincides with the class of nearly parallel networks that was recently shown by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30 225–244] to ensure uniqueness in nonatomic (or Wardrop) routing games. We further show that uniqueness of the link flows holds under somewhat weaker convexity conditions, which apply to the mixed Nash-Wardrop equilibrium problem. We finally propose a generalized continuum-game formulation of the routing problem that allows for a unified treatment of atomic and nonatomic users.