Wardrop Equilibrium Research Articles

From discrete to continuous Wardrop equilibria

The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50's, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of $\Gamma$-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model we obtain this way is very similar to the continuous model of optimal transport with congestion of Carlier, Jimenez and Santambrogio [6] except that it keeps track of the anisotropy of the network.

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Competition yields efficiency in load balancing games

We study a nonatomic congestion game with N parallel links, with each link under the control of a profit maximizing provider. Within this ‘load balancing game’, each provider has the freedom to set a price, or toll, for access to the link and seeks to maximize its own profit. Given prices, a Wardrop equilibrium among users is assumed, under which users all choose paths of minimal and identical effective cost. Within this model we have oligopolistic price competition which, in equilibrium, gives rise to situations where neither providers nor users have incentives to adjust their prices or routes, respectively. In this context, we provide new results about the existence and efficiency of oligopolistic equilibria. Our main theorem shows that, when the number of providers is small, oligopolistic equilibria can be extremely inefficient; however as the number of providers N grows, the oligopolistic equilibria become increasingly efficient (at a rate of 1 / N ) and, as N → ∞ , the oligopolistic equilibrium matches the socially optimal allocation.

Multi-class percentile user equilibrium with flow-dependent stochasticity

Travelers often reserve a buffer time for trips sensitive to arrival time in order to hedge against the uncertainties in a transportation system. To model the effects of such behavior, travelers are assumed to choose routes to minimize the percentile travel time, i.e. the travel time budget that ensures their preferred probability of on-time arrival; in doing so, they drive the system to a percentile user equilibrium (UE), which can be viewed as an extension of the classic Wardrop equilibrium. The stochasticity in the supply of transportation are incorporated by modeling the service flow rate of each road segment as a random variable. Such stochasticity is flow-dependent in the sense that the probability density functions of these random variables, from which the distribution of link travel time are constructed, are specified endogenously with flow-dependent parameters. The percentile route travel time, obtained by directly convolving the link travel time distributions in this paper, is not available in closed form in general and has to be numerically evaluated. To reveal their structural properties, percentile UE solutions are examined in special cases and verified with numerical results. For the general multi-class percentile UE traffic assignment problem, a variational inequality formulation is given and solved using a route-based algorithm. The algorithm makes use of the diagonal elements in the Jacobian of percentile route travel time, which is approximated through recursive convolution. Preliminary numerical experiments indicate that the algorithm is able to achieve highly precise equilibrium solutions.

Equilibrium estimation based on unreliable information in transport networks by adaptive simulation

The goal of this paper is to find Wardrop equilibrium in transport networks at case of uncertainty situations, where the uncertainty comes from lack of information. The uncertainty has been handled by Dempster-Shafer theory, an interval-based solution has been developed for handling the paths with cost. Adaptive simulation tool has been constructed in order to find the equilibrium, which gives only estimated solution, but this is sufficient for large networks as well. A new developed adaptive run length control algorithm helps for the reliability in the simulation run.

Traffic network equilibrium with capacity constraints and generalized Wardrop equilibrium

The traffic network equilibrium problem with capacity constraints is investigated with the help of generalized Lagrangian theory of variational inequality in our paper. It is proved to be equivalent to the generalized Wardrop equilibrium solution. This result is very useful in transportation analysis since the study in Wardrop equilibrium problem is relatively mature. Furthermore, some new concepts are offered.

Wardrop Equilibria with Risk-Averse Users

Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most applications they also have an uncertain component. We extend Wardrop’s network game (1952) by adding random deviations, which are independent of the flow, to the cost functions that model congestion in each arc. We map these uncertainties into a Wardrop equilibrium model with nonadditive path costs. The cost on a path is given by the sum of the congestion on its arcs plus a constant safety margin determined by risk-averse agents. First, we prove that an equilibrium for this game always exists and is essentially unique. Then, we introduce three specific equilibrium models that fall within this framework: the percentile equilibrium where agents select paths that minimize a specified percentile of the uncertain cost; the added-variability equilibrium where agents add a multiple of the variability of the cost of each arc to the expected cost; and the robust equilibrium where agents select paths by solving a robust optimization problem that imposes a limit on the number of arcs that can deviate from the mean. The percentile equilibrium is difficult to compute because minimizing a percentile among all paths is computationally hard. Instead, the added-variability and robust Wardrop equilibria can be computed efficiently in practice: the former reduces to a standard Wardrop equilibrium problem and the latter is found using a column generation approach that repeatedly solves robust shortest path problems, which are polynomially solvable. Through computational experiments of some random and some realistic instances, we explore the benefits and trade-offs of the proposed solution concepts. We show that when agents are risk averse, both the robust and added-variability equilibria better approximate percentile equilibria than the classic Wardrop equilibrium.

Fast Convergence to Wardrop Equilibria by Adaptive Sampling Methods

We study the question of whether a large population of agents in a traffic network is able to converge to an equilibrium quickly. To that end, we consider a round-based variant of the Wardrop model. Every agent is allowed to reroute its traffic once in a while with the aim of finding a path with minimal latency. As a first result we find that using a replication policy which allows agents to imitate others gives rise to a bicriterial approximate equilibrium very quickly. In particular, the time bound depends logarithmically on the ratio between minimum and maximum latency but is otherwise independent of the network size. In the single-commodity case, this bicriteria approximate equilibrium has an intuitive interpretation as a state in which almost all agents are almost happy. This kind of approximate equilibrium, however, is transient. In order to reach a global approximation, we need to add an exploration component which enables the agents to explore the strategy space independently of the other agents. Although it can be shown that, when used exclusively, exploration policies imply an exponential lower bound, applying exploration carefully allows the population to approximate the global Wardrop equilibrium in polynomial time. Since the distributed and concurrent fashion of our policies bears the risk of oscillating behavior, we must take into account the steepness of the latency functions. We show that the relevant parameter is elasticity, a parameter closely related to the polynomial degree. This improves significantly over earlier results which depend on the absolute slope and therefore have a pseudopolynomial flavor.

The Impact of Oligopolistic Competition in Networks

In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria.

Continuum equilibria and global optimization for routing in dense static ad hoc networks

We consider massively dense ad hoc networks and study their continuum limits as the node density increases and as the graph providing the available routes becomes a continuous area with location and congestion dependent costs. We study both the global optimal solution as well as the non-cooperative routing problem among a large population of users where each user seeks a path from its origin to its destination so as to minimize its individual cost. Finally, we seek for a (continuum version of the) Wardrop equilibrium. We first show how to derive meaningful cost models as a function of the scaling properties of the capacity of the network and of the density of nodes. We present various solution methodologies for the problem: (1) the viscosity solution of the Hamilton-Jacobi-Bellman equation, for the global optimization problem, (2) a method based on Green's Theorem for the least cost problem of an individual, and (3) a solution of the Wardrop equilibrium problem using a transformation into an equivalent global optimization problem.

Eliciting Coordination with Rebates

This article considers a mechanism based on rebates that aims at reducing congestion in urban networks. The framework helps select rebate levels so that enough commuters switch to modes that are under used. Indeed, getting a relative small number of drivers to switch to public transportation can significantly improve congestion. This mechanism is modeled by a Stackelberg game in which the transportation authority offers rebates, and participants factor them into the costs of each mode. A new Wardrop equilibrium arises when participants selfishly select one of the modes of transportation with the lowest updated costs. Rebate levels are chosen taking into account not only the potential reduction of the participants' cost, but also the cost of providing those rebates. Part of the budget for rebates may come from the savings that arise from the more efficient use of capacity. We characterize the Stackelberg equilibria of the game, and describe a polynomial-time algorithm to compute the optimal rebates for each mode. In addition, we provide tight results on the worst-case inefficiency of the resulting Wardrop equilibrium, measured by the so-called price of anarchy. Specifically, we describe the tradeoff between the sensitivity of the owner towards rebate costs and the worst-case inefficiency of the system.

Realizing Wardrop equilibria with real-time traffic information

A Wardrop equilibrium for multiple routes from the same origin to the same destination requires equal travel time on each path used. With the advent of real-time traffic data regarding travel times on alternative routes, it becomes important to analyze how best to use the information provided to drivers. In particular, can a Wardrop equilibrium, which is a desired state, be realized? Simulations using a realistic traffic model (the three-phase model) on a two-route example are presented to answer this question. One route (the main line) is a two-lane highway with a stalled vehicle in the right lane and the other route is a low-speed bypass. For a critical incoming flow of vehicles, a phase transition between free flow and congested flow near the stalled vehicle is observed, making this a challenging example. In the first scenario, drivers choose routes selfishly on the basis of current travel times. The result is strong oscillations in travel time because of the inherent delay in the information provided. The second scenario involves a hypothetical control system that limits the number of vehicles on the main line to prevent the free-flow to congested-flow phase transition by diverting sufficient flow to the bypass. The resulting steady state is neither a Wardrop equilibrium nor a system optimum, but an intermediate state in which the main-line travel time is less than on the bypass but the average for all vehicles is close to a minimum. In a third scenario, anticipation is used as a driver-advice system to provide a fair indicator of which route to take. Prediction is based on real-time data comparing the number of vehicles on the main line at the time a vehicle leaves the origin to the actual travel time when it reaches the destination. Steady states that approximate Wardrop equilibria, or at least as close to them as can be expected, are obtained. This approach is also applied to an example with a low-speed boundary condition imposed at the destination in place of a stalled vehicle. The steady state flow approaches a Wardrop equilibrium because there is no abrupt change in travel time due to a phase transition.

Equilibrium analysis of macroscopic traffic oscillations

Using a simple network model with two parallel links connecting a diverge and a merge, this paper studies under what conditions traffic oscillations may be initiated and propagated in a traffic stream, specially at freeway bottlenecks. Drivers are assumed to minimize either the experienced or instantaneous travel times, and in doing so, they settle at a Wardrop (day-to-day) equilibrium or a Boston (within-day) traffic equilibrium, respectively. We prove that the path travel time function in our model is not monotone, and show that this property leads to multiple Wardrop equilibria, of which only one is both stable and efficient. The paper shows that periodic traffic oscillations do not arise from Wardrop equilibria. Trivial oscillations exist at Boston equilibria, which are caused by drivers’ overreaction to traffic conditions. However, periodic oscillations are likely to emerge when (1) transitions between stable and unstable equilibria take place, and more importantly, (2) drivers make decisions based on out-of-date information of traffic conditions. The latter finding is useful in guiding control practice at freeway bottlenecks and work zones to prevent traffic oscillations.

The Impact of Oligopolistic Competition in Networks

In the traffic assignment problem, commuters select the shortest available path to travel from a given origin to a given destination. This system has been studied for over 50 years since Wardrop's seminal work (1952). Motivated by freight companies, which need to ship goods across the network, we study a generalization of the traffic assignment problem in which some competitors control a non-negligible fraction of the total flow. This type of games, usually referred to as atomic games, readily applies to situations in which some competitors have market power. Other applications include telecommunication network service providers, intelligent transportation systems, and scheduling with flexible machines.Our goal is to determine whether these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game when there is no centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency, which depend on the characteristics of cost functions and the market structure in the game. In addition, building upon the work of Catoni and Pallotino (1991), we show examples in which market aggregation (or collusion) can adversely impact the aggregated competitors, even though their market power increases. For example, all Nash equilibria of an atomic network game may be less efficient than the corresponding Wardrop equilibrium. When the market structure is simple enough, we give an optimization formulation of the Nash equilibrium and prove that this counter-intuitive phenomenon does not arise. Finally, we offer a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria.

Wardrop Routing in Wireless Networks

Routing protocols for multihop wireless networks have traditionally used shortest path routing to obtain paths to destinations and do not consider traffic load or delay as an explicit factor in the choice of routes. We focus on static mesh networks and formally establish that if the number of sources is not too large, then it is possible to construct a perfect flow-avoiding routing, which can boost the throughput provided to each user over that of the shortest path routing by a factor of four when carrier sensing can be disabled or a factor of 3.2 otherwise. So motivated, we address the issue of designing a multipath, load adaptive routing protocol that is generally applicable even when there are more sources. We develop a protocol that adaptively equalizes the mean delay along all utilized routes from a source to destination and does not utilize any routes that have greater mean delay. This is the property satisfied by a system in Wardrop equilibrium. We also address the architectural challenges confronted in the software implementation of a multipath, delay-feedback-based, probabilistic routing algorithm. Our routing protocol is 1) completely distributed, 2) automatically load balances flows, 3) uses multiple paths whenever beneficial, 4) guarantees loop-free paths at every time instant even while the algorithm is suntil converging, and 5) amenable to clean implementation. An ns-2 simulation study indicates that the protocol is able to automatically route flows to avoid each other, consistently out-performing shortest path protocols in a variety of scenarios. The protocol has been implemented in user space with a small amount of forwarding mechanism in a modified Linux 2.4.20 kernel. Finally, we discuss a proof-of-concept measurement study of the implementation on a six node testbed.

Coincident Cost Improvement vs. Degradation by Adding Connections to Noncooperative Networks and Distributed Systems

In computer and transportation networks, we consider the situation where each user has its own routing or load balancing decision and seeks to minimize noncooperatively the expected passage time of its packet or job, given the routing or load balancing decisions of other users. Intuitively, it is anticipated that adding connections to such a noncooperative system may bring cost improvement at least to some users. The Braess paradox is, however, the first example of paradoxical cases where it is not always the case. A few studies have been published on the degrees of coincident cost degradation for all users by adding connections. In contrast, it has not been certain whether or to what degree adding connections to a network can bring coincident cost improvement for all users. We believe that this paper is the first one that studies the possible degrees of coincident cost improvement for all users by adding connections to noncooperative systems. There has been found no system in Wardrop equilibrium for which the degree of degradation can increase without bound by adding connections with the number of nodes fixed. We show that there exist systems in Wardrop equilibria and ones in Nash equilibria for both of which the degree of coincident cost improvement can increase without bound for all users by adding connections, even if the number of nodes is fixed. We then study, for models reflecting realistic distributed systems, the effects of adding connections to them on the coincident cost improvement or degradation for all users.

Sensitivity of Wardrop Equilibria

We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by e or removes an edge carrying only an e-fraction of flow. We study how the equilibrium responds to such an e-change. Our first surprising finding is that, even for linear latency functions, for every e>0, there are networks in which an e-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most e. Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an e-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1+e)p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded.

An equivalent optimization formulation for the traffic assignment problem with asymmetric linear costs

In this paper, we present a general formulation for the deterministic traffic assignment problem, using an equivalent optimization problem applicable to the case of asymmetric linear cost functions. We present a resolution approach for this problem in such a way that in equilibrium Wardrop's first principle or Nash equilibrium is satisfied. We conclude that many deterministic traffic assignment problems with asymmetric linear costs can be formulated as an optimization problem whose objective is defined by a line integral, and whose constraints correspond to non-negativity and flows conservation. By adequately defining the integration path, it is feasible to resolve the problem, obtaining Wardrop's equilibrium. This approach can be applied in other economic contexts, including microeconomic theory and consumer surplus analysis.

Cooperative dynamics and Wardrop equilibria

A simple dynamics capturing user behavior in a network is shown to lead to equilibria that can be viewed as approximate, generalized Wardrop equilibria.

Mathematical programs with complementarity constraints in traffic and telecommunications networks

Given a suitably parametrized family of equilibrium models and a higher level criterion by which to measure an equilibrium state, mathematical programs with equilibrium constraints (MPECs) provide a framework for improving or optimizing the equilibrium state. An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose. Here, a Wardrop equilibrium describes the traffic response to each toll design. Communication networks also have a deep literature on equilibrium flows that suggest some MPECs. We focus on mathematical programs with complementarity constraints (MPCCs), a subclass of MPECs for which the lower level equilibrium system can be formulated as a complementarity problem and therefore, importantly, as a nonlinear program (NLP). Although MPECs and MPCCs are typically non-convex, which is a consequence of the upper level objective clashing with the users' objectives in the lower level equilibrium program, the last decade of research has paved the way for finding local solutions of MPCCs via standard NLP techniques.