The Price of Anarchy in Routing Games as a Function of the Demand

07/23/2019 ∙ by Roberto Cominetti, et al. ∙ Universidad Adolfo Ibáñez 0

Most of the literature concerning the price of anarchy has focused on the search on tight bounds for specific classes of games, such as congestion games and, in particular, routing games. Some papers have studied the price of anarchy as a function of some parameter of the model, such as the traffic demand in routing games, and have provided asymptotic results in light or heavy traffic. We study the price of anarchy in nonatomic routing games in the central region of the traffic demand. We start studying some regularity properties of Wardrop equilibria and social optima. We focus our attention on break points, that is, values of the demand where at all equilibria the set of paths used changes. Then we show that, for affine cost functions, local maxima of the price of anarchy can occur only at break points. We prove that their number is finite, but can be exponential in the size of the network.

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1. Introduction

Nonatomic congestion games provide a model for the distribution of traffic along a network where a large number of players operate and each single player controls a negligible fraction of the total traffic. The model is based on a directed graph with one origin and one destination and the cost is identified with the delay incurred to go from origin to destination. The delay on an edge is a nondecreasing function of the mass of players on that edge and the delay of a path is additive over the edges of the path. The social cost is the total delay experienced by the whole traffic on the network. The solution concept that is used in this class of models is the Wardrop equilibrium: in equilibrium flows are split among paths having minimum cost. Therefore, all the paths used in equilibrium have the same cost and the equilibrium social cost is just the equilibrium cost along each used path times the total traffic demand. Equilibria are known to be inefficient, i.e., a social planner would be able to redirect flows along the network in a way that reduces the social cost. The most common measure of inefficiency of equilibria is the price of anarchy, i.e., the ratio of the equilibrium social cost over the optimum social cost. For nonatomic congestion games with affine costs, the value of the price of anarchy is bounded above by , and this bound is sharp (Roughgarden and Tardos, 2002). Moreover, for a large class of cost functions, that include all polynomials, the price of anarchy (PoA) converges to as the traffic demand goes either to or to infinity. In other words, equilibria tend to perfect efficiency both in light and heavy traffic (Colini-Baldeschi et al., frth).

Some studies have empirically shown that in real life for medium traffic demand the price of anarchy tends to oscillate and often does not even reach its worst bound. The shape and number of these oscillations is the object of this paper, where we will provide theoretical results for the behavior of the price of anarchy focusing mainly on networks with affine costs.

1.1. Our contribution

In this paper we consider nonatomic routing games on a network with a single origin-destination and nondecreasing continuous costs. We look at the price of anarchy as a function of the traffic demand and study the properties of this function. We provide some general results and then we focus on affine cost functions. The main idea of the paper is that, in agreement with with existing empirical results, when costs functions are affine, local maxima of the PoA are achieved when the set of paths that are used at equilibrium changes. To study these changes we need some preliminary results concerning regularity properties of the equilibrium. First of all we resort to the classical result in Beckmann et al. (1956) according to which a Wardrop equilibrium is a solution of a convex optimization program and we prove that the value of this program is convex and differentiable as a function of the traffic demand. Then we show that the equilibrium cost is continuously differentiable in a neighborhood of a demand point where strict complementarity conditions hold and that similar results hold also for the optimum under weaker conditions. When the traffic demand grows, there exist points where the set of used paths in all equilibria changes. We call them break points.

We then turn our attention to a class of cost functions that are heavily used in applications, namely, the ones proposed by the Bureau of Public Roads, and we show that for these cost functions we have a scaling law between the equilibrium and optimum flows. This produces a similar scaling for the break points. Moreover, for affine cost functions, the number of break points is finite for every possible finite network. Although it is finite, there exist classes of networks where the number of break points is exponential in the number of paths. The relevance of break points is due to the fact that between break points the PoA is either monotone or it has a unique minimum, therefore, the PoA can have a local maximum only at a break point.

When cost functions are affine, if an equilibrium uses a certain set of paths at two different demand levels, then it uses the same set of paths at all intermediate demands. In the final section we show that this does not hold for less regular cost functions.

1.2. Related Work

The solution concept that is typically used in nonatomic routing games, and that we adopt in this paper, is due to Wardrop (1952) and its mathematical properties were studied first by Beckmann et al. (1956). Algoritms for computing the equilibrium and the optimum solutions were proposed by Tomlin (1966) when cost functions are affine and by Dafermos and Sparrow (1969) for general convex costs. For a recent survey on the topic, we refer to Correa and Stier-Moses (2011).

The recognition that selfish behavior of agents produces social inefficiency goes back at least to Pigou (1920). A measure quantifying this inefficiency was proposed by Koutsoupias and Papadimitriou (2009, 1999). This measure is the ratio of the social cost of the worst equilibrium over the optimum social cost. It was termed price of anarchy by Papadimitriou (2001). A price of anarchy close to one indicates efficiency of the equilibria of the game, whereas a high price of anarchy implies that, in the worst scenario, strategic behavior can lead to relevant social inefficiency. Most of the subsequent literature has studied sharp bounds for the price of anarchy in some specific classes of games. Several papers applied the idea of the price of anarchy to congestion games and in particular to traffic games on networks. In particular Roughgarden and Tardos (2002) showed that for every congestion games with affine cost the price of anarchy is bounded above by . Moreover, this bound is sharp and attained in a traffic game with a simple two-edge parallel network. Roughgarden (2003) generalized this result to polynomial functions of maximum degree and proved that the price of anarchy grows as . Dumrauf and Gairing (2006) refined the result when the cost functions are sums of monomials whose degrees are between and . Roughgarden and Tardos (2004) generalized the analysis to all differentiable cost function such that is convex. Less regular cost functions and different optimizing criteria for the social cost were studied by Correa et al. (2008, 2004, 2007).

Some papers took a more applied view and studied the actual value of the price of anarchy in real life situations. For instance Youn et al. (2008, 2009) dealt with traffic in Boston, London, and New York. They observed that the price of anarchy varies with traffic demand and shows a similar pattern in the three cities: it is when traffic is light, it oscillates in the central region and then goes back to when traffic increases. A similar behavior was observed by O’Hare et al. (2016), who—taking an approach that relates to the one of our paper—showed how an expansion and retraction of the routes used at equilibrium or at a social optimum affect the behavior of the price of anarchy. An analytical justification for the asymptotic behavior of the price of anarchy has been provided by Colini-Baldeschi et al. (2016, 2017, frth, 2019), Wu et al. (2019). Colini-Baldeschi et al. (2016, 2019) considered the case of single origin-destination parallel networks and proved that, in heavy traffic, the price of anarchy converges to one when the cost functions are regularly varying. Their results were extended in various directions in Colini-Baldeschi et al. (2017, frth): general networks have been considered and both the light and heavy traffic asymptotics have been studied. A different technique, called scalability, has been used by Wu et al. (2017, 2019, 2018) to study the case of heavy traffic.

The behavior of the price of anarchy as a function of a different parameter was studied by Cominetti et al. (2019)

. In this case the parameter of interest is the probability that players actually take part in a congestion game.

Colini-Baldeschi et al. (2018) studied the possibility of achieving efficiency in traffic routing games via the use of tolls, when the demand can vary.

In a more applied direction, Monnot et al. (2017) studied the commuting behavior of a large number of Singaporean students and showed that the efficiency of the system is far from the worst case scenario and the price of anarchy is overall low. Gemici et al. (2019) have dealt with the income inequality effects of reducing the price of anarchy via tolls.

1.3. Organization of the paper

The paper is organized as follows. In Section 2, the concepts of equilibrium and optimum are introduced and their properties are studied, as a function of the demand. In Section 3 the price of anarchy is introduced, together with some of its general properties. The behavior of the PoA is studied in Section 4 in the case of affine costs and in Section 5 for general costs.

2. The nonatomic congestion model

We consider a standard nonatomic routing game with a single origin-destination pair. The network is described by directed graph with vertex set , edge set , an origin and a destination . The traffic inflow is given by a positive real number , interpreted as vehicles per hour, to be routed along the set of all simple paths from to . We assume each vertex is accessible from the origin and can reach the destination , so that there is a path passing through . The nonatomic hypothesis means that each vehicle controls a negligible fraction of the total traffic.

We call the set of feasible path-flow

distributions of traffic given by vectors

with nonnegative entries and such that . Every feasible path-flow induces a load profile over the edges, with representing the amount of traffic that travels over the edge . We call the set of all such load profiles. Note that the correspondence between flows and feasible load profiles is not bijective, since in some networks a given load profile could be induced by more than one flow.

Every edge has an associated cost function , which is assumed nondecreasing and of class , i.e., continuous and differentiable with continuous derivative. The value represents the travel time (or unit cost) of traversing the edge when the load is . When traffic distributes according to a path-flow with induced load profile , the cost experienced by traveling on a path is given by

(2.1)

Notice that, with a slight abuse of language, we use the same symbol for the cost function over paths and over edges. The meaning will be clear from the context.

The total cost experienced by all users traveling across the network is called the social cost and is denoted by

(2.2)

2.1. Wardrop equilibrium

A feasible path-flow is called a Wardrop equilibrium if the paths that are actually used have minimum cost. Formally, is an equilibrium iff there exists such that

(2.3)

The quantity is called the equilibrium cost, which is a function of .

As noted by Beckmann et al. (1956), Wardrop equilibria coincide with the optimal solutions of the convex minimization problem

(2.4)

where is the primitive of the edge cost , namely

(2.5)

This follows by noting that Eq. 2.3 are the optimality conditions for , with the equilibrium cost playing the role of a Lagrange multiplier for the constraint . It follows that, for each fixed inflow , an equilibrium flow always exists.

A Wardrop equilibrium always exists. Although it is not necessarily unique, all equilibria induce the same edge travel times . In particular they have the same equilibrium cost , which is simply the shortest distance:

(2.6)

As a matter of fact, as proved by Fukushima (1984), the equilibrium edge costs are the unique optimal solution of the strictly convex dual program

(2.7)

where .

Since in equilibrium all the paths that carry flow have the same cost , it follows that all equilibria have exactly the same social cost, namely

(2.8)

A less known property is the fact that the optimal value function is a smooth convex function with . We summarize the previous discussion in the following proposition.

Proposition 2.1.

The function is convex and differentiable for , with continuous and nondecreasing. Moreover, the equilibrium edge costs are uniquely defined and are continuous as a function of . If the costs are strictly increasing, the equilibrium edge loads are also unique and depend continuously on .

Under additional hypothesis we can prove the differentiability of the equilibrium cost .

Given a vertex , call and the sets of out-edges and in-edges of , respectively, and the set of all paths from to . Then, an equilibrium load profile for a total inflow of is characterized as a solution of the following system of equations and inequalities:

(2.9)
(2.10)
(2.11)
(2.12)
(2.13)

where is the equilibrium cost of edge and

(2.14)

is the equilibrium cost of a shortest path from the origin to vertex .

Remark 2.1.

Writing Eq. 2.10 for each vertex gives a set of linearly dependent equations, which can be made independent by removing (any) one of them.

Equation (2.12) requires that either or is zero. We say that strict complementarity holds if for every edge exactly one of the two expressions is zero while the other is not zero.

Proposition 2.2.

Assume the costs are continuously differentiable with strictly positive derivative. If strict complementarity holds at , then the equilibrium map is continuously differentiable in a neighborhood of . In particular the equilibrium cost is near .

In the sequel we consider the active network comprising all the edges that are on a shortest path from to , that is to say, the edges that satisfy Eq. 2.11 with equality. We also define a break point as a point at which this active network changes.

Definition 2.3.

For each we define the active network as

(2.15)

The inflow rate is a break point for the equilibrium if there exists such that is constant over the intervals and , with .

Remark 2.2.

By definition, an edge that is inactive at satisfies

(2.16)

so that, by continuity of and , the edge remains inactive for near .

Remark 2.3.

Let be any equilibrium for the inflow , and the induced edge loads. From (2.12), every edge that carries a positive flow must be active, that is to say

(2.17)

In particular, every path with is completely included in the active network . Note however that the set on the left of Eq. 2.17 depends on the particular equilibrium that is chosen, whereas is intrinsically defined for each .

Remark 2.4.

If the costs are strictly increasing and strict complementarity holds at , then Eq. 2.17 holds with equality. In this case, by continuity, all active edges remain active for near , which combined with Remark 2.2 implies that the active network is locally constant. In other words, for strictly increasing costs, a break point can occur at only if strict complementarity fails.

Remark 2.5.

Given a selection of equilibrium flows , the values at which the set of used paths changes can be different from the break points, which are defined independently of the flow. This is due to the possible non-uniqueness of equilibrium. For instance, consider the network in Fig. 1.

Figure 1. The set of paths used in equilibrium may change arbitrarily often.

When , the flow

path
flow

is an equilibrium of mass for any choice of in . So, by letting oscillate between and arbitrarily often, we may build equilibrium flows such that the set of used paths change an arbitrary number of times. Nevertheless, the active network at equilibrium in the whole interval is always the set of edges in all four paths, each of which has cost equal to in any equilibrium.

2.2. Optimum flow

A feasible path-flow is called an optimum flow if it minimizes the social cost, that is to say, is an optimal solution of

(2.18)

where . Since is , it follows that the marginal cost functions

(2.19)

are well defined and continuous. If we further assume to be convex, then is nondecreasing and it follows that the optimal flows coincide with the Wardrop equilibria for the marginal costs . We denote the corresponding equilibrium cost, the active network, and the break points.

Proposition 2.1 then yields the following direct consequence:

Corollary 2.4.

Suppose that are and nondecreasing with convex. Then the optimal social cost map is a convex function with continuous and nondecreasing.

2.3. A scaling law for Bpr cost functions

The Bureau of Public Roads (BPR) proposed a class of cost functions that are monomials of degree plus a constant, namely

(2.20)

For BPR costs we have the following scaling law that relates the equilibrium and optimum flows.

Proposition 2.5.

Assume that the cost function are of the BPR class, as in Eq. 2.20. For each , if is an equilibrium, then

(2.21)

is an optimum, which implies the corresponding scaling law on the flows:

(2.22)
Remark 2.6.

Note that when the cost functions are BPR, the break points for the equilibrium are in one-to-one correspondence with the break points for the optimum, namely

(2.23)

3. Price of Anarchy

Since every equilibrium flow has the same equilibrium cost, we can define the price of anarchy (PoA) for each as the ratio between the social cost at an equilibrium flow and the social cost at an optimum flow :

(3.1)
Proposition 3.1.

The function is continuous in and differentiable where the equilibrium cost is differentiable. In particular, if the edge cost functions are with strictly positive derivative, then the function is continuously differentiable at each for which strict complementarity holds.

Proof.

Recall from Sections 2.2 and 2.1 that the social cost at equilibrium is equal to and is continuous on , while the social cost at optimum is differentiable on . This implies the first part of the proposition. The second part is a consequence of Proposition 2.2. ∎

Remark 3.1.

The break points at equilibrium are natural candidates to be points of non-differentiability for the equilibrium cost and the Price of Anarchy . On the other hand the break points at optimum do not contribute to create first order irregularities for the PoA.

4. Networks with affine cost functions

In this section we consider affine cost functions, i.e., cost functions :

(4.1)

with for each .

Proposition 4.1.

Let . For assume that there exists two corresponding equilibrium flows of demand and of demand , which both use the subset . Then, for every , there exists an equilibrium flow that uses .

Remark 4.1.

Proposition 4.1 implies that, in the case of affine cost functions, any given active network can appear over a certain interval, but once it changes it cannot repeat itself afterwards. Furthermore, as it will be clear from the proof, between break points there exists an affine selection of equilibrium, which means that between break points, we can choose an equilibrium flow of the form

(4.2)

where and are two constant vectors in with and .

4.1. Number of break points

Proposition 4.2.

For affine edge costs, the number of break points for the equilibrium is finite.

Proof.

Since the number of all possible subset of is finite and since the active network at equilibrium cannot repeat itself (see Remark 4.1), the result follows. ∎

Even for affine costs, the number of break points can be exponential in the number of paths.

Proposition 4.3.

There exist networks with affine costs where has elements and the number of break points is of the order of .

4.2. Behavior of the Price of Anarchy

We now prove that the social cost at the equilibrium and at the optimum have a very similar form.

Proposition 4.4.

Let and be two consecutive break points for the equilibrium. We have:

  • with when

  • with when , i. e.,

The behavior of the PoA between break points has a very specific form, which implies that local maxima can exist only at breaking points.

Proposition 4.5.

Let be an open interval which does not contain break points. Then

  • The PoA has at most one stationary point in .

  • The PoA does not attain a local maximum in .

Example 4.1.

The network of Fig. 2 provides an example of routing game where the PoA takes value for small and large values of the demand, but also for an intermediate value, as the plot in Fig. 3 shows. The network is obtained by nesting two Wheatstone networks. Using this technique recursively, it is possible to obtain the bounds in Proposition 4.3. The idea of nesting Wheatstone networks has been used by Lianeas et al. (2016), although in a different way.

Figure 2. Network obtained by nesting two Wheatstone networks.
Figure 3. Plot of the PoA for the nested Wheatstone example of Fig. 2.

5. General cost functions

When the cost functions have less regular shapes, the set of paths used at equilibrium can have a recurring behavior.

Proposition 5.1.

The active network at equilibrium can repeat itself over disjoint intervals defined by break points.

Proof.

Consider the network in Fig. 4 with the cost defined in as follows

with , and in the interval

we interpolate in any way that makes

continuous and nondecreasing in the whole .

Figure 4. Wheatsone network with non-affine costs.

Then we have the following regimes:

  • when , the equilibrium flow uses only the path , the load on the two edges , and is zero and the equilibrium cost is ;

  • when , the equilibrium flow uses all the three paths with the following distribution

    path
    flow

    the load on the two edges , is , and the equilibrium cost is ;

  • when , the equilibrium flow splits equally between the two paths , , the load on the two edges , is , and the equilibrium cost is ;

  • when , the equilibrium flow uses all the three paths with the following distribution

    path
    flow

    the load on the two edges , is , and the equilibrium cost is ;

  • when , , the equilibrium flow splits equally between the two paths , , and the load on the two edges , is .

This proves the Proposition, as in the third and fifth analyzed interval the equilibrium uses only the two paths and , while in the second and fourth analyzed interval it uses all three paths. ∎

Remark 5.1.

Note that in the same way one can construct examples of networks with an infinite number of break points for the equilibrium. Furthermore one can make the increasing sequence of such break points to be convergent.

Acknowledgments

Marco Scarsini and Valerio Dose are members of INdAM-GNAMPA. Roberto Cominetti gratefully acknowledges the support of LUISS during a visit in which this research was initiated, as well as the support of the Complex Engineering Systems Institute, ISCI (ICM-FIC: P05-004-F, CONICYT: FB0816) and FONDECYT 1171501. This research project received partial support from the COST action GAMENET, the INdAM-GNAMPA Project 2019 “Markov chains and games on networks,” and the Italian MIUR PRIN 2017 Project ALGADIMAR “Algorithms, Games, and Digital Markets.”

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Appendix A Proofs

a.1. Proofs of Section 2

Proof of Proposition 2.1.

This is a consequence of the convex duality theorem. Indeed, consider the perturbation function , given by

(A.1)

Clearly is a proper closed convex function (Rockafellar, 1997, page 24). Calling

(A.2)

we have and in particular , which we consider as the primal convex minimization problem . From general convex duality, we have that is a convex function, and therefore so is . Moreover, the perturbation function yields a corresponding dual

(A.3)

where is the Fenchel conjugate function, that is,

The convex duality theorem asserts that if is finite with for in some interval around 0, then there is no duality gap and the subdifferential at is nonempty, compact and convex, and it coincides with the optimal solution set of the dual problem.

Now, since and is finite for all , the convex duality theorem applies. It follows that is a finite convex function and . It remains to show that is a singleton. To this end, fix an optimal solution for and recall that this is just a Wardrop equilibrium. Having fixed such an , the dual optimal solutions are precisely the ’s such that

This equation can be written explicitly as

from which it follows that is an optimal solution in the latter supremum. The corresponding optimality conditions are

which imply that is the equilibrium cost for the Wardrop equilibrium, that is, . It follows that so that is not only convex but also differentiable with . The conclusion follows by noting that every convex differentiable function is automatically of class , with nondecreasing.

The continuity of the equilibrium edge costs is a consequence of Berge’s Maximum Theorem (see, e.g., Aliprantis and Border, 2006, Section 17.5). Indeed, the equilibrium edge costs are optimal solutions for the dual program in Eq. 2.7. Since the objective function is jointly continuous in , Berge’s Theorem implies that the optimal solution correspondence is upper-semicontinous. However, in this case the optimal solution is unique, so that the optimal correspondence is single-valued, and, therefore, the equilibrium edge costs are continuous. The last claim about the continuity of the equilibrium edge loads follows directly from this. ∎

Proof of Proposition 2.2.

First observe that we have a unique equilibrium load profile, since in this case the equilibrium is a solution of a strictly convex program, because the cost functions are strictly increasing.

We know that the equilibrium load profile at mass must satisfy Eqs. 2.13, 2.12 and 2.10 for some , .

First, we observe that the edge equilibrium costs are continuous (see Section 2.1) and so also the equilibrium costs of a shortest path to any vertex are continuous, as well as the functions , for all . Since strict complementarity holds at , according to Remark 2.3 the active network is constant in a neighborhood of and for all and near . So, let denote the active network in a neighborhood of , and the corresponding vertices.

Now consider Eqs. 2.13, 2.12 and 2.10 relative to the edges in and remove one dependent equation from Eq. 2.10 (the one relative to vertex ), and add the equation . We obtain the following system:

(A.4)
(A.5)
(A.6)
(A.7)

The equilibrium loads satisfy such system of equations for some , and . We want to apply the implicit function theorem to this system of equations, and, to do this, we have to check that the associated linearized system has a unique solution.

Let , and be respectively the increments in the variables , and for each and . The homogeneous linear system obtained from Eqs. A.7, A.6, A.5 and A.4 is the following:

(A.8)
(A.9)
(A.10)
(A.11)

Strict complementarity on an active link implies that Eq. A.9 is equivalent to

(A.12)

which, together with Eq. A.10, gives

(A.13)

These equation are stationary conditions for the strongly convex (since ) quadratic program

(P)

under the constraints Eq. A.8 and the dependent constraint for the vertex . Indeed, associating a Lagrange multiplier to each of those constraints, we get the Lagrangian

and the equation is precisely equivalent to Eq. A.13.

Hence, under strict complementarity, every solution of Eqs. A.10, A.9 and A.8 corresponds to an optimal solution of (P). Since for all is feasible, it is also the unique optimal solution. It follows that for all , and Eq. A.10 yields for all . Finally, from Eqs. A.11 and A.12 we obtain also for all .

We showed that the linear system Eqs. A.11, A.10, A.9 and A.8 has only the trivial solution, thus the Jacobian of Eqs. A.7, A.6, A.5 and A.4 with respect to is invertible and the implicit function theorem implies the smoothness of the solution. In particular is continuously differentiable. ∎

Proof of Proposition 2.5.

Let and be the load profiles induced, respectively, by the equilibrium flow and the optimum flow . The equilibrium load profile is characterized by the inclusion

(A.14)

which, in the case of BPR cost functions, assumes the form

(A.15)
(A.16)
(A.17)

for each .

Now, observe that the expression in Eq. A.17 identifies a load profile of mass that, scaled by , minimizes the social cost. Hence such flow is equal to the optimum load profile

times a factor . ∎

a.2. Proofs of Section 4

Let be the cardinality of , i.e., the number of paths from origin to destination, and we consider the set of paths indexed by the numbers . We can define an matrix and a vector in the following way:

so that when some flow is using the network, the cost of using path

is given by the -th entry of the vector .

Let be the set of paths used at some equilibrium flow , with equilibrium cost . Then we have that satisfies the following linear system in the variable :

(A.18)

where the matrix and the vectors , are defined in the following way:

Lemma A.1.

The matrix has the following properties:

  • is symmetric with nonnegative entries