Parametrized Nash Equilibria in Atomic Splittable Congestion Games via Weighted Block LaplaciansThis research was carried out in the framework of Matheon supported by Einstein Foundation Berlin.

Parametrized Nash Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplaciansthanks: This research was carried out in the framework of Matheon supported by Einstein Foundation Berlin.

Max Klimm Humboldt-Universität zu Berlin, Germany
11email: {max.klimm,philipp.warode}@hu-berlin.de
   Philipp Warode Humboldt-Universität zu Berlin, Germany
11email: {max.klimm,philipp.warode}@hu-berlin.de
Abstract

We consider atomic splittable congestion games with affine cost functions and develop an algorithm that computes all Nash equilibria of the game parametrized by the players’ demands. That is, given a game where the players’ demand rates are piece-wise linear functions of some parameter , we compute a family of multi-commodity flows parametrized in such that for all the flow is a Nash equilibrium for the corresponding demand rate vector . Our algorithm is based on a novel weighted block Laplacian matrix concept for atomic splittable games. We show that the weighted block Laplacians have similar properties as ordinary weighted graph Laplacians which allows to compute the parametrized Nash equilibria by matrix pivot operations. Our algorithm is output-polynomial on all instances, and each pivot step needs only where is the number of players and is the number of vertices.

Keywords:
Equilibrium Computation Congestion Game Laplacian.
\shortversiononly\NewEnviron

killcontents

1 Introduction

Congestion games are a central topic in algorithmic game theory with applications in traffic, telecommunication, and logistics. In Wardrop’s basic model [27], we are given a network with flow-dependent cost functions and a set of commodities, each specified by a source node, a target node and a fixed flow demand. In this setting, a multi-commodity flow is a Wardrop equilibrium if for each commodity all paths that carry flow have the same costs, and all unused paths do not have smaller costs. Wardrop equilibria naturally model situations where each commodity consists of a continuum of infinitesimal small players that are interested in minimizing their travel costs, e.g., travel times in the context of traffic networks, latencies in the context of telecommunication networks, and monetary costs in the context of logistics networks.

With the rise of navigation systems such as Waze and TomTom and ride sharing platforms such as Lyft and Uber, and in view of the anticipated market penetration of autonomous cars, it is sensible to assume that in the near future several competing companies will control significant portions of the road traffic. For these future traffic scenarios Wardrop equilibria are not sufficient models for road traffic anymore since operators of fleets of cars may be interested in minimizing the overall performance of their fleet and may, thus, be willing to sacrifice the travel time of some of the traffic controlled by them in order to improve the overall performance of their flee, see also Catoni and Pallottino [5]. These effects, however, can be studied within the class of atomic splittable congestion games. Here, each player is associated with a source node, a target node and a fixed demand rate. A strategy of the player is to distribute her demand on the paths from her source to her target. The player is interested in minimizing the overall travel time of the flow controlled by her. As shown by Haurie and Marcotte [13], ordinary non-atomic games can be obtained as the limit of a series of atomic splittable games; in that sense atomic splittable games generalize the class of non-atomic games.

While the existence of equilibria in atomic splittable games can be established by standard fixed point arguments, much less is known regarding the computation of equilibria in atomic splittable congestion games. For affine cost functions, Cominetti et al. [6] showed that an equilibrium can be found by computing the minimum of a convex potential function, see also Huang [16] for a combinatorial algorithm for special graph topologies. Bhaskar and Lolakapuri [4] proposed two algorithms with exponential worst-case complexity that compute -approximate Nash equilibria in singleton games with convex costs. Harks and Timmermans [10] developed a polynomial time algorithm that computes an equilibrium in singleton games with player-specific affine cost functions.

All these approaches above yield a single equilibrium for a fixed vector of player demands. Moreover, the algorithms of Bhaskar and Lolakapuri [3] and Harks and Timmermans [10] work only for singleton games played on a network with two nodes. In actual traffic scenarios, the assumption that the players’ demand vector is fully known and fixed is unrealistic since demands often fluctuate. In this paper, we are interested in understanding how the equilibria in atomic splittable games change as a function of the players’ demand vectors.

1.1 Our results and techniques

We propose an algorithm that computes all equilibria of an atomic splittable congestion game as a function of the players’ demands. We assume that all cost functions are affine. More formally, consider an atomic splittable congestion game played on a graph with cost functions . Further, let be a piece-wise linear function assigning a demand rate vector to each value . Then, we compute functions such that for all the multi-commodity flow is a pure Nash equilibrium of the atomic splittable congestion game with demand rate vector .

Our algorithm is based on the observation that for given strategies of the other players, each player plays a best-reply. Using the well-known correspondence between optimal flows and Wardrop equilibria (cf. Beckmann et al. [1]) we can view the equilibrium strategies of each player as a Wardrop equilibrium in a game where costs are replaced by marginal costs. Wardrop equilibria, in turn, have a close relation to electrical flows and, thus, graph Laplacians. Using this approach, we can reformulate the Nash equilibrium conditions as a system of linear equations of the form , where is a multi-commodity excess vector containing the excess for each player  and each vertex , is a multi-commodity potential vector containing a (shortest-path) potential for each player  and each vertex , and is a vector of offsets. The matrix is a novel structure that we call the total Laplacian of the graph. Roughly speaking, it has the same structure as a graph Laplacian except that all entries are graph Laplacians rather than scalars. We show that the total Laplacian shares many properties with regular Laplacians, i.e., it is symmetric, positive semi-definite, and has a very easy nullspace.

Using these properties, we can show that the parametrized Nash equilibrium problem can be solved by pivot operations on a generalized inverse of . In the non-degenerate case, the overall time complexity of the algorithm is where is the number of breakpoints of the output function. In case of degeneracy, we can replace the pivoting operation by solving a quadratic program, so that also in this case we have an output-polynomial algorithm.

\shortversiononly

Due to space constraints, this extended abstract does not contain any proofs. For the full proofs, we refer to our preprint [klimm2018preprint].

1.2 Further related work

Haurie and Marcotte [13] showed that non-atomic congestion games can be obtained as a sequence of atomic splittable congestion games. Existence of pure Nash equilibria in atomic splittable congestion games follows from standard fixed point arguments (cf. Kakutani [17] and Rosen [24]). Bhaskar et al. [3] showed that the equilibria are not necessarily unique and gave several topological conditions on the network for the uniqueness of equilibria. Richman and Shimkin [23] characterized two-terminal network topologies that are necessary and sufficient for uniqueness. Harks and Timmermans [11] characterized the uniqueness of equilibria in terms of the combinatorial structure of the strategy set. The price of anarchy of atomic splittable congestion games price of anarchy has been studied by Cominetti et al. [6], Harks [9], and Roughgarden and Schoppmann [25]. Catoni and Pallottino [5] provide a paradox of a non-atomic game where replacing the non-atomic players of one commodity by an atomic player with the same demand decreases the overall performance of that commodity. In previous work [20], we developed an algorithm that computes all Wardrop equilibria parametrized by the flow demand. In this work, we generalize this approach towards atomic splittable games. From a mathematical point of view, our algorithm is a homotopy method, for further homotopy methods for computing equilibria, see [8, 14, 15, 18, 22].

2 Preliminaries

An atomic splittable routing game is a tuple where is a directed graph with vertices and edges , the family , …, contains triples each of which consisting of a source node , a sink node , and a demand rate for each of the players, and is a family of strictly increasing affine linear latency functions with for some and .

A feasible strategy for every player is to route her demand between her terminal vertices and . Thus, a strategy for player is a --flow of rate , i.e., a non-negative vector in satisfying

(1)

for every vertex . A strategy profile for all players is a vector containing all flows stacked. We use the notation for the strategy profile where player uses the flow and all other players use their flow as in the strategy profile . The latency experienced by the flow of the players on some edge depends on the total flow where . The cost paid by every player is the total latency experienced by the flow sent by this player, i.e., . We say that is a Nash equilibrium if for every player there is no profitable deviation from , i.e., for all --flows of rate . We define the marginal cost of player on edge given the flow by

We then obtain the following characterization of Nash equilibria, see, e.g., Bhaskar et al. [3] for a reference.

Lemma 1

The strategy profile is a Nash equilibrium flow if and only if, for every player , is a --flow and {flexequation} ∑_e ∈P L^i_e(x) ≤∑_e ∈Q L^i_e (x) for all --paths with for all .

Lemma 1 states that is a Nash equilibrium if and only if all path used by player  are also shortest path for that player with respect to the marginal costs. This enables us to give another characterization based on (shortest path) potentials.

Lemma 2

The flow is a Nash equilibrium if and only if, for every player there is a potential vector such that \shortversion if and if

(2a)
(2b)

for every edge .

As for the whole flow vector we will denote by the vector of all stacked potentials .

We say an edge is active for some player if is satisfied with equality. (In particular all edges used by a player (i.e. ) are active.) Further, we define the sets of players for which is active. We call the family of these sets supports. When the supports are known, computing a Nash equilibrium reduces to finding a feasible flow and some potential vector satisfying for all and all . In order to find an equilibrium, we have to solve a system of equations consisting of flow conservation and potential equalities. In the special case of affine-linear cost functions these equations are linear equations and can be solved explicitly as we will show in more detail in the next section.

3 Weighted Block Laplacians

Let us fix a support . In this section, we work towards expressing the multi-commodity excess vector in a Nash equilibrium for support as a linear system of the form , and we will derive useful properties of the matrix . To this end, for two players and an edge , let \shortversion if player , and otherwise.

We also write as a shorthand for . Further, let be the number of players using the edge in . We introduce the diagonal matrix

as well as the vector . With these definitions, we obtain the following linear equation relating Nash equilibria to their potential vectors.

Lemma 3

Let be a Nash equilibrium and be a corresponding potential. Then, {flexequation} \shortsmashx^i = C^ii   Γ^⊤  π^i - ∑_j ≠i C^ij   Γ^⊤  π^j - ~b^i, where is the vertex-edge incidence matrix.

Proof

Equations (2a) and (2b) imply that, for every player and every edge , either or . The latter is equivalent to

(3)

Summing up all potential differences of players using an edge , we obtain

and, hence,

(4)

With (4) we finally obtain from (3) the equation

(5)
(6)

Let be the vertex-edge incidence matrix of . Then is a vector containing all potential differences for every edge. We then rewrite (6) in vector form

as claimed. ∎

The vector contains the excess, i.e., the difference of out- and in-flow for every vertex. In order to be a feasible flow, we need , , and otherwise. Consider the weighted Laplacian matrices of defined as for every pair of players player . The matrix is a weighted Laplacian matrices of the graph with edge weights if or edge weight otherwise. Since , we also have for all . Let be the stacked excess vector. Then we can express the relation between the stacked potential vector and in the following convenient way

(7)

We call the block matrix the total Laplacian matrix. It is structured as a normal Laplacian matrix, but it contains the “sub-Laplacians” rather than real entries. We proceed to show that it has similar properties as a usual Laplacian matrix with scalar entries.

Lemma 4

The total Laplacian is symmetric and positive semi-definite.

Proof

Since all sub-Laplacian matrices are symmetric, is symmetric by definition.

Let be the block matrix containing all diagonal matrices , i.e.

Every row of contains exactly (possibly) non-zero elements, namely the value as diagonal element and for all as off-diagonal elements for fixed and . If, for this fixed and , we have , then the row contains only zeros. Otherwise, the difference between diagonal element and absolute of the off-diagonal elements is

where for the first inequality we used that . Thus is (weakly) diagonally dominant, symmetric and real-valued implying that is positive semi-definite.

Further, let be a -block diagonal matrix containing times the matrix as block diagonal entries. Then and, thus, is also positive semi-definite. ∎

As we are interested in solving the system of linear equations (7), we want to invert the matrix . Thus, we are interested in the rank of .

Theorem 3.1

Let be the total Laplacian matrix. Then for every vector containing the stacked vectors for , we have \shortversion if and only if .

(8)
Proof

We can use Theorem C.2 from Appendix C. We only have to ensure that satisfies Assumption 1. The parts (i) and (ii) are satisfied by definition and the properties of Laplacians. For part (iii) observe that, for every , we have . This yields

and thus we can apply Theorem C.2. ∎

Theorem 3.1 shows that the nullspace (and thus the rank) of the total Laplacian depends solely on the nullspace of the sub-Laplacians on the diagonal.

Assuming that for every player , the graph contains a single connected component with respect to (this means, we consider two vertices to be connected, if there is a path between these vertices using only edges with ) then by the properties the standard Laplacian , we know that and the nullspace of is the linear hull of the all-one vector . We obtain the following direct corollary.

Corollary 1

If, for every player , there is only one connected component with respect to , then

  1. .

  2. the nullspace of contains only vectors that contain stacked multiples of the all-one vector .

  3. the solution of (7) is unique up to an additive constant for every player potential .

  4. given the support , the Nash equilibrium flows are unique.

The last statement follows from the fact that the flows depend only on the potential difference as it can be seen in equation (3) and thus the additive constant of the potential is irrelevant. Note that this only proves the uniqueness if the support is given, not the uniqueness of the Nash equilibrium.

4 Computing parametrized Nash equilibria

4.1 Potential directions

Recall from equation (7), that for a fixed support , we have where is the stacked excess vector, is the stacked potential vector and is the total Laplacian for . We are interested how the equilibria change as the excess vector change, i.e., we want to solve the equation for . As shown in Corollary 1, the total Laplacian matrix is singular. For the purpose of a compact notation, we will use generalized inverses—for any matrix we denote by a generalized inverse of , that is a matrix that satisfies . Every matrix has at least one generalized inverse and, for every vector , is a solution of the system if the system is consistent, see also [2] for a reference on generalized inverses.

We then obtain . The solution may depends on the choice of the generalized inverse , but by Corollary 1 we know that the solution has to be unique up to additive constants for every player potential. Changing the demands of the players changes the excess vector by which results in a change of the potentials by . We call this vector the potential direction. Moving along this direction in the space of all potentials yields new equilibrium potentials for demands as long as the support does not change.

4.2 Potential space

The equilibrium solution changes only linearly when the excess is changed linearly as long as the support does not change. Therefore, we are interested in the sets of (possible) equilibrium solutions (i.e. potential vectors ) that induce the same support.

We consider the potential space of all possible potential vectors . Given a support , we define for every edge and player the vectors

(9)

where is the column of the vertex-edge incidence matrix corresponding to the edge . This vector enables us to simplify the formula for the player flow on active edges to . Since every equilibrium potential is a shortest path potential with respect to the marginal cost, we know that for every inactive edge we have {flexequation} 0 ≥π^i_w - π^i_v - a_e ¯x_e - b_e = w_e,i^⊤ π - b_e. Thus, given a potential of an equilibrium flow and the corresponding support , we have

(10)

Given any potential vector and any support , we say is consistent with if (10) is satisfied. We define the potential region corresponding to the support as the set {flexequation} R_S := { π ∈R^kn   —   π is consistent with S } . Every region is a convex subset of the potential space and bounded by the hyperplanes with normal vectors and the offsets . We also call these hyperplanes boundaries induced by an edge and a player . See Figure 1 for a graphical representation of a region in the potential space.

\setdimensions\drawline\drawline\drawline\drawline\drawline
Figure 1: A region in the space of vertex potentials with potential direction . The solution curve (red line) hits the boundary induced by edge and player in the point .

By definition, all potentials in some region are consistent with the same support . Thus, given some excess direction , the potential direction as defined in the previous subsection is constant in the whole region.

4.3 The main algorithm

We want to develop an algorithm that, starting from a given equilibrium for demand vector , computes all equilibria for demands where is some vector specifying a direction in the demand space. For a piece-wise linear function is piece-wise constant in . These demand vectors naturally induce corresponding vectors and in the excess space where is piece-wise constant as well. For ease of exposition, in the following, we describe only the basic version of the algorithm with constant . The general case can be easily obtained by concatenating different runs of this algorithm where the final solution of the last run serves as a starting solution of the next.

In order to compute all equilibrium potentials belonging to these excess vectors (or demand vectors, respectively), we construct a homotopy method inspired by a similar method introduced by Katzenelson [18] that computes electrical flows. The algorithm computes a piecewise linear function mapping to a potential vector belonging to an equilibrium flow for demand and a piecewise constant function that returns the associated supports. We call the parametrized curves and solution curve (in the potential space) and support curve, respectively. Note that the solution curve and the support curve are sufficient to reconstruct the equilibrium flows for all .
The basic procedure can be described as follows.

  1. Start with an equilibrium for the excess with support and with potentials for the initial support .111If one may start with an appropriate shortest path potential and appropriate supports. In general, a starting solution can be found with the convex program\shortversion presented in Appendix B (24).

  2. For fixed , compute the potential direction .

  3. Compute the maximal such that for all . Find a new support such that . Continue with 2. in the region .

Given a potential that is consistent with a given support (i.e. ), we define the values

(11)

for every edge and every player and . Then the potential is consistent with if and only if and in the potential at least one boundary is hit (see also Figure 1).

In order to proceed for , there are two possible cases. Either there is a unique minimal (i.e. the solution curve hits a single boundary) or there are multiple inducing the minimal . We refer to the former as unique boundary crossing and to the latter as degenerate boundary crossing. Likewise, we call points in the potential space where more than one boundary hyperplane intersect degenerate points.

4.4 Unique boundary crossing

We introduce the following abuse of notation: We write if , for all edges , and . Likewise, we write if , for all edges , and . Thus, the support () is the same support as except that the status of edge is changed from inactive to active (active to inactive) for player .

Assume the solution curve hits a unique boundary, i.e., there is an unique edge and player such that is minimal. We want analyze what happens if we change the activity status of this edge for this player (either from active to inactive or vice versa). The following theorem gives us a relation between the inverses of the associated total Laplacian matrices. For ease of notation, denote by the total Laplacian matrix that is induced by the support .

Theorem 4.1

Let be two supports with for some edge  and player . Let where is the number of players using edge  including player . Further, let . Then,

  1. the inverses of the total Laplacians satisfy

    (12)
  2. the directions satisfy

    (13)
Proof

We assume without loss of generality that . Further, we assume that was inactive in and is active in for player . (For the other direction we just have to change signs.) Let be the number of players using including player , i.e. . Then we can express the sub-Laplacians for as

With the constant and the vector as defined in (9) we can express the total Laplacian matrix for as

We now claim that

(14)

holds true independently of the choice of the generalized inverse . Let be the matrix obtained from by deleting the rows and columns belonging to the first vertex for every player. Then this matrix is non-singular (by Corollary 1) and strictly positive definite. Let be the vector without the rows belonging to the first vertex for every player. Then it is easy to show that

Finally, we observe that the matrix defined as the matrix with additional zero rows and columns for the first vertex for every player is a generalized inverse of . This together with the aforementioned fact that is unique up to additive constants for every player implies that .

The claim (14) implies that . Thus, we obtain (12) with the Sherman-Morrison-Woodbury formula for generalized inverses (see, e.g., [12, Theorem 18.2.14]).

Using this identity, we get

which proves the second statement. ∎

Theorem 4.1 shows that the generalized inverse of the Laplacian matrix after crossing the boundary can be obtained by a simple matrix multiplication. Further, the second statement ensures that the boundary crossing is well-defined in the following sense:

Corollary 2

Assume the solution curve hits a unique boundary induced by edge and player of the region in the point . Let . Then there is such that {flexequation} π + λL^+_S_2 Δy ∈R_S_2 for all .

Proof

Since by assumption, it is clear that and . Thus, the statement holds for . (This is just the formalization of the fact that lies on the boundary between and .)

Now we need to show that we can move away from in the direction of the second region without directly leaving region . Since we assumed that edge and player is the only pair with at we can move at least an step in the direction without violating any of the inequalities (10) for or . Using (13) from Theorem 4.1 we know that if the direction points towards the boundary in the direction must be directed away from that boundary in region . This implies we can move at least for some positive without leaving . ∎

Corollary 2 states that when the solution curve hits a unique boundary, the solution can proceed in the adjacent region induced by the same support except for changed activity status of the edge for player that induced the boundary.

\shortversiononly

In Appendix A, we discuss a lexicographic rule that deals with degenerate boundary crossings. Alternatively, a new direction vector may also be computed by solving a quadratic program as discussed in Appendix B. \longversiononly\shortversion

5 Degeneracy

5.1 Degeneracy

We now want to consider the case when two or more boundaries are hit at the same time. Note that this situation can occur if multiple edges change their status for one or more players or the status of one edge changes simultaneously for multiple players. In any case, the main idea for this situation is to add a small perturbation to the potential in order to avoid the degenerate point. Observing the perturbed solution we can obtain the right support by finitely many unique boundary crossings. In fact, we will show that this can be done implicitly using a lexicographic rule.

\setdimensions\drawline\drawline\drawline\drawline
Figure 2: The solution curve (red) hits the degenerate point starting from the potential . Adding the small perturbation results in a new, perturbed solution curve (green) that bypasses the degenerate point and moves away from in the same region as the original solution curve.

Assume that, starting from a potential vector , the next potential is a degenerate point. Then, for some , we define the perturbation vector and consider the solution curve starting from the potential . (We will refer to thus solution as the perturbed solution curve.) For almost every the vectors and are linearly independent and thus the perturbed solution curve hits the point where is the minimal distance from to all boundaries. The perturbed solution curve will move on to further points until it does not hit any further boundary intersecting in the degenerate point . See Figure 3 for a visualization of the perturbation and the resulting perturbed solution curve.

Note that, by continuity, for sufficiently small , the perturbed solution curve will only cross boundaries that intersect in the degenerate points rather than other boundaries before reaching the reaching the region where the solution curve moves away from the degenerate point. Thus, we will only consider boundaries that intersect in the degenerate point for the remainder of this subsection.

The next lemma shows that the potentials and the epsilon values can be expressed as a linear function depending on .

Lemma 5

There are matrices and vectors for every player and every edge such that

  1. and for .

  2. for every , and for .

Proof

Denote by the player-edge pair inducing the boundary that is hit by the perturbed solution curve in the -th step. We define the matrices and vectors

Note that we admit -values in the vectors if the denominator is zero. In this case, the perturbed solution curve moves parallel to the boundary induced by and, in particular, the whole vector