Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games

We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow polymatroid, then equilibria are unique. Bidirectional flow polymatroids are introduced as a subclass of polymatroids possessing certain exchange properties. We show that important cases such as base orderable matroids can be recovered as a special case of bidirectional flow polymatroids. On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and nonmatroidal set systems per player, there is an isomorphic game having multiple equilibria. Our results leave a gap between base orderable matroids and general matroids for which we do not know whether equilibria are unique.


Introduction
We revisit the issue of uniqueness of equilibria in atomic splittable congestion games. In this class of games there is a finite set of resources E, a finite set of players N , and each player i ∈ N is associated with a weight d i ≥ 0 and a collection of allowable subsets of resources S i ∈ 2 E . A strategy for player i is a (possibly fractional) distribution x i ∈ R |Si| + of the weight over the allowable subsets S i . Thus, we can compactly represent the strategy space of every player i ∈ N by the following polytope We denote by x = ( x i ) i∈N the overall strategy profile. The induced load under x i at e is defined as x i,e := S∈Si:e∈S x S and the total load on e is then given as x e := i∈N x i,e . Resources have player-specific cost functions c i,e : R + → R + which are assumed to be non-negative, increasing, differentiable and convex. The total cost of player i in strategy distribution x is defined as Each player wants to minimize the total cost on the used resources and a Nash equilibrium is a strategy profile x from which no player can unilaterally deviate and reduce its total cost. Using that the strategy space is compact and cost functions are increasing and convex Kakutanis' fixed point theorem implies the existence of a Nash equilibrium.
Example 1.1. A well-known special case of the above formulation arises when the resources E correspond to edges of a graph G = (V, E) and the allowable subsets S i correspond to the set of s i -t i -paths for some (s i , t i ) ∈ V × V . In this case, we speak of an atomic splittable network congestion game.

Uniqueness of Equilibria
Uniqueness of equilibria is fundamental to predict the outcome of distributed resource allocation: if there are multiple equilibria it is not clear upfront which equilibrium will be selected by the players. An intriguing question in the field of atomic splittable congestion games is the possible non-uniqueness of equilibria. Multiple equilibria x, y exist whenever there exists a player i and resource e such that x i,e = y i,e . A variant on this question is whether or not there exist multiple equilibria such that there exists at least one resource e for which x e = y e . We call this variant "uniqueness up to induced load on the resources".
For non-atomic players and network congestion games on directed graphs, Milchtaich [21] proved that Nash equilibria are not unique when cost functions are playerspecific. Uniqueness is only guaranteed if the underlying graph is two terminal st-nearly-parallel. Richman and Shimkin [26] extended this result to hold for atomic splittable network games. Bhaskar et al. [5] looked at uniqueness up to induced load on the resources. They proved that even when all players experience the same cost on a resource, there can exist multiple equilibria. They further proved that for two players, the Nash equilibrium is unique if and only if the underlying undirected graph is generalized series-parallel. For multiple players of two types (players are of the same type if they have the same weight and share the same origin-destination pair), there is a unique equilibrium if and only if the underlying undirected graph is s-t-seriesparallel. For more than two types of players, there is a unique equilibrium if and only if the underlying undirected graph is generalized nearly-parallel.

Our Results and Outline of the Paper
In this paper we study the uniqueness of equilibria for general set systems (S i ) i∈N . Interesting combinatorial structures of the S i 's beyond paths may be trees, forests, Steiner trees or tours all in a directed or undirected graph or bases of matroids.
As our main result we give a sufficient condition for uniqueness based on the theory of polymatroids. We show that if the strategy space of every player is a polymatroid Graphic matroid on GSP graph Strongly base orderable Base orderable Figure 1: Several well-known classes of matroids and the relations between them. Here GSP is short for generalized series-parallel. References and arguments for the seven inclusions can be found in Appendix A.
base polytope satisfying a special exchange property -we term this class of polymatroids bidirectional flow polymatroids -the equilibria are unique. 1 We demonstrate that bidirectional flow polymatroids are quite general as they contain base-orderable matroids, gammoids, transversal and laminar matroids. For an overview of special cases that follow from our main result, see Figure 1.
The uniqueness result is stated in Section 4. In Section 5 we show that base-orderable matroids are a special case of bidirectional flow polymatroids. Definitions of polymatroid congestion games and bidirectional flow polymatroids are introduced in Sections 2 and 3, respectively. In Section 6 we complement our uniqueness result by showing the following. Consider a game with at least three players for which the set systems S i of all players i ∈ N are not bases of a matroid. Then there exists a game with strategy spaces φ(S i ) isomorphic to S i which admits multiple equilibria. Here, the term isomorphic means that there is no a priori description on how the individual strategy spaces of players interweave in the ground set of resources. Our results leave a gap between general matroids and base orderable matroids for which we do not know whether or not equilibria are unique.
In Section 7 we consider uniqueness of equilibria if the set systems S i correspond to paths in an undirected graph. The instance used for showing multiplicity of equilibria of non-matroid games can be seen as a 3-player game played on an undirected 3vertex cycle graph. From this we can derive a new characterization of uniqueness of equilibria in undirected graphs. If we assume at least three players and if we do not specify beforehand which vertices of the graph serve as sources or sinks, an undirected graph induces unique equilibria if and only if the graph has no cycle of length at least 3.

Further Related Work
Atomic splittable (network) congestion games have been first proposed by Orda et al. [24] and Altman et al. [3] in the context of modeling routing in communication networks. Other applications include traffic and freight networks (cf. Cominetti et al. [8]) and scheduling (cf. Huang [17]). Haurie and Marcotte [16] showed that classical nonatomic congestion games (cf. Beckmann et al. [4] and Wardrop [29]) can be modeled as atomic splittable congestion games by constructing a sequence of games and taking the limit with respect to the number of players. It follows that atomic splittable congestion games are strictly more general as their nonatomic counterpart. Cominetti et al. [8], Harks [12] and Roughgarden and Schoppmann [27] studied the price of anarchy in atomic splittable congestion games.
Integral polymatroid congestion games were introduced in Harks, Klimm and Peis [13] and later they were studied from an optimization perspective in Harks, Oosterwijk and Vredeveld [14]. Polymatroid theory was recently used in the context of nonatomic congestion games, where it is shown that matroid set systems are immune to the Braess paradox, see Fujishige et al. [10].

Polymatroid Congestion Games
In polymatroid congestion games we assume that the strategy space for every player corresponds to a polymatroid base polytope.
In order to define polymatroids we first have to introduce submodular functions. A function ρ : Given a submodular, monotone and normalized function ρ, the pair (E, ρ) is called a polymatroid. The associated polymatroid base polytope is defined as: where x(U ) := e∈U x e for all U ⊆ E.
In a polymatroid congestion game, we associate with every player i a player-specific polymatroid (E, ρ i ) and assume that the strategy space of player i is defined by the (player-specific) polymatroid base polytope P ρi .
From now on, when we mention a polymatroid congestion game, we mean a weighted atomic splittable polymatroid congestion game. We give three examples of polymatroid congestion games: Every queue q has a single server with exponentially distributed service time with mean 1/µ q , where µ q > 0. Each packet is routed to a single server q out of a set of allowable queues, depending on the company. Given a distribution of packets x ∈ R m ≥0 , the mean delay of queue q can be computed as c q (x q ) = 1 µq−xq . In this case the sets S i are uniform rank-1 matroids, which are also called singleton games. . . , n} of players. Each player has to store an amount of d i of divisible goods in different area's. Each area j can be served from any storing facility within a given set S j ∈ E. The sets S j may overlap, even for the same player i. However, due to reliability reasons, a player cannot store more than d j goods in one storing facility. The cost c i,e for using a specific storing facility depends on the total amount of goods that have to be stored in storing facility e. The more goods need to be stored, the larger the cost to use it. In this setting, the strategy space of every player i ∈ N corresponds to the base polytope P di·rki , where rk i is the rank function of a transversal matroid. Example 2.3 (Matroid Congestion Games). Consider an atomic splittable matroid congestion model, where for every i ∈ N the allowable subsets are the base set B i of a matroid M i = (E, I i ). The rank function rk i : 2 E → R of matroid M i is defined as: rk i (S) := max{|U | | U ⊆ S and U ∈ I i } for all S ⊆ E, and is submodular, monotone and normalized [25]. Moreover, the characteristic vectors of the bases in B i are exactly the vertices of the polymatroid base polytope P rki . It follows that the polytope P i := { x ∈ R |Bi| + | B∈Bi x B = d i } corresponds to strategy distributions that lead to load vectors in the following polytope: . Hence matroid congestion models are a special case of polymatroid congestion models. Both the singleton games in Example 2.1 and the transversal games in Example 2.2 are a special case of matroid congestion games.

Bidirectional Flow Polymatroids
We provide a sufficient condition for a class of polymatroid congestion games to have a unique Nash equilibrium. We prove that if the strategy space of every player is the base polytope of a bidirectional flow polymatroid, Nash equilibria are unique. In order to define the class of bidirectional flow polymatroids we first discuss some basic properties of polymatroids. We start with a generalization of the strong exchange property for matroids. Let χ e ∈ Z |E| be the characteristic vector with χ e (e) = 1, and χ e (e ′ ) = 0 for all e ′ = e. Lemma 3.1 (Strong exchange property polymatroids (Murota [22])). Let P ρ be a polymatroid base polytope defined on (E, ρ). Let x, y ∈ P ρ and suppose x e > y e for some e ∈ E. Then there exists an e ′ ∈ E with x e ′ < y e ′ and an ǫ > 0 such that: This exchange property will play an important role in the definition of bidirectional flow polymatroids. Given a strategy x in the base polytope of polymatroid (E, ρ), we are interested in the exchanges that can be made between x e and x e ′ for some resources in e, e ′ ∈ E. For that, we define a directed exchange graph D( x) = (E, V ), where the set of vertices equals the set of resources E. The edge set is We define exchange capacitieŝ c x (e, e ′ ) (following notation of Fujishige [9]), which denotes the maximal amount of load that can be exchanged in x between resources e and e ′ . More formally: We use Lemma 3.1 to prove the following: Lemma 3.2. Let P ρ be a polymatroid base polytope defined on (E, ρ). For x, y ∈ P ρ , there exists a flow in D( x) satisfying all supplies and demands, where a resource e with x e > y e has supply of x e − y e and e with x e < y e has a demand of y e − x e .
Proof. Consider the following algorithm: 1. Let f be the zero flow, a flow where we send zero flow along all edges in D( x).
2. If x = y, then stop and output flow f .
3. Choose any element e ∈ E such that x e > y e . 4. Use Lemma 3.1 to find e ′ ∈ E such that x e ′ < y e ′ and ǫ > 0 with and add α flow to edge (e, e ′ ) in flow f . 5. If α < x e − y e , then go to step 4. Otherwise (α = x e − y e ), go to step 2. This is a slightly changed version of Fujishige [9,Theorem 3.27], where the roles of x and y are switched. The only difference between this lemma and Fujishige [9,Theorem 3.27] is that we do not change y to x with exchanges that only can be made on strategy y (which is proven in Fujishige [9, Theorem 3.27]) but even with exchanges that can be executed on both x and y. It follows that these exchanges can also be translated to a flow in D( x). The difference between the two algorithms is step 4, where our algorithm uses the strong exchange property 3.1, whereas Fujishige's algorithm only requires the weak one, defined in [9, Section 2.2]. Therefore the results proved in Fujishige [9, Theorem 3.27] are still valid for our algorithm.
According to Fujishige [9,Theorem 3.27], the algorithm transforms y into x with at most ⌊|E| 2 /4⌋ elementary transformations described in Lemma 3.1, such that each component y e with y e < x e monotonically increases and each component y e with y e > x e monotonically decreases. Therefore f satisfies all supplies an demands as described in the lemma. Flow f also satisfies all capacity constraints, as every pair of resources (e, e ′ ) is considered at most once, and all exchanges can be done on x. Hence f (e,e ′ ) ≤ĉ x (e, e ′ ), thus f is a flow in D( x) satisfying all supplies and demands.
The flow f mentioned in Lemma 3.2 is a flow from the perspective of strategy x and therefore we call this a directed flow. In the following we define a bidirectional flow. Let P ρ again be a polymatroid base polytope on set E. For any x, y ∈ P ρ define the capacitated graph D( x, y) on vertices E. An edge (e, e ′ ) exist if there is an ǫ > 0 such that x + ǫ(χ ′ e − χ e ) ∈ P ρ and y + ǫ(χ e − χ ′ e ) ∈ P ρ . For edges (e, e ′ ) we define capacitieŝ c x, y (e, e ′ ) as follows: c x, y (e, e ′ ) := max{α| x + α(χ e ′ − χ e ) ∈ P ρ and y + α(χ e − χ ′ e ) ∈ P ρ } A bidirectional flow is a flow in D( x, y) where every resource e with x e > y e has supply of x e − y e and every resource e with x e < y e has a demand of y e − x e . Such a flow might not exists. In that case we say that x and y are conflicting strategies.
We are now ready to define the class of bidirectional flow polymatroids:

A Uniqueness Result
In this section we prove that when the strategy space of every player is the base polytope of a bidirectional flow polymatroid, equilibria are unique. We denote the marginal cost of player i on resource e ∈ E by µ i,e ( x) = c i,e (x e ) + x i,e c ′ i,e (x e ). An equilibrium condition for polymatroid congestion games, a result that follows from [12, Lemma 1], is as follows: Lemma 4.1. Let x be a Nash equilibrium in a polymatroid congestion game. If x i,e > 0, then for all e ′ ∈ E for which there is an ǫ > 0 such that In the rest of this section we will prove the following theorem: If for a polymatroid congestion game, the strategy space for every player is the base polytope of a bidirectional flow polymatroid, then the equilibria of this game are unique.
From now on we assume x = ( x i ) i∈N and y = ( y i ) i∈N are strategy profiles, where strategies x i and y i are taken from the base polytope P ρi of a player-specific bidirectional flow polymatroid. Before we prove Theorem 4.2, we first introduce some new notation. We define E + = {e ∈ E|x e > y e } and E − = {e ∈ E|x e < y e } as the sets of globally overloaded and underloaded resources. Define E = = {e ∈ E|x e = y e } as the set of resources on which the total load does not change. In the same way we define playerspecific sets of locally underloaded and overloaded resources E i,+ = {e ∈ E|x i,e > y i,e } and E i,− = {e ∈ E|x i,e < y i,e }. We also introduce four player sets: We can distinguish between two cases. Either E = E = , thus x e = y e for all resources e ∈ E, or E = E = , which implies that E + and E − are non-empty. Note that the first term in the last expression is non-negative and the second one is non-positive. As the whole equation should be positive, we need that this first term is strictly positive and therefore N + > = ∅.
For each player i we create a graph G( x i , y i ) from graph D( x i , y i ) by adding a supersource s i and a super-sink t i to D( x i , y i ). We add edges from s i to e ∈ E i,+ with capacity x i,e − y i,e and edges from e ∈ E i,− to t i with capacity y i,e − x i,e . Graph G( x i , y i ) is visualized in Figure 2.
Recall that strategies x i and y i are both chosen from the base polytope of a bidirectional flow polymatroid. Therefore there exists a flow f i in D( x i , y i ) where every resource e ∈ E i,+ has a supply of x i,e − y i,e and e ∈ E i,− a demand of y i,e − x i,e . Using f i we define a flow f ′ i in G( x i , y i ) as follows: Proof. If E = E = , then using Lemma 4.3 we have that N + > = ∅, and we pick a player i ∈ N + > . Flow f ′ i can be decomposed into flow carrying s i -t i paths, and we will show that there exists a path in this path decomposition that goes from s i to a vertex e 1 ∈ E i,+ ∩ E + , and, after visiting possibly other vertices, finally goes through a vertex e k ∈ E i,− ∩ E − to t i . To see this consider the cut δ(E + ), following notation by Schrijver [28], as visualized in Figure 2. Recall that i ∈ N + > , hence, e∈E + x i,e − y i,e > 0. Thus, in f ′ i more load enters E + from s i , than leaves E + to t i . This implies that in the flow decomposition of f ′ i there must be a path that goes from s i to a vertex e 1 ∈ E i,+ ∩ E + , crosses cut δ(E + ) an odd number of times to a vertex e k ∈ E i,− ∩ (E − ∪ E = ) before ending in t i . As this is a flow-carrying path in f ′ i , it exists in G( x i , y i ).
If E = E = , pick any player i for which there exists a resource e with x i,e = y i,e and look at the path decomposition of f ′ i . Every path (s i , e 1 , . . . , e k , t i ) in this decomposition is a path such that e 1 ∈ E i,+ and e k ∈ E i,+ . As E = E = , it also holds that e 1 ∈ E i,+ ∩E = and e k ∈ E i,− ∩ E = . As this is a flow-carrying path in Proof of Theorem 4.2. Assume x and y are both Nash equilibria. Using Lemma 4.4 we find a path (s i , e 1 , . . . , e k , t i ) in G( x i , y i ) such that e 1 ∈ E i,+ ∩ (E + ∪ E = ) and e k ∈ E i,− ∩ (E − ∪ E = ). Since every edge (e j , e j+1 ) exists in G( x i , y i ), for all j ∈ {1, . . . , k − 1} we get: Using Lemma 4.1 we obtain for x: and similarly for y: Recall that µ i,e ( x) = c i,e (x e ) + x i,e c ′ i,e (x e ). As e 1 ∈ E i,+ , we have that x i,e1 > y i,e1 . Because c i,e1 is strictly increasing and e 1 ∈ ( Moreover, since c i,e1 is convex, the slope of c i,e1 is non-decreasing and, hence, c ′ i,e1 (x e1 ) ≥ c ′ i,e1 (y e1 ). Putting things together, we get Similarly, as e k ∈ E i,− ∩ (E − ∪ E = ), we have: Combining (3), (4), (5) and (6), we have: This is a contradiction and therefore either strategy x i or y i is not a Nash equilibrium for player i.

Applications
In this section we demonstrate that bidirectional flow polymatroids are general enough to allow for meaningful applications. As described in Example 2.3, matroid congestion games belong to polymatroid congestion games. A subclass of matroids are base orderable matroids introduced by Brualdi [6] and Brualdi and Scrimger [7]. We prove that polymatroids defined by the rank function of a base orderable matroid belong to the class of bidirectional flow polymatroids. Therefore, all matroid congestion games for which the player-specific matroids are base orderable have unique equilibria.
Theorem 5.2. Let rk be the rank function of a base orderable matroid (E, rk). Then, for any d ≥ 0, the polymatroid (E, d · rk) is a bidirectional flow polymatroid.
Proof. Polytope P i in Example 2.3 describes exactly how some player-specific weight d i can be divided over different bases in B i to obtain a feasible strategy x i ∈ P di·rk . In this proof we use the same polytope structures, but remove the player specific index i. Thus polytope P describes how weight d can be divided over bases in B to obtain a feasible strategy x ∈ P d·rk . We call vector x ′ ∈ P a base decomposition of x if it satisfies x e = B∈B;e∈B x ′ B for all e ∈ E. Given two vectors x, y ∈ P d·rk , we look at the differences between two base decompositions x ′ , y ′ ∈ P . We introduce sets B + , B − ⊂ B that will contain respectively the overloaded and underloaded bases: Using these sets we create the complete directed bipartite graph D B ( x, y) on vertices As the total supply equals the total demand, there exists a transshipment t from strategies B ∈ B + to strategies B ′ ∈ B − , such that, when carried out, we obtain y ′ from x ′ . We denote by t (B,B ′ ) the amount of load transshipped from B ∈ B + to B ′ ∈ B − .
In the remainder of the proof, we use transshipment t to construct a flow f in graph D( x, y). As the polymatroid is defined by the rank function of a base orderable matroid, for every pair of bases (B, B ′ ) there exists a bijective function g B,B ′ : B → B ′ such that both B − e + g B,B ′ (e) ∈ B and B ′ + e − g B,B ′ (e) ∈ B for all e ∈ B. Note that when e ∈ B ∩ B ′ , g B,B ′ (e) = e. Define Then we define flow f as: f (e,e ′ ) = (B,B ′ )∈B 2 e,e ′ t B,B ′ for all (e, e ′ ) ∈ E × E. Flow f does satisfy all demands and supplies in D( x, y) as f is created from base decompositions x ′ , y ′ for strategy profiles x and y. Note that: Then x ′′ is a base decomposition of strategy x + f (e,e ′ ) (χ e ′ − χ e ), and thus f (e,e ′ ) ≤ c x, y (e, e ′ ). Therefore f is a bidirectional flow between x and y.
An application of these results can be found in the spanning tree games.  E) with non-negative, increasing, differentiable, convex and player specific edge costs functions c i,e for all e ∈ E and i ∈ N . In a spanning tree game, every player i is associated with a weight d i and a subgraph G i of G. A strategy for player i is to divide it's weight along the spanning trees of G i , to minimize his total costs. If we can design G to be generalized series-parallel then P di·rki is a bidirectional flow polymatroid, where rk i be the rank function for the graphic matroid on subgraph G i , (cf. Figure 1). Theorem 5.2 implies that equilibria will be unique.
For graphic matroids, the generalized series-parallel graph is the maximal graph structure that allows for a bidirectional flow between every pair of strategies.
Theorem 5.4 (Korneyenko [19], Nishizeki [23]). A graph is generalized series-parallel if and only if it does not contain the K 4 as a minor.
Let rk be the rank function for the graphic matroid on the K 4 , we show that there exists two conflicting strategies x, y ∈ P rk , thus there does not exist a flow f in D( x, y).
Example 5.5. Polymatroid (E, rk) based on the rank function of the graphic matroid on the K 4 is not a bidirectional flow polymatroid. Let the resources be numbered as in Figure 3 and look at the strategies x = (1, 1, 0, 0, 0, 1) and y = (0, 0, 1, 1, 1, 0). Graph D( x, y) is depicted in Figure 3. Then there is no flow f in D( x, y) that satisfies all supplies and demands. Resource 1 and 6 have both a supply of 1 and can only exchange load with resource 4 , which only has demand 1. Thus such a flow f does not exist, and (E, rk) is not a bidirectional flow polymatroid.

Non-Matroid Set Systems
We now derive necessary conditions on a given set system (S i ) i∈N so that any atomic splittable congestion game based on (S i ) i∈N admits unique equilibria. We show that the matroid property is a necessary condition on the players' strategy spaces that guarantees uniqueness of equilibria without taking into account how the strategy spaces of different players interweave. 2 To state this property mathematically precisely, we introduce the notion of embeddings of S i in E. An embedding is a map τ := (τ i ) i∈N , where every τ i : Definition 6.1. A family of set systems S i ⊆ 2 Ei , for i ∈ N is said to have the strong uniqueness property if for all embeddings τ , the induced game with isomorphic strategy space φ τ (S) has unique Nash equilibria.
Since for bases of matroids any embedding τ i with isomorphism φ τi has the property that φ τi (S i ) is again a collection of bases of a matroid, we obtain the following immediate consequence of Theorem 4.2. For obtaining necessary conditions we need a certain property of non-matroids stated in the following Lemma. Its proof can be derived from the proof of Lemma 5.1 in [15], or the proof of Lemma 16 in [2].
Theorem 6.4. Let |N | ≥ 3 and assume that for all i ∈ N , S i is a non-matroid set system. Then, (S i ) i∈N does not have the strong uniqueness property.
We can assume w.l.o.g. that each set system S i forms an anti-chain (in the sense that X ∈ S i , X ⊂ Y implies Y ∈ S i ) since cost functions are non-negative and strictly increasing. Let us call a non-empty set system S i ⊆ 2 Ei a non-matroid if S i is an anti-chain and (E i , {X ⊆ S : S ∈ S i }) is not a matroid. LetẼ = i∈N τ i (E i ) denote the set of all resources under the embeddings τ i , i ∈ N . The costs on all resources inẼ \ (τ 1 (E 1 ) ∪ τ 2 (E 2 ) ∪ τ 3 (E 3 )) are set to zero. Also, the demands of all players d i with i ∈ N \ {1, 2, 3} are set to zero. This way, the game is basically determined by the players 1, 2, 3. We set the demands d 1 = d 2 = d 3 = 1.
Let us choose two sets X, Y in S 1 and {a, b, c} ⊆ X ∪ Y as described in Lemma 6.3. Let e := τ 1 (a), f := τ 1 (b) and g := τ 1 (c). We set the costs of all resources in τ 1 (E 1 ) \ (τ 1 (X) ∪ τ 1 (Y )) to some very large cost M (large enough so that player 1 would never use any of these resources). The cost on all resources in (τ 1 (X) ∪ τ 1 (Y )) \ {e, f, g} is set to zero. This way, player 1 always chooses a strategy τ 1 (Z) ⊆ τ 1 (X) ∪ τ 1 (Y ) which, by Lemma 6.3, either contains e, or it contains both f and g. We apply the same construction for player 2 and 3, only changing the role of e to act as f and g, respectively.
Note that the so-constructed game is essentially isomorphic to the routing game illustrated in Figure 4 if we interpret resource e as arc (s 1 , t 1 ), resource f as arc (s 2 , t 2 ), and resource g as arc (s 3 , t 3 ). On every edge there is a player specific cost function, given in Table 1.
Every player has two possible paths: the direct path that uses only one edge, or the indirect path that uses two edges. We show that the game where everyone puts all their weight on the direct path is a Nash equilibrium, as is the game where everybody puts their weight on the indirect path. On the other hand, when all players put their weight on the indirect direct route, player 1 can also not deviate, as: The same inequalities hold for player 2 and 3. And therefore everyone playing the direct route, or everyone playing the indirect route both results in a Nash equilibrium.

A Characterization for Undirected Graphs
In Section 6 we proved that non-matroid set systems in general do not have the strong uniqueness property when there are at least three players, by constructing embeddings τ i that lead to the counter example in Figure 4. This example also gives new insights in uniqueness of equilibria in network congestion games. In the following, we give x 3 x + 1 1 k−2 (x + 1) x + M Player 2 x + 1 x 3 1 k−2 (x + 1) x + M Player 3 x + 1 a characterization of graphs that guarantee uniqueness of Nash equilibria even when player-specific cost functions are allowed.
1. An undirected graph G is said to have the uniqueness property if for any atomic splittable network congestion game on G = (V, E), equilibria are unique.
Note that in the above definition, we do not specify how source-and sink vertices are distributed in V . We obtain the following result which is related to Theorem 3 of Meunier and Pradeau [20], where a similar result is given for non-atomic congestion games with player-specific cost functions.
2. An undirected graph has the uniqueness property if and only if G has no cycle of length 3 or more.
Proof. Let G = (V, E) be the network in an atomic splittable network congestion game. Assume there exists a cycle C in G of length k, with k ≥ 3. Already for three players, we can create a game with multiple equilibria by generalizing the previous example visualized in Figure 4. Pick three clockwise adjacent vertices v 1 , v 2 , v 3 in cycle C and create three players which have equal weight 1. Player 1 has source v 1 and sink v 2 , player 2 has source v 2 and sink v 3 and player 3 has source v 3 and sink v 1 . Let c i,e (x) be the cost function for player i at resource e. Define c i,e (x) as in Table 2.
For the same reason as in Example 4 this game has two Nash equilibria: one where all players send their flow clockwise, another where all players send all flow counter clockwise.
On the other hand, assume no cycle of length 3 or more in G exists, then G is a tree with parallel edges. Thus, for every source s and sink t, there is a unique path from s to t in G modulo parallel edges. Therefore, players only have to decide on how to divide their weight over every set of parallel edges they encounter. As the total cost for a player is just the sum of the costs for all resources separately, players compete only in sets of parallel edges. Atomic splittable congestion games on parallel edges with player-specific cost functions are proven to have a unique Nash equilibrium by Orda et al. [24]. Thus when G does not contain cycles of length 3 or more, Nash equilibria are unique.