Enforcing Efficient Equilibria in Network Design Games via Subsidies

Abstract

The efficient design of networks has been an important engineering task that involves challenging combinatorial optimization problems. Typically, a network designer has to select among several alternatives which links to establish so that the resulting network satisfies a given set of connectivity requirements and the cost of establishing the network links is as low as possible. The Minimum Spanning Tree problem, which is well-understood, is a nice example.

In this paper, we consider the natural scenario in which the connectivity requirements are posed by selfish users who have agreed to share the cost of the network to be established according to a well-defined rule. The design proposed by the network designer should now be consistent not only with the connectivity requirements but also with the selfishness of the users. Essentially, the users are players in a so-called network design game and the network designer has to propose a design that is an equilibrium for this game. As it is usually the case when selfishness comes into play, such equilibria may be suboptimal. In this paper, we consider the following question: can the network designer enforce particular designs as equilibria or guarantee that efficient designs are consistent with users’ selfishness by appropriately subsidizing some of the network links? In an attempt to understand this question, we formulate corresponding optimization problems and present positive and negative results.

Appendix: Proof of Theorem 5

The proof is based on a reduction from Independent Set in 3-regular graphs and uses an inapproximability result due to Berman and Karpinski [7]. Given a 3-regular graph H with n nodes and 3n/2 edges, we construct an instance of a broadcast game consisting of a graph G as follows. The graph G has a node for each node and each edge of H and an additional root-node r. We denote by U the set of nodes of G that correspond to a node of H and by V the set of nodes of G that correspond to an edge of H. For each non-root node of G, there is an edge connecting it with the root; these edges have unit weight. A node of V that corresponds to an edge (u,v) in H is connected with edges to the nodes of U that correspond to the nodes u and v of H. The weight of these edges is $\frac{2+\delta}{3}$ for some δ∈(0,1/12]. Clearly, the subgraph of G induced by the nodes in U∪V (i.e., all nodes besides r) is bipartite; in this subgraph, the nodes of U have degree 3 while the nodes of V have degree 2.

We claim that the graph H has an independent set of size m if and only if the broadcast game has an equilibrium of weight 5n/2−(1−δ)m. Consider a spanning tree T of G and let F be the forest obtained by removing the edges of T that are adjacent to r. We call a branch of T any subgraph consisting of a connected component of F, the edge connecting a node of this connected component to r in T, and r itself.

For any spanning tree of G, each of its branches can belong to one of the following types (see Fig. 7):

• Type A: It consists of a single edge connecting the root to a node in U∪V (see Fig. 7a).

  Fig. 7

  

  Examples with branches considered in the proof of Theorem 5. Black and white nodes denote nodes of U and V, respectively while the grey nodes represent the root r. (a) A branch of type A. (b) A branch of type B. (c) A branch of type C. (d) and (e) Branches of type D. (f) and (g) Branches of type E

• Type B: It consists of an edge connecting the root to a node in U which in turn is connected with its three adjacent nodes of V (see Fig. 7b).

• Type C: It consists of an edge connecting the root to a node in U∪V which is connected to either one or two of its adjacent nodes in G (see Fig. 7c).

• Type D: It is a tree of depth exactly 3 rooted at r (see Figs. 7d and 7e).

• Type E: It is a tree of depth at least 4 rooted at r (see Figs. 7f and 7g).

We will first prove that if T is an equilibrium for the broadcast game in G, then it has a very special structure. In particular, none of its branches rooted at r can be of type C, D, or E. Assume otherwise and let h be such a branch:

• If h is of type C, consider a leaf u of h. The first edge in the path from u to r in h (i.e., the one adjacent to u) is not used by any other player besides the one associated with node u while the second edge of the path is used by at most 3 players (i.e., the players associated with the leaves and the player associated with the node of h which is connected with r). Thus, the cost player u experiences is at least $\frac{2+\delta}{3}+1/3>1$ and, hence, this player has an incentive to change her strategy and use the direct edge from u to r. See Fig. 7c for an example.

• If h is of type D, then it has at most 7 non-root nodes. Consider a leaf u that is at distance 3 from r. If u belongs to U, then its adjacent node in h has degree 2. Thus, the first edge of the path from u to r in h is not used by any other player besides the one associated with u while the second edge in the path is used by at most 2 players. In total, the cost the player associated with node u experiences in these two edges of the path is $\frac{2+\delta}{3}+\frac{2+\delta}{6}>1$ and, hence, this player has an incentive to change her strategy and use the direct edge from u to r (see Fig. 7d). If u belongs to V, its adjacent node in h belongs to U and the next node in the path from u to r belongs to V. Thus, the first edge of the path from u to r in h is not used by any other player besides the one associated with u, the second edge in the path is used by at most 3 players, and the third edge in the path (the one adjacent to r) is used by at most 7 players. In total, the cost the player associated with node u experiences in these three edges of the path is $\frac{2+\delta }{3}+\frac{2+\delta}{9}+1/7>1$ and, hence, this player has an incentive to change her strategy and use the direct edge from u to r (see Fig. 7e).

• If h is of type E, consider a leaf u that is at maximum distance (i.e., at least 4) from r. If u belongs to U, then its adjacent node in h has degree 2. Thus, the first edge of the path from u to r in h is not used by any other player besides the one associated with u while the second edge in the path is used by at most 2 players. In total, the cost the player associated with node u experiences in these two edges of the path is $\frac{2+\delta }{3}+\frac{2+\delta}{6}>1$ and, hence, this player has an incentive to change her strategy and use the direct edge from u to r (see Fig. 7f). If u belongs to V, its next two nodes in the path from u to r in h belong to U and V, respectively. Thus, the first edge of the path from u to r in h is not used by any other player besides the one associated with u, the second edge in the path is used by at most 3 players, and the third edge in the path is used by at most 4 players. In total, the cost the player associated with node u experiences in these three edges of the path is at least $\frac{2+\delta}{3}+\frac{2+\delta}{9}+\frac{2+\delta }{12}>1$ and, hence, this player has an incentive to change her strategy and use the direct edge from u to r as well (see Fig. 7g).

Instead, if T consists only of branches of types A and B, no player has an incentive to deviate. Indeed, assume that a player associated with a node u in a branch h ₁ has an incentive to change her strategy and use a new path. Note that the cost she experiences on the edges of h ₁ she uses is at most 1. Clearly, her new path cannot include an edge incident to the root which does not belong to any branch since the cost experienced in such a path would be at least 1. So, assume that the new path of the player contains the edges of another branch h ₂ in order to connect u to r. Clearly, this path should contain an edge of G that is not contained in any branches (and, hence, it is not used by any player besides the one associated with node u) while the first edge of branch h ₂ that the path contains is used by exactly one player besides the one associated with node u; this follows by the structure of G and by the fact that branches of T are of type A or B (and, hence, the new path enters branch h ₂ through one of its leaves). Thus, the cost the player experiences in the new path is at least $\frac{2+\delta}{3}+\frac {2+\delta}{6}>1$ and, hence, she has no incentive to deviate.

Now, consider a spanning tree T that is an equilibrium for the broadcast game in G and let m be the number of branches of type B it contains. Clearly, the weight of the edges in such a branch is 3+δ while the total weight of the edges in branches of type A equals the number of nodes in U∪V which do not belong to branches of type B, i.e., 5n/2−4m. Therefore, the total weight of the edges of T is 5n/2−(1−δ)m. Let I be the set of nodes of H which correspond to the nodes of the branches of type B that are connected to r in T. Due to the structure of G and T, I is an independent set of H with size m. Also, consider any independent set in H with size m. We can conversely construct a spanning tree of G which consists of branches of type A and B and, hence, is an equilibrium: for each node of U corresponding to a node in I, we create a branch of type B by connecting this node to r and to its three adjacent nodes in V. In this way, we create m branches of type B. Also, we create 5n/2−4m branches of type A by connecting each node of U∪V that does not participate in branches of type B to the root through their direct edges. The cost of this equilibrium tree is again 5n/2−(1−δ)m.

Now, we use the inapproximability result due to Berman and Karpinski [7]. Their result can be thought of as a polynomial-time reduction from the decision version of Satisfiability. The reduction uses a constant ϵ∈(0,1/2). Given an instance ϕ of Satisfiability, they construct an instance of Independent Set which consists of a 3-regular graph H with 284k nodes (for some parameter k) such that

• H has an independent set of size at least (140−ϵ)k if ϕ is satisfiable, and

• H has no independent set of size more than (139+ϵ)k if ϕ is not satisfiable.

Using the particular graphs as input to our reduction, we can view it as a reduction from Satisfiability as well. Given an instance ϕ of Satisfiability, our reduction defines a broadcast game such that

• there exists an equilibrium of total weight at most 570+140δ+(1−δ)ϵ if ϕ is satisfiable, and

• there exists no equilibrium of total weight less than 571+139δ−(1−δ)ϵ if ϕ is not satisfiable.

By selecting ϵ and δ to be arbitrarily small, we conclude that approximating the minimum total weight among all equilibria (and, hence, the price of stability) within a factor better than 571/570 is NP-hard.