Strategic Network Formation Through an Intermediary

Abstract

Settings in which independent self-interested agents form connections with each other are extremely common, and are usually modeled using network formation games. We study a natural extension of network formation games in which the nodes cannot form connections themselves, but instead must do it through an intermediary, and must pay the intermediary to form these connections. The price charged by the intermediary is assumed to be determined by its operating costs, which in turn depend on the total amount of connections it facilitates. We investigate the existence and worst-case efficiency (price of anarchy) of stable solutions in these settings, and especially when the intermediary uses common pricing schemes like proportional pricing or marginal cost pricing. For both these pricing schemes we prove existence of stable solutions and completely characterize their structure, as well as generalize these results to a large class of pricing schemes. Our main results are on bounding the price of anarchy in such settings: we show that while marginal cost pricing leads to an upper bound of only 2, i.e., stable solutions are always close to optimal, proportional pricing also performs reasonably well as long as the operating costs of the intermediary are not too convex.

Appendix: Proof of Lemma 5.3

Lemma 5.3

Consider an instance that achieves the worst-case PoA with intermediary price function p(⋅) and suppose that PoA > 2. Then for all pairs (ij) we have λ_ij = λforsomeλ > 0.

Proof outline

Let τ_y denote min{λ_ij : y_ij > 0} for a given allocation vector y, i.e., τ_y corresponds smallest λ_ij that a pair with positive connection strength can have in y. In the instance that achives the worst-case PoA, let g be an allocation vector corresponding to the minimum social cost and let (f, p(f)) be a stable solution of maximum social cost. This means that the ratio SC(f)/SC(g) equals the worst-case PoA in the instance under consideration.

The proof of the Lemma 5.3 involves a series of steps: The first step (Claim 1 below) involves proving that for the instance under consideration that achieves the worst-case PoA, τ_g ≥ τ_f. The second step (Claim 2 below) involves showing that λ_ij ∈{τ_g, τ_f} for every pair (ij). Given this, the final step (Claim 3 below) involves proving τ_g = τ_f.

The theme of all these steps is to make use of Propositions 1 and 2 to create an instance by perturbing λ_ij values for some intelligently chosen class of pairs, while maintaining the stability of the solution (f, p(f)). Later carefully applying Lemmas 5.1 and 5.2 we show that in the newly created instance, the ratio SC(f)/SC(g) strictly increases. This contradicts our initial assumption that the original instance achieves the worst-case PoA. □

Claim 1

τ_g ≥ τ_f.

Proof

Suppose we had τ_f > τ_g on the contrary. We break the further analysis into two cases:

• 1. Case when p(f) > τ_g: Recall that connection cost incurred by a pair (ij) for a connection strength of y_ij is given by λ_ij(B_ij − y_ij). Now we claim p(f) > τ_g, implies that the pairs in the class T(τ_g) incur strictly positive connection cost in total for the allocation vector f but incur zero connection cost for the allocation vector g. Note: recall that T(τ_g) is the set of all pairs with λ_ij = τ_g.

  Note that p(f) > τ_g implies that the intermediary charges a price higher than τ_g in the solution (f, p(f)). Thus by the first stability condition, f_ij = 0 for (ij) ∈ T(τ_g). This, together with Lemma 5.2 implies B_ij = g_ij for (ij) ∈ T(τ_g). Hence the pairs in T(τ_g) incur zero cost for the allocation vector g. At the same time, the definition of τ_g implies that at least one pair with λ_ij = τ_g has g_ij > 0. Combining it with f_ij = 0 and B_ij = g_ij (as discussed above), we get that the pairs in T(τ_g) incur strictly positive cost in total for the allocation vector f.

  Now let us obtain another instance by increasing λ_ij for all the pairs in T(τ_g) to any fixed Λ in the open interval (p(f), τ_g) while Proposition 2 ensures the stability (f, p(f)). Notice that in this procedure, the connection cost of the pairs in the set T(τ_g) strictly increases for the allocation vector f. However as B_ij = g_ij, they still incur zero connection cost for the allocation vector g even in the transformed instance. Thus during this transformation SC(g) does not change but SC(f) strictly increases. Thus the ratio SC(f)/SC(g) becomes strictly greater in the transformed instance, contradicting the assumption that the original instance achieves the worst-case PoA.

• 2. Case when p(f) ≤ τ_g (and λ¹ ≥ τ_f > τ_g): In this case, we claim that the following holds true

  $$ \frac{\lambda^{1}\cdot {\sum}_{(ij)\in T(\lambda^{1})} (B_{ij} - f_{ij})}{\lambda^{1}\cdot {\sum}_{(ij)\in T(\lambda^{1})} (B_{ij} - g_{ij})} \leq 2 $$

  (11)

  To see it, suppose that the following held true:

  $$ \sum\limits_{(ij)\in T(\lambda^{1})} f_{ij} \geq \sum\limits_{(ij)\in T(\lambda^{1})} g_{ij} $$

  (12)

  If the condition in (12) holds true then it trivially implies the bound in (11). Now suppose that the condition in (12) does not hold. By our assumptions, we have p(f) ≤ τ_g < τ_f < λ¹. Thus the bound given in Lemma 5.1 holds true. It can be shown that the left hand side term in (11) gets maximized when the bound in Lemma 5.1 is met with equality to give us a desired upper bound of 2.

  Now if it were true that PoA > 2 then we can create another instance by reducing λ¹ by a tiny 𝜖 such that 0 < 𝜖 < λ¹ − λ². Note that (f, p(f)) still stays a stable solution in the transformed instance by Proposition 1. However, reduction in λ¹ leads to an increase in the ratio SC(f)/SC(g) by Observation 5.2. This contradicts the assumption that the original instance achieves the worst-case PoA.

Thus assuming τ_f > τ_g leads us to contradictions in both above cases. Hence it proves that τ_g ≥ τ_f. □

Claim 2

λ_ij ∈{τ_f, τ_g} for any pair (ij).

Proof

We already know that τ_g ≥ τ_f. We also know by Lemma 5.2 that B_ij = 0 whenever λ_ij < τ_f ≤ τ_g, thus we can ignore all such pairs. We break the further analysis into two cases:

• 1. Suppose there exists a pair with λ_ij > τ_g: This implies λ¹ > τ_g. Now we claim that the following holds true

  $$\begin{array}{@{}rcl@{}} \frac{\lambda^{1}\cdot {\sum}_{(ij)\in T(\lambda^{1})} (B_{ij} - f_{ij})}{\lambda^{1}\cdot {\sum}_{(ij)\in T(\lambda^{1})} (B_{ij} - g_{ij})} \leq 2 \end{array} $$

  (13)

  To prove our claim, suppose that the following holds:

  $$ \sum\limits_{(ij)\in T(\lambda^{1})} f_{ij} \geq \sum\limits_{(ij)\in T(\lambda^{1})} g_{ij} $$

  (14)

  If the condition in (14) holds true then it trivially implies the bound in (13). Now suppose that the condition in (14) does not hold. We already know that p(f) ≤ τ_f from the first stability condition and the definition of τ_f. Combining this with our assumptions, we have p(f) ≤ τ_f ≤ τ_g < λ¹, giving us p(f) < λ¹. Thus the bound given in Lemma 5.1 holds true. It can be shown that the left hand side term in (13) gets maximized when the bound in Lemma 5.1 is met with equality to give us a desired upper bound of 2.

  Now if it were true that PoA > 2 then we can create another instance by reducing λ¹ by a tiny 𝜖 such that 0 < 𝜖 < λ¹ − λ². Note that (f, p(f)) still stays a stable solution in the transformed instance by Proposition 1. However, reduction in λ¹ leads to an increase in the ratio SC(f)/SC(g) by Observation 5.2. This contradicts the assumption that the original instance achieves the worst-case PoA.

• 2. Suppose there exists a pair with τ_g > λ_ij > τ_f : Note that by definition of τ_g that these pairs satisfy g_ij = 0. Thus for these pairs, B_ij = f_ij. Since we only allow B_ij > 0 in our problem settings, we have f_ij > 0 for all these pairs. This implies that these pairs have zero connection cost in the allocation vector f (because of B_ij = f_ij) but strictly positive connection cost in the allocation vector g (because of g_ij = 0).

  Now Consider all the pairs (uv) which satisfy the condition λ_uv = min{λ_ij : τ_g > λ_ij > τ_f}. By applying Proposition 1, we can reduce λ_uv of all such pairs to τ_f without affecting the stability of (f, p(f)). In this process, the connection cost of all these pairs stays zero in the allocation vector f but strictly decreases in the allocation vector g. Hence in this process SC(f) does not change but SC(g) strictly decreases, increasing the ratio SC(f)/SC(g) leading to a contradiction with the assumption that the original instance achieves the worst-case PoA.

This completes the proof of Claim 2. □

Claim 3

τ_g = τ_f.

Proof

We already proved above in Claim 1 and Claim 2 that in the instance I achieving worst-case efficiency we have λ_ij ∈{τ_f, τ_g} for every pair (ij) and τ_g ≥ τ_f.

Now if τ_g > τ_f, then we can apply the analysis from case (b) of proof of the claim τ_g ≥ τ_f to prove that we can construct another instance by reducing λ¹ = τ_g by a tiny 𝜖, while keeping (f, p(f)) stable, such that ratio SC(f)/SC(g) strictly increases in this transformation. This leads to the contradiction that the original instance could achieve the worst-case PoA hence proving our claim.

Proving Claim 3 completes the last step to prove Lemma 5.3 as discussed in the proof outline of Lemma 5.3. □