A Clash of Models

We stated in Observation 5.2 that k-BE ⊆ BE holds. In the following, we will discuss the limits of several proof techniques to identify the parameters for which k-BE = BE, i.e., both equilibria concepts coincide. Specifically, we ask

𝛼 𝑘 1 2 √𝑛/2 𝑛1−u� 𝑛 12𝑛log 𝑛 1 2 6 𝑘 = 2√𝛼

𝑘 = 2 ⋅ 51+√log u�+ 24log 𝑛 + 3

4.667 ⋅ 3⌈1/u�⌉+ 8

PoA= 𝑂(u�

u�+ 𝑘)

PoA= 𝑂(1)

PoA= u�(√u�)

PoA= 𝑂(5√log u�log 𝑛)

PoA= 𝑂(1)

Figure 5.5: Overview of our results from Theorem 5.15 and Theorem 5.28. The light blue area indicates where buy equilibria and 𝑘-local buy equilibria coincide and the orange lines mark the ranges that are covered by different proofs.

for which combinations of 𝑘 and 𝛼 any 𝑘-local buy equilibrium diameter is smaller than 𝑘. If this is true, then a 𝑘-local operation can achieve the same result as an arbitrary operation. Theorem 5.15 combines the results, which we will prove below.

Theorem 5.15. The equilibrium concepts 𝑘-local buy equilibrium and buy equilibrium coincide for the following parameter combinations and yield the respective price of anarchy results (cf. Figure 5.5): We have k-BE = BE for

⎧ { { { { { { { ⎨ { { { { { { { ⎩ 𝛼 ∈ (0, 1) ∧ 𝑘 ≥ 2 ⇒PoA = O(1), 𝛼 ∈ [1, √𝑛/2] ∧ 𝑘 ≥ 6 ⇒PoA = O(1),

𝛼 ∈ [1, 𝑛1−u�] ∧ 𝜀 ≥ _log(u�)1 ∧ 𝑘 ≥ 4.667 ⋅ 3⌈1/u�⌉+ 8 ⇒PoA = O(3⌈1/u�⌉), 𝛼 ∈ [1, 12𝑛lg 𝑛] ∧ 𝑘 ≥ 2 ⋅ 51+√lg u�+ 24lg(𝑛) + 3 ⇒ PoA = O(5√lg u�lg 𝑛),

𝛼 ≥ 12𝑛log 𝑛 ∧ 𝑘 ≥ 2 ⇒PoA = O(1).

Lemma 5.16. For parameters 0 < 𝛼 < 1 and 𝑘 ≥ 2, it holds k-BE = BE and the price of anarchy is 1.

agents 𝑢, 𝑣 ∈ 𝐺[𝑆] that are not connected by one edge: i.e., 𝑑u�[u�](𝑢, 𝑣) = 2. In

this case, creating an edge {𝑢, 𝑣} is an improving response for 𝑢. Hence, the only equilibrium graph for 𝛼 < 1 is a clique, which is also the optimal solution (cf. [Fab+03]).

Lemma 5.17 ([Dem+07], Theorem 4). For parameters 1 ≤ 𝛼 ≤ √𝑛/2 and 𝑘 ≥ 6, it holds k-BE = BE and the price of anarchy is at most 6.

Proof. In [Dem+07], the authors show that every shortest path tree rooted at some agent 𝑢 has a height of at most 5. For this, they assume the contrary and show the existence of an improving response where an agent at a distance of at least 6 buys an edge towards 𝑢. This operation is available with 𝑘 ≥ 6, hence every 𝑘-local equilibrium has a diameter of at most 5. In this case, we get k-BE = BE and the price of anarchy bound of [Dem+07] applies.

Lemma 5.18 ([Dem+07], Theorem 10). For parameters 1 ≤ 𝛼 < 𝑛1−u�_{, 𝜀 ≥ 1/ lg(𝑛)}

and 𝑘 ≥ 4.667 ⋅ 3⌈1/u�⌉_{+ 8}, it holds k-BE = BE and the price of anarchy is at most

4.667 ⋅ 3⌈1/u�⌉_{+ 8}.

Proof. In Theorem 10 of [Dem+07], the authors use an inductive argument to find some agent 𝑢 and a radius 𝑑 such that the 𝑑-neighborhood of 𝑢 contains more than (𝑛/2)-many agents. For this, they start with their Lemma 3 (for which only 𝑘 ≥ 2 must hold) and apply their Lemma 9 iteratively. They show that the maximal radius 𝑑, for which their Lemma 9 must be applied, is at most 4.667 ⋅ 3⌈1/u�⌉+ 8, which gives a first lower bound for 𝑘. Using this result, they apply their Corollary 7 to show that actually all agents are contained in a ball of radius 4.667 ⋅ 3⌈1/u�⌉_{+ 7, for which they need the operation of creating an}

edge to an agent at distance 4.667 ⋅ 3⌈1/u�⌉+ 8, which is the second lower bound

for 𝑘.

Using both results, they show that the diameter of every equilibrium is at most 4.667⋅3⌈1/u�⌉+8. By the choice of 𝑘, the same holds for 𝑘-local buy equilibria.

We get k-BE = BE and thus the price of anarchy is at most 4.667 ⋅ 3⌈1/u�⌉+ 8.

Lemma 5.19 ([Dem+07], Theorem 12). For parameters 1 ≤ 𝛼 ≤ 12𝑛 log 𝑛 and

𝑘 ≥ 2 ⋅ 51+√lg u� + 24lg(𝑛) + 3, it holds k-BE = BE and the price of anarchy is at most O(5√lg u�lg 𝑛).

Proof. Similar to the proof of their Theorem 10 in [Dem+07], the authors pro- vide a price of anarchy upper bound for a larger range of 𝛼: Again, they use an inductive argument to find an agent 𝑢 and a radius 𝑑 such that the 𝑑-neighborhood of 𝑢 contains more than (𝑛/2)-many agents. For this, they start with looking at the number of agents in any radius (12 lg 𝑛)-neighborhood and then apply their Lemma 11 iteratively. They show that the maximal radius 𝑑, for which their Lemma 11 must be applied, is at most 51+√lg u�, which gives a first lower bound for 𝑘. Using this result, they apply their Corollary 8 to show that actually all agents are contained in a specific ball, for which they need the operation of creating an edge to an agent at distance 2 ⋅ 51+√lg u�+ 24lg(𝑛) + 3, which is the second lower bound for 𝑘.

Using both, they show that in every equilibrium network there is an agent who contains all others in a ball of radius (8 ⋅ 51+√lg u�+ 24lg(𝑛) + 2). With the choice of 𝑘, the same holds for 𝑘-local buy equilibria and we get k-BE = BE as well as a price of anarchy upper bound of O(5√lg u�lg 𝑛).

Lemma 5.20 ([Alb+14], Theorem 3.6). For parameters 12𝑛 log 𝑛 ≤ 𝛼 and 2 ≤ 𝑘, it holds k-BE = BE and the price of anarchy is O(1).

Proof. In [Alb+14], the authors provide a technical proof that characterizes equilibria for 𝛼 ≥ 12𝑛 log 𝑛. The main insight that is used for their bound is that there are different types of agents (see their Lemma 3.4, which uses their Lemma 3.2 and Lemma 3.3) with which they characterize equilibria and show that any buy equilibrium network with girth of at least 12 ⋅ ⌈log 𝑛⌉ has a diameter of less than 6 ⋅ ⌈log(𝑛)⌉ and hence is a tree. In their Lemma 3.5, they prove that the considered big 𝛼 values ensure a girth of at least 12⌈log 𝑛⌉. The result of their Theorem 3.6 then comes from a comparison to the social optimum and gives a price of anarchy upper bound of at most 1.5.

Interestingly, in all used statements, there are only two statements concerning the creation or deletion of edges. For their Lemma 3.3, the operation of creating an edge to an agent in distance 2 is considered, and for their Lemma 3.5, the operation of deleting an edge is considered. Both operations are available with 𝑘 ≥ 2. Hence, for any 𝑘 ≥ 2, we have k-BE = BE and the price of anarchy bound of 1.5 from [Alb+14] applies.