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![Page 1: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/1.jpg)
Network Creation GamesA Game Theoretic Approach to Network Evolution
Network Creation GamesA Game Theoretic Approach to Network Evolution
November 14th, 2013
Andreas Cord-Landwehr
Source: Fotolia, Jürgen Priewe
Network Creation Games Andreas Cord-Landwehr 1
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 2: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/2.jpg)
Motivation: We want to understand the creation of Internet-like networks by selfishagents, without central design.
Concrete Question: How costly is the lack of coordination?
Model Ideaeach network peer perceived as agentpeers work strategically to improve their own utilitiesutility given by connection maintenance costs plus communication costs
Understand the Evolution of the Internet. . . or a little bit of it
Network Creation Games Andreas Cord-Landwehr 2
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 3: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/3.jpg)
Motivation: We want to understand the creation of Internet-like networks by selfishagents, without central design.
Concrete Question: How costly is the lack of coordination?
Model Ideaeach network peer perceived as agentpeers work strategically to improve their own utilitiesutility given by connection maintenance costs plus communication costs
Understand the Evolution of the Internet. . . or a little bit of it
Network Creation Games Andreas Cord-Landwehr 2
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 4: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/4.jpg)
Motivation: We want to understand the creation of Internet-like networks by selfishagents, without central design.
Concrete Question: How costly is the lack of coordination?
Model Ideaeach network peer perceived as agentpeers work strategically to improve their own utilitiesutility given by connection maintenance costs plus communication costs
Understand the Evolution of the Internet. . . or a little bit of it
Network Creation Games Andreas Cord-Landwehr 2
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 5: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/5.jpg)
Definition (Network Creation Game [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03])
players: 0, 1, . . . , n− 1 =: [n]
strategy space of a player i: Si := 2[n]\i
network graph G[s] that results from the joint strategiess := (s0, . . . , sn−1) ∈ S0 × . . .× Sn−1 is given by
1 nodes [n]
2 edgesi, j ∈ [n]× [n] : j ∈ si ∨ i ∈ sj
=⋃n−1
i=0 (i × si)cost of a player i under the joint strategy s, α > 0:
ci(s) = α · |si |+n−1∑i=0
dG[s](i, j)
(Pure) Nash equilibrium: given by joint strategy s ∈ S0 × . . .× Sn−1 s.t. for eachplayer i and for all s′ ∈ S0 × . . .× Sn−1 with s and s′ to differ only in the i-thcomponent it holds ci(s) ≤ ci(s′).
Network Creation Games by Fabrikant et al.The Model
Network Creation Games Andreas Cord-Landwehr 3
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 6: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/6.jpg)
Definition (Network Creation Game [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03])
players: 0, 1, . . . , n− 1 =: [n]
strategy space of a player i: Si := 2[n]\i
network graph G[s] that results from the joint strategiess := (s0, . . . , sn−1) ∈ S0 × . . .× Sn−1 is given by
1 nodes [n]
2 edgesi, j ∈ [n]× [n] : j ∈ si ∨ i ∈ sj
=⋃n−1
i=0 (i × si)cost of a player i under the joint strategy s, α > 0:
ci(s) = α · |si |+n−1∑i=0
dG[s](i, j)
(Pure) Nash equilibrium: given by joint strategy s ∈ S0 × . . .× Sn−1 s.t. for eachplayer i and for all s′ ∈ S0 × . . .× Sn−1 with s and s′ to differ only in the i-thcomponent it holds ci(s) ≤ ci(s′).
Network Creation Games by Fabrikant et al.The Model
Network Creation Games Andreas Cord-Landwehr 3
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 7: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/7.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 8: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/8.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 9: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/9.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 10: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/10.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 11: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/11.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 12: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/12.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 13: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/13.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 14: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/14.jpg)
NCG with α = 1.5
Network Creation Games by Fabrikant et al.Example
Network Creation Games Andreas Cord-Landwehr 4
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 15: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/15.jpg)
Best Reponses are NP-hardGiven s ∈ S0 × . . .× Sn−1 and i ∈ [n] it is NP-hard to compute the best response forplayer i.
Sketch. We perform reduction from Dominating Set (NP-hard). Assume we hadan algorithm BRi(G[s]) that computes the best response edge selection for node igiven strategy profile graph.
Dominating Set (DS): Given undirected, unweighted graph G = (V ,E). Task is tocompute set D ⊆ V of minimal size s.t. ∀ v ∈ V \ D : ∃ u ∈ D ∧ (v, u) ∈ E
Consider instance G = (V ,E) for DS, then:1 add node n + 1 to G and define α := 1.5 (any value in (1, 2) works)2 D← BRn+1(G)
3 return DResult is dominating set (proof this!) ⇒ BR is NP-hard.
Network Creation Games by Fabrikant et al.Best Response Computation
Network Creation Games Andreas Cord-Landwehr 5
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 16: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/16.jpg)
Best Reponses are NP-hardGiven s ∈ S0 × . . .× Sn−1 and i ∈ [n] it is NP-hard to compute the best response forplayer i.
Sketch. We perform reduction from Dominating Set (NP-hard). Assume we hadan algorithm BRi(G[s]) that computes the best response edge selection for node igiven strategy profile graph.
Dominating Set (DS): Given undirected, unweighted graph G = (V ,E). Task is tocompute set D ⊆ V of minimal size s.t. ∀ v ∈ V \ D : ∃ u ∈ D ∧ (v, u) ∈ E
Consider instance G = (V ,E) for DS, then:1 add node n + 1 to G and define α := 1.5 (any value in (1, 2) works)2 D← BRn+1(G)
3 return DResult is dominating set (proof this!) ⇒ BR is NP-hard.
Network Creation Games by Fabrikant et al.Best Response Computation
Network Creation Games Andreas Cord-Landwehr 5
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 17: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/17.jpg)
Social costs: c(G) =∑
i∈V ci = α · |E|+∑
i,j dG(i, j)
LemmaFor any strategy profile s, we have the social cost lower bound:
c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|
Proof. Note that 2|E| many distances are exactly 1 and hence n(n− 1)− 2|E|distances are at least of length 2. This can be used for the following estimation:
c(G[s]) = α|E|+∑i,j
dG[s](i, j)
≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|
Network Creation Games by Fabrikant et al.Social Cost Lower Bound
Network Creation Games Andreas Cord-Landwehr 6
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 18: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/18.jpg)
Social costs: c(G) =∑
i∈V ci = α · |E|+∑
i,j dG(i, j)
LemmaFor any strategy profile s, we have the social cost lower bound:
c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|
Proof. Note that 2|E| many distances are exactly 1 and hence n(n− 1)− 2|E|distances are at least of length 2. This can be used for the following estimation:
c(G[s]) = α|E|+∑i,j
dG[s](i, j)
≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|
Network Creation Games by Fabrikant et al.Social Cost Lower Bound
Network Creation Games Andreas Cord-Landwehr 6
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 19: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/19.jpg)
Social costs: c(G) =∑
i∈V ci = α · |E|+∑
i,j dG(i, j)
LemmaFor any strategy profile s, we have the social cost lower bound:
c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|
Proof. Note that 2|E| many distances are exactly 1 and hence n(n− 1)− 2|E|distances are at least of length 2. This can be used for the following estimation:
c(G[s]) = α|E|+∑i,j
dG[s](i, j)
≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|
Network Creation Games by Fabrikant et al.Social Cost Lower Bound
Network Creation Games Andreas Cord-Landwehr 6
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 20: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/20.jpg)
TheoremFor α < 1 the price of anarchy is 1.
Claim 1: optimal solution is complete graphLooking at social costs
c(G) = α · |E|+∑i,j
dG(i, j)
sum minimized when |E| is maximized.
Claim 2: every Nash equilibrium has diameter of 1Assume there exists an NE that is not a complete graph:
then there is a “non-existing edge” that can be bought by a player i to improve itsprivate costi.e.: if there is a non-existing edge i, j we have d(i, j) ≥ 2by adding j to si we get c(i)− c′(i) > 0 and hence a contradiction, since α < 1
Hence, the only NE is the complete graph.
NCG Price of Anarchy BoundsSimple Bounds (1/3)
Network Creation Games Andreas Cord-Landwehr 7
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 21: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/21.jpg)
TheoremFor α < 1 the price of anarchy is 1.
Claim 1: optimal solution is complete graphLooking at social costs
c(G) = α · |E|+∑i,j
dG(i, j)
sum minimized when |E| is maximized.
Claim 2: every Nash equilibrium has diameter of 1Assume there exists an NE that is not a complete graph:
then there is a “non-existing edge” that can be bought by a player i to improve itsprivate costi.e.: if there is a non-existing edge i, j we have d(i, j) ≥ 2by adding j to si we get c(i)− c′(i) > 0 and hence a contradiction, since α < 1
Hence, the only NE is the complete graph.
NCG Price of Anarchy BoundsSimple Bounds (1/3)
Network Creation Games Andreas Cord-Landwehr 7
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 22: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/22.jpg)
TheoremFor 1 ≤ α < 2 the price of anarchy is 4/3.
Claim 1: optimal solution is complete graphConsider lower bound c(G[s]) ≥ 2n(n− 1) + (α− 2)|E| that is reached for completegraph.Claim 2: every Nash equilibrium has diameter at most 2Assume ∃ NE with nodes u, v s.t. d(u, v) > 2⇒ su ∪ (u, v) is IR for u⇒ contradiction
Compute worst case ratio:
PoA =C(Star)C(Kn)
=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)
α · n(n−1)2 + n(n− 1)
=(n− 1)(α− 2 + 2n)
(n− 1)n(α/2 + 1)
=2n
nα+22
+α− 2
n(α/2 + 2)<
4α + 2 ≤ 4/3
NCG Price of Anarchy BoundsSimple Bounds (2/3)
Network Creation Games Andreas Cord-Landwehr 8
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 23: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/23.jpg)
TheoremFor 1 ≤ α < 2 the price of anarchy is 4/3.
Claim 1: optimal solution is complete graphConsider lower bound c(G[s]) ≥ 2n(n− 1) + (α− 2)|E| that is reached for completegraph.Claim 2: every Nash equilibrium has diameter at most 2Assume ∃ NE with nodes u, v s.t. d(u, v) > 2⇒ su ∪ (u, v) is IR for u⇒ contradiction
Compute worst case ratio:
PoA =C(Star)C(Kn)
=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)
α · n(n−1)2 + n(n− 1)
=(n− 1)(α− 2 + 2n)
(n− 1)n(α/2 + 1)
=2n
nα+22
+α− 2
n(α/2 + 2)<
4α + 2 ≤ 4/3
NCG Price of Anarchy BoundsSimple Bounds (2/3)
Network Creation Games Andreas Cord-Landwehr 8
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 24: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/24.jpg)
TheoremFor 1 ≤ α < 2 the price of anarchy is 4/3.
Claim 1: optimal solution is complete graphConsider lower bound c(G[s]) ≥ 2n(n− 1) + (α− 2)|E| that is reached for completegraph.Claim 2: every Nash equilibrium has diameter at most 2Assume ∃ NE with nodes u, v s.t. d(u, v) > 2⇒ su ∪ (u, v) is IR for u⇒ contradiction
Compute worst case ratio:
PoA =C(Star)C(Kn)
=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)
α · n(n−1)2 + n(n− 1)
=(n− 1)(α− 2 + 2n)
(n− 1)n(α/2 + 1)
=2n
nα+22
+α− 2
n(α/2 + 2)<
4α + 2 ≤ 4/3
NCG Price of Anarchy BoundsSimple Bounds (2/3)
Network Creation Games Andreas Cord-Landwehr 8
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 25: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/25.jpg)
TheoremFor α > n2 all Nash equilibria are trees and the price of anarchy is constant.
Claim 1: for α > n2 all Nash equilibria are trees
assume there exists a NE that contains a circle v0, . . . , vk , vk+1, . . . , v0 andconsider strategy of v0by removing (v0, v1), G still connected, max-distance improvement is n− 1cost change for v0 is at most (n− 1)(n− 1)− n2 < 0removing edge was IR
Claim 2: for trees the PoA is constant
worst case cost of any tree:c(Tree) = α · |E|+
∑i,j dG(i, j) ≤ α(n− 1) + (n− 1)n2 ≤ 2α(n− 1)
optimum cost:c(Star) = α(n− 1) + (n− 1)(n− 2) + 2(n− 1)
NCG Price of Anarchy BoundsSimple Bounds (3/3)
Network Creation Games Andreas Cord-Landwehr 9
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 26: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/26.jpg)
TheoremFor α > n2 all Nash equilibria are trees and the price of anarchy is constant.
Claim 1: for α > n2 all Nash equilibria are trees
assume there exists a NE that contains a circle v0, . . . , vk , vk+1, . . . , v0 andconsider strategy of v0by removing (v0, v1), G still connected, max-distance improvement is n− 1cost change for v0 is at most (n− 1)(n− 1)− n2 < 0removing edge was IR
Claim 2: for trees the PoA is constant
worst case cost of any tree:c(Tree) = α · |E|+
∑i,j dG(i, j) ≤ α(n− 1) + (n− 1)n2 ≤ 2α(n− 1)
optimum cost:c(Star) = α(n− 1) + (n− 1)(n− 2) + 2(n− 1)
NCG Price of Anarchy BoundsSimple Bounds (3/3)
Network Creation Games Andreas Cord-Landwehr 9
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 27: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/27.jpg)
TheoremFor α < n2 the price of anarchy is O(
√α).
Proof. (consider only n2 > α ≥ 2)Claim 1: PoA ∈ O(α|E|+n2
√α
αn+n2 ).1 star is social cost optimum: our cost lower bound is reached when |E| is
minimized⇒ the social optimum has cost Θ(αn + n2)
2 social cost of every NE at most α|E|+ n2 · 2√α
assume ∃ u, v ∈ V with dG(u, v) ≥ 2√α and consider u creating edge to v
u’s costs change by:
≤ α−b√αc∑
i=12i = α− 2
√α(√α+ 1)
2< 0
this is contradiction and hence diameter < 2√α
Using this, we can compute the price of anarchy as:α|E|+
∑i,j d(i, j)
(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2
√α
αn + n2 = O(α|E|+ n2α
αn + n2
)
NCG Price of Anarchy BoundsA bound for α < n2 (1/4)
Network Creation Games Andreas Cord-Landwehr 10
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 28: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/28.jpg)
TheoremFor α < n2 the price of anarchy is O(
√α).
Proof. (consider only n2 > α ≥ 2)Claim 1: PoA ∈ O(α|E|+n2
√α
αn+n2 ).1 star is social cost optimum: our cost lower bound is reached when |E| is
minimized⇒ the social optimum has cost Θ(αn + n2)
2 social cost of every NE at most α|E|+ n2 · 2√α
assume ∃ u, v ∈ V with dG(u, v) ≥ 2√α and consider u creating edge to v
u’s costs change by:
≤ α−b√αc∑
i=12i = α− 2
√α(√α+ 1)
2< 0
this is contradiction and hence diameter < 2√α
Using this, we can compute the price of anarchy as:α|E|+
∑i,j d(i, j)
(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2
√α
αn + n2 = O(α|E|+ n2α
αn + n2
)
NCG Price of Anarchy BoundsA bound for α < n2 (1/4)
Network Creation Games Andreas Cord-Landwehr 10
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 29: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/29.jpg)
TheoremFor α < n2 the price of anarchy is O(
√α).
Proof. (consider only n2 > α ≥ 2)Claim 1: PoA ∈ O(α|E|+n2
√α
αn+n2 ).1 star is social cost optimum: our cost lower bound is reached when |E| is
minimized⇒ the social optimum has cost Θ(αn + n2)
2 social cost of every NE at most α|E|+ n2 · 2√α
assume ∃ u, v ∈ V with dG(u, v) ≥ 2√α and consider u creating edge to v
u’s costs change by:
≤ α−b√αc∑
i=12i = α− 2
√α(√α+ 1)
2< 0
this is contradiction and hence diameter < 2√α
Using this, we can compute the price of anarchy as:α|E|+
∑i,j d(i, j)
(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2
√α
αn + n2 = O(α|E|+ n2α
αn + n2
)
NCG Price of Anarchy BoundsA bound for α < n2 (1/4)
Network Creation Games Andreas Cord-Landwehr 10
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 30: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/30.jpg)
Claim 2: |E| = O(n2/√α)
(Idea: we associate α-many non-edges with each existing edge.)
Consider arbitrary v ∈ V with own edges e1, . . . , el and define sets Ti :
Ti := u ∈ V |ei ∈ shortest canonical path from v to u
shortest canonical path: arbitrary fixed chosen shortest path for every pair (ensuresthat Tis are disjoint)
The Plan: Consider G′ := (V ,E \ ei) and analyze cases:1 Ti and v are connected in G′
2 Ti and v are not connected in G′
Note: for any path v to u in G′ its length is either∞ or less than 2 · diam(G) < 4√α.
NCG Price of Anarchy BoundsA bound for α < n2 (2/4)
Network Creation Games Andreas Cord-Landwehr 11
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 31: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/31.jpg)
Claim 2: |E| = O(n2/√α)
(Idea: we associate α-many non-edges with each existing edge.)
Consider arbitrary v ∈ V with own edges e1, . . . , el and define sets Ti :
Ti := u ∈ V |ei ∈ shortest canonical path from v to u
shortest canonical path: arbitrary fixed chosen shortest path for every pair (ensuresthat Tis are disjoint)
The Plan: Consider G′ := (V ,E \ ei) and analyze cases:1 Ti and v are connected in G′
2 Ti and v are not connected in G′
Note: for any path v to u in G′ its length is either∞ or less than 2 · diam(G) < 4√α.
NCG Price of Anarchy BoundsA bound for α < n2 (2/4)
Network Creation Games Andreas Cord-Landwehr 11
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 32: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/32.jpg)
Remember G′ = (V ,E \ ei)
Case Ti and v are connected in G′: for all u ∈ Ti we have:dG′(v, u)− dG(v, u) < 4
√α
since ei must improve i ’s distance cost by ≥ α it holds:
α ≤∑u∈Ti
(dG′(v, u)− dG(v, u)) < |Ti |4√α
this gives |Ti | ∈ Ω(√α), i.e., there are Ω(
√α) many vertices u
s.t. (v, u) does not existCase Ti and v are not connected in G′: G′ consists of two components
denote endpoint of ei by wfor v there exist (|Ti | − 1)-many nodes w ′ s.t. (v,w ′) 6∈ G′for w ∈ Ti there are (|V \ Ti | − 1)-many nodes that are notconnected to whence, |Ti | − 1+ |V \Ti | − 1 = n− 2 = Ω(
√α) edges do not exist
For both cases, non-edges counted at most twice (for ordered pairs (v,w) and (w, v))⇒ |E| is an O(1/
√α) fraction of the number of vertex pairs
NCG Price of Anarchy BoundsA bound for α < n2 (3/4)
Network Creation Games Andreas Cord-Landwehr 12
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 33: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/33.jpg)
Remember G′ = (V ,E \ ei)
Case Ti and v are connected in G′: for all u ∈ Ti we have:dG′(v, u)− dG(v, u) < 4
√α
since ei must improve i ’s distance cost by ≥ α it holds:
α ≤∑u∈Ti
(dG′(v, u)− dG(v, u)) < |Ti |4√α
this gives |Ti | ∈ Ω(√α), i.e., there are Ω(
√α) many vertices u
s.t. (v, u) does not existCase Ti and v are not connected in G′: G′ consists of two components
denote endpoint of ei by wfor v there exist (|Ti | − 1)-many nodes w ′ s.t. (v,w ′) 6∈ G′for w ∈ Ti there are (|V \ Ti | − 1)-many nodes that are notconnected to whence, |Ti | − 1+ |V \Ti | − 1 = n− 2 = Ω(
√α) edges do not exist
For both cases, non-edges counted at most twice (for ordered pairs (v,w) and (w, v))⇒ |E| is an O(1/
√α) fraction of the number of vertex pairs
NCG Price of Anarchy BoundsA bound for α < n2 (3/4)
Network Creation Games Andreas Cord-Landwehr 12
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 34: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/34.jpg)
Remember G′ = (V ,E \ ei)
Case Ti and v are connected in G′: for all u ∈ Ti we have:dG′(v, u)− dG(v, u) < 4
√α
since ei must improve i ’s distance cost by ≥ α it holds:
α ≤∑u∈Ti
(dG′(v, u)− dG(v, u)) < |Ti |4√α
this gives |Ti | ∈ Ω(√α), i.e., there are Ω(
√α) many vertices u
s.t. (v, u) does not existCase Ti and v are not connected in G′: G′ consists of two components
denote endpoint of ei by wfor v there exist (|Ti | − 1)-many nodes w ′ s.t. (v,w ′) 6∈ G′for w ∈ Ti there are (|V \ Ti | − 1)-many nodes that are notconnected to whence, |Ti | − 1+ |V \Ti | − 1 = n− 2 = Ω(
√α) edges do not exist
For both cases, non-edges counted at most twice (for ordered pairs (v,w) and (w, v))⇒ |E| is an O(1/
√α) fraction of the number of vertex pairs
NCG Price of Anarchy BoundsA bound for α < n2 (3/4)
Network Creation Games Andreas Cord-Landwehr 12
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 35: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/35.jpg)
Summary:
PoA ∈ O(α|E|+n2√α
αn+n2 )
|E| = O(n2/√α)
This gives PoA ∈ O(√α)
NCG Price of Anarchy BoundsA bound for α < n2 (4/4)
Network Creation Games Andreas Cord-Landwehr 13
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 36: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/36.jpg)
Network Creation Games [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03]
Model: Each node v buys a set of edges Sv (every edge costs α), s.t. v minimizessum of buying cost plus connection costs.Average Distance Model: c(v) = α · |sv |+
∑u∈V d(v, u)
α
PoA bound
0 1 2 3√n/2√
n/2 O(n1−ε) 65n 12n lg n ∞
1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(
√log n) < 5 ≤ 1.5
Maximal Distance Model: c(v) = α · |sv |+ maxu∈V d(v, u)
α
PoA bound
0 1n−2 O(n−1/2) 129 n ∞
1 Θ(1) 2√
log n < 4 ≤ 2
Problems with this model:Equilibria quality/price of anarchy are tightly connected to value of α.Computation of Best-Response is NP-hard.
NCG Price of Anarchy BoundsState of the Art PoA Bounds
Network Creation Games Andreas Cord-Landwehr 14
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 37: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/37.jpg)
Network Creation Games [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03]
Model: Each node v buys a set of edges Sv (every edge costs α), s.t. v minimizessum of buying cost plus connection costs.Average Distance Model: c(v) = α · |sv |+
∑u∈V d(v, u)
α
PoA bound
0 1 2 3√n/2√
n/2 O(n1−ε) 65n 12n lg n ∞
1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(
√log n) < 5 ≤ 1.5
Maximal Distance Model: c(v) = α · |sv |+ maxu∈V d(v, u)
α
PoA bound
0 1n−2 O(n−1/2) 129 n ∞
1 Θ(1) 2√
log n < 4 ≤ 2
Problems with this model:Equilibria quality/price of anarchy are tightly connected to value of α.Computation of Best-Response is NP-hard.
NCG Price of Anarchy BoundsState of the Art PoA Bounds
Network Creation Games Andreas Cord-Landwehr 14
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 38: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/38.jpg)
Network Creation Games [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03]
Model: Each node v buys a set of edges Sv (every edge costs α), s.t. v minimizessum of buying cost plus connection costs.Average Distance Model: c(v) = α · |sv |+
∑u∈V d(v, u)
α
PoA bound
0 1 2 3√n/2√
n/2 O(n1−ε) 65n 12n lg n ∞
1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(
√log n) < 5 ≤ 1.5
Maximal Distance Model: c(v) = α · |sv |+ maxu∈V d(v, u)
α
PoA bound
0 1n−2 O(n−1/2) 129 n ∞
1 Θ(1) 2√
log n < 4 ≤ 2
Problems with this model:Equilibria quality/price of anarchy are tightly connected to value of α.Computation of Best-Response is NP-hard.
NCG Price of Anarchy BoundsState of the Art PoA Bounds
Network Creation Games Andreas Cord-Landwehr 14
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 39: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/39.jpg)
Network Creation Games [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03]
Model: Each node v buys a set of edges Sv (every edge costs α), s.t. v minimizessum of buying cost plus connection costs.Average Distance Model: c(v) = α · |sv |+
∑u∈V d(v, u)
α
PoA bound
0 1 2 3√n/2√
n/2 O(n1−ε) 65n 12n lg n ∞
1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(
√log n) < 5 ≤ 1.5
Maximal Distance Model: c(v) = α · |sv |+ maxu∈V d(v, u)
α
PoA bound
0 1n−2 O(n−1/2) 129 n ∞
1 Θ(1) 2√
log n < 4 ≤ 2
Problems with this model:Equilibria quality/price of anarchy are tightly connected to value of α.Computation of Best-Response is NP-hard.
NCG Price of Anarchy BoundsState of the Art PoA Bounds
Network Creation Games Andreas Cord-Landwehr 14
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 40: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/40.jpg)
Further Models1 Basic Network Creation Games
un-owned edgesnodes can swap connected edgesno parameter α in cost functionBR computation is in P, similar structural results as in NCGs
2 Greedy Network Creation Gamesnodes are only allowed to perform one-edge-change operationsequilibria are 3-approximations to NCG equilibriaBR computation is in P
Other Questions1 convergence dynamics2 price of stability3 effects of local knowledge4 . . .
NCG Price of Anarchy BoundsOther Models & Other Questions
Network Creation Games Andreas Cord-Landwehr 15
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity
![Page 41: Network Creation Games - Heinz Nixdorf Institut · PDF fileNetwork Creation Games A Game Theoretic Approach to Network Evolution November14th,2013 Andreas Cord-Landwehr Source:Fotolia,JürgenPriewe](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8c1d97f8b9ac1058d0a8e/html5/thumbnails/41.jpg)
Thank you for your attention!Thank you for your attention!
Andreas Cord-Landwehr
Heinz Nixdorf Institute& Department of Computer ScienceUniversity of Paderborn
Address: Fürstenallee 1133102 PaderbornGermany
Phone: +49 5251 60-6428Fax: +49 5251 60-6482E-mail: [email protected]: http://wwwhni.upb.de/en/alg/
Source: Fotolia, Jürgen Priewe
Network Creation Games Andreas Cord-Landwehr 16
HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity