[PPT] - Voronoi Games on Cycle Graphs Marios Mavronicolas Burkhard Monien PowerPoint Presentation

SLIDE 1

Introduction The Model Characterization of NE Quality of NE Conclusion

Voronoi Games on Cycle Graphs

Marios Mavronicolas Burkhard Monien Vicky G. Papadopoulou Florian Schoppmann

Department of Computer Science International Graduate School Dynamic Intelligent Systems University of Paderborn

August 26, 2008

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 1 / 22

SLIDE 2

Introduction The Model Characterization of NE Quality of NE Conclusion

Competitive Location

Two ice cream vendors on the beach:

◮ Customers typically buy at nearest

vendor

◮ People are spread evenly ◮ What are the optimal positions?

0m 50m 100m Street Beach Water

◮ Vendors have an incentive to

move towards each other

◮ Hotelling’s law (1929): It is

rational for producers to make products as similar as possible

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 2 / 22

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Introduction The Model Characterization of NE Quality of NE Conclusion

Some non-scientific references to Hotelling’s law...

Los Angeles Times, June 8, 2008 (editorial): “Obama and McCain, the same?”

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 3 / 22

[...] With a Republican president experiencing some

f the worst approval ratings ever, it’s no shock

that the party opted for an unusually centrist candidate. Yet Obama, too, represents a break from Democratic

rthodoxy and is reaching out to the

middle . [...] Some might complain that this means voters will have little to choose between in November. We say: Welcome to the middle , candidates. [...]

SLIDE 4

Introduction The Model Characterization of NE Quality of NE Conclusion

Motivation and Framework

Hotelling’s law:

◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions

Classification by Eiselt et al. (1993) with over 100 references:

1. underlying metric measurable space
2. number of players
3. pricing policy (if any)
4. the equilibrium concept
5. customers’ behavior

They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 22

SLIDE 5

Introduction The Model Characterization of NE Quality of NE Conclusion

Motivation and Framework

Hotelling’s law:

◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions

Classification by Eiselt et al. (1993) with over 100 references:

1. underlying metric measurable space
2. number of players
3. pricing policy (if any)
4. the equilibrium concept
5. customers’ behavior

They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 22

SLIDE 6

Introduction The Model Characterization of NE Quality of NE Conclusion

Motivation and Framework

Hotelling’s law:

◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions

Classification by Eiselt et al. (1993) with over 100 references:

1. underlying metric measurable space
2. number of players
3. pricing policy (if any)
4. the equilibrium concept
5. customers’ behavior

They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 22

SLIDE 7

Introduction The Model Characterization of NE Quality of NE Conclusion

The Model

Definition

A Voronoi game (on a connected undirected graph) is specified by:

◮ Graph G = (V , E) ◮ number of players k ≤ |V |

The strategic game is then:

◮ Strategy set of all players is V , set of profiles is S := V k ◮ Utilities:

◮ Define Fv := S → 2[k], Fv(x) := arg mini∈[k] dist(v, si)

mapping each node to the set of nearest players

◮ Utility of a player i in profile s is:

ui(s) :=

v∈V :i∈Fv (s)

1 |Fv(s)|

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 5 / 22

SLIDE 8

Introduction The Model Characterization of NE Quality of NE Conclusion

Nash Equilibrium

Situation (= strategy profile) where no player can improve:

3 3

◮ ui(s−i, s′ i) ≤ ui(s) holds for all players i and all alternative

strategies s′

i

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 6 / 22

SLIDE 9

Introduction The Model Characterization of NE Quality of NE Conclusion

Social Cost and Example

Social cost is total transport cost: SC(s) :=

v∈V

min

i∈[k]{dist(v, si)}

Optimum is OPT := mins∈S {SC(s)} Loss due to selfish behavior captured by prices of anarchy and stability: PoA := max

s is NE

SC(s) OPT PoS := min

s is NE

SC(s) OPT ( 0

0 := 1 and x 0 := ∞ here)

Cycle graph, 6 players:

4 3 4 3

1 1 2

SC = 3, OPT = 2

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 7 / 22

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Introduction The Model Characterization of NE Quality of NE Conclusion

Related Work (1/2)

Eaton and Lipsey (1975): Voronoi games on a continuous circle

s2 s1 s3 s2 s1 s3 s2 s1,s3

◮ A profile is a NE iff “no firm’s whole market is smaller than

any other firm’s half market”

◮ In a cost-maximizing NE: # players k is even, all players are

paired, pairs are equidistantly located

◮ In an optimum: all players equidistantly located

= ⇒ PoA = 2 and PoS = 1

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 8 / 22

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Introduction The Model Characterization of NE Quality of NE Conclusion

Related Work (2/2)

Only one work on discrete Voronoi games (Dürr and Thang, ESA 2007):

◮ A graph without NE, even for k = 2 ◮ Best response dynamics do not

converge in general, not even on cycle graphs

◮ Deciding existence of NE is NP-hard ◮ Bound on ratio of social cost in

worst/best NE Voronoi games on graphs also similar to (but different than) competitive facility location games (Vetta, FOCS 2002)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 9 / 22

SLIDE 12

Introduction The Model Characterization of NE Quality of NE Conclusion

Cycle Graphs C(n, k)

Major difference to circle:

◮ Points equidistant to more than

ne player have non-zero measure

Convenient representation of profile s:

◮ θ0, . . . , θℓ−1 are used nodes (in

counterclockwise orientation)

◮ di = dist(θi, θi+1) ◮ ci is the number of players on θi

d0 = 2 θ0 θ1 θ2 θ3 θ4

◮ Up to rotation/renumbering, ℓ, (di)i∈Zℓ, and (ci)i∈Zℓ suffice

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 10 / 22

SLIDE 13

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 14

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 15

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 16

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 17

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1). Case ci+1 = 1:

… …

Case ci+2 = 2:

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 18

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1). Case ci+1 = 1:

… …

Case ci+2 = 2:

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 19

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 20

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 21

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 22

Introduction The Model Characterization of NE Quality of NE Conclusion

Characterization

Theorem

A profile with minimum utility γ is a NE if and only if ∀i ∈ Zℓ:

1. ci ≤ 2
2. di ≤ 2γ
3. ci = ci+1 =

⇒ ⌊2γ⌋ odd

4. ci = 1, di−1 = di = 2γ =

⇒ 2γ odd

5. ci = ci+1 = 1, di−1 + di = di+1 = 2γ =

⇒ 2γ odd ci = ci−1 = 1, di−1 = di + di+1 = 2γ = ⇒ 2γ odd Sketch of “⇒”-proof (only for γ = 1).

… …

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 22

SLIDE 23

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 24

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 25

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E Case di = 1:

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 26

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E Case di = 1:

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 27

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E Case di = 2:

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 28

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (1/2)

Theorem

C(n, k) has a NE iff k ≤ 2n

3 or k ≥ n.

Proof. “⇒”: Assume 2n

3 < k < n and ∃ NE. Note: n ≥ 4 and k ≥ 3.

First: No two players may have the same strategy. ∀i ∈ Zℓ :

1. Previous slide:

◮ ci ≤ 2 ◮ di ≤ 2, since min utility γ ≤ n

k < 3 2

2. ci = 2 =

⇒ di−1 = 2 and di = 2

3. ci = 2 =

⇒ ci+1 = 2 Hence, if ∃i : ci > 1 then ∀i ∈ Zℓ : ci = 2 and di = 2. E Case di = 2:

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 22

SLIDE 29

Introduction The Model Characterization of NE Quality of NE Conclusion

Existence of NE in C(n, k) (2/2)

Proof (continued). “⇒”: Consequently, ℓ = k and ∀i ∈ Zℓ : ci = 1. Since k > 2n

3 , ∃i ∈ Zℓ : di−1 = di = 1 and di+1 = 2. Then:

but E “⇐”: If k ≥ n, any profile with ℓ = n, |ci − cj| ≤ 1 is a NE. Consider k ≤ 2n

3 . Define p := ⌊ n k ⌋ and q := (n mod k).

Define a profile by ℓ = k and if q ≤ k

2, then

(di)i∈Zℓ = (p, p + 1, . . . , p, p + 1

2q elements

, p, p, . . . , p) , and otherwise (di)i∈Zℓ = (p, p + 1, . . . , p, p + 1

2(k−q) elements

, p + 1, p + 1, . . . , p + 1) . The NE conditions are fulfilled.

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 13 / 22

SLIDE 30

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Stability (1/2)

Definition

A profile with distances (di)i∈Zℓ is called standard if ℓ = k and ∀i ∈ Zk : di ∈ {⌊ n

k ⌋, ⌈ n k ⌉}.

Hence, there is a NE that is standard!

Theorem

Standard profiles have optimal social cost.

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 14 / 22

SLIDE 31

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Stability (2/2)

Lemma

x∗ := (λ0, . . . , λr−1, n − r−1

i=0 λi, 0, . . . , 0) ∈ Nn 0 is the unique

ptimal solution of (*) where r is minimal with r

j=0 λj ≥ n.

Minimize n−1

i=0 i · xi

(*) subject to n−1

i=0 xi = n

0 ≤ xi ≤ λi ∀i ∈ [n − 1]0 where xi ∈ N0 ∀i ∈ [n − 1]0 Proof of Theorem (Sketch). Let λ := (k, 2k, 2k, . . . , 2k) ∈ Nn.

◮ For s ∈ S define x(s) ∈ Nn 0 by

xi(s) := |{u ∈ V : minj{dist(sj, u)} = i}|

◮ x(s) is a feasible solution and SC(s) = n−1 i=0 i · xi(s) ◮ Hence: x(s) optimal solution to (*) =

⇒ SC(s) = OPT

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 15 / 22

SLIDE 32

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Stability (2/2)

Lemma

x∗ := (λ0, . . . , λr−1, n − r−1

i=0 λi, 0, . . . , 0) ∈ Nn 0 is the unique

ptimal solution of (*) where r is minimal with r

j=0 λj ≥ n.

Minimize n−1

i=0 i · xi

(*) subject to n−1

i=0 xi = n

0 ≤ xi ≤ λi ∀i ∈ [n − 1]0 where xi ∈ N0 ∀i ∈ [n − 1]0 Proof of Theorem (Sketch). Let λ := (k, 2k, 2k, . . . , 2k) ∈ Nn.

◮ For s ∈ S define x(s) ∈ Nn 0 by

xi(s) := |{u ∈ V : minj{dist(sj, u)} = i}|

◮ x(s) is a feasible solution and SC(s) = n−1 i=0 i · xi(s) ◮ Hence: x(s) optimal solution to (*) =

⇒ SC(s) = OPT

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 15 / 22

SLIDE 33

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (1/4)

Theorem

Let s ∈ S be a NE, γ := 1

2 · ⌊ 2n k ⌋. Then:

1. If γ is an odd integer ≥ 3, then SC(s) ≤ 9

4 OPT.

2. Otherwise, SC(s) ≤ 2 OPT.

Crucial for proof: How does a worst-case NE look like?

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 22

SLIDE 34

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (2/4)

Lemma

(ℓ∗, d ∗) with ℓ∗ = ⌈ n

µ⌉ and d ∗ = (µ, . . . , µ, n −(ℓ∗ −1)·µ) ∈ Nℓ∗ is

an optimal solution of (*), where n, µ ∈ N, and f is superadditive.

di = 6 f (di) = 62

4 = 9

3 2 1 1 2

Maximize ℓ

i=1 f (di)

(*) subject to ℓ

i=1 di = n

1 ≤ di ≤ µ ∀i ∈ [ℓ] where ℓ, di ∈ N ∀i ∈ [ℓ] f (x) =   

x2 4

if x ∈ N0 and x is even

x2−1 4

if x ∈ N and x is odd,

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 17 / 22

SLIDE 35

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k · f (γ) + q · [f (γ + 1) − f (γ)] SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 36

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k · f (γ) + q · [f (γ + 1) − f (γ)] SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 37

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k · f (γ) + q · γ + 1 2 SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 38

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k · f (γ) + q · γ + 1 2 SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 39

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k 4 · f (2γ) − k 4 + q · γ + 1 2 SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 40

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k 4 · f (2γ) − k 4 + q · γ + 1 2 SC(s) = k 2 · f (2γ) + f (q)

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 41

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (3/4)

If γ = 1

2 · ⌊ 2n k ⌋ is odd, then γ = ⌊ n k ⌋. Let q := (n mod k).

Proof of theorem (only for k even, γ odd integer ≥ 3, q < 2γ). f (γ + 1) = f (γ) + γ + 1 2 (1) f (γ) = f (2γ) 4 − 1 4 (2) f (q) ≤ q · γ 2 (3) OPT = k 4 · f (2γ) − k 4 + q · γ + 1 2 SC(s) ≤ k 2 · f (2γ) + q · γ 2

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 22

SLIDE 42

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Upper Bound (4/4)

Recall: OPT = k 4 · f (2γ) − k 4 + q · γ + 1 2 SC(s) ≤ k 2 · f (2γ) + q · γ 2 ≤ 2 OPT +k 2

◮ Trivial bound: OPT ≥ n − k ◮ n ≥ γk since γ = ⌊ n k ⌋ ◮ Hence, k ≤ OPT γ−1

Hence: SC(s) ≤ 2 OPT + OPT 2γ − 2

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 19 / 22

SLIDE 43

Introduction The Model Characterization of NE Quality of NE Conclusion

Price of Anarchy – Lower Bound

Worst case occurs when n = 3 · k, and k is even.

2 3 1 2 1

The worst NE with SC = 9

1 1 1 1

The optimum with SC = OPT = 4

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 20 / 22

SLIDE 44

Introduction The Model Characterization of NE Quality of NE Conclusion

Conclusion and Ongoing Research

Motivation:

◮ Competitive service providers in a computer network

Results presented in this talk:

◮ (Non-)Existence and characterization of NE for cycle graphs ◮ Here, NE is compatible with social optimum (PoS = 1) ◮ PoA ≤ 2.5, compared to 2 in the continuous case ◮ Analysis non-trivial in discrete case, often due to parity issues

Open question:

◮ Less restrictive classes of graphs ◮ Does the continuous case give an approximation?

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 21 / 22

SLIDE 45

Introduction The Model Characterization of NE Quality of NE Conclusion

References

Harold Hotelling Stability in Competition In: The Economic Journal, 39(153), pp. 41-57, 1929

B. Curtis Eaton and Richard G. Lipsey

The principle of minimum differentiation reconsidered: Some new developments in the theory of spatial competition In: Review of Economic Studies, 42(129), pp. 27–49, 1975 Christoph Dürr and Nguyen Kim Thang: Nash Equilibria in Voronoi Games on Graphs In: Proceedings of ESA’07, pp. 17–28, 2007 * * * Thank you for your attention!

Intern. Grad. School, University of Paderborn

Florian Schoppmann · 22 / 22