Total Latency in Singleton Congestion Games Price of Anarchy Martin - - PowerPoint PPT Presentation

Introduction Unrestricted Restricted Conclusion

Total Latency in Singleton Congestion Games

Price of Anarchy Martin Gairing1 Florian Schoppmann2

1International Computer Science Institute, Berkeley, CA, USA 2International Graduate School Dynamic Intelligent Systems,

University of Paderborn, Paderborn, Germany

December 13, 2007

Total Latency in Singleton Congestion Games 1 / 22

Introduction Unrestricted Restricted Conclusion

Singleton Congestion Games

Defined by tuple Γ =

• n, m, (wi)i∈[n], (Si)i∈[n], (fe)e∈E
• where

◮ n ∈ N is number of players ◮ m ∈ N is number of resources ◮ wi ∈ R>0 is weight of player i ◮ Si ⊆ 2[m] is set of strategies of i ◮ fe : R≥0 → R≥0 is latency function of resource e

Implicitly defined:

◮ Set of pure strategy profiles S := S1 × · · · × Sn ◮ Set of mixed strategy profiles as a subset of ∆(S)

Special Cases:

◮ unweighted: For all i ∈ [n]: wi = 1 ◮ unrestricted (⇒ symmetric): For all i ∈ [n]: Si = [m]

Total Latency in Singleton Congestion Games 2 / 22

… s t

Introduction Unrestricted Restricted Conclusion

Notation

◮ Load on e ∈ E: δe(s) := i∈[n]|si=e wi ◮ Private cost of i ∈ [n]: PCi(P) := s∈S P(s)fsi(δsi(s)) ◮ Social cost is total latency. For a mixed profile S:

SC(P) :=

• s∈S

P(s)

• e∈E

δe(s) · fe(δe(s)) =

• s∈S

P(s)

• i∈[n]
• e∈si

wi · fe(δe(s)) =

• i∈[n]

wi · PCi(P) .

◮ Pure Price of Anarchy for a set G of games:

PoApure(G) := sup

Γ∈G

sup

p is NE of Γ

SCΓ(p) OPT , Mixed price of anarachy defined analogously.

Total Latency in Singleton Congestion Games 3 / 22

Introduction Unrestricted Restricted Conclusion

Motivation

Scenario: Selfish load balancing

◮ Selfish players may choose the machine to process their job on ◮ Player’s cost = time until all jobs on that machine are

processed Note: Only difference to KP-model [Koutsoupias & Papadimitriou, 1999] is the social cost function The case of non-atomic (singleton) congestion games has long been settled [Roughgarden & Tardos, 2003]. What is known for singleton congestion games?

Total Latency in Singleton Congestion Games 4 / 22

Introduction Unrestricted Restricted Conclusion

Related Work

latencies players PoApure: LB and UB PoAmixed: LB and UB unrestricted x ident. 1 2 − 1/m [7] x arb. 9/8 [7] 2 − 1/m [7,6] ax ident. 4/3 [7] 2 − 1/m ax arb. 2 1 + Φ [2] 2.036 1 + Φ [2] xd ident. 1 Bd+1 [5] d

j=0 ajxj

arb. Bd+1 Φd+1

d

[1] Bd+1 Φd+1

d

[1] restricted x ident. 2.012 [8] 2.012 [3] 2.012 [8] 5/2 [4] ax ident. 5/2 [3] 5/2 [8] 5/2 [3] 5/2 [4] d

j=0 ajxj

ident. Υ(d) Υ(d) [1] Υ(d) Υ(d) [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] d

j=0 ajxj

arb. Φd+1

d

Φd+1

d

[1] Φd+1

d

Φd+1

d

[1]

1. Aland, Dumrauf, Gairing, Monien, Schoppmann. STACS 2006 2. Awerbuch, Azar & Epstein. STOC’05 3. Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscardelli. ICALP’06 4. Christodoulou, Koutsoupias. ESA 2005 5. Gairing, Lücking, Mavronicolas, Monien, Rode. ICALP’04 6. Gairing, Monien & Tiemann. SPAA‘05 7. Lücking, Mavronicolas, Monien & Rode. STACS‘04 8. Suri, Tóth & Zhou. SPAA’04 Total Latency in Singleton Congestion Games 5 / 22

Introduction Unrestricted Restricted Conclusion

Related Work

latencies players PoApure: LB and UB PoAmixed: LB and UB unrestricted x ident. 1 2 − 1/m [7] x arb. 9/8 [7] 2 − 1/m [7,6] ax ident. 4/3 [7] 2 − 1/m ax arb. 2 1 + Φ [2] 2.036 1 + Φ [2] xd ident. 1 Bd+1 [5] d

j=0 ajxj

arb. Bd+1 Φd+1

d

[1] Bd+1 Φd+1

d

[1] restricted x ident. 2.012 [8] 2.012 [3] 2.012 [8] 5/2 [4] ax ident. 5/2 [3] 5/2 [8] 5/2 [3] 5/2 [4] d

j=0 ajxj

ident. Υ(d) Υ(d) [1] Υ(d) Υ(d) [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] d

j=0 ajxj

arb. Φd+1

d

Φd+1

d

[1] Φd+1

d

Φd+1

d

[1]

1. Aland, Dumrauf, Gairing, Monien, Schoppmann. STACS 2006 2. Awerbuch, Azar & Epstein. STOC’05 3. Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscardelli. ICALP’06 4. Christodoulou, Koutsoupias. ESA 2005 5. Gairing, Lücking, Mavronicolas, Monien, Rode. ICALP’04 6. Gairing, Monien & Tiemann. SPAA‘05 7. Lücking, Mavronicolas, Monien & Rode. STACS‘04 8. Suri, Tóth & Zhou. SPAA’04 Total Latency in Singleton Congestion Games 5 / 22

Bd := d-th Bell number Φd := positive real root of (x + 1)d = xd+1 Υ(d) :=

(k+1)2d+1−kd+1(k+2)d (k+1)d+1−(k+2)d+(k+1)d−kd+1 , where k = ⌊Φd⌋

Introduction Unrestricted Restricted Conclusion

Unrestricted, Affine, Weighted: Bounding all NE

Lemma

Let P be NE in an unrestricted, affine, weighted game. Then, for all subsets of resources M ⊆ [m]: SC(P) ≤

• i∈[n]

wi · W + (|M| − 1)wi +

j∈M bj aj

• j∈M

1 aj

Proof omitted here.

Total Latency in Singleton Congestion Games 6 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Affine, Weighted: Bounding OPT (1/2)

Lemma

Let s ∈ S be optimal and let M := {e | δe(s) > 0}. Define X := {x ∈ RM

>0 | j∈M xj = W } and let

x∗ ∈ arg minx∈X{

j∈M xj · fj(xj)}. Denote

M∗ = {j ∈ M | x∗

j > 0}. Then,

SC(s) ≥ W 2 + W

2 · j∈M∗ bj aj

• j∈M∗ 1

aj

. Proof. SC(s) =

• j∈M

fj(δj(s)) · δj(s) ≥

• j∈M

fj(x∗

j ) · x∗ j =

• j∈M∗

fj(x∗

j ) · x∗ j

=

• j∈M∗
• aj · x∗

j + bj

• · x∗

j =

• j∈M∗

x∗

j + bj aj 1 aj

· x∗

j

Total Latency in Singleton Congestion Games 7 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Affine, Weighted: Bounding OPT (2/2)

x is an equilibrium in the nonatomic game where each fe(x) is replaced by

d dx (x · fe(x)) = 2aex + be. Hence, for all resources

j ∈ M∗, x∗

j + 1 2 · bj aj 1 aj

=

• k∈M∗(x∗

k + 1 2 · bk ak )

• k∈M∗ 1

ak

= W + 1

2 · k∈M∗ bk ak

• k∈M∗ 1

ak

. We get SC(s) ≥

• j∈M∗

x∗

j + bj aj 1 aj

· x∗

j ≥

• j∈M∗

x∗

j + 1 2 · bj aj 1 aj

· x∗

j

= W + 1

2 · k∈M∗ bk ak

• k∈M∗ 1

ak

·

• j∈M∗

x∗

j =

W 2 + W

2 · k∈M∗ bk ak

• k∈M∗ 1

ak

.

Total Latency in Singleton Congestion Games 8 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Affine, Weighted: Upper Bound

Theorem

Let G be set of unrestricted, affine, unweighted games. Then, PoA(G) < 2.

• Proof. Using |M∗| ≤ n, we get for any NE P:

SC(P) OPT ≤ n2 + n · (|M∗| − 1) + n ·

j∈M∗ bj aj

n2 + n

2 · j∈M∗ bj aj

≤ 1 + n2 · |M∗|−1

|M∗|

+ n

2 · j∈M∗ bj aj

n2 + n

2 · j∈M∗ bj aj

< 2

Total Latency in Singleton Congestion Games 9 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Polynomial, Weighted: Lower Bound (1/3)

Theorem

Let G be class of unrestricted, polynomial (with max degree d), weighted games. Then, PoApure(G) ≥ Bd+1.

• Proof. Construction with parameter k ∈ N:

Resources:

◮ k + 1 disjoint sets M0, . . . , Mk of resources ◮ |Mk| = 1 and |Mj| = 2(j + 1) · Mj+1 for j ∈ [k − 1]0 ◮ For all j ∈ [k]0 and for all e ∈ Mj: fe(x) = xd 2jd

Players:

◮ k disjoint sets of players N1, . . . , Nk ◮ |Nj| = |Mj−1| for j ∈ [k] ◮ All players in Nj have weight wi = 2j−1

Total Latency in Singleton Congestion Games 10 / 22

… 1 1 2 2 2 2 1 1 1 1 1 1

Introduction Unrestricted Restricted Conclusion

Unrestricted, Polynomial, Weighted: Lower Bound (2/3)

Proof (continued). Example for k = 2:

# resources: 8 4 1 … latency: x 20 d x 21 d x 22 d M0 M1 M2 1 1 2 2 2 2 1 1 1 1 1 1

This profile s is a Nash equilibrium with SC(s) =

• j∈[k]

|Mj| · j · 2j · jd = 2k · k!

• j∈[k]

jd+1 j!

Total Latency in Singleton Congestion Games 11 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Polynomial, Weighted: Lower Bound (2/3)

Proof (continued). Example for k = 2:

# resources: 8 4 1 … latency: x 20 d x 21 d x 22 d M0 M1 M2 1 1 2 2 2 2 1 1

This profile s∗ is strategy profile with SC(s∗) =

• j∈[k−1]0

|Mj| · 2j = 2k · k!

• j∈[k−1]0

1 j!

Total Latency in Singleton Congestion Games 11 / 22

Introduction Unrestricted Restricted Conclusion

Unrestricted, Polynomial, Weighted: Lower Bound (3/3)

Proof (continued). Hence, PoApure(G) ≥ ∞

j=1 jd+1 j!

∞

j=0 1 j!

= 1 e

∞

• j=1

jd+1 j! = Bd+1 .

Total Latency in Singleton Congestion Games 12 / 22

Introduction Unrestricted Restricted Conclusion

Improved Lower Bound for Linear Latencies

Instance: 5 resources, 5 jobs, for parameters p ∈ [0, 1], w ∈ R>0 Nash equilibrium:

x x x x w · x w + 4 1 1 1 1 w

1−p 4 1−p 4 1−p 4 1−p 4

p

◮ w1 = w, w2 = · · · = w5 = 1 ◮ Nash equation for jobs 2, . . . , 5:

w · (4 + pw) w + 4 ≤ 1 − p 4 w + 1 ⇔ p ≤ w2 − 8w + 16 5w2 + 4w ≤ 1 With p maximal, w = 3.258, and profile

• n the right:

PoAmixed > 2.036

Total Latency in Singleton Congestion Games 13 / 22

Introduction Unrestricted Restricted Conclusion

Improved Lower Bound for Linear Latencies

Instance: 5 resources, 5 jobs, for parameters p ∈ [0, 1], w ∈ R>0 Nash equilibrium:

x x x x w · x w + 4 1 1 1 1 w

1−p 4 1−p 4 1−p 4 1−p 4

p

◮ w1 = w, w2 = · · · = w5 = 1 ◮ Nash equation for jobs 2, . . . , 5:

w · (4 + pw) w + 4 ≤ 1 − p 4 w + 1 ⇔ p ≤ w2 − 8w + 16 5w2 + 4w ≤ 1 With p maximal, w = 3.258, and profile

• n the right:

PoAmixed > 2.036

x x x x w · x w + 4 1 1 1 1 w

Total Latency in Singleton Congestion Games 13 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Weighted: Lower Bound (1/2)

Theorem

Let d ∈ N and G be the set of restricted, polynomial (of max degree d), weighted games. Then, PoApure(G) ≥ Φd+1

d

. Proof. Consider game with n ∈ N players, n + 1 resources:

… Φd Φ2

d

Φn−1

d

Φn

d

xd Φd+1

d

xd Φ2(d+1)

d

xd Φ3(d+1)

d

xd Φn·(d+1)

d

xd Φn·(d+1)

d

Φ3

d

Let s := (i)n

i=1 and s∗ := (i + 1)n i=1.

Total Latency in Singleton Congestion Games 14 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Weighted: Lower Bound (1/2)

Theorem

Let d ∈ N and G be the set of restricted, polynomial (of max degree d), weighted games. Then, PoApure(G) ≥ Φd+1

d

. Proof. Consider game with n ∈ N players, n + 1 resources:

… Φd Φ2

d

Φn−1

d

Φn

d

xd Φd+1

d

xd Φ2(d+1)

d

xd Φ3(d+1)

d

xd Φn·(d+1)

d

xd Φn·(d+1)

d

Φ3

d

Let s := (i)n

i=1 and s∗ := (i + 1)n i=1.

Total Latency in Singleton Congestion Games 14 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Weighted: Lower Bound (1/2)

Theorem

Let d ∈ N and G be the set of restricted, polynomial (of max degree d), weighted games. Then, PoApure(G) ≥ Φd+1

d

. Proof. Consider game with n ∈ N players, n + 1 resources:

… Φd Φ2

d

Φn−1

d

Φn

d

xd Φd+1

d

xd Φ2(d+1)

d

xd Φ3(d+1)

d

xd Φn·(d+1)

d

xd Φn·(d+1)

d

Φ3

d

Let s := (i)n

i=1 and s∗ := (i + 1)n i=1.

Total Latency in Singleton Congestion Games 14 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Weighted: Lower Bound (2/2)

Proof (continued).

… Φd Φ2

d

Φn−1

d

Φn

d

xd Φd+1

d

xd Φ2(d+1)

d

xd Φ3(d+1)

d

xd Φn·(d+1)

d

xd Φn·(d+1)

d

Φ3

d

s := (i)n

i=1 is a NE as:

PCi(s−i, i + 1) = (Φi

d + Φi+1 d

)d Φ(d+1)·(i+1)

d

= (Φi

d(Φd + 1))d

Φ(d+1)·(i+1)

d

= Φid

d

Φ(d+1)·i

d

= PCi(s) The theorem follows as: SC(s) =

n

• i=1

Φi

d ·

Φid

d

Φi(d+1)

d

= n SC(s∗) = (n − 1) · 1 Φd+1

d

+ 1

Total Latency in Singleton Congestion Games 15 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Unweighted: Lower Bound (1/5)

Theorem

Let d ∈ N and G be the set of restricted, polynomial (of max degree d), unweighted games. Then, PoApure(G) ≥ Υ(d).

• Proof. Recursive construction with parameter k ∈ N:

1 k + 1 k 2k (d + 1)k Level:

Again, we let s be profile where each player uses strategy closer to root and s∗ be profile where each player uses her other strategy.

Total Latency in Singleton Congestion Games 16 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Unweighted: Lower Bound (2/5)

Proof (continued): For any resource on level (d + 1 − i) · k + j, where i ∈ [d + 1] and j ∈ [k − 1]0, let the latency function be fi,j : R≥0 → R≥0, fi,j(x) := d+1

• l=i+1

l l + 1 d·(k−1) ·

• i

i + 1 dj · xd . Resources on level (d + 1) · k have the same latency function f0,0 := f1,k−1 as those on level (d + 1) · k − 1. Note that s is Nash equilibrium:

◮ fi+1,k−1 = fi,0 for i ∈ [d + 1]0 ◮ fi,j(i) = fi,j+1(i + 1)

for all i ∈ [d + 1] and j ∈ [k − 2]0

Total Latency in Singleton Congestion Games 17 / 22

1 k + 1 k 2k (d + 1)k Level:

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Unweighted: Lower Bound (3/5)

Social cost of Nash equilibrium s: SC(s) =

d+1

• i=1

k−1

• j=0

d+1

• l=i+1

lk

• · ij · i · fi,j(i)

=

d+1

• i=1
• id+1 ·

d+1

• l=i+1

l

• ·

id+1 id+1 − (i + 1)d + (i − 1)d+1 · d+1

• l=i

l

• ·

id id − (i − 1)d+1

• ·

d+1

• l=i

ld+1 (l + 1)d k−1 + (d + 1)d+1 · (d + 2)d (d + 2)d − (d + 1)d+1 (by rearranging terms and simplifying geometric series)

Total Latency in Singleton Congestion Games 18 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Unweighted: Lower Bound (4/5)

Repeat for SC(s∗). Then,

SC(s) SC(s∗) is of the form:

d+1

i=0 βi · αk−1 i

d+1

i=0 γi · αk−1 i

where ∀i ∈ [d + 1]: βi, γi ∈ Q, αi =

id+1 (d+2)d · d+1 l=i+1 l and α0 = 1

Hence, for finding limit:

◮ Find i ∈ [d + 1]0 fow which αi is max ◮ Note that αi+1 > αi is equivalent to

(i + 1)d+1 ·

d+1

• l=i+2

l > id+1 ·

d+1

• l=i+1

l = id+1 · (i + 1) ·

d+1

• l=i+2

l , i.e., (i + 1)d > id+1 .

◮ Moreover, α1 = (d+1)! (d+2)d < 1 and αd+1 = (d+1)d+1 (d+2)d

> 1

Total Latency in Singleton Congestion Games 19 / 22

Introduction Unrestricted Restricted Conclusion

Restricted, Polynomial, Unweighted: Lower Bound (5/5)

Let λ := ⌊Φd⌋.

◮ Then, (λ + 1)d > λd+1 but (λ + 2)d < (λ + 1)d+1 ◮ Thus, λ ∈ [d] and αλ+1 is maximal ◮ Using standard calculus we get

lim

k→∞

d+1

i=0 βi · αk−1 i

d+1

i=0 γi · αk−1 i

= βλ+1 γλ+1 .

◮ Inserting gives the desired bound.

Total Latency in Singleton Congestion Games 20 / 22

Introduction Unrestricted Restricted Conclusion

Exponential Growth

Even for singleton congestion games with polynomial latency functions, price of anarchy for is dΘ(d): d Φd Υ(d) Φd+1

d

Bd+1 1 1.618 2.5 2.618 2 2 2.148 9.583 9.909 5 3 2.630 41.54 47.82 15 4 3.080 267.6 277.0 52 5 3.506 1,514 1,858 203 6 3.915 12,345 14,099 877 7 4.309 98,734 118,926 4,140 8 4.692 802,603 1,101,126 21,147 9 5.064 10,540,286 11,079,429 115,975 10 5.427 88,562,706 120,180,803 678,570

Total Latency in Singleton Congestion Games 21 / 22

Introduction Unrestricted Restricted Conclusion

Conclusion

Motivation:

◮ Understanding the dependence of the PoA on network

topology Results presented in this talk:

◮ Collection of upper and lower bounds on PoA ◮ For the unrestricted case, closing the gaps between upper and

lower bounds seems challenging

◮ Surprisingly, both upper bounds on the PoA for general

congestion games with polynomial latency functions are already exact for singleton games and pure NE

Total Latency in Singleton Congestion Games 22 / 22