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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Wardrop Equilibria and Price of Stability in Bottleneck Games With Splittable Traffic

Vladimir Mazalov1 Burkhard Monien2 Florian Schoppmann2 Karsten Tiemann2

1Karelian Research Center, Russian Academy of Sciences, Russia 2University of Paderborn, Germany

December 17, 2006

University of Paderborn · Florian Schoppmann

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Motivation

Communication, traffic, and logistics networks:

◮ Goal: high “network performance” ◮ Decentralized networks due to:

◮ Central coordination inherently

impossible

◮ System of autonomous agents

→ tractable subproblems

◮ . . .

Famous model due to Wardrop (1952): Selfish drivers minimize

• wn travel time

Computer scientist’s questions:

◮ Predictions → Game theory: Equilibria ◮ Quantify loss due to selfishness → “Prices of anarchy/stability”

University of Paderborn · Florian Schoppmann

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Motivation

Communication, traffic, and logistics networks:

◮ Goal: high “network performance” ◮ Decentralized networks due to:

◮ Central coordination inherently

impossible

◮ System of autonomous agents

→ tractable subproblems

◮ . . .

Famous model due to Wardrop (1952): Selfish drivers minimize

• wn travel time

Computer scientist’s questions:

◮ Predictions → Game theory: Equilibria ◮ Quantify loss due to selfishness → “Prices of anarchy/stability”

University of Paderborn · Florian Schoppmann

• Dec. 17, 2006

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Single-Commodity Routing Games

Definition (Bottleneck Game Γ)

Γ = (G, s, t, (fe)e∈E, r)

◮ G = (V , E) multigraph ◮ s, t ∈ V source/destination ◮ fe : R≥0 → R≥0 ∪ {∞}

nonnegative, continuous, and nondecreasing latency function

◮ r ∈ R>0 amount of s-t traffic

(G, s, t, (fe)e∈E) is called a network. P := {all simple paths from s to t} L := {λ ∈ RP

≥0 | p∈P λp = r} set

• f load vectors

s a b t x 1 1 1 p1 p2 1

◮ V = {a, b, c, d} ◮ E = {(s, a), (a, t), . . . } ◮ r = 1 ◮ f(s,a)(x) = x, . . . ◮ p1 = {(s, a), (a, t)}, . . . ◮ P = {p1, p2}

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Motivation Revisited

Classic Wardrop Model:

◮ (Uncountably) infinitely many

players, each having a negligible effect on the system

◮ Mathematically: Nonatomic

anonymous games, zero-sets of players do not matter

s a b t x 1 1 1 p1 p2 1

◮ Wardrop’s (1952) first principle (Wardrop equilibrium):

The journey times on all the routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route. However, when assuming steady streams of flow, one might be interested in throughput, not individual travel time.

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Equilibria in Bottleneck Games

◮ For e ∈ E denote by le(λ) := p∈P|e∈p λp the edge load of e ◮ For p ∈ P denote by ℓp(λ) := (fe(le(λ)))e∈p∞ the path

latency of p.

Definition (Wardrop Equilibrium)

A load vector λ ∈ L is a Wardrop equilibrium (WE) iff for all p ∈ P with λp > 0 it holds that ℓp(λ) = minq∈P{ℓq(λ)}. What is different? Classic Wardrop Games: ℓp(λ) = fe(le(λ))1 =

• e∈p

fe(le(λ)) Bottleneck Games: ℓp(λ) = fe(le(λ))∞ = max

e∈p {fe(le(λ))}

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Measuring Social Utility

Social cost defined as before (→ average path latency): SC(Γ, λ) :=

• p∈P

λp · ℓp(λ) Optimal social cost: OPT(Γ) := min

λ∈L(Γ){SC(Γ, λ)}

Worst-case ratios between stable states and the respective optima (Prices of anarchy/stability) for specific non-empty classes G of games: PoA(G ) := sup

Γ∈G λ WE in Γ

SC(Γ, λ) OPT(Γ)

• PoS(G ) := sup

Γ∈G

• inf

λ WE in Γ

SC(Γ, λ) OPT(Γ)

• University of Paderborn

· Florian Schoppmann

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Measuring Social Utility (II)

Define for a set of latency functions F:

◮ G (F) := class of bottleneck games with latency functions

drawn from F

◮ P(F) := class of all games in G (F) whose graph only

consists of parallel edges First objective: Similar result as Roughgarden/Tardos (2002) for bottleneck games? For instance, let P1 denote set of affine latency functions (with positive coefficients). PoA(G (P1)) =?

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Reusable Results?

The classic model has been studied extensively—which results carry

• ver, which not?

◮ Trivially, no difference on parallel edges ◮ It makes a difference whether non-simple paths are allowed.

s t a p2 3 p1

For e = (s, t): fe(x) := 1 4 − x , f(s,t)(x) := 1 3 − x Equilibrium: λ = (1, 2)

◮ Wardrop equilibria always exist (Schmeidler, 1973)

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Reusable Results?

The classic model has been studied extensively—which results carry

• ver, which not?

◮ Trivially, no difference on parallel edges ◮ It makes a difference whether non-simple paths are allowed.

p3 s t a p2 3 p1

For e = (s, t): fe(x) := 1 4 − x , f(s,t)(x) := 1 3 − x Equilibria: λ = (1, 2, 0, . . . ) λ′ = (0, 1, 2, 0, . . . )

◮ Wardrop equilibria always exist (Schmeidler, 1973)

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Series Parallel Graphs

Recursively defined:

◮ Base case:

s t

◮ An arbitrary multigraph G is series parallel iff it can be

constructed from two series parallel graphs. If

G1 s1 t1 G2 s2 t2 and

are series parallel then

G1 G2 s t G2 t2 G1 s1 and

are series parallel.

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Strong Cuts—From Equilibria to Maximum Flows

Definition (Strong Cut)

Let Γ = (G, s, t, (fe)e∈E, r) be a bottleneck game and λ ∈ L a Wardrop equilibrium. A cut S V is called strong with respect to Γ and λ iff fe(le(λ)) ≥ SC(Γ,λ)

r

for all edges leaving S and le(λ) = 0 for all edges e ∈ E going into S. Note: SC(Γ,λ)

r

= minp∈P{ℓp(λ)} is unique path latency for all used paths in λ

5 5 1 1 s t a b

1 6−x 1 6−x 1 6−x 1 2−x

6

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts

Lemma

Let G be a series parallel graph, Γ = (G, s, t, (fe)e∈E, r), and λ WE

• f Γ. Then a strong cut S V with respect to Γ, λ exists.

Proof (By structural induction). Induction hypothesis: Every series parallel graph G fulfills: If Γ is a bottleneck game on G and λ is WE for Γ, then a strong cut w.r.t. Γ, λ exists.

◮ Base Case: Trivial ◮ Induction Step, Parallel Connection:

G1 G2 s t

For i ∈ {1, 2}: Let ri be traffic through Gi Then strong cuts Si w.r.t. (Gi, s, t, (fe)e∈Ei, ri), λ exist. Combine (S = S1 ∪ S2) to get strong cut for G w.r.t. Γ, λ

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts

Lemma

Let G be a series parallel graph, Γ = (G, s, t, (fe)e∈E, r), and λ WE

• f Γ. Then a strong cut S V with respect to Γ, λ exists.

Proof (By structural induction). Induction hypothesis: Every series parallel graph G fulfills: If Γ is a bottleneck game on G and λ is WE for Γ, then a strong cut w.r.t. Γ, λ exists.

◮ Base Case: Trivial ◮ Induction Step, Parallel Connection:

G1 G2 s t

For i ∈ {1, 2}: Let ri be traffic through Gi Then strong cuts Si w.r.t. (Gi, s, t, (fe)e∈Ei, ri), λ exist. Combine (S = S1 ∪ S2) to get strong cut for G w.r.t. Γ, λ

University of Paderborn · Florian Schoppmann

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts

Lemma

Let G be a series parallel graph, Γ = (G, s, t, (fe)e∈E, r), and λ WE

• f Γ. Then a strong cut S V with respect to Γ, λ exists.

Proof (By structural induction). Induction hypothesis: Every series parallel graph G fulfills: If Γ is a bottleneck game on G and λ is WE for Γ, then a strong cut w.r.t. Γ, λ exists.

◮ Base Case: Trivial ◮ Induction Step, Parallel Connection:

G1 G2 s t

For i ∈ {1, 2}: Let ri be traffic through Gi Then strong cuts Si w.r.t. (Gi, s, t, (fe)e∈Ei, ri), λ exist. Combine (S = S1 ∪ S2) to get strong cut for G w.r.t. Γ, λ

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts (II)

Proof (continued):

◮ Induction Step, Series Connection:

G2 t G1 s a

Consider games Γ1 = (G1, s, a, (fe)e∈E, r) and Γ2 = (G2, a, t, (fe)e∈E, r) λ is a WE either in Γ1 or in Γ2. Hence strong cuts S1 V1 or S2 V2 exist. Obtain strong cut S for G by setting S = S1 or S = V1 ∪ S2

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts (II)

Proof (continued):

◮ Induction Step, Series Connection:

G2 t G1 s a

Consider games Γ1 = (G1, s, a, (fe)e∈E, r) and Γ2 = (G2, a, t, (fe)e∈E, r) λ is a WE either in Γ1 or in Γ2. Hence strong cuts S1 V1 or S2 V2 exist. Obtain strong cut S for G by setting S = S1 or S = V1 ∪ S2

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Existence of Strong Cuts (II)

Proof (continued):

◮ Induction Step, Series Connection:

G2 t G1 s a

Consider games Γ1 = (G1, s, a, (fe)e∈E, r) and Γ2 = (G2, a, t, (fe)e∈E, r) λ is a WE either in Γ1 or in Γ2. Hence strong cuts S1 V1 or S2 V2 exist. Obtain strong cut S for G by setting S = S1 or S = V1 ∪ S2

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Strong Cuts vs. Minimum Cuts

Note: If S V is strong cut w.r.t. Γ, λ, then λ is maximum flow for (G, s, t, k), where for all e ∈ E: ke =

• if fe(0) > SC(Γ,λ)

r

f −1

e

• SC(Γ,λ)

r

• therwise

For simplicity, assume fe’s are strictly increasing from now on. ⇒ Uniqueness of social costs of WE in games on series parallel graphs:

◮ Assume ∃ WE λ, λ′ with SC(Γ, λ) < SC(Γ, λ′) ◮ Capacities on all used edges become larger ◮ Since λ′ maximal flow

r =

• p∈P

λp <

• p∈P

λ′

p = r

E

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Strong Cuts vs. Minimum Cuts

Note: If S V is strong cut w.r.t. Γ, λ, then λ is maximum flow for (G, s, t, k), where for all e ∈ E: ke =

• if fe(0) > SC(Γ,λ)

r

f −1

e

• SC(Γ,λ)

r

• therwise

For simplicity, assume fe’s are strictly increasing from now on. ⇒ Uniqueness of social costs of WE in games on series parallel graphs:

◮ Assume ∃ WE λ, λ′ with SC(Γ, λ) < SC(Γ, λ′) ◮ Capacities on all used edges become larger ◮ Since λ′ maximal flow

r =

• p∈P

λp <

• p∈P

λ′

p = r

E

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

A Uniqueness Result for Series Parallel Graphs

Hence we get:

Theorem

Let Γ be a bottleneck game on a series parallel graph, λ and λ′

• WE. Then SC(Γ, λ) = SC(Γ, λ′).

Question: What may happen if the graph is not series parallel?

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

A Uniqueness Result for Series Parallel Graphs

Hence we get:

Theorem

Let Γ be a bottleneck game on a series parallel graph, λ and λ′

• WE. Then SC(Γ, λ) = SC(Γ, λ′).

Question: What may happen if the graph is not series parallel?

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

The Braess Paradox Graph

Consider the following game Γ on the Braess paradox graph:

s t a b r p1 p3 p2

◮ Strictly increasing latency f : R≥0 → R≥0 for all edges e ∈ E ◮ λ = ( r 2, 0, r 2) and λ′ = (0, r, 0) are both WE ◮ SC(Γ, λ) = r · f (r/2) and SC(Γ, λ′) = r · f (r)

Hence: PoA unbounded for this singleton set of games if f (r) = ∞.

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Games with WE of Different Social Costs

Theorem

Let G = (V , E) be a multigraph whose subgraph induced by all simple paths from s to t is not series parallel. Then there exists a game Γ = (G, s, t, (fe)e∈E, r) which has WE with different social costs.

• Proof. Let G ′ be a graph, B the Braess paradox graph

.

• 1. Valdes (1978): G ′ acyclic with single source & single sink (4S):

G ′ series parallel ⇔ Braess paradox graph B not minor of G ′

• 2. One can show: G ′ is 4S ∧ all edges on simple s-t path ∧ B

not minor ⇒ G ′ acyclic

• 3. Subgraph of G induced by all simple s-t paths is 4S ∧ not

series parallel ⇒ B minor of G ′ (Otherwise: G ′ acyclic by 2. ⇒ G ′ series parallel by 1.).

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

No Good Bounds on Price of Anarchy

Consider the following game Γ = (G, s, t, (fe)e∈E, k), due to Cole, Dodis, Roughgarden (2006):

s k t … … x x x k Layers

◮ All traffic split evenly over “direct” paths: WE with social cost

k · 1 = k

◮ All traffic on “zigzag” path: WE with social cost k2

⇒ PoA(G (P1)) = ∞

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

No Good Bounds on Price of Anarchy

Consider the following game Γ = (G, s, t, (fe)e∈E, k), due to Cole, Dodis, Roughgarden (2006):

s k t … … x x x k Layers

◮ All traffic split evenly over “direct” paths: WE with social cost

k · 1 = k

◮ All traffic on “zigzag” path: WE with social cost k2

⇒ PoA(G (P1)) = ∞

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Price of Stability “Independent of Network Topology”

Theorem

For every bottleneck game Γ = (G, s, t, (fe)e∈E, r) we can find a game Γ′ on parallel edges with latency functions drawn only from {fe | e ∈ E} such that there are WE λ for Γ and λ′ for Γ′ with SC(Γ, λ) OPT(Γ) ≤ SC(Γ′, λ′) OPT(Γ′) . Proof.

◮ ∃ WE λ s.t. there is strong cut

S V . For all e ∈ E leaving S: fe(le(λ)) ≥ SC(Γ, λ) r

s t … …

◮ Use these edges for Γ′ ◮ Then: SC(Γ, λ) = SC(Γ′, λ′) and OPT(Γ) ≥ OPT(Γ′)

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Price of Stability “Independent of Network Topology”

Theorem

For every bottleneck game Γ = (G, s, t, (fe)e∈E, r) we can find a game Γ′ on parallel edges with latency functions drawn only from {fe | e ∈ E} such that there are WE λ for Γ and λ′ for Γ′ with SC(Γ, λ) OPT(Γ) ≤ SC(Γ′, λ′) OPT(Γ′) . Proof.

◮ ∃ WE λ s.t. there is strong cut

S V . For all e ∈ E leaving S: fe(le(λ)) ≥ SC(Γ, λ) r

s t … … s t …

◮ Use these edges for Γ′ ◮ Then: SC(Γ, λ) = SC(Γ′, λ′) and OPT(Γ) ≥ OPT(Γ′)

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Price of Stability for Polynomial Latency Functions

Corollary (Cole, Dodis, Roughgarden (2006))

Let Pd denote the set of all polynomial latency functions with maximum degree d and positive coefficients. Then PoS(G (Pd)) = (d + 1) ·

d

√ d + 1 (d + 1) ·

d

√ d + 1 − d . For those familiar with Roughgarden’s (2002) “anarchy value”: If α(F) exists for a set of functions F, then PoS(G (F)) ≤ α(F).

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Bottleneck Games and M/M/1 Latency Functions

Latency functions of form f (x) =

• 1

c−x

if x < c ∞

• therwise are called

M/M/1 latency functions with capacity c. They arise as the expected delay of M/M/1 queues. We define:

◮ M := {all M/M/1 latency functions} ◮ M≥c := {all functions from M with capacity ≥ c}

We have already seen:

◮ PoA({Γ}) = ∞ if Γ is the depicted

game, where all latency functions are M/M/1 with capacity r

◮ Furthermore, PoS(P(M)) = ∞

(follows by our later result)

s t a b r p1 p3 p2

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Bottleneck Games and M/M/1 Latency Functions (II)

Hence, to further differentiate, we introduce:

◮ G (F, m, r) := class of all games with latency functions from

F, with ≤ m edges, and traffic ≤ r

◮ P(F, m, r) := G (F, m, r) ∩ P(F)

Since PoA meaningless if M/M/1 latency functions are allowed, our goal will be: Determine the exact value for PoS(G (M≥c, m, r)) In the following:

◮ Consider bottleneck games on m parallel edges with M/M/1

lantency functions, then generalize PoS as before

◮ Capacities are, w.l.o.g., c1 ≥ c2 ≥ · · · ≥ cm = 1 ◮ For i ∈ [m]: C ≤i := i j=1 cj

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Social Costs on Parallel Edges

Theorem

Let Γ ∈ P(M) be a bottleneck game on m parallel edges and λ a

• WE. Denote by s = |{i ∈ [m] | λi > 0}| the number of edges used

by λ. Then s = max

• i ∈ [m] | r + i · ci > C ≤i

and SC(Γ, λ) = s · r C ≤s − r Proof.

c1 c2 … c3 cm

◮ In a WE, ci − xi = cj − xj for all

used edges i, j ∈ [m]

◮ The unique “remaining capacity” on

all used edges must be C ≤s−r

s ◮ Hence, the unique path latency on

all used paths is

s C ≤s−r

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Social Costs on Parallel Edges (II)

Theorem

Let Γ ∈ P(M) be a bottleneck game on m parallel edges and λ ∈ L with SC(Γ, λ) = OPT(Γ). Denote by t = |{i ∈ [m] | xi > 0}| the number of edges used by λ. Then: t = max

• i ∈ [m] | r + √ci ·

i

• k=1

√ck > C ≤i

• and

OPT(Γ) = t

i=1

√ci 2 C ≤t − r − t Proof similar as before: Use that λ is a WE in a game Γ′ where all edge latencies fe of Γ are replaced by

d dx (x · fe(x))

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Price of Stability/Anarchy for Games on Parallel Links

Theorem

PoS(P(M≥1, m, r)) = m · r r + 2 · (m − 1) · (√r + 1 − 1) Proof sketch of lower bound. Choose edge capacities: c1 = r + 1, c2 = · · · = cm = 1 WE uses only first edge, optimum use all edges (as r + √cm · m

i=1

√ci = r + √r + 1 + m − 1 > r + m = C ≤m).

r + 1 1 … 1 1 r + 1 1 … 1 1

Equilibrium Optimum

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Price of Stability in General

Corollary

• 1. PoS(G (M≥c, m, r)) = PoS(P(M≥c, m, r))
• 2. The price of anarchy for bottleneck games with M/M/1

latency functions on general graphs is PoS(G (M≥c, m, r)) = m · r

c r c + 2 · (m − 1) ·

r

c + 1 − 1

• Note:

◮ PoS(G (M≥c, m, r)) is increasing in m and r ◮ It converges to m for large r

Previous result (based on Roughgarden, 2002):

1 2 1 2 3

◮ PoS(G (M≥c, ∞, r)) ≤ 1 2 · (1 +

• c/(c − r)), only for c > r

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

Conclusion

Our results:

◮ Games with general latency functions:

◮ Uniqueness result w.r.t. to social cost for bottleneck games on

series parallel graphs

◮ PoS “independent of network topology”

◮ M/M/1 latency functions:

◮ Exact PoA/PoS on parallel edges ◮ = exact price of stability for general graphs

Open Problem: Multi-commodity setting?

◮ Our techniques based on maximum flows ◮ Results for M/M/1 latency functions heavily rely on findings

for parallel edges → New ideas necessary

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Bottleneck Games Properties of WE M/M/1 Latency Functions Conclusion

References

• R. Cole, Y. Dodis, and T. Roughgarden:

Bottleneck Links, Variable Demand, and the Tragedy of the Commons In: Proceedings of SODA’06, pp. 668–677, 2006

• V. Mazalov, B. Monien, F. Schoppmann, and K. Tiemann:

Wardrop Equilibria and Price of Stability for Bottleneck Games with Splittable Traffic In: Proceedings of WINE’06, pp. 331–342, LNCS 4286, Springer * * * Thank you for your attention!

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