Tight Bounds for Cost-Sharing in Weighted Congestion Games Martin - - PowerPoint PPT Presentation

Tight Bounds for Cost-Sharing in Weighted Congestion Games

Martin Gairing

University of Liverpool

Joint work with: Konstantinos Kollias (Stanford University) Grammateia Kotsialou (University of Liverpool)

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 1 / 18

Weighted Congestion Games

Setting: Players with asymmetric demands (weights) compete for resources Each one selects which resources she will use A joint cost Ce(x) = x · ce(x) is induced on resource e (x = total users weight) The joint cost is paid for by the users of e Players try to minimize their individual costs Applications: Scheduling Network design Selfish routing

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 2 / 18

Example: A Routing Game

3 t s 16 · x x3 → 64

1

Players: The two s-t flows with weights 1 and 3 Resources: The two edges with cost functions x3 and 16 · x Joint cost on top edge is 64. How is it shared between red and blue?

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 3 / 18

Cost-Sharing and Equilibria

Cost-Sharing Method: Input: Weights of edge users and a cost function Output: Cost shares for the users Interpretation in Delay Settings: Flows of packets with Poisson rates equal to players’ weights Cost functions are M/M/1 aggregate queueing delays Messing around with packets in the queue changes the aggregate delays of individual players but not the overall aggregate delay Pure Nash Equilibrium (PNE): Focus on each i, fix the paths of others We have a PNE if every i is already using her best path

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 4 / 18

Proportional Sharing

Informal Definition (Proportional Sharing)

α fraction of the total weight → α fraction of the cost

3 t s 16 · x x3 → 16 + 48

1

This is a PNE

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 5 / 18

Proportional Sharing

Informal Definition (Proportional Sharing)

α fraction of the total weight → α fraction of the cost

16 · x → 16 + 48 t s

1

3 x3

This is not a PNE, both want to deviate

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 6 / 18

Proportional Sharing

Informal Definition (Proportional Sharing)

α fraction of the total weight → α fraction of the cost

16 · x → 48 t s

1

3 x3 → 1

This is a PNE

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 7 / 18

Proportional Sharing

Informal Definition (Proportional Sharing)

α fraction of the total weight → α fraction of the cost

16 · x → 16 t s

1

3 x3 → 27

Optimal paths (also a PNE)

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 8 / 18

Shapley Value

Informal Definition (Shapley Value)

The cost share of a player is the average joint cost jump she causes over all possible orderings.

3 t s 16 · x x3 → 64

1

Order: 1 3 c0(x) 1 2 3 4 x 1 63 Order: 1 3 c0(x) 1 2 3 4 x 37 27

Resulting cost shares for x3 are 19 and 45

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 9 / 18

Weighted Shapley Value

Informal Definition (Weighted Shapley Value)

Shapley Value with distribution over orderings. Players have sampling weights and the last one is repeatedly drawn from the remaining set.

3 t s 16 · x x3 → 64

1

Order: 1 3 c0(x) 1 2 3 4 x 1 63 Order: 1 3 c0(x) 1 2 3 4 x 37 27

Suppose sampling weights = weights. Cost shares become 7.5 and 56.5

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 10 / 18

Price of Anarchy and Price of Stability

Price of Anarchy (PoA)

PoA(Ξ,C) = worst case total cost in equilibrium

• ptimal total cost

Price of Stability (PoS)

PoS(Ξ,C) = best case total cost in equilibrium

• ptimal total cost

with Ξ = cost-sharing method, C = set of allowable cost functions

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 11 / 18

Price of Anarchy and Price of Stability

Price of Anarchy (PoA)

PoA(Ξ,C) = worst case total cost in equilibrium

• ptimal total cost

Price of Stability (PoS)

PoS(Ξ,C) = best case total cost in equilibrium

• ptimal total cost

with Ξ = cost-sharing method, C = set of allowable cost functions Thought process:

1 The application gives us C 2 We decide on a cost-sharing method Ξ 3 How bad is the worst cast and best case equilibrium outcome? I for the worst-case network (given Ξ, C) Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 11 / 18

What We Know

Proportional Sharing: PoA

I polynomial costs: PoA = Θ(d/ ln d)d+1

[Awerbuch et al.’05, Christodoulou & Koutsoupias’05, Aland et al.’06]

I Recipe for general C.

[Roughgarden’09, Bhawalkar et al.’10]

unweighted: d ≤ PoS ≤ d + 1

[Christodoulou & Koutsoupias’05, Caragiannis et al.’06, Christodoulou & Gairing’13]

There are games with no PNE

[Libman & Orda’01, Fotakis et al.’04, Goemans et al.’05]

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 12 / 18

What We Know

Proportional Sharing: PoA

I polynomial costs: PoA = Θ(d/ ln d)d+1

[Awerbuch et al.’05, Christodoulou & Koutsoupias’05, Aland et al.’06]

I Recipe for general C.

[Roughgarden’09, Bhawalkar et al.’10]

unweighted: d ≤ PoS ≤ d + 1

[Christodoulou & Koutsoupias’05, Caragiannis et al.’06, Christodoulou & Gairing’13]

There are games with no PNE

[Libman & Orda’01, Fotakis et al.’04, Goemans et al.’05]

Weighted Shapley Values: PoA

[Kollias & Roughgarden’11, Gkatzelis et al.’14]

Existence of PNE only guaranteed by weighted Shapley values

[Gopalakrishnan et al.’13]

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 12 / 18

Our Results: PoA

Assumptions:

1 Convexity: Every cost function c ∈ C is continuous, nondecreasing,

and convex.

2 Closure under dilation: If c(x) ∈ C → c(ax) ∈ C. 3 Consistency: Cost shares depend only on how players contribute to

the joined cost.

4 Fairness: The cost share of a player on a resource is a convex

function of her weight.

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 13 / 18

Our Results: PoA

Assumptions:

1 Convexity: Every cost function c ∈ C is continuous, nondecreasing,

and convex.

2 Closure under dilation: If c(x) ∈ C → c(ax) ∈ C. 3 Consistency: Cost shares depend only on how players contribute to

the joined cost.

4 Fairness: The cost share of a player on a resource is a convex

function of her weight.

Main result 1: General PoA bounds

A recipe for computing the PoA given:

I the set of allowable cost functions, and I the cost sharing method.

Matching upper and lower bounds.

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 13 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P))

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P)) ≤ X

i∈N

X

e∈P ⇤

i

ξce(i, Se(P) ∪ {i})

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P)) ≤ X

i∈N

X

e∈P ⇤

i

ξce(i, Se(P) ∪ {i}) = X

e∈E

X

i∈Se(P ⇤)

ξce(i, Se(P) ∪ {i})

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P)) ≤ X

i∈N

X

e∈P ⇤

i

ξce(i, Se(P) ∪ {i}) = X

e∈E

X

i∈Se(P ⇤)

ξce(i, Se(P) ∪ {i}) ≤ λ · C(P ∗) + µ · C(P)

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P)) ≤ X

i∈N

X

e∈P ⇤

i

ξce(i, Se(P) ∪ {i}) = X

e∈E

X

i∈Se(P ⇤)

ξce(i, Se(P) ∪ {i}) ≤ λ · C(P ∗) + µ · C(P)

Convex Program

min λ 1 − µ s.t. µ ≤ 1 X

i∈T ⇤

ξc(i, T ∪ {i}) ≤ λ · wT ⇤ · c(wT ⇤) + µ · wT · c(wT ), ∀c, T, T ∗

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Recipe (sketch) for PoA

Smoothness argument: P is NE; P ∗ is optimum C(P) = X

i∈N

X

e∈Pi

ξce(i, Se(P)) ≤ X

i∈N

X

e∈P ⇤

i

ξce(i, Se(P) ∪ {i}) = X

e∈E

X

i∈Se(P ⇤)

ξce(i, Se(P) ∪ {i}) ≤ λ · C(P ∗) + µ · C(P)

Convex Program

min λ 1 − µ s.t. µ ≤ 1 X

i∈T ⇤

ξc(i, T ∪ {i}) ≤ λ · wT ⇤ · c(wT ⇤) + µ · wT · c(wT ), ∀c, T, T ∗ Primal solution ⇒ upper bound Lagrangian Dual solution ⇒ construction of matching lower bound

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 14 / 18

Our Results: PoS

Class of weighted shapley values; sampling parameter: λi = wγ

i

Only class of cost sharing method that

I is consistent and I guarantees existence of pure NE. Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 15 / 18

Our Results: PoS

Class of weighted shapley values; sampling parameter: λi = wγ

i

Only class of cost sharing method that

I is consistent and I guarantees existence of pure NE.

Upper bound for γ = 0 (Shapley cost sharing)

For arbitrary C, we have PoS ≤ maxc∈C,x>0

x·c(x) R x

0 c(x0)dx0 .

x

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 15 / 18

Our Results: PoS

Class of weighted shapley values; sampling parameter: λi = wγ

i

Only class of cost sharing method that

I is consistent and I guarantees existence of pure NE.

Upper bound for γ = 0 (Shapley cost sharing)

For arbitrary C, we have PoS ≤ maxc∈C,x>0

x·c(x) R x

0 c(x0)dx0 .

x

Corollary: polynomials with non-negative coefficients and degree at most d PoS ≤ d + 1 asymptotically matches lower bound for unweighted congestion games with proportional sharing

[Christodoulou, Gairing’13]

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 15 / 18

Our Results: PoS

polynomials with non-negative coefficients and degree at most d sampling parameters: λi = wγ

i

Theorem

PoS is at least (a) (2

1 d+1 − 1)−(d+1), for all γ > 0, and

(b) (d + 1)d+1, for all γ < 0.

γ

(2

1 d+1 1)−(d+1)

(d + 1)d+1 d + 1

PoS

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 16 / 18

Our Results: PoS

polynomials with non-negative coefficients and degree at most d sampling parameters: λi = wγ

i

Theorem

PoS is at least (a) (2

1 d+1 − 1)−(d+1), for all γ > 0, and

(b) (d + 1)d+1, for all γ < 0.

γ

(2

1 d+1 1)−(d+1)

(d + 1)d+1 d + 1

PoS PoA Gkatzelis et al.’14

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 16 / 18

Sketch: PoA ≤ maxc∈C,x>0

x·c(x) R x

0 c(x0)dx0

potential function (arbitrary ordering)

[Kollias & Roughgarden’11]

Φ(P) = X

e∈E

Φe(P) = X

e∈E

X

i∈Se(P)

ξce(i, {j : j ≤ i, j ∈ Se(P)}) Key point:

I Split player i into two players of weight wi/2. I We show that this can only reduce the potential. Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 17 / 18

Sketch: PoA ≤ maxc∈C,x>0

x·c(x) R x

0 c(x0)dx0

potential function (arbitrary ordering)

[Kollias & Roughgarden’11]

Φ(P) = X

e∈E

Φe(P) = X

e∈E

X

i∈Se(P)

ξce(i, {j : j ≤ i, j ∈ Se(P)}) Key point:

I Split player i into two players of weight wi/2. I We show that this can only reduce the potential.

We get Φ(P) ≥ X

e∈E

Z fe(P) ce(x)dx ≥ min

e∈E

R fe(P) ce(x)dx fe(P) · ce(fe(P)) · C(P) Combining this with C(P ∗) ≥ Φ(P ∗) ≥ Φ(P) gives the result.

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 17 / 18

Conclusion

Take home point

General tight PoA bounds based on

I set of allowed cost functions I cost sharing method

Shapley cost sharing minimises PoS

I under all cost sharing methods that guarantee pure NE I weight dependent priorities make PoS as bad as PoA Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 18 / 18

Conclusion

Take home point

General tight PoA bounds based on

I set of allowed cost functions I cost sharing method

Shapley cost sharing minimises PoS

I under all cost sharing methods that guarantee pure NE I weight dependent priorities make PoS as bad as PoA

Some open problems: further relax the assumptions does symmetry change things PoS bounds without using potential function Games where weighted Shapley beats Shapley?

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 18 / 18

Conclusion

Take home point

General tight PoA bounds based on

I set of allowed cost functions I cost sharing method

Shapley cost sharing minimises PoS

I under all cost sharing methods that guarantee pure NE I weight dependent priorities make PoS as bad as PoA

Some open problems: further relax the assumptions does symmetry change things PoS bounds without using potential function Games where weighted Shapley beats Shapley? Thanks! Are there any questions?

Gairing, Kollias, Kotsialou Tight Bounds for Cost-Sharing in WCG ICALP 2015 18 / 18