Generalized coverage problems: approximation …dariop/presentations/...Combinatorial resource...

Generalized coverage problems:approximation through game design

Dario PaccagnanJoint work with J. R. Marden (UCSB)

Combinatorial resource allocation

. a set of resources

. a set of agents

Goal: assign resources to agents to maximize a given welfare function

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Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . , n}

allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iai

vrwr (|a|r ) wr : N→ R≥0

System-level objective: maxa∈AW (a)

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

. no constraints on wr (j)

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

. no constraints on wr (j)typically concave

agents: i ∈ {1, . . . , n}

welfare: W (a) =∑

r∈∪iai

1 2 3 4 5

wr (j)

. no constraints on wr (j)typically concave

. to ease the presentationwr (j) = w(j)

Connection with Coverage problems

GMMC problem

- set of weighted resources: R- n collections of sets: {Ai}ni=1 ⊆ 2R

- choose one set per collection to max

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

- set of weighted resources: R- one collection of sets: A ⊆ 2R

- choose n sets from collection to max

W (a) =∑

r∈∪iai

. GMMC subsumes max-n-cover (set w(j) ≡ 1, Ai = Aj for all i , j)

. GMMC subsumes [Che04],[Gair09] (set w(j) ≡ 1)

[Che04] C. Chekuri et al. “Maximum Coverage Problem with Group Budget Constraints and Applications”, APPROX 04

[Gair09] M. Gairing, “Covering Games: Approximation through Non-cooperation”, WINE 09

GMMC problem

- set of weighted resources: R- n collections of sets: {Ai}ni=1 ⊆ 2R

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

W (a) =∑

r∈∪iai

GMMC problem- set of weighted resources: R

- n collections of sets: {Ai}ni=1 ⊆ 2R

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

W (a) =∑

r∈∪iai

GMMC problem- set of weighted resources: R- n collections of sets: {Ai}ni=1 ⊆ 2R

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover- set of weighted resources: R

- one collection of sets: A ⊆ 2R

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

Max-n-cover- set of weighted resources: R- one collection of sets: A ⊆ 2R

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

W (a) =∑

r∈∪iai

W (a) =∑

r∈∪iai

vrw(|a|r )

W (a) =∑

r∈∪iai

[Gair09] M. Gairing, “Covering Games: Approximation through Non-cooperation”, WINE 09 4

Facts: hardness and approximability

max-n-cover problem:

. NP-hard

. NP-hard to approximate within any ratio better than 1− 1/e

. Poly-algorithms achieve 1− 1/e

GMMC problem:

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− (w(n)− w(n − 1))

Issues:- distributedness?- best possible approximation? Ai 6= Aj , w not concave?

[Svir17] M. Sviridenko, J. Vondrak, J. Ward, “Optimal Approximation for Submodular and Supermodular Optimization withBounded Curvature”, MOR 17

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

GMMC problem:

. NP-hard

Issues:

- distributedness?- best possible approximation? Ai 6= Aj , w not concave?

. NP-hard

GMMC problem:

. NP-hard

Issues:- distributedness?

- best possible approximation? Ai 6= Aj , w not concave?

. NP-hard

GMMC problem:

. NP-hard

Issues:- distributedness?- best possible approximation?

Ai 6= Aj , w not concave?

. NP-hard

GMMC problem:

. NP-hard

Main result

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

approx.

w(j) = jd , 40 agents

max-n-cover1− 1

[Pac18a] DP, R.Chandan, J. Marden “Distributed resource allocation through utility design-Part I: optimizing the performance certificates via the price of anarchy”, ArXiv 2018

[Pac18b] DP, J. Marden “- Part II: applications to submodular, supermodular and set covering problems”, ArXiv 2018

Main result

Game theory can be used to produce algorithms that are:

distributed, efficient, match/improve existing approximations

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

Game theory can be used to produce algorithms that are:distributed

, efficient, match/improve existing approximations

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

Game theory can be used to produce algorithms that are:distributed, efficient

, match/improve existing approximations

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

1− c/e

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

GT-approach

1− c/e

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

GT-approach

1− c/e

max-n-cover

1− 1e

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

GT-approach

1− c/e

max-n-cover1− 1

Main result

0 0.2 0.4 0.6 0.8 10.5

approx.

GT-approach

1− c/e

max-n-cover1− 1

Outline

1. Introduction

2. Game-design approach

3. Characterizing the price of anarchy

4. Optimizing the price of anarchy

5. Conclusions and Outlook

The game-theoretic approach

Game design

maxW (a)

Game design

. Distributed algorithm

. Good approximation

. Polytime

maxW (a)

Game design

. Polytime

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

Game design

. Polytime

maxW (a)

Game design

. Polytime

maxW (a)

Game design

. Polytime

utilities

maxW (a)

Game design

. Polytime

utilitiesgame

maxW (a)

Game design

. Polytime

utilitiesgame

equilibria

maxW (a)

Game design

. Polytime

utilitiesgame

equilibria

maxW (a)

Game design

. Polytime

utilitiesgame

equilibria

maxW (a)

Game design

. Polytime

utilitiesgame

equilibria

Utility design and approximation ratio

ui (ai , a−i )

=∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤n

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

ui (ai , a−i ) =∑r∈ai

PoA(f ) =

W (aopt)≤ 1

How to design f ?

Maximize worst-case performance

PoA(f ) =

W (aopt)≤ 1

PoA(f ) =

W (aopt)≤ 1

Given instance I, fix f

→ game Gf = {I, f } → NE(Gf )

PoA(f ) =

W (aopt)≤ 1

Given instance I, fix f → game Gf = {I, f }

→ NE(Gf )

PoA(f ) =

W (aopt)≤ 1

PoA(f ) =

W (aopt)≤ 1

PoA(f ) =

W (aopt)≤ 1

PoA(f ) = infGf : #agents ≤n

W (aopt)≤ 1

PoA(f ) = infGf : #agents ≤n

W (aopt)≤ 1

Utility design reduces to

Given f , characterize or bound PoA(f )

Find f with highest PoA(f )

Characterizing the price of anarchy

The quantity we wish to compute: PoA(f ) = infG∈G

(mina∈NE(G) W (a)

W (aopt)

)- well studied in game theory

- difficult to compute

- bounds are available in special cases

- bounds do not explicitly depend on f

- smoothness not applicable

- tightness?

Theorem (Characterization of PoA(f )) [Pac18a],[Pac18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(n2) constraints

. LP involves all the components w(j) and f (j)

(mina∈NE(G) W (a)

W (aopt)

- well studied in game theory

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

(mina∈NE(G) W (a)

W (aopt)

- tightness?

Proof Sketch - Part 1/4

Idea: transform the definition of PoA itself into a LP

Four steps towards the goal:

1. PoA(f ) is the same of the price of anarchy over a reduced class ofgames where each agent has only two feasible allocations

i.e. we can reduce to Ai = {ei , oi} with ei the worst NE

PoA(f ) =( )

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

2. Relax the previous program

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) =( )

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) =( )

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

1. PoA(f ) is the same of the price of anarchy over a reduced class ofgames where each agent has only two feasible allocationsi.e. we can reduce to Ai = {ei , oi} with ei the worst NE

PoA(f ) =( )

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) = infG∈G

(mina∈NE(G) W (a)

maxa∈AW (a)

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) = infG∈G

(mina∈NE(G) W (a)

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) = infG∈G

(W (e)

s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) = infG∈G

(W (e)

)s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

ui (e) ≥

ui (oi , e−i )

PoA(f ) = infG∈G

(W (e)

)s.t. ui (e) ≥ ui (oi , e−i ) ∀i

PoA(f ) = infG∈G

s.t.∑i

ui (e) ≥∑i

ui (oi , e−i ) ∀i

3. How to describe an instance? Need to describe W (a), ui (a) on A

Variables θ allow to computeW (a), ui (a) in all allocations

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

Issue: #weights is exponential!

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1e1 θo1

θo1o2

θe1e2

θo2e1 θo2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

o2θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1o2e1e2

θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1o2e1e2

θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1o2e1e2

θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

Variables θ allow to computeW (a), ui (a) in all allocations,e.g.

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1o2e1e2

θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

θo1o2e1e2

θo1o2e1

θo1e1 θo1

θo1o2

θo1e1e2

θo2e1e2

θe1e2

θo2e1 θo2

θo1o2e2

θo1e2

θo2e2

W (e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

+ (θe2 + θo2e2

+ θo1e2

+ θo1o2e2

+ (θe1e2 + θo1e1e2

+ θo2e1e2

+ θo1o2e1e2

u1(e1, e2)

= (θe1 + θo1e1

+ θo2e1

+ θo1o2e1

)w(1)f (1)

+ (θe1e2 + θo1e1e2

+ θo1o2e1e2

+ θo2e1e2

)w(2)f (2)

Issue: #weights is exponential!13

4. use reduced variables for W(e), W(o),∑

i ui(e)− ui(oi, e−i)→ define θ(a, x , b) ∈ R≥0 for 1 ≤ a + x + b ≤n, a, x , b∈{1, . . . , n}

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

[af (a+x)w(a+x)−bf (a+x+1)w(a+x+1)]θ(a, x , b)

The program becomes PoA(f ) =

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

The program becomes

PoA(f ) =

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

The program becomes PoA(f ) = infθ(a,x ,b)≥0

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

W (e) =∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b)

W (o) =∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

equil. =∑a,x ,b

The program becomes PoA(f ) = infθ(a,x ,b)≥0

s.t.∑i

ui (e)− ui (oi , e−i ) ≥ 0

W (e) = 1

Proof Sketch - Part 4/4: Primal LP

PoA(f ) =1

W ? = supθ(a,x ,b)

∑a,x ,b

1{b+x≥1}w(b + x)θ(a, x , b)

s.t.∑a,x ,b

[af (a+x)w(a+x)−bf (a+x+1)w(a+x+1)]θ(a, x , b)≥0

∑a,x ,b

1{a+x≥1}w(a + x)θ(a, x , b) = 1

θ(a, x , b) ≥ 0 ∀(a, x , b) ∈ I .

Dual LP

PoA(f ) =1

W ? = infλ∈R≥0, µ∈R

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

+λ[af (a+x)w(a+x)− bf (a+x+1)w(a+x+1)] ≤ 0

∀(a, x , b) ∈

. 2 decision variables, O(n2) constraints

. observe the special structure i.e. minλ,µ µ subject to µ ≥ . . .

. gives PoA for e.g., fsv(j) = 1/j , fmc(j) = 1− w(j − 1)/w(j)

Dual LP

PoA(f ) =1

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

∀(a, x , b) ∈ I

Dual LP

PoA(f ) =1

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

∀(a, x , b) ∈ ∂I

Dual LP

PoA(f ) =1

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

∀(a, x , b) ∈ ∂I

Dual LP

PoA(f ) =1

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

∀(a, x , b) ∈ ∂I

Dual LP

PoA(f ) =1

s.t. 1{b+x≥1}w(b+x)− µ1{a+x≥1}w(a+x)+

∀(a, x , b) ∈ ∂I

PoA: connection with existing literature

Covering Games: Approximation throughNon-Cooperation ?

Martin Gairing

Department of Computer Science, University of Liverpool, U.K.m.gairing@liverpool.ac.uk

Abstract. We propose approximation algorithms under game-theoreticconsiderations. We indroduce and study the general covering problemwhich is a natural generalization of the well-studied max-n-cover prob-lem. In the general covering problem, we are given a universal set ofweighted elements E and n collections of subsets of the elements. Thetask is to choose one subset from each collection such that the totalweight of their union is as large as possible. In our game-theoretic set-ting, the choice in each collection is made by an independent player.For covering an element, the players receive a payo↵ defined by a non-increasing utility sharing function. This function defines the fraction thateach covering player receives from the weight of the elements.We show how to construct a utility sharing function such that every NashEquilibrium approximates the optimal solution by a factor of 1 � 1

also prove that any sequence of unilateral improving steps is polynomiallybounded. This gives rise to a polynomial-time local search approximationalgorithm whose approximation ratio is best possible.

1 Introduction

Motivation and Framework. Large scale distributed systems, like the Inter-net, usually lack a centralized control authority. Instead, they are operated andcontrolled in a distributed fashion by competing entities – modeled as players –which make their decisions in order to optimize their own private utility. Suchsystems are assumed to end up in a Nash equilibrium [21] – a state in whichno player wishes to unilaterally leave her own strategy in order to improve thevalue of her private utility. However, Nash equilibria are often suboptimal solu-tions with respect to the social objective function. The price of anarchy [19] isa measure for the performance degradation. It is defined as the worst-case ratiobetween the values of a social objective function in a Nash equilibrium and inan optimum solution.

As the designer of a distributed system we are faced with the main challengeof how to design the distributed system in order to optimize this social objec-tive function even in the presence of myopic players. However, even if all players

? This work was supported by a fellowship within the Postdoc-Programme of theGerman Academic Exchange Service (DAAD).

Martin Gairing

1 Introduction

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 3, MARCH 2014 571

Generalized Efficiency Bounds inDistributed Resource Allocation

Jason R. Marden and Tim Roughgarden

Abstract—Game theory is emerging as a popular tool for dis-tributed control of multiagent systems. To take advantage of thesegame theoretic tools, the interactions of the autonomous agentsmust be designed within a game-theoretic environment. A centralcomponent of this game-theoretic design is the assignment of a localutility function to each agent. One promising approach to utilitydesign is assigning each agent a utility function according to theagent’s Shapley value. This method frequently results in games thatpossess many desirable features, such as the existence of pure Nashequilibria with near-optimal efficiency. In this paper, we explorethe relationship between the Shapley value utility design and theresulting efficiency of both pure Nash equilibria and coarse corre-lated equilibria. To study this relationship, we introduce a simpleclass of resource allocation problems. Within this class, we derivean explicit relationship between the structure of the resource allo-cation problem and the efficiency of the resulting equilibria. Lastly,we derive a bicriteria bound for this class of resource allocationproblems—a bound on the value of the optimal allocation relativeto the value of an equilibrium allocation with additional agents.

Index Terms—Cost sharing, distributed control, game theory,price of anarchy.

I. INTRODUCTION

R ESOURCE allocation is a fundamental problem thatarises in many application domains ranging from the

social sciences to engineering [1]–[7]. One example is theproblem of routing information through a shared network,where the global objective is to minimize average delay [7].An alternative example is the problem of allocating sensors toa given mission space where the global objective is to maxi-mize coverage area [6]. Regardless of the specific applicationdomain, the central objective is always the same: allocateresources to optimize a given global objective.Research has focused on both centralized and distributed

approaches for resource allocation [2], [5], [8]–[13]. In thispaper, we study distributed algorithms for resource allocation

Manuscript received May 21, 2012; revised March 29, 2013 and August 16,2013; accepted October 15, 2013. Date of publication January 20, 2014; dateof current version February 19, 2014. This paper was presented in part at theProceedings of the 48th IEEE Conference on Decision and Control, December2010. This work was supported by AFOSR grants FA9550-09-1-0538 andFA9550-12-1-0359 and ONR grant N00014-12-1-0643. Recommended byAssociate Editor Ozdaglar.J. R. Marden is with the Department of Electrical, Computer, and Energy

Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: jason.marden@colorado.edu).T. Roughgarden is with the Department of Computer Science, Stanford, Stan-

ford, CA 94305 USA (e-mail: tim@cs.stanford.edu).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2014.2301613

in large-scale engineering systems, where a centralized con-trol approach is undesirable or even infeasible. For example,a centralized control approach may be impossible for theaforementioned sensor allocation problem because of the com-plexity associated with a potentially large number of sensors,the vastness/uncertainty of the mission space, or potentialstealth requirements that restrict communication capabilities.A more desirable control approach is to establish a distributedcontrol algorithm that allows the sensors to allocate themselveseffectively over the mission space without the need for globalintervention [14], [15]. Such an algorithm would eliminate theneed for centralized communication and introduce an inherentrobustness to communication failures, sensor failures, andenvironmental uncertainties. While desirable, establishing sucha distributed control algorithm comes with its share of chal-lenges. Is it possible to characterize the global behavior thatresults from the interactions of a large group of autonomousagents, each acting independently in response to its own localinformation? How can we coordinate the agents behavior toensure that the emergent global behavior is desirable? What dowe give up in terms of efficiency when we transition from acentralized to a distributed control approach?A popular tool for distributed resource allocation is game

theory [11]–[13], [16]–[18]. Game theory is a well-establisheddiscipline in the social sciences used for describing the emer-gent global behavior in social systems such as traffic networks,social networks, and auctions. More generally, “Game theory isa bag of analytical tools designed to help us understand the phe-nomena that we observe when decision-makers interact” [19].The appeal of applying game-theoretic tools to distributed en-gineering systems stems from the fact that the underlying deci-sion-making architecture in social systems and the desired deci-sion-making architecture in distributed engineering systems canbe analyzed using the same mathematical tools. Furthermore,the field of game theory provides a vast array of tools that areextremely valuable for the design and control of distributed en-gineering systems [20]–[22].To take advantage of game theoretic tools for distributed en-

gineering systems, the interactions of the autonomous agentsmust be designed within a game-theoretic environment. Thismeans that the system designer must specify the following ele-ments: i) the set of decision-making agents, ii) a set of actions foreach agent, and iii) a local utility function for each agent. Whilespecifying the agents and their respective actions can be rela-tively straightforward, assigning local utility functions is some-what more opaque. There are many pertinent issues that needto be considered when designing the agents’ utility functions

0018-9286 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.