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Congestion Games and Price of Anarchy:

How to Reduce it and the Impact of Social Ignorance

Dimitris Fotakis

SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL TECHNICAL UNIVERSITY OF ATHENS, GREECE In part, joint work with Vasilis Gkatzelis − CIMS, New York Univ. Alexis Kaporis − ICSD, Univ. of the Aegean George Karakostas − DCS, McMaster Univ. Stavros Kolliopoulos − DI, Kapodistrian Univ. of Athens Paul Spirakis − RA CTI & CEID, Univ. of Patras

Dimitris Fotakis Congestion Games and Price of Anarchy

Setting and Outline

Question What happens when selfish users share resources?

Dimitris Fotakis Congestion Games and Price of Anarchy

Setting and Outline

Question What happens when selfish users share resources? Model Symmetric network congestion games with atomic players. s − t network and players with unit unsplittable demands. Players route on minimum latency s − t paths : Pure Nash Equilibrium (PNE). PNE may fail to optimize performance (total latency).

Dimitris Fotakis Congestion Games and Price of Anarchy

Setting and Outline

Question What happens when selfish users share resources? Model Symmetric network congestion games with atomic players. s − t network and players with unit unsplittable demands. Players route on minimum latency s − t paths : Pure Nash Equilibrium (PNE). PNE may fail to optimize performance (total latency). Objectives Quantify the inefficiency of PNE (under natural assumptions). Mitigate (or eliminate) the inefficiency of PNE.

Dimitris Fotakis Congestion Games and Price of Anarchy

Symmetric Network Congestion Games

Symmetric network congestion game Γ(N, G(V, E), (de)e∈E) : Set N of n players, each controls an unsplittable unit demand. Directed network G(V, E) with source s and sink t. Set E of m edges (resources) where demands are assigned. Common set of players’ actions : set P of s − t paths in G. Non-decreasing latency function de : I N → I R≥0 on each edge e.

s t x 4x2 2x+1 x/2 2x 2 2x 3 players x x x x x x t s 2 players

Dimitris Fotakis Congestion Games and Price of Anarchy

Symmetric Network Congestion Games

Symmetric network congestion game Γ(N, G(V, E), (de)e∈E) : Set N of n players, each controls an unsplittable unit demand. Directed network G(V, E) with source s and sink t. Set E of m edges (resources) where demands are assigned. Common set of players’ actions : set P of s − t paths in G. − Parallel links : each path consists of a single edge. Non-decreasing latency function de : I N → I R≥0 on each edge e. − Linear game: de(x) = aex + be, ae, be ≥ 0, ∀e ∈ E.

s t 2x+3 4x x+1 3x+5 10 players x x x x x x t s 2 players

Dimitris Fotakis Congestion Games and Price of Anarchy

Pure Nash Equilibrium

Configuration σ = (σ1, . . . , σn) : each player i selects path σi ∈ P . Congestion of edge e in σ: σe = |{i ∈ N : e ∈ σi}|. Latency of edge e in σ: de(σe). Latency of player i in σ: ci(σ) =

e∈σi de(σe).

s t x 4x2 2x+1 x/2 2x 2 2x 3 players 9 10 9 x x x x x x t s 3 3 2 players

Dimitris Fotakis Congestion Games and Price of Anarchy

Pure Nash Equilibrium

Configuration σ = (σ1, . . . , σn) : each player i selects path σi ∈ P . Congestion of edge e in σ: σe = |{i ∈ N : e ∈ σi}|. Latency of edge e in σ: de(σe). Latency of player i in σ: ci(σ) =

e∈σi de(σe).

Pure Nash Equilibrium (PNE) Stable state σ where no player can improve own latency unilaterally: ∀i ∈ N, ∀s ∈ P, ci(σ) ≤ ci(σ−i, s)

s t x 4x2 2x+1 x/2 2x 2 2x 3 players 9 10 9 x x x x x x t s 3 3 2 players

Dimitris Fotakis Congestion Games and Price of Anarchy

Pure Nash Equilibrium

Configuration σ = (σ1, . . . , σn) : each player i selects path σi ∈ P . Congestion of edge e in σ: σe = |{i ∈ N : e ∈ σi}|. Latency of edge e in σ: de(σe). Latency of player i in σ: ci(σ) =

e∈σi de(σe).

Pure Nash Equilibrium (PNE) Stable state σ where no player can improve own latency unilaterally: ∀i ∈ N, ∀s ∈ P, ci(σ) ≤ ci(σ−i, s) Potential Function [Ros 73] PNE correspond to local optima of Φ(σ) =

e∈E

σe

i=0 de(i)

s t x 4x2 2x+1 x/2 2x 2 2x 3 players 9 10 9 x x x x x x t s 3 3 2 players

Dimitris Fotakis Congestion Games and Price of Anarchy

Inefficiency due to Selfishness

Optimal Configuration

• = (o1, . . . , on) minimizes total latency : C(o) =

e∈E σede(σe)

x x x x x x t s 2 players Optimal (and PNE): C(o) = 4 2 2

Dimitris Fotakis Congestion Games and Price of Anarchy

Inefficiency due to Selfishness

Optimal Configuration

• = (o1, . . . , on) minimizes total latency : C(o) =

e∈E σede(σe)

Price of Anarchy [Kouts Papa 99] Game Γ : PoA(Γ) = max σ∈PNE(Γ)C(σ)/C(o) Class G : PoA(G) = supΓ∈G PoA(Γ)

x x x x x x t s 2 players Optimal (and PNE): C(o) = 4 2 2 x x x x x x t s PNE: C(σ) = 6, PoA = 1.5 3 3

Dimitris Fotakis Congestion Games and Price of Anarchy

Inefficiency due to Selfishness

Optimal Configuration

• = (o1, . . . , on) minimizes total latency : C(o) =

e∈E σede(σe)

Price of Anarchy [Kouts Papa 99] Game Γ : PoA(Γ) = max σ∈PNE(Γ)C(σ)/C(o) Class G : PoA(G) = supΓ∈G PoA(Γ) Price of Stability [Anshel DKTWR 04] Game Γ : PoS(Γ) = min σ∈PNE(Γ)C(σ)/C(o) Class G : PoS(G) = supΓ∈G PoS(Γ)

x x x x x x t s 2 players Optimal (and PNE): C(o) = 4, PoS = 1 2 2 x x x x x x t s PNE: C(σ) = 6, PoA = 1.5 3 3

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

total latency confs

• f Γ1

PoA, PoS Pure Nash equilibrium PoA(Γ1) PoS(Γ1)

• pt

1 PoA(Γ) = maxσ∈PNE(Γ) C(σ)/C(o) PoS(Γ) = minσ∈PNE(Γ) C(σ)/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

total latency confs

• f Γ1

PoA, PoS Pure Nash equilibrium PoA(Γ1) PoS(Γ1)

• pt

confs

• f Γ2
• pt

PoA(Γ2) PoS(Γ2) 1 PoA(Γ) = maxσ∈PNE(Γ) C(σ)/C(o) PoS(Γ) = minσ∈PNE(Γ) C(σ)/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

total latency confs

• f Γ1

PoA, PoS Pure Nash equilibrium PoA(Γ1) PoS(Γ1)

• pt

confs

• f Γ2
• pt

PoA(Γ2) PoS(Γ2) confs

• f Γ3
• pt

PoS(Γ3) = 1 PoA(Γ3) PoA(Γ) = maxσ∈PNE(Γ) C(σ)/C(o) PoS(Γ) = minσ∈PNE(Γ) C(σ)/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

total latency confs

• f Γ1

PoA, PoS Pure Nash equilibrium PoA(Γ1) PoS(Γ1)

• pt

confs

• f Γ2
• pt

PoA(Γ2) PoS(Γ2) confs

• f Γ3
• pt

PoS(Γ3) = 1 PoA(Γ3) confs

• f Γ4
• pt

PoA(Γ4) = PoS(Γ4) PoA(Γ) = maxσ∈PNE(Γ) C(σ)/C(o) PoS(Γ) = minσ∈PNE(Γ) C(σ)/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

total latency PoA, PoS Pure Nash equilibrium

• pt
• pt

PoA(G)

• pt

1

• pt

PoS(G) Class of games G PoA(G) = supΓ∈G PoA(Γ) PoS(G) = supΓ∈G PoS(Γ)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

Non-atomic games: infinite #players with infinitesimal demand. PoA for non-atomic games with latencies in class D PoA = α(D)

[Rough03]

4/3 for linear latencies, Θ( p

ln p) for polynomials of degree p.

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

Non-atomic games: infinite #players with infinitesimal demand. PoA for non-atomic games with latencies in class D PoA = α(D)

[Rough03]

4/3 for linear latencies, Θ( p

ln p) for polynomials of degree p.

PoA for atomic games with unsplittable demands Linear latencies: PoA = 2.5 [AzarAweEpst05], [ChristKouts05], [AlandDGMS06] Polynomial latencies of degree p: PoA = pΘ(p)

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

Non-atomic games: infinite #players with infinitesimal demand. PoA for non-atomic games with latencies in class D PoA = α(D)

[Rough03]

4/3 for linear latencies, Θ( p

ln p) for polynomials of degree p.

PoA for atomic games with unsplittable demands Linear latencies: PoA = 2.5 [AzarAweEpst05], [ChristKouts05], [AlandDGMS06] Polynomial latencies of degree p: PoA = pΘ(p) Parallel links : PoA = α(D) [L¨

uckMMR04], [CaragKaklKanel07], [Fot07]

Extension-parallel networks : PoA = α(D) [Fotakis08]

Dimitris Fotakis Congestion Games and Price of Anarchy

Price of Anarchy and Stability

Non-atomic games: infinite #players with infinitesimal demand. PoA for non-atomic games with latencies in class D PoA = α(D)

[Rough03]

4/3 for linear latencies, Θ( p

ln p) for polynomials of degree p.

PoA for atomic games with unsplittable demands Linear latencies: PoA = 2.5 [AzarAweEpst05], [ChristKouts05], [AlandDGMS06] Polynomial latencies of degree p: PoA = pΘ(p) Parallel links : PoA = α(D) [L¨

uckMMR04], [CaragKaklKanel07], [Fot07]

Extension-parallel networks : PoA = α(D) [Fotakis08] Price of Stability Linear latencies: PoS = 1 + √ 3/3

[ChristKouts05], [CaragKaklamKanel07]

s − t networks : PoS = α(D)

[Fotakis08]

Dimitris Fotakis Congestion Games and Price of Anarchy

Reducing the Price of Anarchy

Network Design Detection (and elimination) of Braess’s paradox [Rough01] Difficult if the network is operational, computationally hard Tractable for several interesting cases [FotaKapoSpir 09]

Dimitris Fotakis Congestion Games and Price of Anarchy

Reducing the Price of Anarchy

Network Design Detection (and elimination) of Braess’s paradox [Rough01] Difficult if the network is operational, computationally hard Tractable for several interesting cases [FotaKapoSpir 09] Coordination mechanisms Modify the players’ costs [ChristKoutsNanav 04] Scheduling [ImmorLMS 05],[AzarJainMir 08], [Caragiannis 09] Connection games : priority cost-sharing better than fair [CRV08] Significant improvement. Fairness and implementation issues.

Dimitris Fotakis Congestion Games and Price of Anarchy

Reducing the Price of Anarchy

Stackelberg routing Exploit the presence of coordinated players [KorLazOrd 97] Improvement depends on the fraction of coordinated players. No system modifications are required.

Dimitris Fotakis Congestion Games and Price of Anarchy

Reducing the Price of Anarchy

Stackelberg routing Exploit the presence of coordinated players [KorLazOrd 97] Improvement depends on the fraction of coordinated players. No system modifications are required. Resource pricing Introduce economic disincentives (refundable tolls). Tolls increase players’ disutility, large tolls may be required. Tolls known to enforce optimal configuration for non-atomic games.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Routing

Model Both selfish and coordinated players are present. Leader determines paths of coordinated players to optimize performance. Selfish players (followers) seek to minimize their own latency and reach a pure Nash equilibrium.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Routing

Model Both selfish and coordinated players are present. Leader determines paths of coordinated players to optimize performance. Selfish players (followers) seek to minimize their own latency and reach a pure Nash equilibrium. Stackelberg Strategy Algorithm allocating a path to each coordinated player. Objective: lead the selfish players to a good PNE.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Γ(N, G(V, E), (de)e∈E) : k coordinated and n − k selfish players.

Fraction of coordinated players : γ = k/n

Optimal configuration o = (o1, . . . , on)

Poly-time for symmetric network games with convex latencies.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Γ(N, G(V, E), (de)e∈E) : k coordinated and n − k selfish players.

Fraction of coordinated players : γ = k/n

Optimal configuration o = (o1, . . . , on)

Poly-time for symmetric network games with convex latencies.

Coordinated players are assigned to k optimal paths .

Stackelberg strategy selects L ⊆ N, |L| = k = γn Stackelberg configuration s(L) = (oi)i∈L Stackelberg congestion se(L) = |{i ∈ L : e ∈ oi}|

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Γ(N, G(V, E), (de)e∈E) : k coordinated and n − k selfish players.

Fraction of coordinated players : γ = k/n

Optimal configuration o = (o1, . . . , on)

Poly-time for symmetric network games with convex latencies.

Coordinated players are assigned to k optimal paths .

Stackelberg strategy selects L ⊆ N, |L| = k = γn Stackelberg configuration s(L) = (oi)i∈L Stackelberg congestion se(L) = |{i ∈ L : e ∈ oi}|

Game ˜ ΓL(N \ L, G(V, E), (˜ de)e∈E), with ˜ de(x) = de(x + se(L))

Selfish players : worst PNE σ(L) of maximum C(σ(L) + s(L))

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Γ(N, G(V, E), (de)e∈E) : k coordinated and n − k selfish players.

Fraction of coordinated players : γ = k/n

Optimal configuration o = (o1, . . . , on)

Poly-time for symmetric network games with convex latencies.

Coordinated players are assigned to k optimal paths .

Stackelberg strategy selects L ⊆ N, |L| = k = γn Stackelberg configuration s(L) = (oi)i∈L Stackelberg congestion se(L) = |{i ∈ L : e ∈ oi}|

Game ˜ ΓL(N \ L, G(V, E), (˜ de)e∈E), with ˜ de(x) = de(x + se(L))

Selfish players : worst PNE σ(L) of maximum C(σ(L) + s(L))

Price of Anarchy PoAΓ

Str(γ) =

max

σ(L)∈PNE(˜ ΓL)

C(σ(L) + s(L))/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Γ(N, G(V, E), (de)e∈E) : k coordinated and n − k selfish players.

Fraction of coordinated players : γ = k/n

Optimal configuration o = (o1, . . . , on)

Poly-time for symmetric network games with convex latencies.

Coordinated players are assigned to k optimal paths .

Stackelberg strategy selects L ⊆ N, |L| = k = γn Stackelberg configuration s(L) = (oi)i∈L Stackelberg congestion se(L) = |{i ∈ L : e ∈ oi}|

Game ˜ ΓL(N \ L, G(V, E), (˜ de)e∈E), with ˜ de(x) = de(x + se(L))

Selfish players : worst PNE σ(L) of maximum C(σ(L) + s(L))

Price of Anarchy PoAG

Str(γ) = sup Γ∈G

max

σ(L)∈PNE(˜ ΓL)

C(σ(L) + s(L))/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Computing the best strategy is NP-complete even for parallel links and linear latencies [Roughgarden 02].

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Computing the best strategy is NP-complete even for parallel links and linear latencies [Roughgarden 02]. Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c1(o) ≥ · · · ≥ cn(o), then L = {1, . . . , k} .

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Computing the best strategy is NP-complete even for parallel links and linear latencies [Roughgarden 02]. Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c1(o) ≥ · · · ≥ cn(o), then L = {1, . . . , k} . Scale [Rough02] Random set L , |L| = k , with probability 1/ n

k

• .

Each resource e gets γoe coordinated players on expectation.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies

Computing the best strategy is NP-complete even for parallel links and linear latencies [Roughgarden 02]. Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c1(o) ≥ · · · ≥ cn(o), then L = {1, . . . , k} . Scale [Rough02] Random set L , |L| = k , with probability 1/ n

k

• .

Each resource e gets γoe coordinated players on expectation. Cover Atomic games when k large enough (e.g. k ≥ m) . Integer λ : each e gets ≥ min{λ, oe} coordinated players. L is computed greedily so that min{λ, oe} ≤ se(L) ≤ oe for all e.

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 18 players x 2x+16 2x+16 2x+16 2x+16 2x+16 s t 18 players 11 2 2 1 1 1 Optimal Total lat.: 255 s t 18 players 18 Worst PNE Total lat.: 324

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 18 players x 2x+16 2x+16 2x+16 2x+16 2x+16 s t 18 players 11 2 2 1 1 1 Optimal Total lat.: 255 s t 18 players 18 Worst PNE Total lat.: 324 s t 18 players 12 2 2 1 1 LLF, 6 players Total lat.: 260

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 18 players x 2x+16 2x+16 2x+16 2x+16 2x+16 s t 18 players 11 2 2 1 1 1 Optimal Total lat.: 255 s t 18 players 18 Worst PNE Total lat.: 324 s t 18 players 12 2 2 1 1 LLF, 6 players Total lat.: 260 s t 18 players 1+12 1 1 1 1 1 Cover, 6 players Total lat.: 259

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 18 players x 2x+16 2x+16 2x+16 2x+16 2x+16 s t 18 players 11 2 2 1 1 1 Optimal Total lat.: 255 s t 18 players 18 Worst PNE Total lat.: 324 s t 18 players 12 2 2 1 1 LLF, 6 players Total lat.: 260 s t 18 players 1+12 1 1 1 1 1 Cover, 6 players Total lat.: 259

Scale : I E[C(s + σ)] ≥ 284.98 − With probability ≥ 0.572, C(s + σ) ≥ 292 − With probability ≥ 0.883, C(s + σ) ≥ 279 − With probability ≥ 0.987, C(s + σ) ≥ 268

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 18 players x 2x+16 2x+16 2x+16 2x+16 2x+16 s t 18 players 11 2 2 1 1 1 Optimal Total lat.: 255 s t 18 players 18 Worst PNE Total lat.: 324 s t 18 players 12 2 2 1 1 LLF, 6 players Total lat.: 260 s t 18 players 1+12 1 1 1 1 1 Cover, 6 players Total lat.: 259 s t 18 players 12 2 1 1 1 1 Optimal strategy 6 players Total lat: 256

Scale : I E[C(s + σ)] ≥ 284.98 − With probability ≥ 0.572, C(s + σ) ≥ 292 − With probability ≥ 0.883, C(s + σ) ≥ 279 − With probability ≥ 0.987, C(s + σ) ≥ 268

Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 6 3 1 4 Optimal, total lat: 147

Upper: 6x13 Middle: 1x15 Lower: 3x18

s t 10 players 8 7 2 1 3 PNE, total lat.: 148

Upper: 7x15 Middle: 1x15 Lower: 2x14 Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 6 3 1 4 Optimal, total lat: 147

Upper: 6x13 Middle: 1x15 Lower: 3x18

s t 10 players 7 7 3 3 LLF, 3 pl., total lat.: 149!

Upper: 7x14 Middle: 0 Lower: 3x17

s t 10 players 8 7 2 1 3 PNE, total lat.: 148

Upper: 7x15 Middle: 1x15 Lower: 2x14 Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 6 3 1 4 Optimal, total lat: 147

Upper: 6x13 Middle: 1x15 Lower: 3x18

s t 10 players 7 7 3 3 LLF, 3 pl., total lat.: 149!

Upper: 7x14 Middle: 0 Lower: 3x17

s t 10 players 1+1 1 Cover, 3 pl., total lat.: 148

Upper: (1+6)x15 Middle: (1+0)x15 Lower: (1+1)x14

2+1 2+6 1+6 s t 10 players 8 7 2 1 3 PNE, total lat.: 148

Upper: 7x15 Middle: 1x15 Lower: 2x14 Dimitris Fotakis Congestion Games and Price of Anarchy

Stackelberg Strategies: Examples

s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 6 3 1 4 Optimal, total lat: 147

Upper: 6x13 Middle: 1x15 Lower: 3x18

s t 10 players 7 7 3 3 LLF, 3 pl., total lat.: 149!

Upper: 7x14 Middle: 0 Lower: 3x17

s t 10 players 1+1 1 Cover, 3 pl., total lat.: 148

Upper: (1+6)x15 Middle: (1+0)x15 Lower: (1+1)x14

2+1 2+6 1+6 s t 10 players 8 7 2 1 3 PNE, total lat.: 148

Upper: 7x15 Middle: 1x15 Lower: 2x14

Scale : I E[C(s + σ)] ≥ 148.0083 − With probability 119

120, C(s + σ) = 148

− With probability

1 120, C(s + σ) = 149

Dimitris Fotakis Congestion Games and Price of Anarchy

Work on Non-Atomic Games

PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoALLF is 1/γ for general and 4/(3 + γ) for linear latencies.

Dimitris Fotakis Congestion Games and Price of Anarchy

Work on Non-Atomic Games

PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoALLF is 1/γ for general and 4/(3 + γ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08]

Dimitris Fotakis Congestion Games and Price of Anarchy

Work on Non-Atomic Games

PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoALLF is 1/γ for general and 4/(3 + γ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08] PoA of LLF and Scale PoALLF ≤ γ + (1 − γ)α(D) for parallel links [Swamy07] PoALLF ≤ 1 + 1/γ for series-parallel networks [Sw07], [CorStMos07] PoA of LLF and Scale for linear congestion games [KarakKolliop06]

Dimitris Fotakis Congestion Games and Price of Anarchy

Work on Non-Atomic Games

PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoALLF is 1/γ for general and 4/(3 + γ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08] PoA of LLF and Scale PoALLF ≤ γ + (1 − γ)α(D) for parallel links [Swamy07] PoALLF ≤ 1 + 1/γ for series-parallel networks [Sw07], [CorStMos07] PoA of LLF and Scale for linear congestion games [KarakKolliop06] Non-PoA Results FPTAS for parallel links with polynomial latencies [KumMar02] Smallest fraction of coordinated players for optimality [KapSpir06] Smallest fraction of coordinated players to improve [SharWilliam07]

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Work on Atomic Games

Games with Linear Latencies and Arbitrary Actions Upper bounds on PoA of LLF, Scale, Cover, and combinations. Nearly matching lower bound on PoA of LLF . Lower bound on PoA of any randomized Stackelberg strategy.

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Work on Atomic Games

Games with Linear Latencies and Arbitrary Actions Upper bounds on PoA of LLF, Scale, Cover, and combinations. Nearly matching lower bound on PoA of LLF . Lower bound on PoA of any randomized Stackelberg strategy. Games on Parallel Links with Arbitrary Latencies Same upper bounds on PoA of LLF as for non-atomic games. For arbitrary latencies, PoALLF ≤ 1/γ For latencies in class D, PoALLF ≤ γ + (1 − γ)α(D) For linear latencies, PoALLF ≤ (4 − γ)/3 .

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bounds

Notation : Stackelberg configuration : s, se Worst Nash equilibrium : σ, σe Worst configuration : f = s + σ, fe = se + σe Approach similar to [AzarAwerEpst 05], [ChristKouts 05].

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bounds

Notation : Stackelberg configuration : s, se Worst Nash equilibrium : σ, σe Worst configuration : f = s + σ, fe = se + σe Approach similar to [AzarAwerEpst 05], [ChristKouts 05]. Nash inequality for selfish player i ci(f) ≤

e∈oi(ae(fe + 1) + be)

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bounds

Notation : Stackelberg configuration : s, se Worst Nash equilibrium : σ, σe Worst configuration : f = s + σ, fe = se + σe Approach similar to [AzarAwerEpst 05], [ChristKouts 05]. Nash inequality for selfish player i ci(f) ≤

e∈oi(ae(fe + 1) + be)

Optimal action for coordinated player j cj(f) =

e∈oj(aefe + be)

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bounds

Notation : Stackelberg configuration : s, se Worst Nash equilibrium : σ, σe Worst configuration : f = s + σ, fe = se + σe Approach similar to [AzarAwerEpst 05], [ChristKouts 05]. Nash inequality for selfish player i ci(f) ≤

e∈oi(ae(fe + 1) + be)

Optimal action for coordinated player j cj(f) =

e∈oj(aefe + be)

Putting everything together C(f) ≤

e∈E(ae(oefe + oe − se) + beoe)

Bound rhs in terms of C(f) and C(o) using strategy’s properties.

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bound for LLF

PoALLF ≤ min 20 − 11γ 8 , 3 − 2γ + √5 − 4γ 2

• 1

1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of LLF 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper and Lower Bound for LLF

5(2 − γ) 4 + γ − ε ≤ PoALLF ≤ min 20 − 11γ 8 , 3 − 2γ + √5 − 4γ 2

• 1

1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of LLF 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper and Lower Bound for Scale

I E[C(f)] C(o) ≤ max 5 − 3γ 2 , 5 − 4γ 3 − 2γ

• 1

1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of Scale 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper and Lower Bound for Scale

I E[C(f)] C(o) ≤ max 5 − 3γ 2 , 5 − 4γ 3 − 2γ

• I

E[C(f)] C(o) ≥        5 − 5γ + 2γ2 2 − ε γ ∈ [0, 1/2) 2 1 + γ − ε γ ∈ [1/2, 1]

1 1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of Scale 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bound for Cover

PoA of Cover tends to PoA of non-atomic game as λ grows Linear latencies : PoACover ≤ 4λ−1

3λ−1

Linear latencies without offset : PoACover ≤ 1 +

1 2λ

Parallel links, linear latencies, no offset : PoACover ≤ 1 +

1 4(λ+1)2−1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bound for Cover-Scale

If n/m large and k ≥ m, Cover-Scale achieves better PoA ! Cover assigns λm players so that min{λ, oe} ≤ sC

e ≤ oe for each e,

λ ≤⌊k/m⌋ any integer. Scale assigns k − λm players randomly wrt o − sC .

n/m = 10 k ≥ m γ = k/n ≥ 0.1 λ = 1 − − Scale − − Cover-Scale 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of Cover-Scale 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Linear Games: Upper Bound for LLF-Cover

If n/m large and k ≥ m, LLF-Cover achieves better PoA ! LLF assigns k − λm players to the largest latency actions in o. Cover assigns λm players so that min{λ, oe − sL

e } ≤ sC e ≤ oe − sL e

for each e, λ ≤⌊k/m⌋ any integer.

n/m = 10 k ≥ m γ = k/n ≥ 0.1 λ = 1 − − LLF − − LLF-Cover 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Price of Anarchy of LLF-Cover 0.2 0.4 0.6 0.8 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls

Economic (dis)incentives (tolls) to improve total latency. Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency.

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls

Economic (dis)incentives (tolls) to improve total latency. Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ(N, G(V, E), (de)e∈E) Toll function τ : E → I R≥0 assigns toll τe ≥ 0 to every edge e.

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls

Economic (dis)incentives (tolls) to improve total latency. Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ(N, G(V, E), (de)e∈E) Toll function τ : E → I R≥0 assigns toll τe ≥ 0 to every edge e. Modified congestion game with tolls Γτ(N, G(V, E), (de)e∈E) Cost of edge e in configuration σ : de(σe) = de(σe) + τe Cost of player i in configuration σ : ci(σ) =

e∈σi(de(σe) + τe)

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls

Economic (dis)incentives (tolls) to improve total latency. Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ(N, G(V, E), (de)e∈E) Toll function τ : E → I R≥0 assigns toll τe ≥ 0 to every edge e. Modified congestion game with tolls Γτ(N, G(V, E), (de)e∈E) Cost of edge e in configuration σ : de(σe) = de(σe) + τe Cost of player i in configuration σ : ci(σ) =

e∈σi(de(σe) + τe)

Refundable tolls increase players’ cost but not total latency Admin seeks to minimize total latency C(σ) =

e∈E σede(σe)

Tolls τ such that optimal o of Γ is some (the worst) PNE of Γτ

Dimitris Fotakis Congestion Games and Price of Anarchy

Enforceable Congestions

Configuration f weakly enforceable by tolls τ Every configuration σ with σe = fe on all e ∈ E is a PNE of Γτ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Enforceable Congestions

Configuration f weakly enforceable by tolls τ Every configuration σ with σe = fe on all e ∈ E is a PNE of Γτ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1 Configuration f strongly enforceable by tolls τ Configuration σ is a PNE of Γτ iff σe = fe for all e ∈ E Strongly enforceable : weakly enforceable and unique PNE of Γτ (Strongly) optimal tolls τ strongly enforce optimal o : PoA = 1

Dimitris Fotakis Congestion Games and Price of Anarchy

Enforceable Congestions

Configuration f weakly enforceable by tolls τ Every configuration σ with σe = fe on all e ∈ E is a PNE of Γτ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1 Configuration f strongly enforceable by tolls τ Configuration σ is a PNE of Γτ iff σe = fe for all e ∈ E Strongly enforceable : weakly enforceable and unique PNE of Γτ (Strongly) optimal tolls τ strongly enforce optimal o : PoA = 1 Weak and strong optimality equivalent for non-atomic games.

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls: Example

s t 10 players 7 6 3 1 4 Optimal Total lat.: 147 s t 10 players x x 3x+3 x+1 x+2 s t 10 players 8 7 2 1 3 PNE Total lat.: 148

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls: Example

s t 10 players 7 6 3 1 4 Optimal Total lat.: 147 s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 6 3 1 4 Total lat.: 147

Upper: 6x(13+3) Middle: 1x(15+2) Lower: 3x(18+0)

s t 10 players x x+3 3x+3 x+1+2 x+2 s t 10 players 8 7 2 1 3 PNE Total lat.: 148

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls: Example

s t 10 players 7 6 3 1 4 Optimal Total lat.: 147 s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 7 3 3 Total lat.: 149

Upper: 7x(14+3) Middle: 0 Lower: 3x(17+0)

s t 10 players 7 6 3 1 4 Total lat.: 147

Upper: 6x(13+3) Middle: 1x(15+2) Lower: 3x(18+0)

s t 10 players x x+3 3x+3 x+1+2 x+2 Optimal weakly enforceable s t 10 players 8 7 2 1 3 PNE Total lat.: 148

Upper: 7x(15+3) Middle: 1x(15+2) Lower: 2x(14+0) Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls: Example

s t 10 players 7 6 3 1 4 Optimal Total lat.: 147 s t 10 players x x 3x+3 x+1 x+2 s t 10 players 7 7 3 3 Total lat.: 149

Upper: 7x(14+3) Middle: 0 Lower: 3x(17+0)

s t 10 players 7 6 3 1 4 Total lat.: 147

Upper: 6x(13+3) Middle: 1x(15+2) Lower: 3x(18+0)

s t 10 players x x+3 3x+3 x+1+2 x+2 Optimal weakly enforceable s t 10 players 8 7 2 1 3 PNE Total lat.: 148

Upper: 7x(15+3) Middle: 1x(15+2) Lower: 2x(14+0)

s t 10 players 7 6 3 1 4 Total lat.: 147

Upper: 6x(13+5) Middle: 1x(15+3) Lower: 3x(18+0)

s t 10 players x x+5 3x+3 x+1+3 x+2 Optimal strongly enforceable

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls for Non-Atomic Games

Marginal cost tolls for optimal : de(oe) = de(oe) + oed′

e(oe)

Optimal of Γ iff Nash equilibrium of Γτ Not weakly enforce optimal for atomic games.

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls for Non-Atomic Games

Marginal cost tolls for optimal : de(oe) = de(oe) + oed′

e(oe)

Optimal of Γ iff Nash equilibrium of Γτ Not weakly enforce optimal for atomic games. Optimal tolls for heterogeneous players Players have different latency vs. tolls valuation. Existence for s − t networks and computation for finite #types

[ColeDodisRough 03]

Existence of moderate tolls for s − t networks and computation for series-parallel nets and infinite #types [Flei 04] Efficient computation follows from LP duality [FJM 04], [KaraKoll 04]

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls for Atomic Games

Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1.2 for any tolls. ∃ non-symmetric games not admitting optimal tolls!

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls for Atomic Games

Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1.2 for any tolls. ∃ non-symmetric games not admitting optimal tolls! Simple optimal tolls for parallel links Efficiently computable tolls reducing PoA to 2 + ε Not strongly optimal for series-parallel networks.

Dimitris Fotakis Congestion Games and Price of Anarchy

Refundable Tolls for Atomic Games

Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1.2 for any tolls. ∃ non-symmetric games not admitting optimal tolls! Simple optimal tolls for parallel links Efficiently computable tolls reducing PoA to 2 + ε Not strongly optimal for series-parallel networks. Optimal tolls for s − t networks other than parallel links?

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost-Balancing Tolls

Cost-balancing tolls τ for f ∀p ∈ P with mine∈p{fe} > 0 and ∀p′ ∈ P,

• e∈p

(de(fe) + τe) ≤

• e∈p′

(de(fe) + τe) Any used path becomes min-cost path.

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost-Balancing Tolls

Cost-balancing tolls τ for f ∀p ∈ P with mine∈p{fe} > 0 and ∀p′ ∈ P,

• e∈p

(de(fe) + τe) ≤

• e∈p′

(de(fe) + τe) Any used path becomes min-cost path. Any configuration σ with σe = fe for all e ∈ E is a PNE of Γτ Configuration f weakly enforceable by cost-balancing tolls for it.

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost-Balancing Tolls

Cost-balancing tolls τ for f ∀p ∈ P with mine∈p{fe} > 0 and ∀p′ ∈ P,

• e∈p

(de(fe) + τe) ≤

• e∈p′

(de(fe) + τe) Any used path becomes min-cost path. Any configuration σ with σe = fe for all e ∈ E is a PNE of Γτ Configuration f weakly enforceable by cost-balancing tolls for it. Which configurations admit cost-balancing tolls?

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Results on Cost-Balancing Tolls

Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!).

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Results on Cost-Balancing Tolls

Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Results on Cost-Balancing Tolls

Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal.

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Results on Cost-Balancing Tolls

Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal. Heterogenous players : ∃ parallel-link games not admitting strongly optimal tolls.

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Results on Cost-Balancing Tolls

Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal. Heterogenous players : ∃ parallel-link games not admitting strongly optimal tolls. Complexity of computing best optimal tolls NP-hard even for linear games on series-parallel networks. For 2-player linear games on series-parallel networks, NP-hard to distinguish between PoA = 1 and PoA ≥ 1.2.

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o.

x 2x x x x x s x x 2x x x x t 3 players 8x 8 5 5

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o. Eo = {e ∈ E : oe > 0} : G(V, Eo) is directed acyclic graph ( DAG ) ∀ edge e ∈ Eo , edge length ℓe = de(oe)

x 2x x x x x s x x 2x x x x t 3 players 8x 8 5 5

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o. Eo = {e ∈ E : oe > 0} : G(V, Eo) is directed acyclic graph ( DAG ) ∀ edge e ∈ Eo , edge length ℓe = de(oe)

1 2 1 1 s 1 1 2 1 t 8

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o. Eo = {e ∈ E : oe > 0} : G(V, Eo) is directed acyclic graph ( DAG ) ∀ edge e ∈ Eo , edge length ℓe = de(oe) Longest path tree from s in linear time ∀ vertex u, ℓu = length of longest s − u path in Go

1 2 1 1 s 1 1 2 1 t 8 1 1 1 3 4 4 4 8

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o. Eo = {e ∈ E : oe > 0} : G(V, Eo) is directed acyclic graph ( DAG ) ∀ edge e ∈ Eo , edge length ℓe = de(oe) Longest path tree from s in linear time ∀ vertex u, ℓu = length of longest s − u path in Go ∀e = (u, v) ∈ Eo , τe = ℓv − (ℓu + de(oe)) ∀e ∈ Eo , τe = τ max ≥ ℓt

1 2 1 1 s 1 1 2 1 t 8 1 1 1 3 4 4 4 8 1 3 2 τmax τmax

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

Game Γ(N, G(V, E), (de)e∈E) and acyclic optimal o. Eo = {e ∈ E : oe > 0} : G(V, Eo) is directed acyclic graph ( DAG ) ∀ edge e ∈ Eo , edge length ℓe = de(oe) Longest path tree from s in linear time ∀ vertex u, ℓu = length of longest s − u path in Go ∀e = (u, v) ∈ Eo , τe = ℓv − (ℓu + de(oe)) ∀e ∈ Eo , τe = τ max ≥ ℓt Non-negative tolls ∀e = (u, v) ∈ Eo , ℓv ≥ ℓu + de(oe)

1 2 1 1 s 1 1 2 1 t 8 1 1 1 3 4 4 4 8 1 3 2 τmax τmax

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

∀e = (u, v) ∈ Eo , τe = ℓv − (ℓu + de(oe)) ∀e ∈ Eo , τe = τ max ≥ ℓt Cost-balancing tolls for o ∀used p ∈ P :

e∈p(de(oe) + τe) = e∈p(ℓv(e) − ℓu(e)) = ℓt

Otherwise, ∃e ∈ p \ Eo with τe = τ max ≥ ℓt

1 2 1 1 s 1 1 2 1 t 8 1 1 1 3 4 4 4 8 1 3 2 τmax τmax

Dimitris Fotakis Congestion Games and Price of Anarchy

Computing Cost-Balancing Tolls

∀e = (u, v) ∈ Eo , τe = ℓv − (ℓu + de(oe)) ∀e ∈ Eo , τe = τ max ≥ ℓt Cost-balancing tolls for o ∀used p ∈ P :

e∈p(de(oe) + τe) = e∈p(ℓv(e) − ℓu(e)) = ℓt

Otherwise, ∃e ∈ p \ Eo with τe = τ max ≥ ℓt Moderate tolls Amount of tolls paid by any player in o ≤ ℓt Sufficiently large τ max : tolls paid by any player in any PNE ≤ ℓt

1 2 1 1 s 1 1 2 1 t 8 1 1 1 3 4 4 4 8 1 3 2 τmax τmax

Dimitris Fotakis Congestion Games and Price of Anarchy

Social Ignorance in Congestion Games

Motivation Players have partial information which depends on their social context. How does the social context affect inefficiency?

Dimitris Fotakis Congestion Games and Price of Anarchy

Social Ignorance in Congestion Games

Motivation Players have partial information which depends on their social context. How does the social context affect inefficiency? Ideas from Previous Work A Bayesian approach to load balancing [GairMonTiem 08] Social graph : player knows neighbors’ weights and probability distribution for others’ [KoutsPanaSpir 07] Some players are ignorant of the presence of others [KarKimViglXia 07] Social context affects individual costs [AshlKrysTennen 08]

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Graphical Congestion Games [BiloFaneFlamMosca 08] Social graph G = (N, R). Each player / vertex has:

Full information about his social neighbors. No information whatsoever about the remaining players.

Players select strategies based on presumed costs.

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Graphical Congestion Games [BiloFaneFlamMosca 08] Social graph G = (N, R). Each player / vertex has:

Full information about his social neighbors. No information whatsoever about the remaining players.

Players select strategies based on presumed costs. Consequences Linear graphical games admit potential function (and PNE). PoA ≤ n(∆(G) + 1) PoS ≤ n

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Work on Graphical Congestion Games

Question Which parameter of the social graph characterizes inefficiency

• f PNE and the Nash dynamics (for weighted players too) ?

Dimitris Fotakis Congestion Games and Price of Anarchy

Our Work on Graphical Congestion Games

Question Which parameter of the social graph characterizes inefficiency

• f PNE and the Nash dynamics (for weighted players too) ?

Our Answer The independence number α(G) of the social graph G : PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

α(G) ≤ PoS ≤ 2α(G) Convergence to PNE not slower due to social ignorance (what about faster?).

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Configuration σ = (σ1, . . . , σn) Presumed congestion of player i on e in σ: σi

e = 1 + |{j ∈ N : e ∈ σj ∧ {i, j} ∈ R}|

G(N, R)

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Configuration σ = (σ1, . . . , σn) Presumed congestion of player i on e in σ: σi

e = 1 + |{j ∈ N : e ∈ σj ∧ {i, j} ∈ R}|

Ge(Ne, Re) i j

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Configuration σ = (σ1, . . . , σn) Presumed congestion of player i on e in σ: σi

e = 1 + |{j ∈ N : e ∈ σj ∧ {i, j} ∈ R}|

Presumed latency of i on e in σ: de(σi

e)

Presumed latency of i in σ: pi(σ) =

e∈σi de(σi e)

Ge(Ne, Re) i j

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Pure Nash Equilibrium (PNE) for Graphical Games No player can improve own presumed latency unilaterally: ∀i ∈ N, ∀s ∈ P, pi(σ) ≤ pi(σ−i, s)

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Pure Nash Equilibrium (PNE) for Graphical Games No player can improve own presumed latency unilaterally: ∀i ∈ N, ∀s ∈ P, pi(σ) ≤ pi(σ−i, s) Potential Function for Linear Latencies Φ(σ) = P(σ) + U(σ) 2 where P(σ) = n

i=1 pi(σ) and U(σ) = n i=1

• e∈σi de(1)

Dimitris Fotakis Congestion Games and Price of Anarchy

Graphical Congestion Games

Pure Nash Equilibrium (PNE) for Graphical Games No player can improve own presumed latency unilaterally: ∀i ∈ N, ∀s ∈ P, pi(σ) ≤ pi(σ−i, s) Potential Function for Linear Latencies Φ(σ) = P(σ) + U(σ) 2 where P(σ) = n

i=1 pi(σ) and U(σ) = n i=1

• e∈σi de(1)

Price of Anarchy for Graphical Games Game Γ : PoA(Γ) = max σ∈PNE(Γ)C(σ)/C(o)

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost Approximation

Main Lemma For any configuration σ, C(σ) ≤ α(G)P(σ)

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost Approximation

Main Lemma For any configuration σ, C(σ) ≤ α(G)P(σ) Intuition For edge e and configuration σ, let Ge(Ne(σ), Re(σ)) be social subgraph induced by σe players on e in σ.

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost Approximation

Main Lemma For any configuration σ, C(σ) ≤ α(G)P(σ) Intuition For edge e and configuration σ, let Ge(Ne(σ), Re(σ)) be social subgraph induced by σe players on e in σ. Worst case when Ge is made up of α(Ge) disjoint cliques each of size k = σe/α(Ge).

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost Approximation

Main Lemma For any configuration σ, C(σ) ≤ α(G)P(σ) Intuition For edge e and configuration σ, let Ge(Ne(σ), Re(σ)) be social subgraph induced by σe players on e in σ. Worst case when Ge is made up of α(Ge) disjoint cliques each of size k = σe/α(Ge). Total actual cost for e in σ : Ce(σ) = aeσ2

e + beσe

Total presumed cost for e in σ : Pe(σ) = σe[aek + be]

Dimitris Fotakis Congestion Games and Price of Anarchy

Cost Approximation

Main Lemma For any configuration σ, C(σ) ≤ α(G)P(σ) Intuition For edge e and configuration σ, let Ge(Ne(σ), Re(σ)) be social subgraph induced by σe players on e in σ. Worst case when Ge is made up of α(Ge) disjoint cliques each of size k = σe/α(Ge). Total actual cost for e in σ : Ce(σ) = aeσ2

e + beσe

Total presumed cost for e in σ : Pe(σ) = σe[aek + be] Since σe = kα(Ge), Ce(σ) ≤ α(Ge)Pe(σ)

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Stackelberg strategies Approximability of the best Stackelberg strategy.

Atomic games on parallel links: (F)PTAS ? (Symmetric) network games even with linear latencies?

How much can LLF (or Scale) increase PoA and PoS ?

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Stackelberg strategies Approximability of the best Stackelberg strategy.

Atomic games on parallel links: (F)PTAS ? (Symmetric) network games even with linear latencies?

How much can LLF (or Scale) increase PoA and PoS ? Tolls for Atomic Games Which s − t network games admit strongly optimal tolls?

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Stackelberg strategies Approximability of the best Stackelberg strategy.

Atomic games on parallel links: (F)PTAS ? (Symmetric) network games even with linear latencies?

How much can LLF (or Scale) increase PoA and PoS ? Tolls for Atomic Games Which s − t network games admit strongly optimal tolls? Impact of Social Ignorance Realistic models for congestion games where players have limited social interaction, and thus limited information.

One should model that after a strategy is realized, the player becomes aware of its actual cost .

Does sparse social graphs facilitate convergence to PNE ?

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Bursty Players and Risk Aversion Which way to work if A wants to be there by 9:00?

s t x/2 1 n+1 players n n

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Bursty Players and Risk Aversion Which way to work if A wants to be there by 9:00?

With probability ≈ 37%, A gets there at 8:30 With probability ≈ 37%, A gets there at 9:00 With probability ≈ 26%, A gets there after 9:30! s t x/2 1 n+1 players n n

Dimitris Fotakis Congestion Games and Price of Anarchy

Directions for Further Research

Bursty Players and Risk Aversion Which way to work if A wants to be there by 9:00?

With probability ≈ 37%, A gets there at 8:30 With probability ≈ 37%, A gets there at 9:00 With probability ≈ 26%, A gets there after 9:30!

Reasonable and technically manageable model for congestion games with stochastic and / or risk averse players.

s t x/2 1 n+1 players n n

Dimitris Fotakis Congestion Games and Price of Anarchy