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The price of anarchy of finite congestion games Kapelushnik Lior Based on the articles: “ The price of anarchy of finite congestion games ” by Christodoulou & Koutsoupias “ the price of routing unsplittable flow ” by Awerbuch, Azar & Epstein Agenda Congestion games description

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The price of anarchy of finite congestion games l.jpg

The price of anarchy of finite congestion games

Kapelushnik Lior

Based on the articles:

“The price of anarchy of finite congestion games”

by Christodoulou & Koutsoupias

“the price of routing unsplittable flow”

by Awerbuch, Azar & Epstein


Agenda l.jpg
Agenda

  • Congestion games description

  • Price of anarchy definitions

  • Linear latency functions PoA upper and lower bounds

  • Polynomial latency functions PoA upper and lower bounds


Network congestion game l.jpg
Network congestion game

  • A directed graph G=(V,E)

  • For each edge exists a latency function

  • n users, user j have request

  • Request j assigned to path which is the strategy of player j

  • Users are none cooperative


Network game example l.jpg
Network game example

s1

t1

t3

t2

s2

s3


Network game example agent paths strategies l.jpg
Network game exampleagent paths (strategies)

Agent 2 path 2

s1

t1

t3

t2

s2

Agent 2 path 1

s3


Network congestion game6 l.jpg
Network congestion game

  • Consider a strategy profile

  • Denote Ne(A) as the load on e in A

  • Mark as the possible paths for user i

  • Player j cost is the total cost of it’s strategy path

  • Strategy A is a NE if no player has reason to deviate from his strategy


Congestion game l.jpg
Congestion game

  • More generalized than a network congestion game

  • N players, a set of facilities E

  • Each player i has a strategy chosen from several sets of facilities

  • Facility j have cost (latency) function


Congestion game definitions l.jpg
Congestion game definitions

  • Symmetric game (single-commodity) – all players choose from the same strategies

  • Asymmetric game (multi-commodity) – different players may have different strategy options

  • Mixed strategy for player i – a probability distribution over


Congestion game example l.jpg
Congestion game example

f1

f2

f3

f4

F5

F6


Congestion game example10 l.jpg
Congestion game example

f1

f2

f3

f4

F5

F6


Congestion game example11 l.jpg
Congestion game example

f1

f2

f3

f4

F5

F6


Congestion game example12 l.jpg
Congestion game example

f1

f2

f3

f4

F5

F6


Congestion games l.jpg
Congestion games

  • Every network congestion game is a congestion game

  • Each edge will be represented by a facility

  • Each strategy path of a player will be replaced by the set of edges in the path


Network to congestion game l.jpg
Network to congestion game

e1

e2

fe1

fe2

fe3

s

t

e3

Instead of possible path strategies (e1->e2) and (e3)

The congestion game strategies are

({fe1,fe2}, {fe3}}


Social cost l.jpg
Social cost

  • Two possible definitions

  • Definition 1:

  • Definition 2:

    or for weighted requests

  • When considering PoA the social cost definition of sum is equivalent to average (just divide by n)


Price of anarchy l.jpg
Price of anarchy

  • The worst-case ratio between the social cost of a NE and the optimal social cost

  • Definition 1:

  • Definition 2:


Linear latency function l.jpg
Linear latency function

  • If an equivalent problem can be described with function

  • Duplicate an edge times


Avg social cost poa l.jpg
Avg. social cost PoA

  • Asymmetric case

    • Unweighted requests

      • pure strategy

      • Mixed strategy

    • Weighted requests

      • Pure + mixed strategies

  • Symmetric case

    • Unweighed pure strategy


Upper poa bounds l.jpg
Upper PoA bounds

  • Sketch of proof

  • Compare agent’s delay to the delay that would be encountered at the optimal path

  • Combine the bounds and transform to a relation between a total NE delay and the total optimal delay


Upper bound unweighted requests fe x x l.jpg
Upper bound unweighted requests, fe(x)=x

  • Lemma:

    for a pair of nonnegative integers a,b


Upper bound unweighted requests fe x x21 l.jpg
Upper bound unweighted requests, fe(x)=x

  • In a NE A and an optimal P allocation

  • The inequality holds since moving from a NE does not decrease the cost

  • Summing for all players we get


Upper bound unweighted requests fe x x cont l.jpg
Upper bound unweighted requests, fe(x)=x cont

  • Using the lemma we get

  • And thus

  • And the upper bound is proven


Upper bound weighted requests l.jpg
Upper bound weighted requests

  • Notations:

    • J(e) – set of agents using e

    • P – NE strategies profile

    • Qj – request j path in P

    • X* - value X in optimal state

    • l – load vector of a system


Upper bound weighted requests24 l.jpg
Upper bound weighted requests

Lemma 1: (follows from Cauchy-Schwartz inequality)

Lemma 2: for any


Upper bound weighted requests25 l.jpg
Upper bound weighted requests

since deviation from NE doesn’t decrease cost

Multiplying by Wj we get


Upper bound weighted requests26 l.jpg
Upper bound weighted requests

Summing for all agents we get

Changing summation order we get

Using

lemma 2

we get


Upper bound weighted requests27 l.jpg
Upper bound weighted requests

Using lemma 1 in previous expression


Lower bounds unweighted requests congestion game l.jpg
Lower bounds, unweighted requests, congestion game

  • Assume N≥3 agents, 2N facilities fe(x)=x

  • Facilities

  • Agent i strategies

  • Optimal allocation: each agent i chooses

  • Worst NE agent i choose

  • The cost for each agent is 2 in the optimal allocation and 5 in the NE, PoA is 5/2




Lower bounds weighted requests network game31 l.jpg
Lower bounds, weighted requests, network game

V

fe(x)=0

Agent 1

U

Agent 4

fe(x)=x

Agent 3

Agent 2

W

Optimal cost,

player 1 use UV, player 2 use UW,

Player 3 use VW, player 4 use WV

Total cost:


Lower bounds weighted requests network game32 l.jpg
Lower bounds, weighted requests, network game

Agent 2

V

fe(x)=0

Agent 3

U

fe(x)=x

Agent 1

Agent 4

W

NE cost,

player 1 use UWV, player 2 use UVW,

Player 3 use VUW, player 4 use WUV

Total cost:


Linear congestion symmetric games lower bound of poa l.jpg
Linear congestion symmetric games lower bound of PoA

  • The upper bound for asymmetric games with avg. social cost also holds for symmetric games

  • The lower bound both max and avg. social cost is (5N-2)/(2N+1)

  • Next is a game description which achieves this PoA for N players


Lower bound game construction for symmetric games l.jpg
Lower bound game construction for symmetric games

  • The facilities will be in N sets of the same size P1,P2,…,Pn

  • Each Pi is a pure strategy and in optimal allocation each player i plays Pi

  • Each Pi contains facilities

  • At NE player i plays alone facilites of each Pj

  • At NE each pair of players play together facilities of each Pj


Lower bound game construction for symmetric games cont l.jpg
Lower bound game construction for symmetric games cont

  • At NE A,

  • We want that at NE no players will switch to Pj

  • For NE we need

  • Which proves the PoA of (5N-2)/(2N+1)


Max social cost poa l.jpg
Max social cost PoA

  • Unweighted pure strategy cases only

  • Symmetric case

    • Lower bound already shown

  • Asymmetric case


Asymmetric case upper bound l.jpg
Asymmetric case upper bound

  • Let A be a NE, P optimal allocation, w.l.o.g Max(A)=c1(A), the NE imply

  • Denote the players in A that use facilities of P1

  • The avg. social cost lower bound showed


Asymmetric case upper bound38 l.jpg
Asymmetric case upper bound

  • Combining the last 2 inequalities

  • substitute in the first inequality


Asymmetric case lower bound l.jpg
Asymmetric case lower bound

k

k

k

V1

V2

V3

Vk

Vk+1

k

  • One player wants to go from V1 to Vk+1

  • For each i k-1 players wants to go from Vi to Vi+1

  • Between Vi and Vi+1 one path with length 1, k-1 paths with length k

  • Opt – all players use different paths, cost is k

  • NE – all players use the 1 length paths cost is PoA is


Symmetric case upper bound l.jpg
Symmetric case upper bound

  • Let A be a NE, P optimal allocation, w.l.o.g Max(A)=c1(A), the NE imply

  • Summing for all possible j in P and using the lemma

  • Using the avg. social cost lower bound


Polynomial latency function l.jpg
Polynomial latency function

  • The latency functions are polynomials of bounded degree p

  • The proofs for PoA of linear latency functions are quite similar to those of polynomial latencies


Polynomial latencies cost poa l.jpg
Polynomial latencies cost PoA

  • For polynomials of degree p, nonnegative coefficients

  • Avg. social cost

    weighted requests, unweighted requests, symmetric games, asymmetric games, pure strategies, mixed strategies

  • Max social cost

    • Pure strateties symmetric games

    • Pure strateties asymmetric games


Upper bound unweighted requests polynomial latencies l.jpg
Upper bound unweighted requests polynomial latencies

  • Instead of the lemma for linear functions

    for a pair of nonnegative integers a,b

  • A new lemma is used, if f(x) polynomial in x with nonnegative coefficients, of degree p, for nonegative x and y

  • Where


Upper bound unweighted requests polynomial latencies cont l.jpg
Upper bound unweighted requests polynomial latencies cont

  • In a NE A and an optimal P allocation

  • The inequality holds since moving from a NE does not decrease the cost

  • Summing for all players we get


Upper bound unweighted requests polynomial latencies cont45 l.jpg
Upper bound unweighted requests polynomial latencies cont

  • Using the lemma we get

  • And thus

  • And the upper bound is proven


Lower bound game construction for symmetric games46 l.jpg
Lower bound game construction for symmetric games

  • The facilities will be in N sets of the same size P1,P2,…,Pn

  • Each Pi is a pure strategy and in optimal allocation each player i plays Pi

  • Each Pj contains N facilities

  • At NE player i plays


Lower bound game construction for symmetric games cont47 l.jpg
Lower bound game construction for symmetric games cont

  • At NE A,

  • We want that at NE no players will switch to Pj

  • For NE we need to select N such that

  • For opt

  • The PoA is

  • Which proves the PoA of when choosing N that satisfies the equation


Asymmetric case lower bound almost like in the linear case l.jpg
Asymmetric case lower boundalmost like in the linear case

k

k

k

V1

V2

V3

V(k^p)

V(k^p+1)

  • One player wants to go from V1 to

  • For each i k-1 players wants to go from Vi to Vi+1

  • Between Vi and Vi+1 a path with length 1, k-1 paths with length

  • Opt – all players use different paths, cost is

  • NE – all players use the 1 length paths cost is


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