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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
- Price of anarchy definitions
- Linear latency functions PoA upper and lower bounds
- Polynomial latency functions PoA upper and lower bounds

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 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

- 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

- 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 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

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

- 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

- The worst-case ratio between the social cost of a NE and the optimal social cost
- Definition 1:
- Definition 2:

Linear latency function

- If an equivalent problem can be described with function
- Duplicate an edge times

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

- 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

- 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’

- Using the lemma we get
- And thus
- And the upper bound is proven

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 requests

Summing for all agents we get

Changing summation order we get

Using

lemma 2

we get

Upper bound weighted requests

Using lemma 1 in previous expression

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 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 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

- 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

- 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’

- 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

- Unweighted pure strategy cases only
- Symmetric case
- Lower bound already shown
- Asymmetric case

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 bound

- Combining the last 2 inequalities
- substitute in the first inequality

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

- 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

- 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

- 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

- 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’

- 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 cont’

- Using the lemma we get
- And thus
- And the upper bound is proven

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 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 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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