Beyond selfish routing:Network Formation

Games

Network Formation Games NFGs model the various ways in

which selfish agents might create/use networks

They aim to capture two competing issues for agents: to minimize the cost they incur in

creating/using the network to ensure that the network provides

them with a high quality of service

Motivations NFGs can be used to model:

social network formation (edge represent social relations)

how subnetworks connect in computer networks

formation of P2P networks connecting users to each other for downloading files (local connection games)

how users try to share costs in using an existing network (global connection games)

Setting What is a stable network?

we use a NE as the solution concept we refer to networks corresponding to Nash

Equilibria as being stable How to evaluate the overall quality of a

network? we consider the social cost: the sum of

players’ costs Our goal: to bound the efficiency loss

resulting from selfishness

A warming-up network game: The ISPs dilemma

two Internet Service Providers (ISP): ISP1 e ISP2

ISP1 wants to send traffic from s1 to t1

ISP2 wants to send traffic from s2 to t2

s1

s2

t1

t2

C S

Long links incur a per-user cost of 1 to the ISP owning the link

C, S: peering points

Each ISPi can use two paths: either passing through C or not

A (bad) DSEs1

s2

t1

t2

C S

C, S: peering points

avoidC

throughC

avoidC

2, 2 5, 1

throughC

1, 5 4, 4

Cost Matrix

ISP1

ISP2

PoA is (4+4)/(2+2)=2

Dominant Strategy Equilibrium

Our case study:Global Connection

Games

The model

G=(V,E): directed graph, k players ce: non-negative cost of e E Player i has a source node si and a sink node ti

Strategy for player i: a path Pi from si to ti

The cost of Pi for player i in S=(P1,…,Pk) is shared with all the other players using (part of) it:

costi(S) = ce/keePi

this cost-sharing scheme is called fair or Shapley cost-sharing mechanism

The model Given a strategy vector S, the constructed network will be N(S)= i Pi The cost of the constructed network will be shared among all players as follows:

Notice that each user has a favorable effect on the performance of other users (so-called cross monotonicity), as opposed to the congestion model of selfish routing

cost(S)= costi(S) = ce/ke= ceePii i eN(S)

Goals

Remind that given a strategy vector S, N(S) is stable if S is a NE

To evaluate the overall quality of a network, we consider its cost as the social cost, i.e., the sum of all players’ costs

A network is optimal or socially efficient if it minimizes the social cost

Be optimist!: The price of stability (PoS)

Definition (Schulz & Moses, 2003): Given a game G and a social-choice minimization (resp., maximization) function f (i.e., the sum of all players’ payoffs), let S be the set of NE, and let OPT be the outcome of G optimizing f. Then, the Price of Stability (PoS) of G w.r.t. f is:

Ssf

inf)(G

(OPT)

)(sup.,resp

(OPT)

)(

f

sf

f

sf

Ss

Some remarks PoA and PoS are

1 for minimization problems 1 for maximization problems

PoA and PoS are small when they are close to 1

PoS is at least as close to 1 than PoA In a game with a unique NE, PoA=PoS Why to study the PoS?

sometimes a nontrivial bound is possible only for PoS

PoS quantifies the necessary degradation in quality under the game-theoretic constraint of stability

Addressed issues in GCG Does a stable network always exist? Can we bound the price of anarchy

(PoA)? Can we bound the price of stability

(PoS)? Does the repeated version of the game

always converge to a stable network?

An example

1

1

11 3

3

3 2

4 5.5

s1

s2

t1 t2

An example

1

1

11 3

3

3 2

4 5.5

s1

s2

t1 t2

optimal network has cost 12cost1=7cost2=5

is it stable?

An example

1

1

11 3

3

3 2

4 5.5

s1

s2

t1 t2

PoA 13/12, PoS ≤ 13/12

cost1=5cost2=8

is it stable?…yes, and has cost 13!

…no!, player 1 can decrease its cost

An example

1

1

11 3

3

3 2

4 5.5

s1

s2

t1 t2

the social cost is 12.5 PoS ≤ 12.5/12

cost1=5cost2=7.5

…a better NE…

Price of Anarchy: a lower bound

s1,…,sk t1,…,tk

k

1

optimal network has cost 1

best NE: all players use the lower edge

worst NE: all players use the upper edge

PoS is 1

PoA is k

Upper-bounding the PoA

The price of anarchy in the global connectiongame with k players is at most k.

Theorem (Prove by yourself)

Proof: Let OPT=(P1*,…,Pk

*) denote the optimal network, and let i be a shortest path in G between si and ti. Let w(i) be the length of such a path, and let S be any NE. Observe that costi(S)≤w(i) (otherwise the player i would change). Then:

cost(S) = costi(S) ≤ w(i) ≤ w(Pi*) ≤

k·costi(OPT) = k· cost(OPT).

i=1

k

i=1

k

i=1

k

i=1

k

PoS for GCG: a lower bound

1+. . .

1 1/2 1/(k-1) 1/k1/3

0 0 0 0 0

s1 sks2 s3sk-1

t1,…,tk

>o: small value

1+. . .

1 1/2 1/(k-1) 1/k1/3

0 0 0 0 0

s1 sks2 s3sk-1

t1,…,tk

>o: small value

The optimal solution has a cost of 1+

is it stable?

PoS for GCG: a lower bound

1+. . .

1 1/2 1/(k-1) 1/k1/3

0 0 0 0 0

s1 sks2 s3sk-1

t1,…,tk

>o: small value

…no! player k can decrease its cost…

is it stable?

PoS for GCG: a lower bound

1+. . .

1 1/2 1/(k-1) 1/k1/3

0 0 0 0 0

s1 sks2 s3sk-1

t1,…,tk

>o: small value

…no! player k-1 can decrease its cost…

is it stable?

PoS for GCG: a lower bound

1+. . .

1 1/2 1/(k-1) 1/k1/3

0 0 0 0 0

s1 sks2 s3sk-1

t1,…,tk

>o: small value

A stable network

social cost: 1/j = Hk ln k + 1 k-th harmonic number

j=1

k

PoS for GCG: a lower bound

Any instance of the global connection game has a pure Nash equilibrium, and best responsedynamic always converges.

Theorem 1

The price of stability in the global connectiongame with k players is at most Hk, the k-thharmonic number.

Theorem 2

To prove them we use the Potential function method

For any finite game, an exact potential function is a function that maps every strategy vector S to some real value and satisfies the following condition:

Definition

S=(s1,…,sk), s’isi, let S’=(s1,…,s’i,…,sk), then

(S)-(S’) = costi(S)-costi(S’).

A game that does possess an exact potential function

is called potential game

Every potential game has at least one pure Nash equilibrium, namely the strategy vector S that minimizes (S).

Lemma 1

Proof: consider any move by a player i that results in a new strategy vector S’. Since (S) is minimum, we have:

(S)-(S’) = costi(S)-costi(S’)

0

costi(S) costi(S’)

player i cannot decrease its cost, thus S is a NE.

In any finite potential game, best response dynamic always converges to a Nash equilibrium

Lemma 2

Proof: best response dynamic simulates local search on .

Observation: any state S with the property that (S) cannot be decreased by altering any one strategy in S is a NE by the same argument. This implies the following:

Convergence in potential games

Suppose that we have a potential game with potential function , and assume that for any outcome S we have

Lemma 3

Proof:Let S’ be the strategy vector minimizing (i.e., S’ is a NE)

we have:

(S’) (S*)

cost(S)/A (S) B cost(S)

for some A,B>0. Then the price of stability is at most AB.

Let S* be the strategy vector minimizing the social cost

B cost(S*) cost(S’)/A

PoS ≤ cost(S’)/cost(S*) ≤ A·B.

…turning our attention to the global connection game…

Let be the following function mapping any strategy vector S to a real value [Rosenthal 1973]:

(S) = eN(S) e(S)

where

e(S) = ce · H = ce · (1+1/2+…+1/ke).

ke

Let S=(P1,…,Pk), let P’i be an alternative path for some player i, and define a new strategy vector S’=(S-i,P’i).Then:

Lemma 4 ( is a potential function)

(S) - (S’) = costi(S) – costi(S’).

Proof:It suffices to notice that:• If edge e is used one more time in S:

(S+e)=(S)+ce/(ke+1)• If edge e is used one less time in S: (S-e)=(S) - ce/ke

(S) -(S’) = (S) -(S-Pi+P’i) = (S) –

((S) - ePicosti(S) + eP’i

costi(S’))= costi(S) – costi(S’).

Any instance of the global connection game has a pure Nash equilibrium, and best responsedynamic always converges.

Theorem 1

Proof: From Lemma 4, a GCG is a potential game, and from Lemma 1 and 2 best response dynamic converges to a pure NE.

Lemma 5 (Bounding )

cost(S) (S) Hk cost(S)

For any strategy vector S, we have:

(S) = eN(S) e(S) = eN(S) ce· Hke

Proof: Indeed:

(S) cost(S) = eN(S) ce

and (S) ≤ Hk· cost(S) = eN(S) ce· Hk.

The price of stability in the global connectiongame with k players is at most Hk, the k-thharmonic number

Theorem 2

Proof: From Lemma 4, a GCG is a potential game, and from Lemma 5 and Lemma 3 (with A=1 and B=Hk), its PoS is at most Hk.