Download - Beyond selfish routing: Network Games
Beyond selfish routing:Network Games
Network Games NGs model the various ways in
which selfish agents strategically interact in using a network
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 NGs 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 ISP 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 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 (for positive s.c.f. f)
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
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 best possible NE:
Addressed issues in GCG Does a stable network always exist? Does the repeated version of the game
always converge to a stable network? How long does it take to converge to a
stable network? Can we bound the price of anarchy
(PoA)? Can we bound the price of stability
(PoS)?
Any instance of the global connection game has a pure Nash equilibrium, and best responsedynamic always converges.
Theorem 1
The price of anarchy in the global connectiongame with k players is at most k.
Theorem 2
The price of stability in the global connectiongame with k players is at most Hk, the k-thharmonic number.
Theorem 3
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:
The potential function method
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
…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 (recall that ke is the number of players using e)
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 3 ( 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) - ePi ce/ke + eP’i
ce/(ke+1))= 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 3, a GCG is a potential game, and from Lemma 1 and 2 best response dynamic converges to a pure NE.
Existence of a 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 2
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
Suppose that we have a potential game with potential function , and assume that for any outcome S we have
Lemma 4
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.
Lemma 5 (Bounding )
cost(S) (S) Hk cost(S)
For any strategy vector S in the GCG, 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 3
Proof: From Lemma 3, a GCG is a potential game, and from Lemma 5 and Lemma 4 (with A=1 and B=Hk), its PoS is at most Hk.
Upper-bounding the PoS