Download Global connection 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:
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
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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?



How to evaluate the overall quality of a
network?

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we use a NE as the solution concept
we refer to networks corresponding to Nash
Equilibria as being stable
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 C, S: peering points
s1
two Internet Service Providers (ISP):
ISP1 e ISP2
ISP1 wants to send traffic from s1 to t1
t2
C
ISP2 wants to send traffic from s2 to t2
s2
Long links incur a per-user cost of 1 to the ISP owning the link
Each ISP can use two paths: either passing through C or not
S
t1
A (bad) DSE
C, S: peering points
s1
t2
Cost Matrix
C
ISP2
ISP1
avoid
C
through
C
avoid
C
2, 2
5, 1
through
C
1, 5
4, 4
S
t1
s2
 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/ke
eP
i
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:
cost(S)=

i costi(S) = i eP
 ce/ke=eN(S)
 ce
i
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
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:
f ( s) 
f ( s) 
 resp ., sup

 G ( f )  inf
sS f (OPT) 
sS f (OPT) 
Some remarks

PoA and PoS are (for positive s.c.f. f)

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 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
3
s2
3
1
s1
3
2
4
1
1
1
t1
5.5
t2
An example
3
s2
s1
1
3
3
2
4
optimal network has cost 12
cost1=7
cost2=5
is it stable?
1
1
1
t1
5.5
t2
An example
3
s2
s1
3
1
3
2
1
1
1
4
t1
t2
5.5
…no!, player 1 can decrease its cost
cost1=5
cost2=8
is it stable? …yes, and has cost 13!
 PoA  13/12, PoS ≤ 13/12
An example
3
s2
s1
3
1
3
2
4
…a best possible NE:
1
1
1
t1
t2
5.5
cost1=5
cost2=7.5
the social cost is 12.5  PoS = 12.5/12
Addressed issues in GCG
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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)?
Theorem 1
Any instance of the global connection game has
a pure Nash equilibrium, and best response
dynamic always converges.
Theorem 2
The price of anarchy in the global connection
game with k players is at most k.
Theorem 3
The price of stability in the global connection
game with k players is at most Hk, the k-th
harmonic number.
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:
S=(s1,…,sk), s’isi, 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
Lemma 1
Every potential game has at least one pure Nash
equilibrium, namely the strategy vector S that
minimizes (S).
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.
Convergence in potential games
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:
Lemma 2
In any finite potential game, best response dynamic
always converges to a Nash equilibrium
Proof: best response dynamic simulates local search on .
…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) = eN(S) e(S)
where (recall that ke is the number of players using e)
e(S) = ce · H k= ce · (1+1/2+…+1/ke).
e
Lemma 3 ( is a potential function)
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:
(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) - ePi ce/ke + eP’i ce/(ke+1))= costi(S) – costi(S’).
Existence of a NE
Theorem 1
Any instance of the global connection game has
a pure Nash equilibrium, and best response
dynamic always converges.
Proof: From Lemma 3, a GCG is a potential game, and
from Lemma 1 and 2 best response dynamic converges to
a pure NE.
Price of Anarchy: a lower
bound
k
s1,…,sk
t1,…,tk
1
optimal network has cost 1
best NE: all players use the lower edge
PoS is 1
worst NE: all players use the upper edge
PoA is k


Upper-bounding the PoA
Theorem 2
The price of anarchy in the global connection
game with k players is at most k.
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:
k
k
k
i=1
i=1
i=1
cost(S) = costi(S) ≤  w(i) ≤  w(Pi*) ≤
k
 k·costi(OPT) = k· cost(OPT).
i=1
PoS for GCG: a lower bound
>o: small value
t1,…,tk
s1
1
s2
0
1/3
1/2
s3
0
1/(k-1)
1/k
sk-1
...
0
0
0
sk
1+
PoS for GCG: a lower bound
>o: small value
t1,…,tk
s1
1
s2
0
1/3
1/2
s3
0
1/(k-1)
1/k
sk-1
...
0
0
The optimal solution has a cost of 1+
is it stable?
0
sk
1+
PoS for GCG: a lower bound
>o: small value
t1,…,tk
s1
1
s2
0
1/3
1/2
s3
0
1/(k-1)
1/k
sk-1
...
0
0
…no! player k can decrease its cost…
is it stable?
0
sk
1+
PoS for GCG: a lower bound
>o: small value
t1,…,tk
s1
1
s2
0
1/3
1/2
s3
0
1/(k-1)
1/k
sk-1
...
0
0
…no! player k-1 can decrease its cost…
is it stable?
0
sk
1+
PoS for GCG: a lower bound
>o: small value
t1,…,tk
s1
1
s2
0
1/3
1/2
s3
0
1/(k-1)
1/k
sk-1
...
0
0
sk
1+
0
A stable network
k
social cost:  1/j = Hk  ln k + 1
j=1
k-th harmonic number
Lemma 4
Suppose that we have a potential game with potential
function , and assume that for any outcome S we have
cost(S)/A  (S)  B cost(S)
for some A,B>0. Then the price of stability is at most AB.
Proof:
Let S’ be the strategy vector minimizing  (i.e., S’ is a NE)
Let S* be the strategy vector minimizing the social cost
we have:
cost(S’)/A  (S’)  (S*)  B cost(S*)
 PoS ≤ cost(S’)/cost(S*) ≤ A·B.
Lemma 5 (Bounding  )
For any strategy vector S in the GCG, we have:
cost(S)  (S)  Hk cost(S)
Proof: Indeed:
(S) = eN(S) e(S) = eN(S) ce· Hke
 (S)  cost(S) = eN(S) ce
and (S) ≤ Hk· cost(S) = eN(S) ce· Hk.
Upper-bounding the PoS
Theorem 3
The price of stability in the global connection
game with k players is at most Hk, the k-th
harmonic number
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.