Beyond selfish routing:
Network Games
Network Games

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NGs model the various ways in which
selfish users (i.e., players) strategically
interact in using a network
They aim to capture two competing
issues for players:


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
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NGs can be used to model:
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formation of social network (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?
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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
the social cost is given by the sum of players’
costs
Our goal: to bound the efficiency loss
resulting from selfishness
Our case study:
Global Connection Games
The model
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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
Let ke denote the number of users using edge e.
The cost of Pi for player i in S=(P1,…,Pk) is shared
with all the other players using (part of) it:
costi(S) = eP
 ce/ke
i
this cost-sharing scheme is called
fair or Shapley cost-sharing mechanism
The model


Given a strategy vector S, the constructed
network N(S) is given by the union of all paths Pi
Then, the cost of the constructed network
(social-choice function) is the following:
C(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
Bounding the loss of efficiency
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Remind that a network is optimal or socially
efficient if it minimizes the social cost
We use the PoA to estimate the loss of
efficiency we may have in the worst case, as
given by the ratio between the cost of a worst
stable network and the cost of an optimal
network
What about the ratio between the costs of a
best stable network and of an optimal
network?
Be optimist!:
The price of stability (PoS)

Definition (Schulz & Moses, 2003): Given a game G and
a social-choice minimization (resp., maximization)
function C (i.e., the sum of all players’ payoffs),
let S be the set of NE, and let OPT be the
outcome of G optimizing C. Then, the Price of
Stability (PoS) of G w.r.t. C is:
C ( s) 
C ( s) 
 resp ., sup

PoSG(C) = inf
sS C (OPT)
sS C (OPT) 

Some remarks

PoA and PoS are (for positive s.c.f. C)
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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
Homework: find a worst possible NE
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
C(S) =  costi(S) ≤  w(i) ≤  w(Pi*) ≤
k
 k·costi(OPT) = k· C(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
1/3
1/2
0
s3
0
1/(k-1)
1/k
sk-1
...
0
0
sk
1+
0
The only stable network
k
social cost: C(S)=  1/j = Hk  ln k + 1 k-th harmonic number
j=1
Lemma 4
Suppose that we have a potential game with potential
function , and assume that for any outcome S we have
C(S)/A  (S)  B C(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:
C(S’)/A  (S’)  (S*)  B C(S*)
 PoS ≤ C(S’)/C(S*) ≤ A·B.
Lemma 5 (Bounding  )
For any strategy vector S in the GCG, we have:
C(S)  (S)  Hk C(S).
Proof: Indeed:
(S) = eN(S) e(S) = eN(S) ce· Hke
 (S)  C(S) = eN(S) ce
and (S) ≤ Hk· C(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.