# Slides - Microsoft Research ```Vasilis Syrgkanis
Cornell University
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
Can we efficiently compute some Pure Nash
Equilibrium of such games?
 Not in the “general case”: PLS-hard

Can we efficiently compute a “good” Pure Nash
Equilibrium so that we can propose it to the
players?
 Yes if players choose Spanning Trees or, in
general, a base of some player-specific Matroid

Formal Definitions

Algorithm for Computing a Good Pure Nash
Equilibrium

PLS-hardness Results



N players and F facilities
For each player i a set of strategies S i  2 F
Given a strategy profile s  iN Si , denote:
 n f s  : the number of players using f
 
 r f n f : a decreasing cost function
 r n  : the cost of player i
SC ( s )   C s    n r n  : the social cost
 Ci ( s) 

f si
iN
f
f
i
f F
f
f
f

Anshelevich et al., FOCS ’04
If rf (n f ) has the form c f (n f ) / n f and c f ( n f ) is a nondecreasing concave function then there exists a PNE with
social cost H nOPT (the worst PNE can be as bad as nOPT )

Fabrikant et al., STOC ‘04 - Ackermann et al., J ACM
‘08
It is PLS-complete to compute any PNE in Network Congestion
Games with rf (n f )  a f n f  b f
There exists a polynomial time greedy algorithm that computes a
PNE with social cost at most H nOPT for Matroid Cost Sharing
Games with rf (n f ) of the form c f (n f ) / n f , where c f ( n f ) is a
non-decreasing concave function.
Note: Computing the best PNE or the PNE minimizing the
potential function is NP-hard even in very simple matroid costsharing games.
For general decreasing cost functions the algorithm computes a
PNE with social cost at most the potential of the optimal strategy
profile.
It is PLS-complete to compute any PNE in Undirected Network
Cost Sharing Games with rf (n f ) of the form c f (n f ) / n f , where
c f (n f ) is a non-decreasing concave function.
If we extend to Directed Networks then it is PLS-complete even in
the case when rf (n f ) have the form c f / n f

Formal Definitions

Algorithm for Computing a Good Pure Nash
Equilibrium

PLS-hardness Results
The Price of Stability of a game is the fraction of the
Social Cost of the best PNE over the Optimal Social
Cost.
 If a game admits a Potential  such that for
some quantity a and for any strategy profile s :
SC(s)  s   aSC(s)
Then, the PoS is at most a
 Let PNEg .m. be the global minimum of  and OPT
the socially optimal outcome. Then:
SC ( PNEg .m. )  PNEg .m.   OPT   aSC (OPT )


Cost Sharing Games are Congestion Games and
 s  
nf
 r k 
f F k 1

If rf (n f ) 
cf
nf
, then  s  
f
c
f F
f
H n f , SC s    c f
SC ( s)  s   H n SC ( s)  PoS  H n
f s

Compute a PNE that is as good as the upper bound
on the PoS produced by the Potential Method
r1 n1 
r2 n2 
r3 n3 

Recall the proof of the Potential Method

Attempt 1: Compute Socially Optimal PNE
 [ADKTWR04]: NP-hard even when rf n f   1 / n f

Attempt 2: Compute global Potential Minimizer PNEg .m.
 [CCLNO07]: NP-hard even when rf n f   1 / n f
SC ( PNEg .m. )  PNEg .m.   OPT   aSC (OPT )


Consider the following generalization of the greedy
set cover approximation algorithm:
 At each iteration pick the facility that has
minimum cost if all currently unassigned players
were assigned to it.
 Assign all possible players to the chosen facility.
When rf (n f )  c f / n f , the above is exactly the
greedy H n - approximation algorithm for Set Cover
c2  20
c3  30
c1  12
c1  10

Theorem 1
The greedy algorithm computes a PNE with social
cost at most the potential of the socially optimal
solution

Corollary
The greedy algorithm computes a PNE with social
cost at most the best upper bound on the PoS given
by the Potential Method
SC ( PNEg .m. ) 
SC(PNE
PNEgreedy

OPT
OPT aSC
aSC(OPT )
g .m. )
r1
……
ri
Nj
……
rj

……
 rk  N k
rk
……
Nk
rm
 
rj N j  rk  N k


Matroid Cost Sharing:
Si is the set of bases of a matroid F, I i
 e.g. Spanning Trees on a set of nodes (possibly
different nodes for each player)
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7

Algorithm 2:
 Build player’s strategies incrementally starting
from empty sets
 At each iteration pick the facility that has
minimum cost if it is added to the strategy of all
possible players
 Add the facility to the strategy of all possible
players
Player 1
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Player 2
8
5
10
12
10
7
Player 3

Theorem 2
Algorithm 2 computes a PNE of any Matroid Cost Sharing
Game with social cost at most the potential of the
optimal strategy profile.
 Prove outcome is PNE:
A base of a matroid is minimum iff it is locally
minimum under (1,1) exchanges
 Prove efficiency guarantee:
Construct a 1-1 mapping of the facilities of a player in
the greedy strategy and those in the optimum:
When algorithm adds a facility to a player its
corresponding optimal facility was also an option

Formal Definitions

Algorithm for Computing a Good Pure Nash
Equilibrium

PLS-hardness Results
Theorem 3
Computing a PNE in General Cost Sharing Games
where rf (n f )  c f / n f is PLS-complete
 Proof: Reduction from MAX CUT
 MAX CUT: Given a weighted graph G find a cut
that cannot be increased by switching a node from
one side to the other.
 Given an instance of MAX CUT create a Cost
Sharing Game such that any PNE is a locally
maximum cut.

Create a player for each node of the graph
A
B
s
,
s
Each player has two strategies: i i
A
s
Given a strategy profile if a player is playing i then
assign the corresponding node to partition A, else to
B
Create incentives for players to play opposite strategies
s
B
j
wi,j
siB
f i ,2j
…
f i1,j
j
s Aj
ci1, j  ci2. j  2wi , j
…
…
siA
i
…



s1A
s 2B
s1B
f11, 2
2
w1, 2
1
w2 ,3
w1, 3
f1,22
s 2A
s3B
3
s3A

C s
f 21,3
f11,3
f1,23
f 22,3

  wi, B  w
Ci siA , si  wi, A  wi
i
B
i
, s i
i

Network Cost Sharing Games
Given a directed graph G  (V , E ) , Si is the set of
paths in a graph between two nodes si , ti 
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Theorem 5
Computing a PNE in Network Cost Sharing Games
where rf (n f )  c f / n f is PLS-complete.

Theorem 6
Computing a PNE in Undirected Network Cost
Sharing Games where rf (n f )  c f n f  / n f and c f n f 
is an non decreasing concave function.
s1
s1A
s 2B
s2
f11, 2
f1,22
s 2A
s3B
s3
s1B
f11,3
f 21,3
f1,23
s3A
t1
f 22,3
t2
t3
s1A
2 wij
s1
s 2B
s1B
f11, 2
f1,22
s 2A
f
s2
f1,23
H
6H
f11,3
f1,23
t1
f 21,3
f11,3
s3A
f1,22
s3
s3B
H
1
1, 2
6H
f 21,3
6H
f 22,3
t2
6H
t3
f 22,3
 D'
s1
s1A
s 2B
 D'
D
f
D
s3
D
t1
f 21,3
f11,3
f1,23
f 22,3
H
D
6H
f1,23
 D'
s3B
D
f11,3
 D'
f1,22
s3A
f1,22
 D'
f11, 2
s 2A
H
1
1, 2
 D'
s2
D
s1B
f 21,3
6H
D
6H
f 22,3
D
D
t2
6H
D
D
t3
D

Despite NP-completeness of computing Social Cost
Minimizer and Potential Minimizer we manage to
compute a PNE with very good social cost.
 Computing equilibria with social cost at most the
potential of the optimal might prove useful in
other games too

For Network Games our results show that the
decreasing cost function case is not easier than the
increasing one
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