Approximate pure Nash equilibria!
in weighted congestion games
Max Klimm
Combinatorial Optimization and Graph Algorithms!
Technische Universität Berlin
Christoph Hansknecht
Combinatorial Optimization and Graph Algorithms!
Technische Universität Berlin!
Alexander Skopalik
Heinz Nixdorf Institute!
University of Paderborn!
Introduction
cb(x) = x3
ca(x) =
t2
cd(x) = (x+1)3
x3
s1
cg(x) =
t1
ce(x) = x3
(x+1)3
s2
cf(x) = x3
2
Introduction
x3
(x+1)3
x3
Nash equilibrium!
Strategy combination such that no
player can unilaterally improve
x3
(x+1)3
x3
3
Introduction
x3
(x+1)3
x3
Nash equilibrium!
Strategy combination such that no
player can unilaterally improve
x3
(x+1)3
x3
3
24, 24
29, 29
29, 29
24, 24
Congestion games
1
2
set of resources R
set of players N
4
Congestion games
ca
cb
cd
1
ce
ch
cf
ck
cg
2
cl
set of resources R
with cost functions cr :
set of players N
≥0 ➝
4
Congestion games
ca
cb
S1
cd
1
ce
ch
cf
ck
cg
2
cl
set of resources R
with cost functions cr :
set of players N
≥0 ➝
with strategies Si ⊆ 2R
4
Congestion games
ca
cb
S1
cd
1
ce
ch
cf
ck
cg
S2
2
cl
set of resources R
with cost functions cr :
set of players N
≥0 ➝
with strategies Si ⊆ 2R
4
Congestion games
ca
cb
S1
cd
1
ce
ch
cf
ck
cg
S2
2
cl
set of resources R
with cost functions cr :
set of players N
≥0 ➝
with strategies Si ⊆ 2R
Ungewichtetes congestion game private costs: "i(s) = ∑ r∈si cr(!j∈N : r∈sj!)$
4
Congestion games
[Rosenthal, IJGT `73]
Theorem Congestion games have a Nash equilibrium.
5
Weighted congestion games
ca
cb
S1
cd
1
ce
ch
cf
ck
cg
S2
2
cl
Set of Resources R
with cost functions cr :
Set of players N
≥0 ➝
with strategies Si ⊆ 2R
congestion game private costs: "i(s) = ∑r∈si cr(!j∈N : r∈sj!)$
7
Weighted congestion games
ca
ce
ch
cb
cf
ck
cd
cg
cl
Set of Resources R
with cost functions cr :
S1
1
d1 > 0
2
d2 > 0
S2
Set of players N
≥0 ➝
with strategies Si ⊆ 2R
congestion game private costs: "i(s) = ∑r∈si cr(!j∈N : r∈sj!)$
weighted congestion game private costs: "i(s) = ∑r∈si di cr(∑j∈N : r∈sj dj)
7
Weighted congestion games
x3
(x+1)3
x3
x3
(x+1)3
x3
8
24, 24
29, 29
29, 29
24, 24
Weighted congestion games
x3
(x+1)3
x3
d1 = 1
x3
(x+1)3
x3
d2 = 2
9
62, 162
66, 160
66, 160
62, 162
Further counterexamples
[Fotakis et al.,TCS `05]
10|12|14
d1 = 1
d2 = 2 s
1|2|8
1|40|79
2|10|12
16|18|20
12|120|228
10
t
Further counterexamples
[Fotakis et al.,TCS `05]
10|12|14
d1 = 1
d2 = 2 s
1|2|8
1|40|79
2|10|12
t
16|18|20
12|120|228
[Goemans et al., FOCS `05]
x+33
d1 = 1
d2 = 2 s
3x2
6x2
13x
x2+44
47x
11
t
Further counterexamples
[Fotakis et al.,TCS `05]
10|12|14
d1 = 1
d2 = 2 s
1|2|8
1|40|79
2|10|12
t
16|18|20
12/11-approximate equilibrium
12|120|228
[Goemans et al., FOCS `05]
x+33
d1 = 1
d2 = 2 s
3x2
6x2
13x
x2+44
47x
12
t
Further counterexamples
[Fotakis et al.,TCS `05]
10|12|14
d1 = 1
d2 = 2 s
1|2|8
1|40|79
2|10|12
t
16|18|20
12/11-approximate equilibrium
12|120|228
[Goemans et al., FOCS `05]
x+33 = 35
d1 = 1
d2 = 2 s
3x2
6x2
13x = 26
t
x2+44
61/60-approximate equilibrium
47x = 47
13
Previous work
[Caragiannis et al.,EC `12]
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation factors!
!
!
quadratic!
2$
cubic!
6$
polynomials of max. degree Δ!
∆!
Our results
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation factors!
!
!
quadratic!
2$
cubic!
6$
polynomials of max. degree Δ!
∆+1
Our results
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation factors!
!
quadratic!
cubic!
polynomials of max. degree Δ!
!
4/3$
1.785…$
∆+1
Our results
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation factors!
!
quadratic!
cubic!
!
4/3$
1.785…$
polynomials of max. degree Δ!
∆+1$
concave!
3/2
Our results
!
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation
factors!
!
!
2 players
!
all games
quadratic!
1.054…$
4/3$
cubic!
1.074…$
1.785…$
polynomials of max. degree Δ!
∆+1$
concave!
3/2
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
t1
t2
t4
t3
s1,s3
s2,s4
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
t1
t2
cr(x)
t4
t3
s1,s3
x
s2,s4
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
t1
c(1)
t2
c(1)+c(2)
cr(x)
c(1)
t4
c(1)
c(1)+c(2)$
+c(3)+c(4)
c(1)+c(2)
s1,s3
c(1)
t3
x
s2,s4
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
t1
c(1)
t2
c(1)+c(2)
cr(x)
c(1)
t4
c(1)
c(1)+c(2)$
+c(3)+c(4)
c(1)+c(2)
s1,s3
c(1)
t3
x
s2,s4
The potential function argument
[Rosenthal, IJGT `73]
‣
Let P(s) = ∑r∈R Pr(s), where ∑k=1,…,dr(s) cr(k)
t1
c(1)
t2
c(1)+c(2)
cr(x)
c(1)
t4
c(1)
c(1)+c(2)$
+c(3)+c(4)
c(1)+c(2)
s1,s3
c(1)
t3
x
s2,s4
P(tn,s-n) - P(s) =∑r∈tn cr(dr(tn,s-n)) - ∑r∈sn cr(dr(s)) ="i(ti,s-i) - "i(s)
Weighted players
cr(x)
cr(x)
≠
x
x
Weighted players
cr(x)
cr(x)
≠
x
Observation For a convex function,!
‣ the potential is minimized for non-increasing players!
‣ the potential is maximized for non-decreasing players
x
Weighted players
cr(x)
cr(x)
≠
>
x
Observation For a convex function,!
‣ the potential is minimized for non-increasing players!
‣ the potential is maximized for non-decreasing players
x
Bounding the approximation factor
cr(x)
cr(x)
!min =
x
i
x
i
cr(x)
x
i
Bounding the approximation factor
cr(x)
cr(x)
!min =
i
x
x
i
cr(x)
x
i
Bounding the approximation factor
cr(x)
cr(x)
!max =
x
i
i
cr(x)
ii
x
x
Main result
[Hansknecht K. Skopalik, `14]
Theorem Weighted congestion game have an α-approximate
pure Nash equilibrium, where α ≤ min {1+μmax, 1/(1-μmin)}.
Main result
[Hansknecht K. Skopalik, `14]
Theorem Weighted congestion game have an α-approximate
pure Nash equilibrium, where α ≤ min {1+μmax, 1/(1-μmin)}.
Proof:
‣
Show that either the maximizing order is an αapproximate potential.
Bounding
min
μ
!min =
i
x
x
i
cr(x)
x
i
Bounding
min
μ
!min =
i
x
x
i
cr(x)
x
i
Bounding
min
μ
!min =
i
x
x
i
cr(x)
x
i
Our results
!
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation
factors!
!
!
2 players
!
all games
quadratic!
1.054…$
4/3$
cubic!
1.074…$
1.785…$
polynomials of max. degree Δ!
∆+1$
concave!
3/2
Our results
!
!
Functions ! ! ! ! ! ! ! ! ! ! ! Approximation
factors!
!
!
2 players
!
all games
quadratic!
1.054…$
4/3$
cubic!
1.074…$
1.785…$
polynomials of max. degree Δ!
∆+1$
concave!
3/2
Thank you.