Congestion Games (Review and more
definitions), Potential Games, Network
Congestion games, Total Search Problems,
PPAD, PLS completeness, easy congestion
problems, hard congestion problems
Credit to slides by Carmine Ventre (part of cogestion game slide)
Added/modified by Michal Feldman and Amos Fiat
(General) Congestion Games

A

B
Roma
Milano
resources

players

road 1
road 4

road 2
road 3
Napoli




driver A, driver B
strategies: which roads I
use for reach my
destination?

Salerno
roads 1,2,3,4
A wants to go in Salerno
e.g. SA={{1,2},{3,4}}
B wants to go in Napoli
e.g. SB={{1,4},{2,3}}
what about the payoffs?
Payoffs in (G)CG: an example
A
SIRF (Small Index Road First)
B
Roma
Milano


road 1
road 4

road 2

A choose path 1,2
B choose path 1,4
uA = - (c1(2) + c2(1)) = - 4
uB = - (c1(2) + c4(1)) = - 5
road 3
Napoli
Salerno
B
Costs for the roads
c1(1)=2
c1(2)= 3
c2(1)=1
c2(2)= 4
c3(1)=4
c3(2)= 6
c4(1)=2
c4(2)= 5
{1,4}
{2,3}
{1,2}
(-4,-5)
(-6,-8)
{3,4}
(-9,-7)
(-8,-7)
A
Congestion games: special cases

Symmetric CG


Si are all the same and payoffs are identical symmetric
function of n-1 variables
Network CG

Each player has a starting and terminal node and
the strategies are the paths in the network
Rosenthal’s result



The class of GCG is “nice”
A pure Nash equilibrium exists
Introduces the potential function:
Potential functions
Several kind of potential functions:




Ordinal potential function
Weighted potential function
(Exact) potential function
Generalized ordinal potential function
Ordinal potential function
A function P (from S to R) is an OPF for a game G
if for every player i
ui(s-i, x) - ui(s-i, z) > 0 iff P(s-i, x) - P(s-i, z) > 0
for every x, z in Si and for every s-i
BoS =
(2,1)
(0,0)
(0,0)
(1,2)
P1 =
4
0
0
2
u1(B,B) – u1(S,B) > 0 implies that P1(B,B) – P1(S,B) > 0
u1(B,S) – u1(S,S) < 0 implies that P1(B,S) – P1(S,S) < 0
P1(B,B) – P1(S,B) > 0 implies that u1(B,B) – u1(S,B) > 0 and that u2(B,B) –
u2(S,B) > 0
P1 is an ordinal potential function for BoS game
Weighted potential function
A function P (from S to R) is a w-PF for a game G
if for every player i
ui(s-i, x) - ui(s-i, z) = wi (P(s-i, x) - P(s-i, z))
for every x, z in Si and for every s-i
C
A
PD
G’ ==
D
B
C
A
D
B
(1,1)
(3,1)
(9,0)
(4,0)
(0,9)
(2,4)
(6,6)
(1,0)
P32 ==
u1(A,A)
(C,C) –– uu11(B,A)
(D,C) == 31 -=22=(P
1/3
(P3(A,A)
– P2(D,C))
– P3(B,A))
2(C,C)
u1(A,B)
(C,D) –– uu11(B,B)
(D,D) == 43 -=12=(P
1/3
(P3(A,B)
– P2(D,D))
– P3(B,B))
2(C,D)
u2(B,A)
(D,C) –– uu22(B,B)
(D,D) == 43 -=02=(P
1/2
(P3(B,A)
– P2(D,D))
– P3(B,B))
2(D,C)
u2(A,A)
(C,C) –– uu22(A,B)
(C,D) == 11 -=02=(P
1/2
(P3(A,A)
– P2(C,D))
– P3(A,B))
2(C,C)
11
2
3/2
9
3/2
8
0
P3 is a (1/3,1/2)P2 is a (2,2)-potential
potential function for
function for PD game
the game G’
Weighted potential function
m identical machines, n jobs, and {wi}i=1..n the
weights of the jobs
Jobs seek to minimize the makespan on their
machine,
Take
P( s) 
l / 2
2
j
j 1,, m
s  s1ss  sn
l j
w
i:s i  j
i
If job i moves from
machine j to j’, then the
difference in load is
and the difference in the
potential function is
Weighted potential function (example)
m identical machines, {wi}i=1..n job weights
P ( s )   l 2j / 2
j 1,, m
If job i moves from machine j to j’,
then the difference in load is
s  s1ss  sn
l ' j '  l j '  wi
l j
l ' j  l j  wi
w
i:s i  j
i
l ' j ' l j  l j '  l j  wi

l '   l '   l   l  
P( s' )  P( s) 
2
j'
2
j
j'
2
 2l j ' wi  wi2  2l j wi  wi2 / 2 

w l
i
j'
 l j  wi 
2

2
j
Ordinal potential function (example)
m machines, {wij}i=1..n,j=1..m job weights
wij – weight of job i on machine j
P( s) 
4 j
l
j 1,, m
s  s1ss  sn
l j
w
i:s i  j
ij
If job i moves from machine j to j’,
then the difference in load is
l ' j '  l j '  wij '
l ' j  l j  wij
l ' j ' l j  l j '  l j  wij '
P( s' )  P( s)  4
l ' j ' 
4
l ' j 

 4
l j ' 
4
l j 

(Exact) potential function
A function P (from S to R) is an (exact) PF for a
game G if it is a w-potential function for G with wi =
1 for every i
C
PD =
D
C
D
(1,1)
(9,0)
(0,9)
(6,6)
u1(C,C) – u1(D,C) = P4(C,C) – P4(D,C)
u1(C,D) – u1(D,D) = P4(C,D) – P4(D,D)
u2(D,C) – u2(D,D) = P4(D,C) – P4(D,D)
u2(C,C) – u2(C,D) = P4(C,C) – P4(C,D)
P4 =
4
3
3
0
P4 is a potential
function for PD game
Generalized ordinal potential function
A function P (from S to R) is an GOPF for a game
G if for every player i
ui(s-i, x) - ui(s-i, z) > 0 implies P(s-i, x) - P(s-i, z) > 0
for every x, z in Si and for every s-i in S-i
G’’ =
A
B
A
B
(1,0)
(2,0)
(2,0)
(0,1)
P5 =
0
3
1
2
P5 is a generalized ordinal potential function for the game G’’
P5 is not an ordinal potential function for the game G’’
P5(A,B) – P5(A,A) > 0 implies that u1(A,B) – u1(A,A) > 0 but not that u2(A,B) –
u2(A,A) > 0
Potential games




A game that admits an OPF is called an
ordinal potential game
A game that admits a weighted PF is
called a weighted potential game
A game that admits an exact PF is called a
potential game
In such games a Nash equilibrium is a
local maximum for the potential
Equilibria in Potential Games
Thm (MS96)
Let G be an ordinal potential game (P is an OPF).
A strategy profile s in S is a pure equilibrium point
for G iff for every player i it holds
P(s) ≥ P(s-i, x) for every x in Si
Therefore, if P has maximal value in S, then G has
a pure Nash equilibrium.
Corollary Every finite OP game has a pure Nash
equilibrium.
An example
Nash
equilibrium
C
PD =
D
C
D
(1,1)
(9,0)
(0,9)
(6,6)
P4 maximal
value
P4 =
4
3
3
0
Thm (MS96)
C
D
C
(4,4)
(3,3)
D
(3,3)
(0,0)
PD(P4) =
FIP: an important property




A path in S is a sequence of states s.t. between
every consecutive pair of states there is only one
deviator
A path is an improvement path w.r.t. G if each
deviator has a strict advantage
ui(sk) > ui(sk-1)
G has the FIP if every improvement path is finite
Clearly if G has the FIP then G has at least one pure
equilibrium

Every improvement path terminates in an equilibrium point
FIP: an important property (2)
Lemma Every finite OP game has the FIP.
The converse is true? “No”
G’’ =
A
B
A
(1,0)
(2,0)
B
(2,0)
(0,1)
• G’’ has the FIP ((B,A) is an
equilibrium)
• any OPF must satisfies the
following impossible relations:
P(A,A) < P(B,A) < P(B,B) <
P(A,B) = P(A,A)
Lemma Let G be a finite game. Then, G has the FIP iff
G has a generalized ordinal potential function.
FIP: an important property (3)
Lemma Let G be a finite game. Then, G has the FIP iff
G has a generalized ordinal potential function.
Proof: Consider the graph of strict improvement
moves. Give v the potential function value = the length
of the longest path to a sink
Congestion vs (Exact) Potential Games
Thm Every congestion game is an (exact)
potential game.
Thm Every finite (exact) potential game is
isomorphic to a congestion game.
Congestion vs (Exact) Potential Games
Thm Every finite (exact) potential game is
isomorphic to a congestion game.
• Potential game:
• n players, k pure strategies each, potential P
• Congestion game:
• n players, k pure strategies each, 2kn resources
resource associated with {0,1}kn vector
Congestion vs (Exact) Potential Games
Thm Every finite (exact) potential game is
isomorphic to a congestion game.
•Congestion game:
player 1 ≤ i ≤n plays pure strategy 0 ≤ q ≤ k-1:
uses all resources rb where bit bik+q = 1 (2kn-1 res.)
For every strategy vector s (for both games)
b(s) where b(s)ik+q = 1 iff player i uses q in s
For n agents, the cost of rb(s) is P(s), for less
agents the cost is 0
For agent i, let b’ be such that b’jk+q= 0 if user j≠i
uses strategy q, 1 otherwise.
Cost of resource rb’ with one user ui(s)-P(s)
Final project part 1
Improve the construction (use less resources)
Why?
Computing NE in congestion games


Sometimes easy (polytime) – symmetric
network congestion games
Sometimes hard (PLS complete) – general
congestion games, symmetric congestion
games, general network congestion games
Symmetric network congestion games
(easy)




Graph G=(V,E), source vertex v, sink t
n players need to find a path from v to t.
Congestion on edge e is given by c(e,k)
where k is the number of players using the
edge (c ≥ 0)
Cost to player is sum of costs of edges player
uses.
Symmetric network congestion games
(easy) (s –strategy, v – source, confusing)



Convert to min-cost flow
Replace edge e in G with n parallel links, of
capacity 1, and with costs c(e,1), c(e,2), …
c(e,n)
The cost of a min cost flow of value n,
(strategy s), from v to t is equal to the
potential function
Hardness results: Total Search Problems – TFNP


A search problem S is a set of inputs IS *
Such that:

For every x  IS there is an associated set of solutions
Sx  



| x| k
Recognizing if y is a solution can be done in polytime
A search problem is total if Sx is not empty, for all x
r-Nash (r player Nash, normal form) is total, needs
accuracy parameter as input (true Nash can be
irrational)
Total Search Problems – TFNP


A search problem S is a set of inputs IS *
Such that:

For every x  IS there is an associated set of solutions
Sx  



| x| k
Recognizing if y is a solution can be done in polytime
A search problem is total if Sx is not empty, for all x
r-Nash (r player Nash, normal form) is total, needs
accuracy parameter as input (true Nash can be
irrational)
Subclasses of TFNP – PPAD and PLS


PPAD: Polynomial Parity Argument, Directed Version (or just
that Papadimitriou has two p’s).
End of Line problem:








n input bit circuits P and S (predecessor and successor)
S(00…0) ≠ 00….0, P(00…0)=00…0
Find the end of the line (S(x)=x) or a loop (P(S(x))=x.
PPAD – class of total search problems reducible to “end of line”.
In particular – includes Brower Fixed point (discrete version)
PPAD complete – natural notion.
Brower is complete, 2 Player Nash is also complete.
We saw how to use Brower to solve Nash, other direction also
works (major recent result – we will return to it when I
understand it).
Subclasses of TFNP – PPAD and PLS


PLS: Polynomial Local Search
A local search problem P belongs to PLS if:



For every instance I, a polytime algorithm A computes
an initial feasible solution
A polytime algorithm B that computes, for every
instance I and every feasible solution S, the objective
function value c(S)
A polytime algorithm C that determines if S is locally
optimal for instance I, and if not gives a better solution
in the neighborhood of I
PLS reductions


PLS: Polynomial Local Search
A PLS reduction from Problem P to Problem Q:




A polytime function f that takes instances of P {IP} to
instances of Q {IQ}
A polytime function g that maps pairs (SQ,IP) to
solutions SP of IP. (SQ is a solution of f(IP)).
For all instances IP of P, if SQ is a local optimum of f(IP)
then g(SQ,IP) is a local optimum of IP.
P is PLS-Complete – all problems in PLS are
PLS reducible to P
PLS Complete problem – max cut
(political party game)


Undirected graph G, weights on edges
Find a partition of the vertices so that the sum of
the weights of edges crossing the cut is
maximized (minimize disagreements within the
party, high weight – high disagreement)
NE in a congestion game is PLS complete
(even a symmetric congestion game)


For every edge e of weight w: two resources: reR
and reL
Every player u has two pure strategies:



Use all resources (reL) for e = (u,v)
Use all resources (reR) for e = (u,v)
If re{L,R} is used by one player only, the cost is zero, if
used by two, the cost is the weight of e