Issues on the border of
economics and computation
‫נושאים בגבול כלכלה וחישוב‬
Congestion Games, Potential
Games and Price of Anarchy
Liad Blumrosen ©
1
Course Outline
• 1st part:
equilibrium analysis of games, inefficiency of
equilibria, dynamics that lead to equilibria.
• 2nd part:
market design, electronic commerce, algorithmic
mechanism design.
• Book: Algorithmic Game Theory
– By Nisan, Roughgarden, Tardos and Vazirani.
– Available online:
http://www.cambridge.org/journals/nisan/downloads/Nisan_Non-printable.pdf
Today’s Outline
• Congestion games.
– Equilibrium.
– Convergence to equilibrium.
• Potential games.
• Inefficiency of equilibria:
– Price of anarchy
– Price of stability
– Example: congestion games.
Reminder: Nash Equilibrium
• Consider a game:
– Si is the set of (pure) strategies for player i
• S = S1 x S2 x … x Sn
– s = (s1,s2,…,sn )  S is a vector of strategies
– Ui : S  R is the payoff function for player i.
• Notation: given a strategy vector s, let
s-i = (s1,…,si-1,si,…,sn)
– The vector i where the i’th item is omitted.
• s is a Nash equilibrium if for every i,
ui)si,s-i) ≥ ui(si’,s-i) for every si’  Si
Externalities
• A standard assumption in classic economics
assume no externalities
– You only care about what you consume.
• In reality, people care about the consumption of
others:
Congestion games
• A special class of games that model externalities:
ui(consuming A) = f( #agents consuming A )
– “Congestion games” (aka as “network externalities).
• Can model both negative and positive
externalities.
– Despite the name that hints for negative externalities.
• Examples:
– Congestion on roads, in restaurants. (negative)
– Fax, social network, fashion, standards (file formats,
etc.). (positive)
Congestion games
• Definition: congestion games (‫)משחקי גודש‬
– A set of players 1,…,n
– A set of resources M = {1,…,m}
– Si is the set of (pure) strategies of player i
i
Si  2 M
• i.e., si  Si is a subset of M.
– Cost for the players that use resource j  M depends
on the number of players using j :
cj(nj)
– For s=(s1,…,sn), let
nj(s) = the number of players using resource j
– The total cost ci for player i:
ci (s)   c j n j (s)
jsi
Congestion games
• Note:
– it only matters how many players use resource j.
• Not their identities.
– Cost structure is symmetric, asymmetry is via the Si’s.
– Externalities may be positive, negative or both.
– Payoffs: today, we will mostly talk about costs, and
players aim to minimize their cost.
• As opposed to maximizing utility: c() = -u()
• The models are game-theoretically equivalent.
• There are differences when we talk about approximation, etc.
Congestion games
• Why are we interested in congestion games?
–
–
–
–
Model some interesting real problems.
Have nice equilibrium properties.
Have nice dynamics properties.
Good example for price-of-anarchy and price-ofstability.
Example 1: network cong. game
E
B
c(n)=n/2
c(n)=1
c(n)=4n
A
c(n)=n
D
c(n)=10
c(n)=n2
C
• Resources: the edges.
• Consider the strategy
profile:
• Pure strategies: subsets of edges.
• Travel time on each edges: f(congestion) s1= {AC,CD}
s2 = {AC,CB}
•
Player 1 wants to travel AD
– S1={ {AB,BD} , {AC,CD}, {AC,CB,BD} }
• Player 2 wants to travel AB
– S1={ {AC,CB} , {AB} }
•
•
c1(s)=4+10
c2(s)=4+1
Equilibria in congestion game
• Structure of Nash equilibria in congestion games:
Theorem: In every congestion game there exists a
pure Nash equilibrium.
– (At least one…)
First observed by Rosenthal (1973).
“A class of games possessing pure-strategies Nash equilibria”
Pure eq. proof (slide 1 of 2)
• Proof:
Consider the following
function (potential function):
m n j (s)
( s)    c j (k )
j 1 k 1
• Economic meaning: unclear….
– Assume that player i deviates from si to ti:
• Recall that si and ti are subsets of resources
• Let ΔΦ be:
(ti , si )  (si , si )   c j n j ( s )  1 
jti \ si
• Let Δc be:
c(ti , si )  c(si , si ) 
jsi \ ti
j
j
 c n (s)  1   c n (s)
jti \ si
 ΔΦ= Δc.
 c n (s)
j
j
jsi \ ti
j
j
Pure eq. proof (slide 2 of 2)
• Proof:
m n j (s)
( s)    c j (k )
j 1 k 1
– Now, consider a pure-strategy profile
s*  argmins Φ(s)
– From the previous slide, we can conclude that
s* is a Nash equilibrium
– Why?
Equilibria in congestion game
• The proof leads to another conclusion:
– Start with some arbitrary strategic behavior of the
players;
– at each step some player improves its payoff (“betterresponse” dynamic);
a pure equilibrium will be reached.
Why?
– Each improvement strictly improves potential.
– there is a finite number of strategy profiles.
– Potential is increasing  no strategy profile is repeated.
 Better response dynamic converges to a pureNash equilibrium in any congestion game.
Potential games
• We saw that congestion games:
– Always have a pure Nash equilibrium
– Best-response dynamics leads to such an equilibrium.
• But the proof seems to be more general, it works
whenever we have such a potential function.
• We now define such games: potential games.
Potential games
• Definition: (exact) potential game
A game is an exact potential game if there is a
function Φ:SR such that
s  S
ti
(ti , si )  (si , si )  c(ti , si )  c(si , si )
(*)
• Definition: (ordinal) potential game
The same, but with instead of (*)
c(ti , si )  c( si , si )  0
(ti , si )  ( si , si )  0
Example: prisoners dilemma
• Consider the prisoners dilemma:
Cooperate
Defect
Cooperate
-1, -1
-5, 0
Defect
0, -5
-3,-3
• Let’s present it via costs instead of utilities…
Example: prisoners dilemma
• Consider the prisoners dilemma:
Cooperate
Defect
Cooperate
1, 1
5, 0
Defect
0, 5
3,3
5
4
4
3
• Is this an exact potential game?
• Goal: assign a number to each entry, such that:
Δ potential= Δ utilities.
Example: prisoners dilemma
• Consider the prisoners dilemma:
Cooperate
Defect
Cooperate
1, 1
5, 0
Defect
0, 5
3,3
• We can build a graph:
– V = strategy profiles
– E = moving from one vertex to another is a best
response
• The game is a potential game iff this graph has no cycles.
– How can we find the (ordinal) potential function?
– No cycles: finite improvement paths.
Example: prisoners dilemma
– Cycles in the local improvement graph 
no potential function.
Tail
Heads
Tail
-1,1
1,-1
Heads
1,-1
-1,1
• If Φ exists:
Φ(TT) < Φ)HT) < Φ)HH) < Φ)TH) < Φ)TT)
Eq. in potential games
• Theorem: every (finite) potential game has a purestrategy equilibrium.
• Theorem: in every (finite) potential game bestresponse dynamic converges to an equilibrium.
• Proof: As before.
Potential games and cong. games
• What other games have this nice property other
than congestion games?
• Answer: none.
• Theorem (Monderer & Shapley):
every exact potential game is a congestion game.
(we already saw the converse)
Outline
• Congestion games.
– Equilibrium.
– Convergence to equilibrium.
• Potential games.
• Inefficiency of equilibria:
– Price of anarchy
– Price of stability
– Example: congestion games.
Quality of equilibria
• We saw: congestion games admit pure Nash
equilibria
• Are these equilibria “good” for the society?
Approximately good?
• We will need to:
– specify some objective function.
– Define “approximation”.
– Deal with multiplicity of equilibria.
Price of anarchy/stability
• Price of anarchy:
Cost of worst Nash eq.
Optimal cost
• Price of stability:
Cost of best Nash eq.
Optimal cost
• When talking about cost minimization,
POA and POS ≥1
• Concepts are not restricted to pure equilibria
• (similar concepts available for other types of equilibria)
Examples
Cooperate
Defect
Cooperate
1, 1
5, 0
Defect
0, 5
3,3
• Optimization goal: social welfare (=sum of payoffs)
• Optimal cost: 1+1=2
• Cost of worst NE = cost of best NE = 6
– One Nash equilibrium.
• POA = POS = 3
Examples
Ballet
•
•
•
•
Football
Ballet
2, 1
5, 5
Football
5, 5
1,4
Optimization goal: social welfare
Two pure equilibria: (Ballet, Ballet), (Football, Football)
Optimal cost:
2+1=3
Cost of worst NE
1+4 = 5
– POA=5/3
• Cost of best NE
– POS=1
1+2 = 3
Approximation measurements
• Several approximation concepts in the design of
algorithms:
– Approximation ratio (approximation algorithms): what is the price of
limited computational resources.
– Competitive ratio (online algorithms): what is the price for not
knowing the future.
– Price of anarchy: the price of lack of coordination
– Price of stability: price of selfish decision making with some
coordination.
Price of stability in cong. games
Theorem: in congestion games with linear cost
function, POS ≤ 2
– Objective: cost minimization.
– Linear cost: cj(nj)=ajnj+bj for some aj,bj≥0
Meaning: in such games there exists pure Nash
equilibria with cost which is at most double the
optimal cost.
Also known:
POA in linear congestion games ≤ 2.5
Price of stability – proof (1 of 2)
Proof: let Φ = potential function from previous slides.
• Consider a strategy profile s  S.
• We first compare: Φ(s) and c(S) = ΣiN ci(s)
( s )    c j (k )    a j k  b j 
m n j (s)
m n j (s)
j 1 k 1
j 1 k 1
 n j ( s)n j ( s)  1

  
a j  n j ( s)b j 
2
j 1 

Φ)s)≤ c(s) ≤ 2Φ(s)
m
c( s )   c j n j ( s )    n j ( s )  c j n j ( s ) 
n
i 1 jsi
m
j 1

  n j ( s)  a j n j ( s)  b j    n j ( s)  a j  n j (s)b j
m
m
j 1
j 1
2

Price of stability – proof (2 of 2)
Proof: for every strategy profile s, we have
Φ)s)≤ c(s) ≤ 2Φ(s)
Let s* = argmins Φ(s).
As argued before, s* is a pure Nash equilibrium.
Let sopt be the optimal solution, c(sopt) = mins c(s)
Then, c(s*)
≤ 2Φ(s*)
≤ 2Φ(sopt)
 POS ≤ c(s*)/c(sopt) ≤ 2
≤ 2c(sopt)
Summary
• We discussed a class of games:
congestion games.
• Model environments with externalities.
• Equivalent to the class of potential games.
• Admits a pure Nash equilibrium
• Best-response dynamic convergence to such a
Nash equilibrium.
• We discussed the POA and POS in congestion
games.