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Issues on the border of economics and computation

וחישוב כלכלה בגבול נושאיםCongestion Games, Potential Games and Price of Anarchy

Liad Blumrosen ©

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 ( גודש (משחקי– A set of players 1,…,n– A set of resources M = {1,…,m} – Si is the set of (pure) strategies of player i

• 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:

MiSi 2

isj

jji sncsc )()(

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-of-

stability.

Example 1: network cong. game

• Resources: the edges.• Pure strategies: subsets of edges.• Travel time on each edges: f(congestion)

• Player 1 wants to travel AD– S1={ {AB,BD} , {AC,CD}, {AC,CB,BD} }

• Player 2 wants to travel AB– S1={ {AC,CB} , {AB} }

A

B

C

D

E

c(n)=1c(n)=n/2

c(n)=n2 c(n)=10

c(n)=4n

• Consider the strategy profile:s1= {AC,CD}

s2 = {AC,CB}

• c1(s)=4+10

• c2(s)=4+1

c(n)=n

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)

– Assume that player i deviates from si to ti:• Recall that si and ti are subsets of resources

• Let ΔΦ be:

• Let Δc be:

ΔΦ= Δc.

m

j

sn

kj

j

kcs1

)(

1

)()(

),(),( iiii ssst

),(),( iiii sscstc

• Proof:Consider the following function (potential function):

• Economic meaning: unclear….

iiii tsj

jjstj

jj sncsnc\\

)(1)(

iiii tsj

jjstj

jj sncsnc\\

)(1)(

Pure eq. proof (slide 2 of 2)

– Now, consider a pure-strategy profile s* argmins Φ(s)

– From the previous slide, we can conclude that s* is a Nash equilibrium

– Why?

• Proof:

m

j

sn

kj

j

kcs1

)(

1

)()(

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 (“better-

response” 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 pure-Nash 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 gameA game is an exact potential game if there is a function Φ:SR such that

• Definition: (ordinal) potential game

The same, but with instead of (*)

Ss it

),(),(),(),( iiiiiiii sscstcssst

0),(),( iiii ssst0),(),( iiii sscstc

(*)

Example: prisoners dilemma

• Consider the prisoners dilemma:

Cooperate Defect

Cooperate -1, -1 -5, 0Defect 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, 0Defect 0, 5 3,3

• Is this an exact potential game?• Goal: assign a number to each entry, such that:

Δ potential= Δ utilities.

5

4

4

3

Example: prisoners dilemma• Consider the prisoners dilemma:

Cooperate Defect

Cooperate 1, 1 5, 0Defect 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.

• If Φ exists: Φ(TT) < Φ(HT) < Φ(HH) < Φ(TH) < Φ(TT)

-1,1 1,-1

1,-1 -1,1

Tail Heads

Tail

Heads

Eq. in potential games

• Theorem: every (finite) potential game has a pure-strategy equilibrium.

• Theorem: in every (finite) potential game best-response 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:

• Price of stability:

Cost of worst Nash eq.

Optimal cost

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

• 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

Cooperate Defect

Cooperate 1, 1 5, 0Defect 0, 5 3,3

Examples

• 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 1+2 = 3– POS=1

Ballet Football

Ballet 2, 1 5, 5Football 5, 5 1,4

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

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

Theorem: in congestion games with linear cost function, POS ≤ 2– Objective: cost minimization.– Linear cost: cj(nj)=ajnj+bj for some aj,bj≥0

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) = ΣiN ci(s)

m

j

sn

kjj

m

j

sn

kj

jj

bkakcs1

)(

11

)(

1

)()(

m

jjjj

jj bsnasnsn

1

)(2

1)()(

m

jjjj

n

i sjjj sncsnsncsc

i 11

)()()()(

m

jjjjj

m

jjjjj bsnasnbsnasn

1

2

1

)()()()(

Φ(s)≤ c(s) ≤ 2Φ(s)

Price of stability – proof (2 of 2)

Proof: for every strategy profile s, we have

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*)

POS ≤ c(s*)/c(sopt) ≤ 2

Φ(s)≤ c(s) ≤ 2Φ(s)

≤ 2Φ(s*) ≤ 2Φ(sopt) ≤ 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.