Upload
esther-gardner
View
219
Download
0
Tags:
Embed Size (px)
Citation preview
1
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.