Networks and Games
Christos H. Papadimitriou
UC Berkeley
christos
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• Goal of TCS (1950-2000): Develop a mathematical understanding of the
capabilities and limitations of the von Neumann computer and its software –the dominant and most novel computational artifacts of that time
(Mathematical tools: combinatorics, logic)
• What should Theory’s goals be today?
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The Internet
• Huge, growing, open, end-to-end• Built and operated by 15.000 companies in
various (and varying) degrees of competition• The first computational artefact that must be
studied by the scientific method• Theoretical understanding urgently needed• Tools: math economics and game theory,
probability, graph theory, spectral theory
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Today:
• Nash equilibrium• The price of anarchy• Vickrey shortest paths• Power Laws• Collaborators: Alex Fabrikant,
Joan Feigenbaum, Elias Koutsoupias, Eli Maneva, Milena Mihail, Amin Saberi, Rahul Sami, Scott Shenker
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Game Theorystrategies
strategies3,-2
payoffs
(NB: also, many players)
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1,-1 -1,1
-1,1 1,-1
0,0 0,1
1,0 -1,-1
3,3 0,4
4,0 1,1
matching pennies prisoner’s dilemma
chicken
e.g.
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concepts of rationality
• undominated strategy (problem: too weak)• (weakly) dominating srategy (alias “duh?”) (problem: too strong, rarely exists)• Nash equilibrium (or double best response) (problem: may not exist) • randomized Nash equilibrium
Theorem [Nash 1952]: Always exists....
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is it in P?
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The critique of mixed Nash equilibrium
• Is it really rational to randomize?
(cf: bluffing in poker, tax audits)
• If (x,y) is a Nash equilibrium, then any y’ with the same support is as good as y
(corollary: problem is combinatorial!)
• Convergence/learning results mixed
• There may be too many Nash equilibria
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The price of anarchy
cost of worst Nash equilibrium“socially optimum” cost
[Koutsoupias and P, 1998]
Also: [Spirakis and Mavronikolas 01,Roughgarden 01, Koutsoupias and Spirakis 01]
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Selfishness can hurt you!
x
1
0
1
x
delays
Social optimum: 1.5
Anarchical solution: 2
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Worst case?
Price of anarchy
= “2” (4/3 for linear delays)
[Roughgarden and Tardos, 2000,Roughgarden 2002]
The price of the Internet architecture?
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Mechanism design(or inverse game theory)
• agents have utilities – but these utilities are known only to them
• game designer prefers certain outcomes depending on players’ utilities
• designed game (mechanism) has designer’s goals as dominating strategies (or other rational outcomes)
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e.g., Vickrey auction
• sealed-highest-bid auction encourages gaming and speculation
• Vickrey auction: Highest bidder wins,
pays second-highest bid
Theorem: Vickrey auction is a truthful mechanism.
Theorem: It maximizes social benefit and auctioneer expected revenue.
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e.g., shortest path auction
6
6
3
4
5
11
10
3
pay e its declared cost c(e),plus a bonus equal to dist(s,t)|c(e) = - dist(s,t)
ts
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Problem:
ts
11
1
1
1
10
Theorem [Elkind, Sahai, Steiglitz, 03]: This is
inherent for truthful mechanisms.
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But…
• …in the Internet (the graph of autonomous systems) VCG overcharge would be only about 30% on the average [FPSS 2002]
• Could this be the manifestation of rational behavior at network creation?
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• Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant (conjectured: ~1/d)
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The monster’s tail
• [Faloutsos3 1999] the degrees of the Internet are power law distributed
• Both autonomous systems graph and router graph
• Eigenvalues: ditto!??!
• Model?
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The world according to Zipf
• Power laws, Zipf’s law, heavy tails,…
• i-th largest is ~ i-a (cities, words: a = 1, “Zipf’s Law”)
• Equivalently: prob[greater than x] ~ x -b
• (compare with law of large numbers)
• “the signature of human activity”
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Models
• Size-independent growth (“the rich get richer,” or random walk in log paper)
• Carlson and Doyle 1999: Highly optimized tolerance (HOT)
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Our model [with Fabrikant and Koutsoupias, 2002]:
minj < i [ dij + hopj]
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Theorem:
• if < const, then graph is a star
degree = n -1
• if > n, then there is exponential concentration of degrees
prob(degree > x) < exp(-ax)
• otherwise, if const < < n, heavy tail:
prob(degree > x) > x -b
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Heuristically optimized tradeoffs
• Power law distributions seem to come from tradeoffs between conflicting objectives (a signature of human activity?)
• cf HOT, [Mandelbrot 1954]
• Other examples?
• General theorem?
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PS: eigenvalues
Theorem [with Mihail, 2002]: If the di’s obey a power law, then the nb largest eigenvalues are almost surely very close to
d1, d2, d3, …
Corollary: Spectral data-mining methods are of dubious value in the presence of large features