Christos H. Papadimitriou UC Berkeley christos Networks and Games Christos H. Papadimitriou UC Berkeley christos

(Mathematical tools: combinatorics, logic) 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? cmu, nov 20 2003

cmu, nov 20 2003

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 cmu, nov 20 2003

Today: Nash equilibria and 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 cmu, nov 20 2003

(NB: also, many players) Game Theory strategies strategies 3,-2 payoffs (NB: also, many players) cmu, nov 20 2003

e.g. 1,-1 -1,1 3,3 0,4 4,0 1,1 0,0 0,1 1,0 -1,-1 matching pennies prisoner’s dilemma 1,-1 -1,1 3,3 0,4 4,0 1,1 chicken 0,0 0,1 1,0 -1,-1 cmu, nov 20 2003

Nash equilibrium Definition: Double best response Problem: may not exist Idea: Randomized Nash equilibrium Theorem [Nash 1951]: Always exists. Problem: but is it in P? Problem: too many! cmu, nov 20 2003

Exploring the multiplicity: “The price of anarchy” cost of worst Nash equilibrium p. of A = “socially optimum” cost cmu, nov 20 2003

Selfishness can hurt you! delays x 1 Social optimum: 1.5 x 1 Anarchical equilibrium: 2 cmu, nov 20 2003

The price of the Internet architecture? Worst case? Price of anarchy = “2” (4/3 for linear delays) [Roughgarden and Tardos, 2000, Roughgarden 2002] The price of the Internet architecture? cmu, nov 20 2003

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) cmu, nov 20 2003

e.g., Vickrey auction sealed-highest-bid auction encourages gaming and speculation Vickrey auction: Highest bidder wins, pays second-highest bid Participants are incentivized to tell the truth: Truthful mechanism cmu, nov 20 2003

e.g., shortest path auction 3 6 5 s 4 t 6 10 3 11 pay e its declared cost c(e), plus a bonus equal to dist(s,t)|c(e) = - dist(s,t) cmu, nov 20 2003

Problem: s t Theorem [Elkind, Sahai, Steiglitz, 03]: This is 1 1 1 1 1 s 10 t Theorem [Elkind, Sahai, Steiglitz, 03]: This is inherent in any truthful mechanism. cmu, nov 20 2003

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? cmu, nov 20 2003

Naaah… Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant (conjectured: ~1/d) cmu, nov 20 2003

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? cmu, nov 20 2003

The world according to Zipf Power laws, Zipf’s law, heavy tails,… “the signature of human activity” vs cmu, nov 20 2003

Models Size-independent growth (“the rich get richer,” or random walk in log paper) Carlson and Doyle 1999: Highly optimized tolerance (HOT) cmu, nov 20 2003

Our model [with Fabrikant and Koutsoupias, 2002]: minj < i [  dij + hopj] cmu, nov 20 2003

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 cmu, nov 20 2003

Heuristically optimized tradeoffs Power law distributions seem to come from tradeoffs between conflicting objectives (a signature of human activity?) cf HOT, file sizes, [Mandelbrot 1954] Other examples? General theorem? cmu, nov 20 2003

Also: 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 cmu, nov 20 2003

PS: How does traffic grow PS: How does traffic grow? (or: why is the Internet backbone overprovisioned?) Trees: n2 Expanders (and most degree-balanced sparse graphs): ~ n The Internet? Theorem (with Mihail and Saberi, 2003): “Scale-free graph models” are almost certainly expanders cmu, nov 20 2003