Networks and Games Christos H. Papadimitriou UC Berkeley christos

sonoma state, november 6, Goal of TCS ( ): 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?

sonoma state, november 6, 20033

4 The Internet Huge, growing, open, end-to-end Built and operated by 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

sonoma state, november 6, 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

sonoma state, november 6, Game Theory strategies 3,-2 payoffs (NB: also, many players)

sonoma state, november 6, ,-1-1,1 1,-1 0,00,00,10,1 1,01,0-1,-1 3,33,30,40,4 4,04,01,11,1 matching penniesprisoner’s dilemma chicken e.g.

sonoma state, november 6, 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

sonoma state, november 6, is it in P?

sonoma state, november 6, 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

sonoma state, november 6, 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]

sonoma state, november 6, Selfishness can hurt you! x x delays Social optimum: 1.5 Anarchical solution: 2

sonoma state, november 6, Worst case? Price of anarchy = “2” (4/3 for linear delays) [Roughgarden and Tardos, 2000, Roughgarden 2002] The price of the Internet architecture?

sonoma state, november 6, 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)

sonoma state, november 6, 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.

sonoma state, november 6, e.g., shortest path auction pay e its declared cost c(e), plus a bonus equal to dist(s,t)| c(e) =  - dist(s,t) ts

sonoma state, november 6, Problem: ts Theorem [Elkind, Sahai, Steiglitz, 03]: This is inherent for truthful mechanisms.

sonoma state, november 6, 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?

sonoma state, november 6, Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant ( conjectured: ~1/d )

sonoma state, november 6, The monster’s tail [Faloutsos ] the degrees of the Internet are power law distributed Both autonomous systems graph and router graph Eigenvalues: ditto!??! Model?

sonoma state, november 6, 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”

sonoma state, november 6, Models Size-independent growth (“the rich get richer,” or random walk in log paper) Carlson and Doyle 1999: Highly optimized tolerance (HOT)

sonoma state, november 6, Our model [with Fabrikant and Koutsoupias, 2002]: min j < i [   d ij + hop j ]

sonoma state, november 6, 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

sonoma state, november 6, 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?

sonoma state, november 6, PS: eigenvalues Theorem [with Mihail, 2002]: If the d i ’s obey a power law, then the n b largest eigenvalues are almost surely very close to  d 1,  d 2,  d 3, … Corollary: Spectral data-mining methods are of dubious value in the presence of large features