Networks and Games

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

2. 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? rosser lecture, nov 13 2003

3. rosser lecture, nov 13 2003

4. 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 rosser lecture, nov 13 2003

5. 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 rosser lecture, nov 13 2003

6. Game Theory strategies strategies 3,-2 payoffs (NB: also, many players) rosser lecture, nov 13 2003

7. matching pennies prisoner’s dilemma e.g. chicken rosser lecture, nov 13 2003

8. 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. . . . rosser lecture, nov 13 2003

9. is it in P? rosser lecture, nov 13 2003

10. 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 rosser lecture, nov 13 2003

11. 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] rosser lecture, nov 13 2003

12. Selfishness can hurt you! delays x 1 Social optimum: 1.5 0 x 1 Anarchical solution: 2 rosser lecture, nov 13 2003

13. Worst case? Price of anarchy = “2” (4/3 for linear delays) [Roughgarden and Tardos, 2000, Roughgarden 2002] The price of the Internet architecture? rosser lecture, nov 13 2003

14. 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) rosser lecture, nov 13 2003

15. 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. rosser lecture, nov 13 2003

16. 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) rosser lecture, nov 13 2003

17. Problem: 1 1 1 1 1 s 10 t Theorem [Elkind, Sahai, Steiglitz, 03]: This is inherent for truthful mechanisms. rosser lecture, nov 13 2003

18. 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? rosser lecture, nov 13 2003

19. Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant (conjectured: ~1/d) rosser lecture, nov 13 2003

20. 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? rosser lecture, nov 13 2003

21. 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” rosser lecture, nov 13 2003

22. Models • Size-independent growth (“the rich get richer,” or random walk in log paper) • Carlson and Doyle 1999: Highly optimized tolerance (HOT) rosser lecture, nov 13 2003

23. Our model [with Fabrikant and Koutsoupias, 2002]: minj < i [  dij + hopj] rosser lecture, nov 13 2003

24. 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 rosser lecture, nov 13 2003

25. Heuristically optimized tradeoffs • Power law distributions seem to come from tradeoffs between conflicting objectives (asignature of human activity?) • cf HOT, [Mandelbrot 1954] • Other examples? • General theorem? rosser lecture, nov 13 2003

26. 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 rosser lecture, nov 13 2003

27. PS: How does traffic grow? • 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 rosser lecture, nov 13 2003