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

2. (Mathematical tools: combinatorics, logic)
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?
rosser lecture, nov

3. rosser lecture, nov

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
rosser lecture, nov

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

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

7. 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
rosser lecture, nov

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

9. is it in P?
rosser lecture, nov

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

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

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

13. 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?
rosser lecture, nov

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

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

16. e.g., shortest path auction3 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

17. Problem: s t Theorem [Elkind, Sahai, Steiglitz, 03]: This is1 1 1 1 1 s 10 t
Theorem [Elkind, Sahai, Steiglitz, 03]: This is
inherent for truthful mechanisms.
rosser lecture, nov

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

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

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

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

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

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

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

25. 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?
rosser lecture, nov

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

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