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

2. jhu, sep 11 20032 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?

3. jhu, sep 11 20033

4. 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

5. jhu, sep 11 20035 Today: Nash equilibrium The price of anarchy Vickrey shortest paths Congestion games Collaborators: Alex Fabrikant, Joan Feigenbaum, Elias Koutsoupias, Eli Maneva, Milena Mihail, Amin Saberi, Rahul Sami, Scott Shenker

6. jhu, sep 11 20036 Game Theory strategies 3,-2 payoffs (NB: also, many players)

7. jhu, sep 11 20037 1,-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.

8. jhu, sep 11 20038 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.......

9. jhu, sep 11 20039 is it in P?

10. jhu, sep 11 200310 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

11. jhu, sep 11 200311 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]

12. jhu, sep 11 200312 Selfishness can hurt you! x 1 0 1 x delays Social optimum: 1.5 Anarchical solution: 2

13. jhu, sep 11 200313 Worst case? Price of anarchy = “2” (4/3 for linear delays) [Roughgarden and Tardos, 2000, Roughgarden 2002] The price of the Internet architecture?

14. jhu, sep 11 200314 Simple net creation game (with Fabrikant, Maneva, Shenker PODC 03) Players: Nodes V = {1, 2, …, n} Strategies of node i: all possible subsets of {[i,j]: j  i} Result is undirected graph G = (s 1,…,s n ) Cost to node i: c i [G] =  | s i | +  i dist G (i,j) delay costs cost of edges

15. jhu, sep 11 200315 Nash equilibria? (NB: Best response is NP-hard…) If  < 1, then the only Nash equilibrium is the clique If 1 <  < 2 then social optimum is clique, Nash equilibrium is the star (price of anarchy = 4/3)

16. jhu, sep 11 200316 Nash equilibria (cont.)  > 2? The price of anarchy is at least 3 Upper bound:  Conjecture: For large enough , all Nash equlibria are trees. If so, the price of anarchy is at most 5. General w i : Are the degrees of the Nash equilibria proportional to the w i ’s?

17. jhu, sep 11 200317 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)

18. jhu, sep 11 200318 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.

19. jhu, sep 11 200319 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

20. jhu, sep 11 200320 Problem: ts 11 1 1 1 10 Theorem [Elkind, Sahai, Steiglitz, 03]: This is inherent for truthful mechanisms.

21. jhu, sep 11 200321 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?

22. jhu, sep 11 200322 Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant ( conjectured: ~1/d )

23. jhu, sep 11 200323 Question: Are there interesting classes of games with pure Nash equilibria?

24. jhu, sep 11 200324 e.g.: the party affiliation game n players-nodes Strategies: +1, -1 Payoff [i]: sgn(  j s[i]*s[j]*w[i,j]) Theorem: A pure Nash equilibrium exists Proof: Potential function  i,j s[i]*s[j]*w[i,j] 3 3 1 -9 -2

25. jhu, sep 11 200325 PLS-complete (that is, as hard as any problem in which we need to find a local optimum) [Schaeffer and Yannakakis 1995]

26. jhu, sep 11 200326 Congestion games [joint work with Alex Fabrikant] n players resources E delay functions Z Z strategies: subsets of E -payoff[i]:  e in s[i] delay[e,c(e)]

27. jhu, sep 11 200327 1, 4 1 2, 2, 4, 5, 6 3 3 5, 6 delay fcn: 10, 32, 42, 43, 45, 46

28. jhu, sep 11 200328 Theorem [Rosenthal 1972]: Pure equilibrium exists Proof: Potential function =  e  j = 1 c[e] delay[e,j] (“pseudo-social cost”) Complexity?

29. jhu, sep 11 200329 Special cases Network game vs Abstract game Symmetric (single commodity)

30. jhu, sep 11 200330 Network, non-symmetric Abstract, non-symmetric Abstract, symmetric Network, symmetric polynomial PLS-complete

31. jhu, sep 11 200331 Algorithm idea: 10,31,42,45 1, 45 1, 31 1, 42 1, 10 capacity cost min-cost flow finds equilibrium delay fcn

32. jhu, sep 11 200332 Also… Same algorithm approximates equilibrium in non-atomic game (as in [Roughgarden 2003]) “Price of anarchy” is unbounded, and NP-hard to compute Other games with guaranteed pure equilibria?