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? jhu, sep 11 2003

3. jhu, sep 11 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 jhu, sep 11 2003

5. 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 jhu, sep 11 2003

6. Game Theory strategies strategies 3,-2 payoffs (NB: also, many players) jhu, sep 11 2003

7. matching pennies prisoner’s dilemma e.g. chicken jhu, sep 11 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. . . . jhu, sep 11 2003

9. is it in P? jhu, sep 11 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 jhu, sep 11 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] jhu, sep 11 2003

12. Selfishness can hurt you! delays x 1 Social optimum: 1.5 0 x 1 Anarchical solution: 2 jhu, sep 11 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? jhu, sep 11 2003

14. 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 = (s1,…,sn) • Cost to node i: ci[G] =  | si | + i distG(i,j) delay costs cost of edges jhu, sep 11 2003

15. 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) jhu, sep 11 2003

16. 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 wi : Are the degrees of the Nash equilibria proportional to the wi’s? jhu, sep 11 2003

17. 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) jhu, sep 11 2003

18. 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. jhu, sep 11 2003

19. 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) jhu, sep 11 2003

20. Problem: 1 1 1 1 1 s 10 t Theorem [Elkind, Sahai, Steiglitz, 03]: This is inherent for truthful mechanisms. jhu, sep 11 2003

21. 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? jhu, sep 11 2003

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

23. Question: • Are there interesting classes of games with pure Nash equilibria? jhu, sep 11 2003

24. 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 -2 1 -9 jhu, sep 11 2003

25. PLS-complete(that is, as hard as any problem in which we need to find a local optimum)[Schaeffer and Yannakakis 1995] jhu, sep 11 2003

26. 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)] jhu, sep 11 2003

27. delay fcn: 10, 32, 42, 43, 45, 46 2, 3 1, 4 5, 6 1 3 2, 4, 5, 6 jhu, sep 11 2003

28. Theorem [Rosenthal 1972]: Pure equilibrium exists Proof: Potential function = e j = 1c[e] delay[e,j] (“pseudo-social cost”) Complexity? jhu, sep 11 2003

29. Special cases • Network game vs Abstract game • Symmetric (single commodity) jhu, sep 11 2003

30. Abstract, non-symmetric Abstract, symmetric • Network, non-symmetric PLS-complete polynomial • Network, symmetric jhu, sep 11 2003

31. Algorithm idea: 1, 45 delay fcn 1, 42 10,31,42,45 1, 31 1, 10 capacity cost min-cost flow finds equilibrium jhu, sep 11 2003

32. 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? jhu, sep 11 2003