Networks and Games Christos H. Papadimitriou UC Berkeley christos

jhu, sep 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?

jhu, sep

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

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

jhu, sep ,-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.

jhu, sep 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 is it in P?

jhu, sep 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 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 Selfishness can hurt you! x x delays Social optimum: 1.5 Anarchical solution: 2

jhu, sep 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 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

jhu, sep 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 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?

jhu, sep 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 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 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

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

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

jhu, sep 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]

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

jhu, sep 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 , 4 1 2, 2, 4, 5, , 6 delay fcn: 10, 32, 42, 43, 45, 46

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

jhu, sep Special cases Network game vs Abstract game Symmetric (single commodity)

jhu, sep Network, non-symmetric Abstract, non-symmetric Abstract, symmetric Network, symmetric polynomial PLS-complete

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

jhu, sep 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?