Networks and Games Christos H Papadimitriou UC Berkeley

Networks and Games Christos H. Papadimitriou UC Berkeley christos jhu, sep 11 2003

• 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 2

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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 4

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 5

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

e. g. matching pennies prisoner’s dilemma 1, -1 -1, 1 3, 3 0, 4 -1, 1 1, -1 4, 0 1, 1 chicken 0, 0 0, 1 1, 0 -1, -1 jhu, sep 11 2003 7

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 8

is it in P? jhu, sep 11 2003 9

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 10

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 11

Selfishness can hurt you! delays x 1 Social optimum: 1. 5 0 x 1 jhu, sep 11 2003 Anarchical solution: 2 12

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 13

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

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 15

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 16

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 17

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 18

e. g. , shortest path auction 6 s 4 3 3 5 6 t 10 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 19

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

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 21

• 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 22

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

e. g. : the party affiliation game • n players-nodes • Strategies: +1, -1 3 • Payoff [i]: sgn( j s[i]*s[j]*w[i, j]) 3 -2 1 -9 Theorem: A pure Nash equilibrium exists Proof: Potential function i, j s[i]*s[j]*w[i, j] jhu, sep 11 2003 24

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 25

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 26

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

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

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

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

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

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 32