Networks and Games

Christos H. Papadimitriou

UC Berkeley

christos

jhu, sep 11 2003 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 Theorystrategies

strategies3,-2

payoffs

(NB: also, many players)

jhu, sep 11 2003 7

1,-1 -1,1

-1,1 1,-1

0,0 0,1

1,0 -1,-1

3,3 0,4

4,0 1,1

matching pennies prisoner’s dilemma

chicken

e.g.

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....

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is it in P?

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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]

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Selfishness can hurt you!

x

1

0

1

x

delays

Social optimum: 1.5

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 costscost 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

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

jhu, sep 11 2003 20

Problem:

ts

11

1

1

1

10

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 1

-9

-2

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

1, 4

1

2,

2, 4, 5, 6

3

3

5, 6

delay fcn: 10, 32, 42, 43, 45, 46

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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

Network, non-symmetric

Abstract, non-symmetric

Abstract, symmetric

Network, symmetric

polynomial

PLS-complete

jhu, sep 11 2003 31

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 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?