Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

New Market Models

and Algorithms

Markets

Stock Markets

Internet

Revolution in definition of markets


New markets defined byGoogle AmazonYahoo!Ebay


Massive computational power available

for running these markets in a

centralized or distributed manner


Massive computational power available

for running these markets in a

centralized or distributed manner

Important to find good models and

algorithms for these markets

Theory of Algorithms

Powerful tools and techniques

developed over last 4 decades.

Theory of Algorithms

Powerful tools and techniques

developed over last 4 decades.

Recent study of markets has contributed

handsomely to this theory as well!

Adwords Market

Created by search engine companiesGoogleYahoo!MSN

Multi-billion dollar market

Totally revolutionized advertising, especially

by small companies.

New algorithmic and game-theoretic questions

Monika Henzinger, 2004: Find an on-line

algorithm that maximizes Google’s revenue.

The Adwords Problem:

N advertisers; Daily Budgets B1, B2, …, BN Each advertiser provides bids for keywords he is interested

in.

Search Engine



in.

Search Enginequeries (online)



in.

Search EngineSelect one Ad

Advertiser pays his bid

queries (online)



in.



queries (online)

Maximize total revenue

Online competitive analysis - compare with best offline allocation



in.



queries (online)

Maximize total revenue

Example – Assign to highest bidder: only ½ the offline revenue

Example:

$1 $0.99

$1 $0

Book

CD

Bidder1 Bidder 2

B1 = B2 = $100

Queries: 100 Books then 100 CDs

Bidder 1 Bidder 2

Algorithm Greedy

LOST

Revenue100$

Example:

$1 $0.99

$1 $0

Book

CD

Bidder1 Bidder 2

B1 = B2 = $100

Queries: 100 Books then 100 CDs

Bidder 1 Bidder 2

Optimal Allocation

Revenue199$

Generalizes online bipartite matching

Each daily budget is $1, and

each bid is $0/1.

Online bipartite matching

advertisers queries


advertisers queries


advertisers queries


advertisers queries


advertisers queries


advertisers queries


advertisers queries


Karp, Vazirani & Vazirani, 1990:

1-1/e factor randomized algorithm.



1-1/e factor randomized algorithm. Optimal!



1-1/e factor randomized algorithm. Optimal!

Kalyanasundaram & Pruhs, 1996:

1-1/e factor algorithm for b-matching:

Daily budgets $b, bids $0/1, b>>1

Adwords Problem

Mehta, Saberi, Vazirani & Vazirani, 2005:

1-1/e algorithm, assuming budgets>>bids.

Adwords Problem

Mehta, Saberi, Vazirani & Vazirani, 2005:

1-1/e algorithm, assuming budgets>>bids.

Optimal!

New Algorithmic Technique

Idea: Use both bid and

fraction of left-over budget

New Algorithmic Technique

Idea: Use both bid and

fraction of left-over budget

Correct tradeoff given by

tradeoff-revealing family of LP’s

Historically, the study of markets

has been of central importance,

especially in the West

A Capitalistic Economy

depends crucially on pricing mechanisms,

with very little intervention, to ensure:

Stability Efficiency Fairness

Do markets even have inherentlystable operating points?

General Equilibrium TheoryOccupied center stage in Mathematical

Economics for over a century

Do markets even have inherentlystable operating points?

Leon Walras, 1874

Pioneered general

equilibrium theory

Supply-demand curves

Irving Fisher, 1891

Fundamental

market model

Fisher’s Model, 1891

milkcheese

winebread

¢¢

$$$$$$$$$$$$$$$$$$

$$

$$$$$$$$

People want to maximize happiness – assume

linear utilities.Find prices s.t. market clears

Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i

on obtaining one unit of j Total utility of i,

i ij ijj

U u xiju

]1,0[

x

xuuij

ijj iji

Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i

on obtaining one unit of j Total utility of i,

Find prices s.t. market clears, i.e.,

all goods sold, all money spent.

i ij ijj

U u xiju

xuu ijj iji

Arrow-Debreu Theorem, 1954

Celebrated theorem in Mathematical Economics

Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

Kenneth Arrow

Nobel Prize, 1972

Gerard Debreu

Nobel Prize, 1983

Arrow-Debreu Theorem, 1954

.

Highly non-constructive

Adam Smith

The Wealth of Nations

2 volumes, 1776.

‘invisible hand’ of the market

What is needed today?

An inherently algorithmic theory of

market equilibrium

New models that capture new markets

Beginnings of such a theory, within

Algorithmic Game Theory

Started with combinatorial algorithms

for traditional market models

New market models emerging

Combinatorial Algorithm for Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual schema

Primal-Dual Schema

Highly successful algorithm design

technique from exact and

approximation algorithms

Exact Algorithms for Cornerstone Problems in P:

Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

Approximation Algorithms

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .

No LP’s known for capturing equilibrium allocations for Fisher’s model

Eisenberg-Gale convex program, 1959

DPSV: Extended primal-dual schema to

solving nonlinear convex programs

2s

1s

2t

1t

A combinatorial market

2s

1s

2t

1t


)(ec

2s

1s

2t

1t


)1(m

)2(m

)(ec


Given: Network G = (V,E) (directed or undirected)Capacities on edges c(e)Agents: source-sink pairs

with money m(1), … m(k)

Find: equilibrium flows and edge prices

1 1( , ),...( , )k ks t s t

Flows and edge prices

f(i): flow of agent i p(e): price/unit flow of edge e

Satisfying: p(e)>0 only if e is saturated flows go on cheapest paths money of each agent is fully spent

Equilibrium

Kelly’s resource allocation model, 1997

Mathematical framework for understanding

TCP congestion control

Highly successful theory

TCP Congestion Control

f(i): source rate prob. of packet loss (in TCP Reno)

queueing delay (in TCP Vegas) p(e):


f(i): source rate prob. of packet loss (in TCP Reno)

queueing delay (in TCP Vegas)

Kelly: Equilibrium flows are proportionally fair:

only way of adding 5% flow to someone’s

dollar is to decrease 5% flow from

someone else’s dollar.

p(e):

primal process: packet rates at sources

dual process: packet drop at links

AIMD + RED converges to equilibrium

in the limit


Kelly & V., 2002: Kelly’s model is a

generalization of Fisher’s model.

Find combinatorial polynomial time

algorithms!

Jain & V., 2005:

Strongly polynomial combinatorial algorithm

for single-source multiple-sink market

Single-source multiple-sink market

Given: Network G = (V,E), s: sourceCapacities on edges c(e)Agents: sinks

with money m(1), … m(k)

Find: equilibrium flows and edge prices

1,..., kt t

Flows and edge prices

f(i): flow of agent i p(e): price/unit flow of edge e

Satisfying: p(e)>0 only if e is saturated flows go on cheapest paths money of each agent is fully spent

Equilibrium

s

t1

t2

2

2

110$

10$

s

t1

t2

2

2

1 10$

10$

$5

$5

s

t1

t2

2

2

1 10$

10$

120$

s

t1

t2

2

2

1 120$

10$

$10

$40

$30

Jain & V., 2005:

Strongly polynomial combinatorial algorithm

for single-source multiple-sink market

Ascending price auctionBuyers: sinks (fixed budgets, maximize flow)Sellers: edges (maximize price)

Auction of k identical goods

p = 0; while there are >k buyers:

raise p; end; sell to remaining k buyers at price p;

s

t1

t2

t3

t4

Find equilibrium prices and flows

s

t1

t2

t3

t4

Find equilibrium prices and flows

m(1)

m(2)

m(3)m(4)

cap(e)

s

t1

t2

t3

t4

min-cut separating from all the sinkss

60

s

t1

t2

t3

t4

p

60

s

t1

t2

t3

t4

p

60

Throughout the algorithm:

s itc(i): cost of cheapest path from to

sink demands flow ( )

( )( )

m if i

c iit

s

t1

t2

t3

t4

p

: ( )i c i p

60

sink demands flow ( )

( )m i

f ip

it

Auction of edges in cut

p = 0; while the cut is over-saturated:

raise p; end; assign price p to all edges in the cut;

s

t1

t2 t3

t4

pp 0

0(2)c p

60 50 (2) 10f

s

t1

t2 t3

t4

p0

p

0(2)c p

60 50 0(1) (3) (4)c c c p p

s

t1

t2 t3

t4

p0

p1

60 50 20

0(2)c p

0 1(1) (3)c c p p

(1) (3) 30f f

s

t1

t2 t3

t4

p0

p1

p

60 50 20

s

t1

t2 t3

t4

p0

p1 p

2

60 50 200 1 2(4)c p p p

(4) 20f

s

t1

t2 t3

t4

p0

p1 p

2 nested cuts

60 50 20

Flow and prices will:

Saturate all red cutsUse up sinks’ moneySend flow on cheapest paths

s

t1

t2

t3

t4

Implementation

s

t1

t2

t3

t4

t

s

t1

t2

t3

t4

t

Capacity of edge =tt i

( )( )

( )

m if i

c i

s

t1

t2

t3

t4

t

min s-t cut

60

s

t1

t2

t3

t4

t

p

60

s

t1

t2

t3

t4

t

p

60

s

t1

t2

t3

t4

t

p tt i

Capacity of edge =

( )( )

m if i

p

: ( )i c i p

s

t1

t2 t3

t4

t

pp 0

0(2)c p

60 50

f(2)=10

s

t1

t2 t3

t4

t

p0

p

60 50

s

t1

t2 t3

t4

t

p0

p1

0(2)c p

0 1(1) (3) (4)c c c p p

60 50 20

s

t1

t2 t3

t4

t

p0

p1

p

s

t1

t2 t3

t4

t

p0

p1 p

2

0 1 2(4)c p p p

Eisenberg-Gale Program, 1959

max ( ) log

. .

:

: 1

: 0

ii

i ij ijj

iji

ij

m i u

s t

i u

j

ij

u xx

x

Lagrangian variables: prices of goods

Using KKT conditions:

optimal primal and dual solutions

are in equilibrium

Convex Program for Kelly’s Model

max ( ) log ( )

. .

: ( )

: ( ) ( )

, : 0

i

pip

pi

m i f i

s t

i f i f

e flow e c e

i p f

JV Algorithm

primal-dual alg. for nonlinear convex program

“primal” variables: flows

“dual” variables: prices of edges

algorithm: primal & dual improvements

Allocations Prices

Rational!!

Irrational for 2 sources & 3 sinks

s1 t1

1

s2

t2

1

t21 2

$1 $1

$1

Irrational for 2 sources & 3 sinks

s1 t1

1

s2

t2

1

t2

31

3

3

Equilibrium prices

Max-flow min-cut theorem!

Other resource allocation markets

2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents)

Branching market (for broadcasting)

s1 s2

)1(m3s

)2(m)(ec

(3)m


s1 s2

)1(m3s

)2(m)(ec

(3)m


s1 s2

)1(m3s

)2(m)(ec

(3)m


s1 s2

)1(m3s

)2(m)(ec

(3)m


Given: Network G = (V, E), directed edge capacities sources, money of each source

Find: edge prices and a packing

of branchings rooted at sources s.t. p(e) > 0 => e is saturated each branching is cheapest possible money of each source fully used.

S V

Eisenberg-Gale-type program for branching market

max ( ) log ii Sm i b

s.t. packing of branchings

Other resource allocation markets

2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents) Spanning trees Network coding

Eisenberg-Gale-Type Convex Program

max ( ) log iim i u

s.t. packing constraints

Eisenberg-Gale Market

A market whose equilibrium is captured

as an optimal solution to an

Eisenberg-Gale-type program

Theorem: Strongly polynomial algs for

following markets :2 source-sink pairs, undirected (Hu, 1963)spanning tree (Nash-William & Tutte, 1961)2 sources branching (Edmonds, 1967 + JV, 2005)

3 sources branching: irrational

Theorem: Strongly polynomial algs for

following markets :2 source-sink pairs, undirected (Hu, 1963)spanning tree (Nash-William & Tutte, 1961)2 sources branching (Edmonds, 1967 + JV, 2005)

3 sources branching: irrational

Open: (no max-min theorems):2 source-sink pairs, directed2 sources, network coding

EG[2]: Eisenberg-Gale markets with 2 agents

Theorem: EG[2] markets are rational.

Chakrabarty, Devanur & V., 2006:

EG[2]: Eisenberg-Gale markets with 2 agents

Theorem: EG[2] markets are rational.

Combinatorial EG[2] markets: polytope

of feasible utilities can be described via

combinatorial LP.

Theorem: Strongly poly alg for Comb EG[2].

Chakrabarty, Devanur & V., 2006:

EG

Rational

Comb EG[2]

SUA

EG[2]

3-source branching

Fisher

2 s-s undir

2 s-s dir

Single-source

Efficiency of Markets

‘‘price of capitalism’’ Agents:

different abilities to control prices idiosyncratic ways of utilizing resources

Q: Overall output of market when forced

to operate at equilibrium?

Efficiency

( )( ) min

max ( )I

equilibrium utility Ieff M

utility I

Efficiency

Rich classification!

( )( ) min

max ( )I

equilibrium utility Ieff M

utility I

1/(2 1)k

Market EfficiencySingle-source 1

3-source branching

k source-sink undirected

2 source-sink directed arbitrarily

small

1/ 2

. . 1/( 1)l b k

Other properties:

Fairness (max-min + min-max fair) Competition monotonicity

Open issues

Strongly poly algs for approximatingnonlinear convex programsequilibria

Insights into congestion control protocols?

Documents

Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms