Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Markets and the Primal-Dual...

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Markets and

the Primal-Dual Paradigm

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 – and still growing!

Totally revolutionized advertising, especially

by small companies.

Historically, the study of markets

has been of central importance,

especially in the West

Historically, the study of markets

has been of central importance,

especially in the West

General Equilibrium TheoryOccupied center stage in Mathematical

Economics for over a century

Leon Walras, 1874

Pioneered general

equilibrium theory

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

General Equilibrium Theory

Also gave us some algorithmic resultsConvex programs, whose optimal solutions capture equilibrium allocations,

e.g., Eisenberg & Gale, 1959 Nenakov & Primak, 1983

Cottle and Eaves, 1960’s: Linear complimentarity

Scarf, 1973: Algorithms for approximately computing fixed points

An almost entirely non-algorithmic theory!

General Equilibrium Theory

What is needed today?

An inherently algorithmic theory of

market equilibrium

New models that capture new markets

and are easier to use than traditional models

Beginnings of such a theory, within

Algorithmic Game Theory

Started with combinatorial algorithms

for traditional market models

New market models emerging

A central tenet

Prices are such that demand equals supply, i.e.,

equilibrium prices.

A central tenet

Prices are such that demand equals supply, i.e.,

equilibrium prices.

Easy if only one good

Supply-demand curves

Irving Fisher, 1891

Defined a fundamental

market model

utility

Utility function

amount of milk

utility

Utility function

amount of bread

utility

Utility function

amount of cheese

Total utility of a bundle of goods

= Sum of utilities of individual goods

For given prices,

1p 2p3p

For given prices,find optimal bundle of goods

1p 2p3p

Fisher market

Several goods, fixed amount of each good

Several buyers,

with individual money and utilities

Find equilibrium prices of goods, i.e., prices s.t., Each buyer gets an optimal bundle No deficiency or surplus of any good

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using the 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

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

Eisenberg-Gale convex program, 1959

DPSV: Extended primal-dual schema to

solving a nonlinear convex program

Fisher’s Model

n buyers, money m(i) for buyer i k goods (unit amount of each good) : utility derived by i

on obtaining one unit of j Total utility of i,

i ij ijj

U u xiju

ijj iji

Fisher’s Model

n buyers, money m(i) for buyer i k goods (unit amount of each good) : utility derived by i

on obtaining one unit of j Total utility of i,

Find market clearing prices

i ij ijj

U u xiju

ijj iji

An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.

An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.

Equilibrium prices are unique!

At prices p, buyer i’s most

desirable goods, S =

Any goods from S worth m(i)

constitute i’s optimal bundle

arg max ijj

Bang-per-buck

For each buyer, most desirable goods, i.e.

arg max ijj

Max flow

infinite capacities

Max flow

p: equilibrium prices iff both cuts saturated

Idea of algorithm

“primal” variables: allocations

“dual” variables: prices of goods

Approach equilibrium prices from below:start with very low prices; buyers have surplus money iteratively keep raising prices and decreasing surplus

Idea of algorithm

Iterations:

execute primal & dual improvements

Allocations Prices

Two important considerations

The price of a good never exceeds

its equilibrium priceInvariant: s is a min-cut

Max flow

p: low prices

its equilibrium priceInvariant: s is a min-cut

Identify tight sets of goods

: ( ) ( ( ))S A p S m S

its equilibrium priceInvariant: s is a min-cutIdentify tight sets of goods

Rapid progress is madeBalanced flows

Network N

buyers

bang-per-buck edges

Balanced flow in N

W.r.t. flow f, surplus(i) = m(i) – f(i,t)

Balanced flow

surplus vector: vector of surpluses w.r.t. f.

Balanced flow

surplus vector: vector of surpluses w.r.t. f.

A flow that minimizes l2 norm of surplus vector.

Property 1

f: max flow in N.

R: residual graph w.r.t. f.

If surplus (i) < surplus(j) then there is no

path from i to j in R.

Property 1

surplus(i) < surplus(j)

Property 1

surplus(i) < surplus(j)

Property 1

Circulation gives a more balanced flow.

Property 1

Theorem: A max-flow is balanced iff

it satisfies Property 1.

Pieces fit just right!

Balanced flows Invariant

Bang-per-buck

edgesTight sets

How primal-dual schema is adapted

to nonlinear setting

A convex program

whose optimal solution is equilibrium allocations.

A convex program

Constraints: packing constraints on the xij’s

A convex program

Constraints: packing constraints on the xxij’s

Objective fn: max utilities derived.

A convex program

Constraints: packing constraints on the xxij’s

Objective fn: max utilities derived. Must satisfy

If utilities of a buyer are scaled by a constant,

optimal allocations remain unchangedIf money of buyer b is split among two new buyers,

whose utility fns same as b, then union of optimal

allocations to new buyers = optimal allocation for b

Money-weighed geometric mean of utilities

1/ ( )( )( ) im im i

Eisenberg-Gale Program, 1959

max ( ) log

i ij ijj

Eisenberg-Gale Program, 1959

max ( ) log

i ij ijj

prices pj

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

ij iij

u ui j

u ui j x

Therefore, buyer i buys from

i.e., gets an optimal bundle

arg max ijj

Therefore, buyer i buys from

i.e., gets an optimal bundle

Can prove that equilibrium prices

are unique!

arg max ijj

Will relax KKT conditions

e(i): money currently spent by i

w.r.t. a special allocation

surplus money of i( ) ( )i m i e i

Will relax KKT conditions

e(i): money currently spent by i

w.r.t. a balanced flow in N

surplus money of i( ) ( )i m i e i

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

ij iij

u ui j

u ui j x

Potential function

2 2 21 2 ... n

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Potential function

2 2 21 2 ... n

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).( ( ))

ipoly m i

Point of departure

KKT conditions are satisfied via a

continuous process Normally: in discrete steps

Point of departure

KKT conditions are satisfied via a

continuous process Normally: in discrete steps

Open question: strongly polynomial algorithm?

Another point of departure

Complementary slackness conditions:

involve primal or dual variables, not both.

KKT conditions: involve primal and dual

variables simultaneously.

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

ij iij

u ui j

u ui j x

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( ) ( )

ij ijij jiij

u ui j

u xu ui j x

p m i m i

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Only exception: Edmonds, 1965: algorithm

for weight matching.

Only exception: Edmonds, 1965: algorithm

for weight matching.

Otherwise primal objects go tight and loose.

Difficult to account for these reversals

in the running time.

Our algorithm

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

Deficiencies of linear utility functions

Typically, a buyer spends all her money

on a single good

Do not model the fact that buyers get

satiated with goods

utility

Concave utility function

amount of j

Concave utility functions

Do not satisfy weak gross substitutability

Do not satisfy weak gross substitutabilityw.g.s. = Raising the price of one good cannot lead to a

decrease in demand of another good.

Do not satisfy weak gross substitutabilityw.g.s. = Raising the price of one good cannot lead to a

decrease in demand of another good.

Open problem: find polynomial time algorithm!

utility

Piecewise linear, concave

amount of j

utility

PTAS for concave function

amount of j

Piecewise linear concave utility

Does not satisfy weak gross substitutability

utility

Piecewise linear, concave

amount of j

rate = utility/unit amount of j

amount of j

Differentiate

amount of j

money spent on j

Spending constraint utility function

$20 $40 $60

Spending constraint utility function

Happiness derived is

not a function of allocation only

but also of amount of money spent.

$20 $40 $100

Extend model: assume buyers have utility for money

Theorem: Polynomial time algorithm for

computing equilibrium prices and allocations for

Fisher’s model with spending constraint utilities.

Furthermore, equilibrium prices are unique.

Satisfies weak gross substitutability!

Old pieces become more complex+ there are new pieces

But they still fit just right!

Don Patinkin, 1956

Money, Interest, and Prices.

An Integration of Monetary and Value Theory

Pascal Bridel, 2002: Euro. J. History of Economic Thought,

Patinkin, Walras and the ‘money-in-the-utility- function’ tradition

An unexpected fallout!!

A new kind of utility functionHappiness derived is

An unexpected fallout!!

A new kind of utility functionHappiness derived is

Has applications in

Google’s AdWords Market!

A digression

The view 5 years ago: Relevant Search Results

Business world’s view now :

(as Advertisement companies)

Bids for different keywords

DailyBudgets

So how does this work?

An interesting algorithmic question!

Monika Henzinger, 2004: Find an on-line

algorithm that maximizes Google’s revenue.

AdWords Allocation Problem

Search Engine

Whose ad to put

How to maximize revenue?

LawyersRus.com

Sue.com

TaxHelper.com

asbestos

Search results

AdWords Problem

Mehta, Saberi, Vazirani & Vazirani, 2005:

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

AdWords Problem

Optimal!

AdWords Problem

Optimal!

Spending

constraint

utilities

AdWords

Market

AdWords market

Assume that Google will determine equilibrium price/click for keywords

AdWords market

Assume that Google will determine equilibrium price/click for keywords

How should advertisers specify their

utility functions?

Choice of utility function

Expressive enough that advertisers get

close to their ‘‘optimal’’ allocations

Efficiently computable

Easy to specify utilities

linear utility function: a business will

typically get only one type of query

throughout the day!

concave utility function: no efficient

algorithm known!

throughout the day!

concave utility function: no efficient

algorithm known!Difficult for advertisers to

define concave functions

Easier for a buyer

To say how much money she should spend

on each good, for a range of prices,

rather than how happy she is

with a given bundle.

Online shoe business

Interested in two keywords: men’s clog women’s clog

Advertising budget: $100/day

Expected profit:men’s clog: $2/clickwomen’s clog: $4/click

Considerations for long-term profit

Try to sell both goods - not just the most

profitable good

Must have a presence in the market,

even if it entails a small loss

If both are profitable,better keyword is at least twice as profitable ($100, $0)otherwise ($60, $40)

If neither is profitable ($20, $0)

If only one is profitable, very profitable (at least $2/$) ($100, $0)otherwise ($60, $0)

$60 $100

men’s clog

rate = utility/click

$60 $100

women’s clog

rate = utility/click

$80 $100

rate = utility/$

AdWords market

Suppose Google stays with auctions but

allows advertisers to specify bids in

the spending constraint model

AdWords market

the spending constraint model expressivity!

AdWords market

the spending constraint model expressivity!

Good online algorithm for

maximizing Google’s revenues?

Goel & Mehta, 2006:

A small modification to the MSVV algorithm

achieves 1 – 1/e competitive ratio!

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?

Equilibrium exists (under mild conditions)

Equilibrium utilities and prices are unique

Rational

With small denominators

Spending constraint utilities satisfy

Equilibrium exists (under mild conditions)

Equilibrium utilities and prices are unique

Rational

With small denominators

Linear utilities also satisfy

Proof follows fromEisenberg-Gale Convex Program, 1959

max ( ) log

i ij ijj

For spending constraint utilities,proof follows from algorithm,

and not a convex program!

Is there an LP whose optimal solutions

capture equilibrium allocations

for Fisher’s linear case?

Use spending constraint algorithm to solve

piecewise linear, concave utilities

Algorithms & Game Theorycommon origins

von Neumann, 1928: minimax theorem for

2-person zero sum games von Neumann & Morgenstern, 1944:

Games and Economic Behavior von Neumann, 1946: Report on EDVAC

Dantzig, Gale, Kuhn, Scarf, Tucker …

utility

Piece-wise linear, concave

amount of j

Differentiate ijg

Start with arbitrary prices, adding up to

total money of buyers.

( ) ( )ij ijj

xh x g

money spent on j

( ) ( )ij ijj

xh x g

Run algorithm on these utilities to get new prices.

( ) ( )ij ijj

xh x g

( ) ( )ij ijj

xh x g

Fixed points of this procedure are equilibrium

prices for piecewise linear, concave utilities!

( ) ( )ij ijj

xh x g

Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Markets and the Primal-Dual...

Documents

Primal blueprintintro

Primal Ladis

Primal fear

Vijay V. Vazirani - Georgia Institute of Technology

Nora Gedgaudas - Primal Health ~ Primal Food Recipes

Umesh V. Vazirani University of California, Berkeley · Umesh V. Vazirani University of California, Berkeley Quantum Mechanics & Quantum Computation Lecture 10: Quantum Factoring

Farcry primal

primal healthcare

The general maximum matching algorithm of micali and vazirani

Quantum Algorithms & Complexity Umesh Vazirani U.C. Berkeley

Primal Body Primal Mind Sample

Primal-Dual Approaches in Online Algorithms, Algorithmic

Primal Power

Barriers in Hamiltonian Complexity Umesh V. Vazirani U.C. Berkeley

Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Markets and the Primal-Dual Paradigm

Market Equilibrium under Separable, Piecewise-Linear ...vazirani/plc.pdfSeparable, Piecewise-Linear, Concave Utilities Vijay V. Vazirani Mihalis Yannakakisy Abstract We consider Fisher

Vijay V. Vazirani Georgia Tech

Umesh V. Vazirani University of California, Berkeley · PDF fileUmesh V. Vazirani University of California, Berkeley Quantum Mechanics & Quantum Computation Lecture 5: Quantum Gates

Primal Body, Primal Mind - Excerpts

Primal Pictures