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

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Markets and

the Primal-Dual Paradigm

The new face of computing

A paradigm shift inthe notion of a “market”!

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

General Equilibrium Theory

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

equilibrium allocations,

e.g., Eisenberg & Gale, 1959

Nenakov & Primak, 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

Scarf, 1973: Algorithms for approximately computing fixed points

New markets defined by Internet companies, e.g., Google Yahoo! Amazon eBay

Massive computing power available for running markets in a distributed or centralized manner

A deep theory of algorithms with many powerful techniques

Today’s reality

What is needed today?

An inherently-algorithmic theory of

markets and market equilibria




Beginnings of such a theory, within

Algorithmic Game Theory




Beginnings of such a theory, within

Algorithmic Game Theory

Natural starting point:

algorithms for traditional market models


An inherently-algorithmic theory of markets and market equilibria

Beginnings of such a theory, within Algorithmic Game Theory

Natural starting point: algorithms for traditional market models

New market models emerging!

Theory of algorithms

Interestingly enough, recent study of

markets has contributed handsomely to

this theory!

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


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

]1,0[

x

xuuij

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

]1,0[

x

xuuij

ijj iji

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

j

u

p

Bang-per-buck

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

i iu

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

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

j

j iji

ij i

j

ij iij

j

j p

j p x

u ui j

p m i

u ui j x

p m i

Therefore, buyer i buys from

only,

i.e., gets an optimal bundle

arg max ijj

j

u

p

Therefore, buyer i buys from

only,

i.e., gets an optimal bundle

Can prove that equilibrium prices

are unique!

arg max ijj

j

u

p

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

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

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

j

j iji

ij i

j

ij iij

j

j p

j p x

u ui j

p m i

u ui j x

p m i

e(i)

e(i)

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??

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!

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

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

arg max ijj

j

u

p

Max flow

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

infinite capacities

Max flow

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

p: equilibrium prices iff both cuts saturated

Two important considerations

The price of a good never exceeds

its equilibrium priceInvariant: s is a min-cut

Max flow

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

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

m p

buyers

goods

bang-per-buck edges

Balanced flow in N

m p

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

i

Balanced flow

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

Balanced flow


A flow that minimizes l2 norm of surplus vector.

Balanced flow



Must be a max-flow.

Balanced flow



Must be a max-flow.

All balanced flows have same surplus vector.

Balanced flow



Makes surpluses as equal as possible.

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

i

surplus(i) < surplus(j)

j

R:

Property 1

i

surplus(i) < surplus(j)

j

R:

Property 1

i

Circulation gives a more balanced flow.

j

R:

Property 1

Theorem: A max-flow is balanced iff

it satisfies Property 1.



w.r.t. a special allocation




w.r.t. a balanced flow in N


Pieces fit just right!

Balanced flows Invariant

Bang-per-buck

edgesTight sets

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( )

j

j iji

ij i

j

ij iij

j

j p

j p x

u ui j

p m i

u ui j x

p m i

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( ) ( )

j

j iji

ij i

j

ij ijij jiij

j

j p

j p x

u ui j

p m i

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

rate = utility/unit amount of j

amount of j

Differentiate

rate

amount of j


money spent on j

rate


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

rate

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, 1922-1995

Considered utility functions that are

a function of allocations and prices.

An unexpected fallout!!


A new kind of utility functionHappiness derived is




A new kind of utility functionHappiness derived is



Has applications in

Google’s AdWords Market!

A digression

AdWords Market

Created by search engine companiesGoogleYahoo!MSN

Multi-billion dollar market – and still growing!

Totally revolutionized advertising, especially

by small companies.

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?

AdWords Allocation Problem

Search Engine

Whose ad to put

How to maximize revenue?

LawyersRus.com

Sue.com

TaxHelper.com

asbestos

Search results

Ads

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




Efficiently computable

Easy to specify utilities

linear utility function: a business will

typically get only one type of query

throughout the day!



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 what are “good” allocations,

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

2

1

rate = utility/click

$60 $100

women’s clog

rate

2

4

rate = utility/click

$80 $100

money

rate

0

1

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!

Open

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

. .

:

: 1

: 0

ii

i ij ijj

iji

ij

m i u

s t

i u

j

ij

u xx

x

For spending constraint utilities,proof follows from algorithm,

and not a convex program!

Open

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

Open

utility

Piece-wise linear, concave

amount of j

ijf

rate


amount of j

Differentiate ijg

Start with arbitrary prices, adding up to

total money of buyers.

( ) ( )ij ijj

xh x g

p

rate

money spent on j


( ) ( )ij ijj

xh x g

p



Run algorithm on these utilities to get new prices.

( ) ( )ij ijj

xh x g

p




( ) ( )ij ijj

xh x g

p




Fixed points of this procedure are equilibrium

prices for piecewise linear, concave utilities!

( ) ( )ij ijj

xh x g

p

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 …

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