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Markets and the Primal-Dual Paradigm . Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani. Markets. Stock Markets. Internet. Revolution in definition of markets. Revolution in definition of markets New markets defined by Google Amazon Yahoo! Ebay .

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Algorithmic game theory and internet computing l.jpg

Markets and

the Primal-Dual Paradigm

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani


Markets l.jpg

Markets


Stock markets l.jpg

Stock Markets


Internet l.jpg

Internet


Slide5 l.jpg

  • Revolution in definition of markets


Slide6 l.jpg

  • Revolution in definition of markets

  • New markets defined by

    • Google

    • Amazon

    • Yahoo!

    • Ebay


Slide7 l.jpg

  • Revolution in definition of markets

  • Massive computational power available

    for running these markets in a

    centralized or distributed manner


Slide8 l.jpg

  • Revolution in definition of markets

  • 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 l.jpg

Theory of Algorithms

  • Powerful tools and techniques

    developed over last 4 decades.


Theory of algorithms10 l.jpg

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 l.jpg

AdWords Market

  • Created by search engine companies

    • Google

    • Yahoo!

    • MSN

  • Multi-billion dollar market – and still growing!

  • Totally revolutionized advertising, especially

    by small companies.


Historically the study of markets l.jpg

Historically, the study of markets

  • has been of central importance,

    especially in the West


Historically the study of markets15 l.jpg

Historically, the study of markets

  • has been of central importance,

    especially in the West

General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century


Leon walras 1874 l.jpg

Leon Walras, 1874

  • Pioneered general

    equilibrium theory


Arrow debreu theorem 1954 l.jpg

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 l.jpg

Kenneth Arrow

  • Nobel Prize, 1972


Gerard debreu l.jpg

Gerard Debreu

  • Nobel Prize, 1983


General equilibrium theory l.jpg

General Equilibrium Theory

  • Also gave us some algorithmic results

    • Convex 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


General equilibrium theory21 l.jpg

General Equilibrium Theory

An almost entirely non-algorithmic theory!


What is needed today l.jpg

What is needed today?

  • An inherently algorithmictheory of

    market equilibrium

  • New models that capture new markets

    and are easier to use than traditional models


Slide23 l.jpg

  • Beginnings of such a theory, within

    Algorithmic Game Theory

  • Started with combinatorial algorithms

    for traditional market models

  • New market models emerging


A central tenet l.jpg

A central tenet

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

    equilibrium prices.


A central tenet25 l.jpg

A central tenet

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

    equilibrium prices.

  • Easy if only one good


Supply demand curves l.jpg

Supply-demand curves


Irving fisher 1891 l.jpg

Irving Fisher, 1891

  • Defined a fundamental

    market model


Slide30 l.jpg

Utility function

utility

amount ofmilk


Slide31 l.jpg

Utility function

utility

amount ofbread


Slide32 l.jpg

Utility function

utility

amount ofcheese


Total utility of a bundle of goods l.jpg

Total utility of a bundle of goods

= Sum of utilities of individual goods


For given prices l.jpg

For given prices,


For given prices find optimal bundle of goods l.jpg

For given prices,find optimal bundle of goods


Fisher market l.jpg

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 l.jpg

Combinatorial Algorithm for Linear Case of Fisher’s Model

  • Devanur, Papadimitriou, Saberi & V., 2002

    Using the primal-dual schema


Primal dual schema l.jpg

Primal-Dual Schema

  • Highly successful algorithm design

    technique from exact and

    approximation algorithms


Exact algorithms for cornerstone problems in p l.jpg

Exact Algorithms for Cornerstone Problems in P:

  • Matching (general graph)

  • Network flow

  • Shortest paths

  • Minimum spanning tree

  • Minimum branching


Approximation algorithms l.jpg

Approximation Algorithms

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .


Slide43 l.jpg

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


Slide44 l.jpg

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

  • Eisenberg-Gale convex program, 1959


Slide45 l.jpg

  • 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 l.jpg

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,


Fisher s model47 l.jpg

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


An easier question l.jpg

An easier question

  • Given prices p, are they equilibrium prices?

  • If so, find equilibrium allocations.


An easier question49 l.jpg

An easier question

  • Given prices p, are they equilibrium prices?

  • If so, find equilibrium allocations.

  • Equilibrium prices are unique!


Bang per buck l.jpg

Bang-per-buck

  • At prices p, buyer i’s most

    desirable goods, S =

  • Any goods from S worth m(i)

    constitute i’s optimal bundle


Slide51 l.jpg

m(1)

p(1)

m(2)

p(2)

m(3)

p(3)

m(4)

p(4)

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


Slide52 l.jpg

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

infinite capacities


Slide53 l.jpg

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

p: equilibrium prices iff both cuts saturated


Idea of algorithm l.jpg

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 algorithm55 l.jpg

AllocationsPrices

Idea of algorithm

  • Iterations:

    execute primal & dual improvements


Two important considerations l.jpg

Two important considerations

  • The price of a good never exceeds

    its equilibrium price

    • Invariant: s is a min-cut


Slide57 l.jpg

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

p: low prices


Two important considerations58 l.jpg

Two important considerations

  • The price of a good never exceeds

    its equilibrium price

    • Invariant: s is a min-cut

    • Identify tight sets of goods


Two important considerations59 l.jpg

Two important considerations

  • The price of a good never exceeds

    its equilibrium price

    • Invariant: s is a min-cut

    • Identify tight sets of goods

  • Rapid progress is made

    • Balanced flows


Slide60 l.jpg

Network N

buyers

p

m

bang-per-buck edges

goods


Slide61 l.jpg

Balanced flow in N

p

m

i

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


Balanced flow l.jpg

Balanced flow

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


Balanced flow63 l.jpg

Balanced flow

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

  • A flow that minimizes l2 norm of surplus vector.


Property 1 l.jpg

Property 1

  • f: max flow in N.

  • R: residual graph w.r.t. f.

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

    pathfrom i to j in R.


Slide65 l.jpg

Property 1

R:

i

j

surplus(i) < surplus(j)


Slide66 l.jpg

Property 1

R:

i

j

surplus(i) < surplus(j)


Slide67 l.jpg

Property 1

R:

i

j

Circulation gives a more balanced flow.


Property 168 l.jpg

Property 1

  • Theorem: A max-flow is balanced iff

    it satisfies Property 1.


Pieces fit just right l.jpg

Pieces fit just right!

Invariant

Balanced flows

Bang-per-buck

edges

Tight sets


How primal dual schema is adapted to nonlinear setting l.jpg

How primal-dual schema is adaptedto nonlinear setting


A convex program l.jpg

A convex program

  • whose optimal solution is equilibrium allocations.


A convex program72 l.jpg

A convex program

  • whose optimal solution is equilibrium allocations.

  • Constraints: packing constraints on the xij’s


A convex program73 l.jpg

A convex program

  • whose optimal solution is equilibrium allocations.

  • Constraints: packing constraints on the xij’s

  • Objective fn: max utilities derived.


A convex program74 l.jpg

A convex program

  • whose optimal solution is equilibrium allocations.

  • Constraints: packing constraints on the xij’s

  • Objective fn: max utilities derived. Must satisfy

    • If utilities of a buyer are scaled by a constant,

      optimal allocations remain unchanged

    • If 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 l.jpg

Money-weighed geometric mean of utilities


Eisenberg gale program 1959 l.jpg

Eisenberg-Gale Program, 1959


Eisenberg gale program 195977 l.jpg

Eisenberg-Gale Program, 1959

prices pj


Kkt conditions l.jpg

KKT conditions


Slide79 l.jpg

  • Therefore, buyer i buys from

    only,

    i.e., gets an optimal bundle


Slide80 l.jpg

  • Therefore, buyer i buys from

    only,

    i.e., gets an optimal bundle

  • Can prove that equilibrium prices

    are unique!


Will relax kkt conditions l.jpg

Will relax KKT conditions

  • e(i): money currently spent by i

    w.r.t. a special allocation

  • surplus money of i


Will relax kkt conditions82 l.jpg

Will relax KKT conditions

  • e(i): money currently spent by i

    w.r.t. a balanced flow in N

  • surplus money of i


Kkt conditions83 l.jpg

KKT conditions

e(i)

e(i)


Potential function l.jpg

Potential function

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).


Potential function85 l.jpg

Potential function

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).


Point of departure l.jpg

Point of departure

  • KKT conditions are satisfied via a

    continuous process

  • Normally: in discrete steps


Point of departure87 l.jpg

Point of departure

  • KKT conditions are satisfied via a

    continuous process

  • Normally: in discrete steps

  • Open question: strongly polynomial algorithm?


Another point of departure l.jpg

Another point of departure

  • Complementary slackness conditions:

    involve primal or dual variables, not both.

  • KKT conditions: involve primal and dual

    variables simultaneously.


Kkt conditions89 l.jpg

KKT conditions


Kkt conditions90 l.jpg

KKT conditions


Primal dual algorithms so far l.jpg

Primal-dual algorithms so far

  • Raise dual variables greedily. (Lot of effort spent

    on designing more sophisticated dual processes.)


Primal dual algorithms so far92 l.jpg

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.


Primal dual algorithms so far93 l.jpg

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.

  • Otherwise primal objects go tight and loose.

    Difficult to account for these reversals

    in the running time.


Our algorithm l.jpg

Our algorithm

  • Dual variables (prices) are raised greedily

  • Yet, primal objects go tight and loose

    • Because of enhanced KKT conditions


Deficiencies of linear utility functions l.jpg

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


Slide96 l.jpg

Concave utility function

utility

amount ofj


Concave utility functions l.jpg

Concave utility functions

  • Do not satisfy weak gross substitutability


Concave utility functions98 l.jpg

Concave utility functions

  • Do not satisfy weak gross substitutability

    • w.g.s. = Raising the price of one good cannot lead to a

      decrease in demand of another good.


Concave utility functions99 l.jpg

Concave utility functions

  • Do not satisfy weak gross substitutability

    • w.g.s. = Raising the price of one good cannot lead to a

      decrease in demand of another good.

  • Open problem:find polynomial time algorithm!


Slide100 l.jpg

Piecewise linear, concave

utility

amount ofj


Slide101 l.jpg

PTAS for concave function

utility

amount ofj


Piecewise linear concave utility l.jpg

Piecewise linear concave utility

  • Does not satisfy weak gross substitutability


Slide103 l.jpg

Piecewise linear, concave

utility

amount ofj


Slide104 l.jpg

rate = utility/unit amount of j

rate

amount ofj

Differentiate


Slide105 l.jpg

rate = utility/unit amount of j

rate

amount ofj

money spent on j


Slide106 l.jpg

Spending constraint utility function

rate = utility/unit amount of j

rate

$20

$40

$60

money spent onj


Spending constraint utility function l.jpg

Spending constraint utility function

  • Happiness derived is

    not a function of allocation only

    but also of amount of money spent.


Slide108 l.jpg

Extend model: assume buyers have utility for money

rate

$20

$40

$100


Slide110 l.jpg

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 l.jpg

Satisfies weak gross substitutability!


Old pieces become more complex there are new pieces l.jpg

Old pieces become more complex+ there are new pieces


But they still fit just right l.jpg

But they still fit just right!


Don patinkin 1956 l.jpg

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 l.jpg

An unexpected fallout!!


An unexpected fallout116 l.jpg

An unexpected fallout!!

  • A new kind of utility function

    • Happiness derived is

      not a function of allocation only

      but also of amount of money spent.


An unexpected fallout117 l.jpg

An unexpected fallout!!

  • A new kind of utility function

    • Happiness derived is

      not a function of allocation only

      but also of amount of money spent.

  • Has applications in

    Google’s AdWords Market!


A digression l.jpg

A digression


Slide119 l.jpg

The view 5 years ago: Relevant Search Results


Slide121 l.jpg

Business world’s view now :

(as Advertisement companies)


So how does this work l.jpg

So how does this work?

Bids for

different

keywords

Daily

Budgets


An interesting algorithmic question l.jpg

An interesting algorithmic question!

  • Monika Henzinger, 2004: Find an on-line

    algorithm that maximizes Google’s revenue.


Adwords allocation problem l.jpg

AdWords Allocation Problem

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com


Adwords problem l.jpg

AdWords Problem

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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


Adwords problem126 l.jpg

AdWords Problem

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

    Optimal!


Adwords problem127 l.jpg

AdWords Problem

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

    Optimal!


Slide128 l.jpg

Spending

constraint

utilities

AdWords

Market


Adwords market129 l.jpg

AdWords market

  • Assume that Google will determine equilibrium price/click for keywords


Adwords market130 l.jpg

AdWords market

  • Assume that Google will determine equilibrium price/click for keywords

  • How should advertisers specify their

    utility functions?


Choice of utility function l.jpg

Choice of utility function

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations


Choice of utility function132 l.jpg

Choice of utility function

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations

  • Efficiently computable


Choice of utility function133 l.jpg

Choice of utility function

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations

  • Efficiently computable

  • Easy to specify utilities


Slide134 l.jpg

  • linear utility function: a business will

    typically get only one type of query

    throughout the day!


Slide135 l.jpg

  • linear utility function: a business will

    typically get only one type of query

    throughout the day!

  • concave utility function: no efficient

    algorithm known!


Slide136 l.jpg

  • linear utility function: a business will

    typically get only one type of query

    throughout the day!

  • concave utility function: no efficient

    algorithm known!

    • Difficult for advertisers to

      define concave functions


Easier for a buyer l.jpg

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 l.jpg

Online shoe business

  • Interested in two keywords:

    • men’s clog

    • women’s clog

  • Advertising budget: $100/day

  • Expected profit:

    • men’s clog: $2/click

    • women’s clog: $4/click


Considerations for long term profit l.jpg

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


Slide140 l.jpg

  • 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)


Slide141 l.jpg

men’s clog

rate = utility/click

rate

2

1

$60

$100


Slide142 l.jpg

women’s clog

rate = utility/click

4

rate

2

$60

$100


Slide143 l.jpg

money

rate = utility/$

rate

1

0

$80

$100


Adwords market144 l.jpg

AdWords market

  • Suppose Google stays with auctions but

    allows advertisers to specify bids in

    the spending constraint model


Adwords market145 l.jpg

AdWords market

  • Suppose Google stays with auctions but

    allows advertisers to specify bids in

    the spending constraint model

    • expressivity!


Adwords market146 l.jpg

AdWords market

  • Suppose Google stays with auctions but

    allows advertisers to specify bids in

    the spending constraint model

    • expressivity!

  • Good online algorithm for

    maximizing Google’s revenues?


Slide147 l.jpg

  • Goel & Mehta, 2006:

    A small modification to the MSVV algorithm

    achieves 1 – 1/e competitive ratio!


Slide148 l.jpg

Open

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?


Spending constraint utilities satisfy l.jpg

Spending constraint utilities satisfy

  • Equilibrium exists (under mild conditions)

  • Equilibrium utilities and prices are unique

  • Rational

  • With small denominators


Linear utilities also satisfy l.jpg

Linear utilities also satisfy

  • Equilibrium exists (under mild conditions)

  • Equilibrium utilities and prices are unique

  • Rational

  • With small denominators


Proof follows from eisenberg gale convex program 1959 l.jpg

Proof follows fromEisenberg-Gale Convex Program, 1959


For spending constraint utilities proof follows from algorithm and not a convex program l.jpg

For spending constraint utilities,proof follows from algorithm, and not a convex program!


Slide153 l.jpg

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 l.jpg

Use spending constraint algorithm to solve piecewise linear, concave utilities

Open


Algorithms game theory common origins l.jpg

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 …


Slide157 l.jpg

Piece-wise linear, concave

utility

amount ofj


Slide158 l.jpg

rate = utility/unit amount of j

rate

amount ofj

Differentiate


Slide159 l.jpg

  • Start with arbitrary prices, adding up to

    total money of buyers.


Slide160 l.jpg

rate = utility/unit amount of j

rate

money spent on j


Slide161 l.jpg

  • Start with arbitrary prices, adding up to

    total money of buyers.

  • Run algorithm on these utilities to get new prices.


Slide162 l.jpg

  • Start with arbitrary prices, adding up to

    total money of buyers.

  • Run algorithm on these utilities to get new prices.


Slide163 l.jpg

  • Start with arbitrary prices, adding up to

    total money of buyers.

  • Run algorithm on these utilities to get new prices.

  • Fixed points of this procedure are equilibrium

    prices for piecewise linear, concave utilities!


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