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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

Markets and

the Primal-Dual Paradigm

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

slide6
Revolution in definition of markets
  • New markets defined by
    • Google
    • Amazon
    • Yahoo!
    • Ebay
slide7
Revolution in definition of markets
  • Massive computational power available

for running these markets in a

centralized or distributed manner

slide8
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
Theory of Algorithms
  • Powerful tools and techniques

developed over last 4 decades.

theory of algorithms10
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
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
Historically, the study of markets
  • has been of central importance,

especially in the West

historically the study of markets15
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
Leon Walras, 1874
  • Pioneered general

equilibrium theory

arrow debreu theorem 1954
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
Kenneth Arrow
  • Nobel Prize, 1972
gerard debreu
Gerard Debreu
  • Nobel Prize, 1983
general equilibrium theory
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
General Equilibrium Theory

An almost entirely non-algorithmic theory!

what is needed today
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
Beginnings of such a theory, within

Algorithmic Game Theory

  • Started with combinatorial algorithms

for traditional market models

  • New market models emerging
a central tenet
A central tenet
  • Prices are such that demand equals supply, i.e.,

equilibrium prices.

a central tenet25
A central tenet
  • Prices are such that demand equals supply, i.e.,

equilibrium prices.

  • Easy if only one good
irving fisher 1891
Irving Fisher, 1891
  • Defined a fundamental

market model

slide30

Utility function

utility

amount ofmilk

slide31

Utility function

utility

amount ofbread

slide32

Utility function

utility

amount ofcheese

total utility of a bundle of goods
Total utility of a bundle of goods

= Sum of utilities of individual goods

fisher market
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
Combinatorial Algorithm for Linear Case of Fisher’s Model
  • Devanur, Papadimitriou, Saberi & V., 2002

Using the primal-dual schema

primal dual schema
Primal-Dual Schema
  • Highly successful algorithm design

technique from exact and

approximation algorithms

exact algorithms for cornerstone problems in p
Exact Algorithms for Cornerstone Problems in P:
  • Matching (general graph)
  • Network flow
  • Shortest paths
  • Minimum spanning tree
  • Minimum branching
approximation algorithms
Approximation Algorithms

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .

slide44
No LP’s known for capturing equilibrium allocations for Fisher’s model
  • Eisenberg-Gale convex program, 1959
slide45
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
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
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
An easier question
  • Given prices p, are they equilibrium prices?
  • If so, find equilibrium allocations.
an easier question49
An easier question
  • Given prices p, are they equilibrium prices?
  • If so, find equilibrium allocations.
  • Equilibrium prices are unique!
bang per buck
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

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

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

infinite capacities

slide53

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
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

AllocationsPrices

Idea of algorithm
  • Iterations:

execute primal & dual improvements

two important considerations
Two important considerations
  • The price of a good never exceeds

its equilibrium price

    • Invariant: s is a min-cut
slide57

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

p: low prices

two important considerations58
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
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

Network N

buyers

p

m

bang-per-buck edges

goods

slide61

Balanced flow in N

p

m

i

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

balanced flow
Balanced flow
  • surplus vector: vector of surpluses w.r.t. f.
balanced flow63
Balanced flow
  • surplus vector: vector of surpluses w.r.t. f.
  • A flow that minimizes l2 norm of surplus vector.
property 1
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

Property 1

R:

i

j

surplus(i) < surplus(j)

slide66

Property 1

R:

i

j

surplus(i) < surplus(j)

slide67

Property 1

R:

i

j

Circulation gives a more balanced flow.

property 168
Property 1
  • Theorem: A max-flow is balanced iff

it satisfies Property 1.

pieces fit just right
Pieces fit just right!

Invariant

Balanced flows

Bang-per-buck

edges

Tight sets

a convex program
A convex program
  • whose optimal solution is equilibrium allocations.
a convex program72
A convex program
  • whose optimal solution is equilibrium allocations.
  • Constraints: packing constraints on the xij’s
a convex program73
A convex program
  • whose optimal solution is equilibrium allocations.
  • Constraints: packing constraints on the xij’s
  • Objective fn: max utilities derived.
a convex program74
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

slide79
Therefore, buyer i buys from

only,

i.e., gets an optimal bundle

slide80
Therefore, buyer i buys from

only,

i.e., gets an optimal bundle

  • Can prove that equilibrium prices

are unique!

will relax kkt conditions
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
Will relax KKT conditions
  • e(i): money currently spent by i

w.r.t. a balanced flow in N

  • surplus money of i
potential function
Potential function

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

potential function85
Potential function

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

point of departure
Point of departure
  • KKT conditions are satisfied via a

continuous process

  • Normally: in discrete steps
point of departure87
Point of departure
  • KKT conditions are satisfied via a

continuous process

  • Normally: in discrete steps
  • Open question: strongly polynomial algorithm?
another point of departure
Another point of departure
  • Complementary slackness conditions:

involve primal or dual variables, not both.

  • KKT conditions: involve primal and dual

variables simultaneously.

primal dual algorithms so far
Primal-dual algorithms so far
  • Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

primal dual algorithms so far92
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
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
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
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

Concave utility function

utility

amount ofj

concave utility functions
Concave utility functions
  • Do not satisfy weak gross substitutability
concave utility functions98
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
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

Piecewise linear, concave

utility

amount ofj

slide101

PTAS for concave function

utility

amount ofj

piecewise linear concave utility
Piecewise linear concave utility
  • Does not satisfy weak gross substitutability
slide103

Piecewise linear, concave

utility

amount ofj

slide104

rate = utility/unit amount of j

rate

amount ofj

Differentiate

slide105

rate = utility/unit amount of j

rate

amount ofj

money spent on j

slide106

Spending constraint utility function

rate = utility/unit amount of j

rate

$20

$40

$60

money spent onj

spending constraint utility function
Spending constraint utility function
  • Happiness derived is

not a function of allocation only

but also of amount of money spent.

slide110
Theorem: Polynomial time algorithm for

computing equilibrium prices and allocations for

Fisher’s model with spending constraint utilities.

Furthermore, equilibrium prices are unique.

don patinkin 1956
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 fallout116
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
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!

slide121

Business world’s view now :

(as Advertisement companies)

so how does this work
So how does this work?

Bids for

different

keywords

Daily

Budgets

an interesting algorithmic question
An interesting algorithmic question!
  • Monika Henzinger, 2004: Find an on-line

algorithm that maximizes Google’s revenue.

adwords allocation problem
AdWords Allocation Problem

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com

adwords problem
AdWords Problem
  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

adwords problem126
AdWords Problem
  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

Optimal!

adwords problem127
AdWords Problem
  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

Optimal!

slide128

Spending

constraint

utilities

AdWords

Market

adwords market129
AdWords market
  • Assume that Google will determine equilibrium price/click for keywords
adwords market130
AdWords market
  • Assume that Google will determine equilibrium price/click for keywords
  • How should advertisers specify their

utility functions?

choice of utility function
Choice of utility function
  • Expressive enough that advertisers get

close to their ‘‘optimal’’ allocations

choice of utility function132
Choice of utility function
  • Expressive enough that advertisers get

close to their ‘‘optimal’’ allocations

  • Efficiently computable
choice of utility function133
Choice of utility function
  • Expressive enough that advertisers get

close to their ‘‘optimal’’ allocations

  • Efficiently computable
  • Easy to specify utilities
slide134
linear utility function: a business will

typically get only one type of query

throughout the day!

slide135
linear utility function: a business will

typically get only one type of query

throughout the day!

  • concave utility function: no efficient

algorithm known!

slide136
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
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
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
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
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

men’s clog

rate = utility/click

rate

2

1

$60

$100

slide142

women’s clog

rate = utility/click

4

rate

2

$60

$100

slide143

money

rate = utility/$

rate

1

0

$80

$100

adwords market144
AdWords market
  • Suppose Google stays with auctions but

allows advertisers to specify bids in

the spending constraint model

adwords market145
AdWords market
  • Suppose Google stays with auctions but

allows advertisers to specify bids in

the spending constraint model

    • expressivity!
adwords market146
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
Goel & Mehta, 2006:

A small modification to the MSVV algorithm

achieves 1 – 1/e competitive ratio!

slide148
Open

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?

spending constraint utilities satisfy
Spending constraint utilities satisfy
  • Equilibrium exists (under mild conditions)
  • Equilibrium utilities and prices are unique
  • Rational
  • With small denominators
linear utilities also satisfy
Linear utilities also satisfy
  • Equilibrium exists (under mild conditions)
  • Equilibrium utilities and prices are unique
  • Rational
  • With small denominators
for spending constraint utilities proof follows from algorithm and not a convex program
For spending constraint utilities,proof follows from algorithm, and not a convex program!
slide153
Open

Is there an LP whose optimal solutions

capture equilibrium allocations

for Fisher’s linear case?

algorithms game theory common origins
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 …
slide158

rate = utility/unit amount of j

rate

amount ofj

Differentiate

slide161
Start with arbitrary prices, adding up to

total money of buyers.

  • Run algorithm on these utilities to get new prices.
slide162
Start with arbitrary prices, adding up to

total money of buyers.

  • Run algorithm on these utilities to get new prices.
slide163
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!