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Algorithmic Game Theory and Internet Computing

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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 Theoryand Internet Computing

the Primal-Dual Paradigm

Vijay V. Vazirani

- Revolution in definition of markets
- New markets defined by
- Amazon
- Yahoo!
- Ebay

- Revolution in definition of markets
- Massive computational power available
for running these markets in a

centralized or distributed manner

- 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

- 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 companies
- Yahoo!
- 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 MathematicalEconomics 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 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

- Convex programs, whose optimal solutions capture

General Equilibrium Theory

An almost entirely non-algorithmic theory!

What is needed today?

- An inherently algorithmictheory 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

Irving Fisher, 1891

- Defined a fundamental
market model

Total utility of a bundle of goods

= Sum of utilities of individual goods

For given prices,find optimal bundle of goods

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 in P:

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

- 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 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 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,
- Find market clearing prices

An easier question Fisher’s model

- Given prices p, are they equilibrium prices?
- If so, find equilibrium allocations.

An easier question Fisher’s model

- Given prices p, are they equilibrium prices?
- If so, find equilibrium allocations.
- Equilibrium prices are unique!

Bang-per-buck Fisher’s model

- At prices p, buyer i’s most
desirable goods, S =

- Any goods from S worth m(i)
constitute i’s optimal bundle

Max flow Fisher’s model

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 Fisher’s model

- “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

Two important considerations Fisher’s model

- The price of a good never exceeds
its equilibrium price

- Invariant: s is a min-cut

Two important considerations Fisher’s model

- The price of a good never exceeds
its equilibrium price

- Invariant: s is a min-cut
- Identify tight sets of goods

Two important considerations Fisher’s model

- 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

Balanced flow Fisher’s model

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

Balanced flow Fisher’s model

- surplus vector: vector of surpluses w.r.t. f.
- A flow that minimizes l2 norm of surplus vector.

Property 1 Fisher’s model

- 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.

Property 1 Fisher’s model

- Theorem: A max-flow is balanced iff
it satisfies Property 1.

How primal-dual schema is adapted Fisher’s modelto nonlinear setting

A convex program Fisher’s model

- whose optimal solution is equilibrium allocations.

A convex program Fisher’s model

- whose optimal solution is equilibrium allocations.
- Constraints: packing constraints on the xij’s

A convex program Fisher’s model

- whose optimal solution is equilibrium allocations.
- Constraints: packing constraints on the xij’s
- Objective fn: max utilities derived.

A convex program Fisher’s model

- 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

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

Money-weighed geometric mean Fisher’s modelof utilities

Eisenberg-Gale Program, 1959 Fisher’s model

Eisenberg-Gale Program, 1959 Fisher’s model

prices pj

KKT conditions Fisher’s model

- Therefore, buyer Fisher’s model i buys from
only,

i.e., gets an optimal bundle

- Therefore, buyer Fisher’s model i buys from
only,

i.e., gets an optimal bundle

- Can prove that equilibrium prices
are unique!

Will relax KKT conditions Fisher’s model

- e(i): money currently spent by i
w.r.t. a special allocation

- surplus money of i

Will relax KKT conditions Fisher’s model

- e(i): money currently spent by i
w.r.t. a balanced flow in N

- surplus money of i

Potential function Fisher’s model

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Potential function Fisher’s model

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Point of departure Fisher’s model

- KKT conditions are satisfied via a
continuous process

- Normally: in discrete steps

Point of departure Fisher’s model

- KKT conditions are satisfied via a
continuous process

- Normally: in discrete steps
- Open question: strongly polynomial algorithm?

Another point of departure Fisher’s model

- Complementary slackness conditions:
involve primal or dual variables, not both.

- KKT conditions: involve primal and dual
variables simultaneously.

KKT conditions Fisher’s model

KKT conditions Fisher’s model

Primal-dual algorithms so far Fisher’s model

- Raise dual variables greedily. (Lot of effort spent
on designing more sophisticated dual processes.)

Primal-dual algorithms so far Fisher’s model

- 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

Primal-dual algorithms so far Fisher’s model

- 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
- Otherwise primal objects go tight and loose.
Difficult to account for these reversals

in the running time.

Our algorithm Fisher’s model

- Dual variables (prices) are raised greedily
- Yet, primal objects go tight and loose
- Because of enhanced KKT conditions

Deficiencies of linear utility functions Fisher’s model

- Typically, a buyer spends all her money
on a single good

- Do not model the fact that buyers get
satiated with goods

Concave utility functions Fisher’s model

- Do not satisfy weak gross substitutability

Concave utility functions Fisher’s model

- 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.

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

Concave utility functions Fisher’s model

- 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.

- w.g.s. = Raising the price of one good cannot lead to a
- Open problem:find polynomial time algorithm!

Piecewise linear concave utility Fisher’s model

- Does not satisfy weak gross substitutability

Spending constraint utility function Fisher’s model

rate = utility/unit amount of j

rate

$20

$40

$60

money spent onj

Spending constraint utility function Fisher’s model

- Happiness derived is
not a function of allocation only

but also of amount of money spent.

Theorem: Fisher’s model 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! Fisher’s model

Old pieces become more complex Fisher’s model+ there are new pieces

But they still fit just right! Fisher’s model

Don Patinkin, 1956 Fisher’s model

- 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

- Euro. J. History of Economic Thought,

An unexpected fallout!! Fisher’s model

An unexpected fallout!! Fisher’s model

- A new kind of utility function
- Happiness derived is
not a function of allocation only

but also of amount of money spent.

- Happiness derived is

An unexpected fallout!! Fisher’s model

- A new kind of utility function
- Happiness derived is
not a function of allocation only

but also of amount of money spent.

- Happiness derived is
- Has applications in
Google’s AdWords Market!

A digression Fisher’s model

The view 5 years ago: Relevant Search Results Fisher’s model

Business world’s view now : Fisher’s model

(as Advertisement companies)

An interesting algorithmic question! Fisher’s model

- Monika Henzinger, 2004: Find an on-line
algorithm that maximizes Google’s revenue.

AdWords Allocation Problem Fisher’s model

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com

AdWords Problem Fisher’s model

- Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids

AdWords Problem Fisher’s model

- Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids

Optimal!

AdWords Problem Fisher’s model

- Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids

Optimal!

AdWords market Fisher’s model

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

AdWords market Fisher’s model

- Assume that Google will determine equilibrium price/click for keywords
- How should advertisers specify their
utility functions?

Choice of utility function Fisher’s model

- Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations

Choice of utility function Fisher’s model

- Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations

- Efficiently computable

Choice of utility function Fisher’s model

- Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations

- Efficiently computable
- Easy to specify utilities

- linear utility function: Fisher’s model a business will
typically get only one type of query

throughout the day!

- linear utility function: Fisher’s model a business will
typically get only one type of query

throughout the day!

- concave utility function: no efficient
algorithm known!

- linear utility function: Fisher’s model 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

- Difficult for advertisers to

Easier for a buyer Fisher’s model

- 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 Fisher’s model

- 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 Fisher’s model

- 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, Fisher’s model
- 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)

AdWords market Fisher’s model

- Suppose Google stays with auctions but
allows advertisers to specify bids in

the spending constraint model

AdWords market Fisher’s model

- Suppose Google stays with auctions but
allows advertisers to specify bids in

the spending constraint model

- expressivity!

AdWords market Fisher’s model

- 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?

- Goel & Mehta, 2006: Fisher’s model
A small modification to the MSVV algorithm

achieves 1 – 1/e competitive ratio!

Open Fisher’s model

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?

Spending constraint utilities satisfy Fisher’s model

- Equilibrium exists (under mild conditions)
- Equilibrium utilities and prices are unique
- Rational
- With small denominators

Linear utilities also satisfy Fisher’s model

- Equilibrium exists (under mild conditions)
- Equilibrium utilities and prices are unique
- Rational
- With small denominators

Proof follows from Fisher’s modelEisenberg-Gale Convex Program, 1959

For spending constraint utilities, Fisher’s modelproof follows from algorithm, and not a convex program!

Open Fisher’s model

Is there an LP whose optimal solutions

capture equilibrium allocations

for Fisher’s linear case?

Use spending constraint algorithm Fisher’s model to solve piecewise linear, concave utilities

Open

Algorithms & Game Theory Fisher’s modelcommon 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 …

- Start with arbitrary prices, adding up to Fisher’s model
total money of buyers.

- Start with arbitrary prices, adding up to Fisher’s model
total money of buyers.

- Run algorithm on these utilities to get new prices.

- Start with arbitrary prices, adding up to Fisher’s model
total money of buyers.

- Run algorithm on these utilities to get new prices.

- Start with arbitrary prices, adding up to Fisher’s model
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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