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Algorithmic Game Theory and Internet Computing

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Markets and

the Primal-Dual Paradigm

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

- Revolution in definition of markets

- 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

- Powerful tools and techniques
developed over last 4 decades.

- Powerful tools and techniques
developed over last 4 decades.

- Recent study of markets has contributed
handsomely to this theory as well!

- Created by search engine companies
- Yahoo!
- MSN

- Multi-billion dollar market – and still growing!
- Totally revolutionized advertising, especially
by small companies.

- has been of central importance,
especially in the West

- has been of central importance,
especially in the West

General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century

- Pioneered general
equilibrium theory

- Celebrated theorem in Mathematical Economics
- Established existence of market equilibrium under
very general conditions using a deep theorem from

topology - Kakutani fixed point theorem.

- Nobel Prize, 1972

- Nobel Prize, 1983

- 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

An almost entirely non-algorithmic theory!

- 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

- Prices are such that demand equals supply, i.e.,
equilibrium prices.

- Prices are such that demand equals supply, i.e.,
equilibrium prices.

- Easy if only one good

- Defined a fundamental
market model

Utility function

utility

amount ofmilk

Utility function

utility

amount ofbread

Utility function

utility

amount ofcheese

= Sum of utilities of individual goods

- 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

- Devanur, Papadimitriou, Saberi & V., 2002
Using the primal-dual schema

- Highly successful algorithm design
technique from exact and

approximation algorithms

- Matching (general graph)
- Network flow
- Shortest paths
- Minimum spanning tree
- Minimum branching

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

- 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,

- 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

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

- 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

m(1)

p(1)

m(2)

p(2)

m(3)

p(3)

m(4)

p(4)

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

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

infinite capacities

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

- “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

AllocationsPrices

- Iterations:
execute primal & dual improvements

- The price of a good never exceeds
its equilibrium price

- Invariant: s is a min-cut

Max flow

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

p: low prices

- The price of a good never exceeds
its equilibrium price

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

- 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

Network N

buyers

p

m

bang-per-buck edges

goods

Balanced flow in N

p

m

i

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

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

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

- 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

R:

i

j

surplus(i) < surplus(j)

Property 1

R:

i

j

surplus(i) < surplus(j)

Property 1

R:

i

j

Circulation gives a more balanced flow.

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

Invariant

Balanced flows

Bang-per-buck

edges

Tight sets

- whose optimal solution is equilibrium allocations.

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

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

- 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,

prices pj

- Therefore, buyer i buys from
only,

i.e., gets an optimal bundle

- Therefore, buyer i buys from
only,

i.e., gets an optimal bundle

- Can prove that equilibrium prices
are unique!

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

- surplus money of i

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

- surplus money of i

e(i)

e(i)

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

- KKT conditions are satisfied via a
continuous process

- Normally: in discrete steps

- KKT conditions are satisfied via a
continuous process

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

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

- KKT conditions: involve primal and dual
variables simultaneously.

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

- 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

- 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.

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

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

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

Concave utility function

utility

amount ofj

- Do not satisfy weak gross substitutability

- 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

- 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

amount ofj

PTAS for concave function

utility

amount ofj

- Does not satisfy weak gross substitutability

Piecewise linear, concave

utility

amount ofj

rate = utility/unit amount of j

rate

amount ofj

Differentiate

rate = utility/unit amount of j

rate

amount ofj

money spent on j

Spending constraint utility function

rate = utility/unit amount of j

rate

$20

$40

$60

money spent onj

- Happiness derived is
not a function of allocation only

but also of amount of money spent.

Extend model: assume buyers have utility for money

rate

$20

$40

$100

Theorem: Polynomial time algorithm for

computing equilibrium prices and allocations for

Fisher’s model with spending constraint utilities.

Furthermore, equilibrium prices are unique.

- 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,

- 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

- 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!

The view 5 years ago: Relevant Search Results

Business world’s view now :

(as Advertisement companies)

Bids for

different

keywords

Daily

Budgets

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

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com

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

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

Optimal!

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

Optimal!

Spending

constraint

utilities

AdWords

Market

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

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

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

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

- Efficiently computable

- 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!

- linear utility function: a business will
typically get only one type of query

throughout the day!

- concave utility function: no efficient
algorithm known!

- 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

- Difficult for advertisers to

- 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.

- 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

- 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)

men’s clog

rate = utility/click

rate

2

1

$60

$100

women’s clog

rate = utility/click

4

rate

2

$60

$100

money

rate = utility/$

rate

1

0

$80

$100

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

the spending constraint model

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

the spending constraint model

- expressivity!

- 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:
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

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

Is there an LP whose optimal solutions

capture equilibrium allocations

for Fisher’s linear case?

Open

- 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 …

Piece-wise linear, concave

utility

amount ofj

rate = utility/unit amount of j

rate

amount ofj

Differentiate

- Start with arbitrary prices, adding up to
total money of buyers.

rate = utility/unit amount of j

rate

money spent on j

- Start with arbitrary prices, adding up to
total money of buyers.

- Run algorithm on these utilities to get new prices.

- Start with arbitrary prices, adding up to
total money of buyers.

- Run algorithm on these utilities to get new prices.

- 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!