- By
**oleg** - Follow User

- 91 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Algorithmic Game Theory and Internet Computing' - oleg

Download Now**An Image/Link below is provided (as is) to download presentation**

Download Now

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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.

- 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

General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century

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

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.

- Easy if only one good

Total utility of a bundle of goods

= Sum of utilities of individual 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

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

- 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

- 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

- 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!

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

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

Two important considerations

- The price of a good never exceeds

its equilibrium price

- Invariant: s is a min-cut

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 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

Balanced flow

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

Balanced flow

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

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.

A convex program

- whose optimal solution is equilibrium allocations.

A convex program

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

A convex program

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

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

Eisenberg-Gale Program, 1959

prices pj

Therefore, buyer i buys from

only,

i.e., gets an optimal bundle

- Can prove that equilibrium prices

are unique!

Will relax KKT conditions

- e(i): money currently spent by i

w.r.t. a special allocation

- surplus money of i

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

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Potential function

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).

Point of departure

- KKT conditions are satisfied via a

continuous process

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

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

- Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

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 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

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

Concave utility functions

- Do not satisfy weak gross substitutability

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 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!

Piecewise linear concave utility

- Does not satisfy weak gross substitutability

Spending constraint utility function

- Happiness derived is

not a function of allocation only

but also of amount of money spent.

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

- 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!!

- A new kind of utility function
- Happiness derived is

not a function of allocation only

but also of amount of money spent.

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!

(as Advertisement companies)

An interesting algorithmic question!

- Monika Henzinger, 2004: Find an on-line

algorithm that maximizes Google’s revenue.

AdWords Allocation Problem

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com

AdWords Problem

- Mehta, Saberi, Vazirani & Vazirani, 2005:

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

Optimal!

AdWords Problem

- Mehta, Saberi, Vazirani & Vazirani, 2005:

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

Optimal!

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

Choice of utility function

- Expressive enough that advertisers get

close to their ‘‘optimal’’ allocations

- Efficiently computable

Choice of utility function

- 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!

- 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

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

- 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

- 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)

AdWords market

- Suppose Google stays with auctions but

allows advertisers to specify bids in

the spending constraint model

AdWords market

- Suppose Google stays with auctions but

allows advertisers to specify bids in

the spending constraint model

- expressivity!

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?

Open

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?

Spending constraint utilities satisfy

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

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!

Use spending constraint algorithm to solve piecewise linear, concave utilities

Open

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 …

Start with arbitrary prices, adding up to

total money of buyers.

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!

Download Presentation

Connecting to Server..