Algorithmic game theory and internet computing l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 163

Algorithmic Game Theory and Internet Computing PowerPoint PPT Presentation


  • 62 Views
  • Uploaded on
  • Presentation posted in: General

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 .

Download Presentation

Algorithmic Game Theory and Internet Computing

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

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


Markets and

the Primal-Dual Paradigm

Algorithmic Game Theoryand 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


  • 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

    • Google

    • 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


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


Supply-demand curves


Irving Fisher, 1891

  • Defined a fundamental

    market model


Utility function

utility

amount ofmilk


Utility function

utility

amount ofbread


Utility function

utility

amount ofcheese


Total utility of a bundle of goods

= Sum of utilities of individual goods


For given prices,


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

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


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


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


AllocationsPrices

Idea of algorithm

  • Iterations:

    execute primal & dual improvements


Two important considerations

  • 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


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


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)


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.


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.


Property 1

  • Theorem: A max-flow is balanced iff

    it satisfies Property 1.


Pieces fit just right!

Invariant

Balanced flows

Bang-per-buck

edges

Tight sets


How primal-dual schema is adaptedto nonlinear setting


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


Money-weighed geometric mean of utilities


Eisenberg-Gale Program, 1959


Eisenberg-Gale Program, 1959

prices pj


KKT conditions


  • 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!


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


KKT conditions

e(i)

e(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


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.


KKT conditions


KKT conditions


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 function

utility

amount ofj


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

amount ofj


PTAS for concave function

utility

amount ofj


Piecewise linear concave utility

  • 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


Spending constraint utility function

  • 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.


Satisfies weak gross substitutability!


Old pieces become more complex+ there are new pieces


But they still fit just right!


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!!


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!


A digression


The view 5 years ago: Relevant Search Results


Business world’s view now :

(as Advertisement companies)


So how does this work?

Bids for

different

keywords

Daily

Budgets


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


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!


Spending

constraint

utilities

AdWords

Market


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!


  • 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)


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


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?


  • Goel & Mehta, 2006:

    A small modification to the MSVV algorithm

    achieves 1 – 1/e competitive ratio!


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


Proof follows fromEisenberg-Gale Convex Program, 1959


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


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

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 …


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!


  • Login