algorithmic game theory and internet computing vijay v. vazirani markets and the primal-dual...
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Markets and
the Primal-Dual Paradigm
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The new face of computing
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A paradigm shift inthe notion of a “market”!
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Historically, the study of markets
has been of central importance,
especially in the West
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Historically, the study of markets
has been of central importance,
especially in the West
General Equilibrium TheoryOccupied center stage in Mathematical
Economics for over a century
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General Equilibrium Theory
Also gave us some algorithmic resultsConvex programs, whose optimal solutions capture
equilibrium allocations,
e.g., Eisenberg & Gale, 1959
Nenakov & Primak, 1983
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General Equilibrium Theory
Also gave us some algorithmic resultsConvex programs, whose optimal solutions capture equilibrium allocations,
e.g., Eisenberg & Gale, 1959 Nenakov & Primak, 1983
Scarf, 1973: Algorithms for approximately computing fixed points
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New markets defined by Internet companies, e.g., Google Yahoo! Amazon eBay
Massive computing power available for running markets in a distributed or centralized manner
A deep theory of algorithms with many powerful techniques
Today’s reality
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What is needed today?
An inherently-algorithmic theory of
markets and market equilibria
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What is needed today?
An inherently-algorithmic theory of
markets and market equilibria
Beginnings of such a theory, within
Algorithmic Game Theory
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What is needed today?
An inherently-algorithmic theory of
markets and market equilibria
Beginnings of such a theory, within
Algorithmic Game Theory
Natural starting point:
algorithms for traditional market models
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What is needed today?
An inherently-algorithmic theory of markets and market equilibria
Beginnings of such a theory, within Algorithmic Game Theory
Natural starting point: algorithms for traditional market models
New market models emerging!
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Theory of algorithms
Interestingly enough, recent study of
markets has contributed handsomely to
this theory!
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A central tenet
Prices are such that demand equals supply, i.e.,
equilibrium prices.
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A central tenet
Prices are such that demand equals supply, i.e.,
equilibrium prices.
Easy if only one good
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Supply-demand curves
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Irving Fisher, 1891
Defined a fundamental
market model
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utility
Utility function
amount of milk
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utility
Utility function
amount of bread
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utility
Utility function
amount of cheese
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Total utility of a bundle of goods
= Sum of utilities of individual goods
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For given prices,
1p 2p3p
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For given prices,find optimal bundle of goods
1p 2p3p
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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
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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using the primal-dual schema
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Primal-Dual Schema
Highly successful algorithm design
technique from exact and
approximation algorithms
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Exact Algorithms for Cornerstone Problems in P:
Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching
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Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
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No LP’s known for capturing equilibrium allocations for Fisher’s model
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No LP’s known for capturing equilibrium allocations for Fisher’s model
Eisenberg-Gale convex program, 1959
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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
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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,
i ij ijj
U u xiju
]1,0[
x
xuuij
ijj iji
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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
i ij ijj
U u xiju
]1,0[
x
xuuij
ijj iji
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At prices p, buyer i’s most
desirable goods, S =
Any goods from S worth m(i)
constitute i’s optimal bundle
arg max ijj
j
u
p
Bang-per-buck
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A convex program
whose optimal solution is equilibrium allocations.
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A convex program
whose optimal solution is equilibrium allocations.
Constraints: packing constraints on the xij’s
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A convex program
whose optimal solution is equilibrium allocations.
Constraints: packing constraints on the xxij’s
Objective fn: max utilities derived.
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A convex program
whose optimal solution is equilibrium allocations.
Constraints: packing constraints on the xxij’s
Objective fn: max utilities derived. Must satisfy
If utilities of a buyer are scaled by a constant,
optimal allocations remain unchangedIf 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
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Money-weighed geometric mean of utilities
1/ ( )( )( ) im im i
i iu
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Eisenberg-Gale Program, 1959
max ( ) log
. .
:
: 1
: 0
ii
i ij ijj
iji
ij
m i u
s t
i u
j
ij
u xx
x
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KKT conditions
1. : 0
2. : 0 1
3. , :( )
4. , : 0( )
j
j iji
ij i
j
ij iij
j
j p
j p x
u ui j
p m i
u ui j x
p m i
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Therefore, buyer i buys from
only,
i.e., gets an optimal bundle
arg max ijj
j
u
p
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Therefore, buyer i buys from
only,
i.e., gets an optimal bundle
Can prove that equilibrium prices
are unique!
arg max ijj
j
u
p
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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
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Idea of algorithm
Iterations:
execute primal & dual improvements
Allocations Prices
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Will relax KKT conditions
e(i): money currently spent by i
w.r.t. a special allocation
surplus money of i( ) ( )i m i e i
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KKT conditions
1. : 0
2. : 0 1
3. , :( )
4. , : 0( )
j
j iji
ij i
j
ij iij
j
j p
j p x
u ui j
p m i
u ui j x
p m i
e(i)
e(i)
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Potential function
2 2 21 2 ... n
Will show that potential drops by an inverse polynomial
factor in each phase (polynomial time).
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Potential function
2 2 21 2 ... n
Will show that potential drops by an inverse polynomial
factor in each phase (polynomial time).( ( ))
ipoly m i
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Point of departure
KKT conditions are satisfied via a
continuous process Normally: in discrete steps
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Point of departure
KKT conditions are satisfied via a
continuous process Normally: in discrete steps
Open question: strongly polynomial algorithm??
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An easier question
Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.
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An easier question
Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.
Equilibrium prices are unique!
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m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
For each buyer, most desirable goods, i.e.
arg max ijj
j
u
p
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Max flow
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
infinite capacities
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Max flow
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: equilibrium prices iff both cuts saturated
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Two important considerations
The price of a good never exceeds
its equilibrium priceInvariant: s is a min-cut
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Max flow
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: low prices
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Two important considerations
The price of a good never exceeds
its equilibrium priceInvariant: s is a min-cut
Identify tight sets of goods
: ( ) ( ( ))S A p S m S
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Two important considerations
The price of a good never exceeds
its equilibrium priceInvariant: s is a min-cutIdentify tight sets of goods
Rapid progress is madeBalanced flows
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Network N
m p
buyers
goods
bang-per-buck edges
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Balanced flow in N
m p
W.r.t. flow f, surplus(i) = m(i) – f(i,t)
i
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
A flow that minimizes l2 norm of surplus vector.
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
A flow that minimizes l2 norm of surplus vector.
Must be a max-flow.
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
A flow that minimizes l2 norm of surplus vector.
Must be a max-flow.
All balanced flows have same surplus vector.
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
A flow that minimizes l2 norm of surplus vector.
Makes surpluses as equal as possible.
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Property 1
f: max flow in N.
R: residual graph w.r.t. f.
If surplus (i) < surplus(j) then there is no
path from i to j in R.
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Property 1
i
surplus(i) < surplus(j)
j
R:
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Property 1
i
surplus(i) < surplus(j)
j
R:
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Property 1
i
Circulation gives a more balanced flow.
j
R:
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Property 1
Theorem: A max-flow is balanced iff
it satisfies Property 1.
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Will relax KKT conditions
e(i): money currently spent by i
w.r.t. a special allocation
surplus money of i( ) ( )i m i e i
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Will relax KKT conditions
e(i): money currently spent by i
w.r.t. a balanced flow in N
surplus money of i( ) ( )i m i e i
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Pieces fit just right!
Balanced flows Invariant
Bang-per-buck
edgesTight sets
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Another point of departure
Complementary slackness conditions:
involve primal or dual variables, not both.
KKT conditions: involve primal and dual
variables simultaneously.
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KKT conditions
1. : 0
2. : 0 1
3. , :( )
4. , : 0( )
j
j iji
ij i
j
ij iij
j
j p
j p x
u ui j
p m i
u ui j x
p m i
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KKT conditions
1. : 0
2. : 0 1
3. , :( )
4. , : 0( ) ( )
j
j iji
ij i
j
ij ijij jiij
j
j p
j p x
u ui j
p m i
u xu ui j x
p m i m i
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Primal-dual algorithms so far
Raise dual variables greedily. (Lot of effort spent
on designing more sophisticated dual processes.)
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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.
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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.
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Our algorithm
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
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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
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utility
Concave utility function
amount of j
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Concave utility functions
Do not satisfy weak gross substitutability
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Concave utility functions
Do not satisfy weak gross substitutabilityw.g.s. = Raising the price of one good cannot lead to a
decrease in demand of another good.
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Concave utility functions
Do not satisfy weak gross substitutabilityw.g.s. = Raising the price of one good cannot lead to a
decrease in demand of another good.
Open problem: find polynomial time algorithm!
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utility
Piecewise linear, concave
amount of j
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utility
PTAS for concave function
amount of j
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Piecewise linear concave utility
Does not satisfy weak gross substitutability
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utility
Piecewise linear, concave
amount of j
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rate
rate = utility/unit amount of j
amount of j
Differentiate
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rate
amount of j
rate = utility/unit amount of j
money spent on j
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rate
rate = utility/unit amount of j
money spent on j
Spending constraint utility function
$20 $40 $60
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Spending constraint utility function
Happiness derived is
not a function of allocation only
but also of amount of money spent.
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$20 $40 $100
Extend model: assume buyers have utility for money
rate
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Theorem: Polynomial time algorithm for
computing equilibrium prices and allocations for
Fisher’s model with spending constraint utilities.
Furthermore, equilibrium prices are unique.
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Satisfies weak gross substitutability!
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Old pieces become more complex+ there are new pieces
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But they still fit just right!
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Don Patinkin, 1922-1995
Considered utility functions that are
a function of allocations and prices.
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An unexpected fallout!!
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An unexpected fallout!!
A new kind of utility functionHappiness derived is
not a function of allocation only
but also of amount of money spent.
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An unexpected fallout!!
A new kind of utility functionHappiness derived is
not a function of allocation only
but also of amount of money spent.
Has applications in
Google’s AdWords Market!
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A digression
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AdWords Market
Created by search engine companiesGoogleYahoo!MSN
Multi-billion dollar market – and still growing!
Totally revolutionized advertising, especially
by small companies.
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The view 5 years ago: Relevant Search Results
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Business world’s view now :
(as Advertisement companies)
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Bids for different keywords
DailyBudgets
So how does this work?
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AdWords Allocation Problem
Search Engine
Whose ad to put
How to maximize revenue?
LawyersRus.com
Sue.com
TaxHelper.com
asbestos
Search results
Ads
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AdWords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids
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AdWords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids
Optimal!
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AdWords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids
Optimal!
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Spending
constraint
utilities
AdWords
Market
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AdWords market
Assume that Google will determine equilibrium price/click for keywords
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AdWords market
Assume that Google will determine equilibrium price/click for keywords
How should advertisers specify their
utility functions?
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Choice of utility function
Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations
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Choice of utility function
Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations
Efficiently computable
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Choice of utility function
Expressive enough that advertisers get
close to their ‘‘optimal’’ allocations
Efficiently computable
Easy to specify utilities
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linear utility function: a business will
typically get only one type of query
throughout the day!
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linear utility function: a business will
typically get only one type of query
throughout the day!
concave utility function: no efficient
algorithm known!
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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
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Easier for a buyer
To say what are “good” allocations,
for a range of prices,
rather than how happy she is
with a given bundle.
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Online shoe business
Interested in two keywords: men’s clog women’s clog
Advertising budget: $100/day
Expected profit:men’s clog: $2/clickwomen’s clog: $4/click
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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
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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)
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$60 $100
men’s clog
rate
2
1
rate = utility/click
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$60 $100
women’s clog
rate
2
4
rate = utility/click
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$80 $100
money
rate
0
1
rate = utility/$
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AdWords market
Suppose Google stays with auctions but
allows advertisers to specify bids in
the spending constraint model
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AdWords market
Suppose Google stays with auctions but
allows advertisers to specify bids in
the spending constraint model expressivity!
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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?
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Goel & Mehta, 2006:
A small modification to the MSVV algorithm
achieves 1 – 1/e competitive ratio!
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Open
Is there a convex program that
captures equilibrium allocations for
spending constraint utilities?
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Equilibrium exists (under mild conditions)
Equilibrium utilities and prices are unique
Rational
With small denominators
Spending constraint utilities satisfy
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Equilibrium exists (under mild conditions)
Equilibrium utilities and prices are unique
Rational
With small denominators
Linear utilities also satisfy
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Proof follows fromEisenberg-Gale Convex Program, 1959
max ( ) log
. .
:
: 1
: 0
ii
i ij ijj
iji
ij
m i u
s t
i u
j
ij
u xx
x
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For spending constraint utilities,proof follows from algorithm,
and not a convex program!
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Open
Is there an LP whose optimal solutions
capture equilibrium allocations
for Fisher’s linear case?
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Use spending constraint algorithm to solve
piecewise linear, concave utilities
Open
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utility
Piece-wise linear, concave
amount of j
ijf
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rate
rate = utility/unit amount of j
amount of j
Differentiate ijg
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Start with arbitrary prices, adding up to
total money of buyers.
( ) ( )ij ijj
xh x g
p
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rate
money spent on j
rate = utility/unit amount of j
( ) ( )ij ijj
xh x g
p
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Start with arbitrary prices, adding up to
total money of buyers.
Run algorithm on these utilities to get new prices.
( ) ( )ij ijj
xh x g
p
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Start with arbitrary prices, adding up to
total money of buyers.
Run algorithm on these utilities to get new prices.
( ) ( )ij ijj
xh x g
p
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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!
( ) ( )ij ijj
xh x g
p
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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 …
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