Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
Markets and the Primal-Dual Paradigm

• Vijay V. Vazirani

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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 MathematicalEconomics for over a century
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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

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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
• Scarf, 1973 Algorithms for approximately
  computing
• fixed points

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Todays reality

• 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

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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 function
utility
amount of milk
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Utility function
utility
amount of bread
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Utility function
utility
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,
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For given prices,find optimal bundle of goods
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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
Fishers 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 LPs known for capturing equilibrium
  allocations for Fishers model

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• No LPs known for capturing equilibrium
  allocations for Fishers model
• Eisenberg-Gale convex program, 1959

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• No LPs known for capturing equilibrium
  allocations for Fishers model
• Eisenberg-Gale convex program, 1959
• DPSV Extended primal-dual schema to
• solving a nonlinear convex
  program

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Fishers 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,

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Fishers 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

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Bang-per-buck

• At prices p, buyer is most
• desirable goods, S
• Any goods from S worth m(i)
• constitute is optimal bundle

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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 xijs

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A convex program

• whose optimal solution is equilibrium
  allocations.
• Constraints packing constraints on the xijs
• Objective fn max utilities derived.

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A convex program

• whose optimal solution is equilibrium
  allocations.
• Constraints packing constraints on the xijs
• 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

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Money-weighed geometric mean of utilities
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Eisenberg-Gale Program, 1959
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KKT conditions
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• Therefore, buyer i buys from
• only,
• i.e., gets an optimal bundle

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• Therefore, buyer i buys from
• only,
• i.e., gets an optimal bundle
• Can prove that equilibrium prices
• are unique!

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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

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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

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KKT conditions
e(i)
e(i)
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Potential function
Will show that potential drops by an inverse
polynomial factor in each phase (polynomial
time).
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Potential function
Will show that potential drops by an inverse
polynomial factor in each phase (polynomial
time).
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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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For each buyer, most desirable goods, i.e.

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Max flow
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
p(4)
infinite capacities
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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
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Two important considerations

• The price of a good never exceeds
• its equilibrium price
• Invariant s is a min-cut

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Max flow
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
p(4)
p low prices
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Two important considerations

• The price of a good never exceeds
• its equilibrium price
• Invariant s is a min-cut
• Identify tight sets of goods

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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

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Network N
buyers
p
m
bang-per-buck edges
goods
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Balanced flow in N
p
m
i
W.r.t. flow f, surplus(i) m(i) f(i,t)
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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) lt surplus(j) then there is no
• path from i to j in R.

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Property 1
R
i
j
surplus(i) lt surplus(j)
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Property 1
R
i
j
surplus(i) lt surplus(j)
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Property 1
R
i
j
Circulation gives a more balanced flow.
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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

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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

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Pieces fit just right!
Invariant
Balanced flows
Bang-per-buck edges
Tight 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
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KKT conditions
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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 loose
• Because 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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Concave utility function
utility
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 substitutability
• w.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 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!

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Piecewise linear, concave
utility
amount of j
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PTAS for concave function
utility
amount of j
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Piecewise linear concave utility

• Does not satisfy weak gross substitutability

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Piecewise linear, concave
utility
amount of j
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Differentiate
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rate
amount of j
money spent on j
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Spending constraint utility function
rate utility/unit amount of j
rate
20
40
60
money spent on j
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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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Extend model assume buyers have utility for
money
rate
20
40
100
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• Theorem Polynomial time algorithm for
• computing equilibrium prices and allocations for
• Fishers 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 function
• Happiness 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 function
• Happiness derived is
• not a function of allocation only
• but also of amount of money spent.
• Has applications in
• Googles AdWords Market!

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A digression
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AdWords Market

• Created by search engine companies
• Google
• Yahoo!
• 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 worlds view now (as Advertisement
companies)
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So how does this work?
Bids for different keywords
Daily Budgets
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AdWords Allocation Problem
LawyersRus.com
asbestos
Search results
Search Engine
Sue.com
Ads
Whose ad to put How to maximize revenue?
TaxHelper.com
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AdWords Problem

• Mehta, Saberi, Vazirani Vazirani, 2005
• 1-1/e algorithm, assuming budgetsgtgtbids

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AdWords Problem

• Mehta, Saberi, Vazirani Vazirani, 2005
• 1-1/e algorithm, assuming budgetsgtgtbids
• Optimal!

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AdWords Problem

• Mehta, Saberi, Vazirani Vazirani, 2005
• 1-1/e algorithm, assuming budgetsgtgtbids
• 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
• mens clog
• womens clog
• Advertising budget 100/day
• Expected profit
• mens clog 2/click
• womens 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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mens clog
rate utility/click
rate
2
1
60
100
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womens clog
rate utility/click
4
rate
2
60
100
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money
rate utility/
rate
1
0
80
100
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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 Googles 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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Spending constraint utilities satisfy

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

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Linear utilities also satisfy

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

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Proof follows fromEisenberg-Gale Convex Program,
1959
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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 Fishers linear case?

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Use spending constraint algorithm to
solve piecewise linear, concave utilities
Open
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Piece-wise linear, concave
utility
amount of j
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Differentiate
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• Start with arbitrary prices, adding up to
• total money of buyers.

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rate
money spent on j
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• Start with arbitrary prices, adding up to
• total money of buyers.
• Run algorithm on these utilities to get new
  prices.

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• Start with arbitrary prices, adding up to
• total money of buyers.
• Run algorithm on these utilities to get new
  prices.

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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!

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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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