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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 .

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Algorithmic game theory and internet computing l.jpg

Markets and

the Primal-Dual Paradigm

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani






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  • 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 l.jpg
Theory of Algorithms

  • Powerful tools and techniques

    developed over last 4 decades.


Theory of algorithms10 l.jpg
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 l.jpg
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 l.jpg
Historically, the study of markets

  • has been of central importance,

    especially in the West


Historically the study of markets15 l.jpg
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 l.jpg
Leon Walras, 1874

  • Pioneered general

    equilibrium theory


Arrow debreu theorem 1954 l.jpg
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.


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Kenneth Arrow

  • Nobel Prize, 1972


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Gerard Debreu

  • Nobel Prize, 1983


General equilibrium theory l.jpg
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 theory21 l.jpg
General Equilibrium Theory

An almost entirely non-algorithmic theory!


What is needed today l.jpg
What is needed today?

  • An inherently algorithmictheory of

    market equilibrium

  • New models that capture new markets

    and are easier to use than traditional models


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A central tenet l.jpg
A central tenet

  • Prices are such that demand equals supply, i.e.,

    equilibrium prices.


A central tenet25 l.jpg
A central tenet

  • Prices are such that demand equals supply, i.e.,

    equilibrium prices.

  • Easy if only one good



Irving fisher 1891 l.jpg
Irving Fisher, 1891

  • Defined a fundamental

    market model


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Utility function

utility

amount ofmilk


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Utility function

utility

amount ofbread


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Utility function

utility

amount ofcheese


Total utility of a bundle of goods l.jpg
Total utility of a bundle of goods

= Sum of utilities of individual goods



For given prices find optimal bundle of goods l.jpg
For given prices,find optimal bundle of goods


Fisher market l.jpg
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 l.jpg
Combinatorial Algorithm for Linear Case of Fisher’s Model

  • Devanur, Papadimitriou, Saberi & V., 2002

    Using the primal-dual schema


Primal dual schema l.jpg
Primal-Dual Schema

  • Highly successful algorithm design

    technique from exact and

    approximation algorithms


Exact algorithms for cornerstone problems in p l.jpg
Exact Algorithms for Cornerstone Problems in P:

  • Matching (general graph)

  • Network flow

  • Shortest paths

  • Minimum spanning tree

  • Minimum branching


Approximation algorithms l.jpg
Approximation Algorithms in P:

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .



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Fisher s model l.jpg
Fisher’s Model 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 model47 l.jpg
Fisher’s Model 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 l.jpg
An easier question Fisher’s model

  • Given prices p, are they equilibrium prices?

  • If so, find equilibrium allocations.


An easier question49 l.jpg
An easier question Fisher’s model

  • Given prices p, are they equilibrium prices?

  • If so, find equilibrium allocations.

  • Equilibrium prices are unique!


Bang per buck l.jpg
Bang-per-buck Fisher’s model

  • At prices p, buyer i’s most

    desirable goods, S =

  • Any goods from S worth m(i)

    constitute i’s optimal bundle


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m Fisher’s model(1)

p(1)

m(2)

p(2)

m(3)

p(3)

m(4)

p(4)

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


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Max flow Fisher’s model

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

infinite capacities


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Max flow Fisher’s model

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 l.jpg
Idea of algorithm Fisher’s model

  • “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


Idea of algorithm55 l.jpg

Allocations Fisher’s modelPrices

Idea of algorithm

  • Iterations:

    execute primal & dual improvements


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Two important considerations Fisher’s model

  • The price of a good never exceeds

    its equilibrium price

    • Invariant: s is a min-cut


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Max flow Fisher’s model

p(1)

m(1)

p(2)

m(2)

p(3)

m(3)

m(4)

p(4)

p: low prices


Two important considerations58 l.jpg
Two important considerations Fisher’s model

  • The price of a good never exceeds

    its equilibrium price

    • Invariant: s is a min-cut

    • Identify tight sets of goods


Two important considerations59 l.jpg
Two important considerations Fisher’s model

  • 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 Fisher’s modelN

buyers

p

m

bang-per-buck edges

goods


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Balanced flow in Fisher’s modelN

p

m

i

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


Balanced flow l.jpg
Balanced flow Fisher’s model

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


Balanced flow63 l.jpg
Balanced flow Fisher’s model

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

  • A flow that minimizes l2 norm of surplus vector.


Property 1 l.jpg
Property 1 Fisher’s model

  • 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.


Slide65 l.jpg

Property 1 Fisher’s model

R:

i

j

surplus(i) < surplus(j)


Slide66 l.jpg

Property 1 Fisher’s model

R:

i

j

surplus(i) < surplus(j)


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Property 1 Fisher’s model

R:

i

j

Circulation gives a more balanced flow.


Property 168 l.jpg
Property 1 Fisher’s model

  • Theorem: A max-flow is balanced iff

    it satisfies Property 1.


Pieces fit just right l.jpg
Pieces fit just right! Fisher’s model

Invariant

Balanced flows

Bang-per-buck

edges

Tight sets


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How primal-dual schema is adapted Fisher’s modelto nonlinear setting


A convex program l.jpg
A convex program Fisher’s model

  • whose optimal solution is equilibrium allocations.


A convex program72 l.jpg
A convex program Fisher’s model

  • whose optimal solution is equilibrium allocations.

  • Constraints: packing constraints on the xij’s


A convex program73 l.jpg
A convex program Fisher’s model

  • whose optimal solution is equilibrium allocations.

  • Constraints: packing constraints on the xij’s

  • Objective fn: max utilities derived.


A convex program74 l.jpg
A convex program Fisher’s model

  • 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 l.jpg
Money-weighed geometric mean Fisher’s modelof utilities



Eisenberg gale program 195977 l.jpg
Eisenberg-Gale Program, 1959 Fisher’s model

prices pj


Kkt conditions l.jpg
KKT conditions Fisher’s model


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  • Therefore, buyer Fisher’s model i buys from

    only,

    i.e., gets an optimal bundle

  • Can prove that equilibrium prices

    are unique!


Will relax kkt conditions l.jpg
Will relax KKT conditions Fisher’s model

  • 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 Fisher’s model

  • e(i): money currently spent by i

    w.r.t. a balanced flow in N

  • surplus money of i


Kkt conditions83 l.jpg
KKT conditions Fisher’s model

e(i)

e(i)


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Potential function Fisher’s model

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).


Potential function85 l.jpg
Potential function Fisher’s model

Will show that potential drops by an inverse polynomial

factor in each phase (polynomial time).


Point of departure l.jpg
Point of departure Fisher’s model

  • KKT conditions are satisfied via a

    continuous process

  • Normally: in discrete steps


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Point of departure Fisher’s model

  • KKT conditions are satisfied via a

    continuous process

  • Normally: in discrete steps

  • Open question: strongly polynomial algorithm?


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Another point of departure Fisher’s model

  • Complementary slackness conditions:

    involve primal or dual variables, not both.

  • KKT conditions: involve primal and dual

    variables simultaneously.


Kkt conditions89 l.jpg
KKT conditions Fisher’s model


Kkt conditions90 l.jpg
KKT conditions Fisher’s model


Primal dual algorithms so far l.jpg
Primal-dual algorithms so far Fisher’s model

  • Raise dual variables greedily. (Lot of effort spent

    on designing more sophisticated dual processes.)


Primal dual algorithms so far92 l.jpg
Primal-dual algorithms so far Fisher’s model

  • 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 far93 l.jpg
Primal-dual algorithms so far Fisher’s model

  • 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 Fisher’s model

  • Dual variables (prices) are raised greedily

  • Yet, primal objects go tight and loose

    • Because of enhanced KKT conditions


Deficiencies of linear utility functions l.jpg
Deficiencies of linear utility functions Fisher’s model

  • 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 Fisher’s model

utility

amount ofj


Concave utility functions l.jpg
Concave utility functions Fisher’s model

  • Do not satisfy weak gross substitutability


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Concave utility functions Fisher’s model

  • 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 functions99 l.jpg
Concave utility functions Fisher’s model

  • 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!


Slide100 l.jpg

Piecewise linear, concave Fisher’s model

utility

amount ofj


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PTAS for concave function Fisher’s model

utility

amount ofj


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Piecewise linear concave utility Fisher’s model

  • Does not satisfy weak gross substitutability


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Piecewise linear, concave Fisher’s model

utility

amount ofj


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rate = utility/unit amount of Fisher’s model j

rate

amount ofj

Differentiate


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rate = utility/unit amount of Fisher’s model j

rate

amount ofj

money spent on j


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Spending constraint utility function Fisher’s model

rate = utility/unit amount of j

rate

$20

$40

$60

money spent onj


Spending constraint utility function l.jpg
Spending constraint utility function Fisher’s model

  • 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 Fisher’s model

rate

$20

$40

$100


Slide110 l.jpg

Theorem: Fisher’s model Polynomial time algorithm for

computing equilibrium prices and allocations for

Fisher’s model with spending constraint utilities.

Furthermore, equilibrium prices are unique.



Old pieces become more complex there are new pieces l.jpg
Old pieces become more complex Fisher’s model+ there are new pieces



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Don Patinkin, 1956 Fisher’s model

  • 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 l.jpg
An unexpected fallout!! Fisher’s model


An unexpected fallout116 l.jpg
An unexpected fallout!! Fisher’s model

  • A new kind of utility function

    • Happiness derived is

      not a function of allocation only

      but also of amount of money spent.


An unexpected fallout117 l.jpg
An unexpected fallout!! Fisher’s model

  • 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 l.jpg
A digression Fisher’s model



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Business world’s view now : Fisher’s model

(as Advertisement companies)


So how does this work l.jpg
So how does this work? Fisher’s model

Bids for

different

keywords

Daily

Budgets


An interesting algorithmic question l.jpg
An interesting algorithmic question! Fisher’s model

  • Monika Henzinger, 2004: Find an on-line

    algorithm that maximizes Google’s revenue.


Adwords allocation problem l.jpg
AdWords Allocation Problem Fisher’s model

LawyersRus.com

asbestos

Search results

SearchEngine

Sue.com

Ads

Whose ad to put

How to maximize

revenue?

TaxHelper.com


Adwords problem l.jpg
AdWords Problem Fisher’s model

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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


Adwords problem126 l.jpg
AdWords Problem Fisher’s model

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

    Optimal!


Adwords problem127 l.jpg
AdWords Problem Fisher’s model

  • Mehta, Saberi, Vazirani & Vazirani, 2005:

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

    Optimal!


Slide128 l.jpg

Spending Fisher’s model

constraint

utilities

AdWords

Market


Adwords market129 l.jpg
AdWords market Fisher’s model

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


Adwords market130 l.jpg
AdWords market Fisher’s model

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

  • How should advertisers specify their

    utility functions?


Choice of utility function l.jpg
Choice of utility function Fisher’s model

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations


Choice of utility function132 l.jpg
Choice of utility function Fisher’s model

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations

  • Efficiently computable


Choice of utility function133 l.jpg
Choice of utility function Fisher’s model

  • Expressive enough that advertisers get

    close to their ‘‘optimal’’ allocations

  • Efficiently computable

  • Easy to specify utilities


Slide134 l.jpg


Slide135 l.jpg

  • linear utility function: Fisher’s model a business will

    typically get only one type of query

    throughout the day!

  • concave utility function: no efficient

    algorithm known!


Slide136 l.jpg

  • linear utility function: Fisher’s model 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 l.jpg
Easier for a buyer Fisher’s model

  • 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 l.jpg
Online shoe business Fisher’s model

  • 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 l.jpg
Considerations for long-term profit Fisher’s model

  • 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


Slide140 l.jpg

  • If both are profitable, Fisher’s model

    • 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)


Slide141 l.jpg

men’s clog Fisher’s model

rate = utility/click

rate

2

1

$60

$100


Slide142 l.jpg

women’s clog Fisher’s model

rate = utility/click

4

rate

2

$60

$100


Slide143 l.jpg

money Fisher’s model

rate = utility/$

rate

1

0

$80

$100


Adwords market144 l.jpg
AdWords market Fisher’s model

  • Suppose Google stays with auctions but

    allows advertisers to specify bids in

    the spending constraint model


Adwords market145 l.jpg
AdWords market Fisher’s model

  • Suppose Google stays with auctions but

    allows advertisers to specify bids in

    the spending constraint model

    • expressivity!


Adwords market146 l.jpg
AdWords market Fisher’s model

  • 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?


Slide147 l.jpg

  • Goel & Mehta, 2006: Fisher’s model

    A small modification to the MSVV algorithm

    achieves 1 – 1/e competitive ratio!


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Open Fisher’s model

Is there a convex program that

captures equilibrium allocations for

spending constraint utilities?


Spending constraint utilities satisfy l.jpg
Spending constraint utilities satisfy Fisher’s model

  • Equilibrium exists (under mild conditions)

  • Equilibrium utilities and prices are unique

  • Rational

  • With small denominators


Linear utilities also satisfy l.jpg
Linear utilities also satisfy Fisher’s model

  • Equilibrium exists (under mild conditions)

  • Equilibrium utilities and prices are unique

  • Rational

  • With small denominators


Proof follows from eisenberg gale convex program 1959 l.jpg
Proof follows from Fisher’s modelEisenberg-Gale Convex Program, 1959


For spending constraint utilities proof follows from algorithm and not a convex program l.jpg
For spending constraint utilities, Fisher’s modelproof follows from algorithm, and not a convex program!


Slide153 l.jpg
Open Fisher’s model

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 l.jpg
Use spending constraint algorithm Fisher’s model to solve piecewise linear, concave utilities

Open


Algorithms game theory common origins l.jpg
Algorithms & Game Theory Fisher’s modelcommon 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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Piece-wise linear, concave Fisher’s model

utility

amount ofj


Slide158 l.jpg

rate = utility/unit amount of Fisher’s model j

rate

amount ofj

Differentiate


Slide159 l.jpg


Slide160 l.jpg

rate = utility/unit amount of Fisher’s model j

rate

money spent on j


Slide161 l.jpg


Slide162 l.jpg


Slide163 l.jpg

  • Start with arbitrary prices, adding up to Fisher’s model

    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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