algorithmic game theory and internet computing vijay v. vazirani polynomial time algorithms for...
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Polynomial Time Algorithms
For Market Equilibria
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Markets
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Stock Markets
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Internet
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Revolution in definition of markets
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Revolution in definition of markets
New markets defined byGoogle AmazonYahoo!Ebay
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Revolution in definition of markets
Massive computational power available
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Revolution in definition of markets
Massive computational power available
Important to find good models and
algorithms for these markets
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Adwords Market
Created by search engine companiesGoogleYahoo!MSN
Multi-billion dollar market
Totally revolutionized advertising, especially
by small companies.
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How will this market evolve??
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The study of market equilibria has occupied
center stage within Mathematical Economics
for over a century.
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The study of market equilibria has occupied
center stage within Mathematical Economics
for over a century.
This talk: Historical perspective
& key notions from this theory.
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2). Algorithmic Game Theory
Combinatorial algorithms for
traditional market models
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3). New Market Models
Resource Allocation Model of Kelly, 1997
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3). New Market Models
Resource Allocation Model of Kelly, 1997
For mathematically modeling
TCP congestion control
Highly successful theory
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A Capitalistic Economy
Depends crucially on
pricing mechanisms to ensure:
Stability Efficiency Fairness
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Adam Smith
The Wealth of Nations
2 volumes, 1776.
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Adam Smith
The Wealth of Nations
2 volumes, 1776.
‘invisible hand’ of the market
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Supply-demand curves
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Leon Walras, 1874
Pioneered general
equilibrium theory
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Irving Fisher, 1891
First fundamental
market model
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Fisher’s Model, 1891
milkcheese
winebread
¢¢
$$$$$$$$$$$$$$$$$$
$$
$$$$$$$$
People want to maximize happiness – assume
linear utilities.Find prices s.t. market clears
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Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is 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, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i
on obtaining one unit of j Total utility of i,
Find prices s.t. market clears, i.e.,
all goods sold, all money spent.
i ij ijj
U u xiju
xuu ijj iji
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Arrow-Debreu Model, 1954Exchange Economy
Second fundamental market model
Celebrated theorem in Mathematical Economics
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Kenneth Arrow
Nobel Prize, 1972
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Gerard Debreu
Nobel Prize, 1983
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Arrow-Debreu Model
n agents, k goods
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Arrow-Debreu Model
n agents, k goods
Each agent has: initial endowment of goods,
& a utility function
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Arrow-Debreu Model
n agents, k goods
Each agent has: initial endowment of goods,
& a utility function Find market clearing prices, i.e., prices s.t. if
Each agent sells all her goodsBuys optimal bundle using this moneyNo surplus or deficiency of any good
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Utility function of agent i
Continuous, monotonic and strictly concave
For any given prices and money m,
there is a unique utility maximizing bundle
for agent i.
: kiu R R
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Agents: Buyers/sellers
Arrow-Debreu Model
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Initial endowment of goods Agents
Goods
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Agents
Prices
Goods
= $25 = $15 = $10
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Incomes
Goods
Agents
=$25 =$15 =$10
$50
$40
$60
$40
Prices
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Goods
Agents1 2: ( , , )i nU x x x R
Maximize utility
$50
$40
$60
$40
=$25 =$15 =$10Prices
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Find prices s.t. market clears
Goods
Agents
$50
$40
$60
$40
=$25 =$15 =$10Prices
1: ( , )i nU x x R
Maximize utility
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Observe: If p is market clearing
prices, then so is any scaling of p
Assume w.l.o.g. that sum of
prices of k goods is 1.
k-1 dimensional
unit simplex
:k
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Arrow-Debreu Theorem
For continuous, monotonic, strictly concave
utility functions, market clearing prices
exist.
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Proof
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
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Proof
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
Will illustrate main idea via Brouwer’s Fixed
Point Theorem (buggy proof!!)
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Brouwer’s Fixed Point Theorem
Let be a non-empty, compact, convex set
Continuous function
Then
:f S S
nS R
: ( )x S f x x
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Brouwer’s Fixed Point Theorem
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Idea of proof
Will define continuous function
If p is not market clearing, f(p) tries to
‘correct’ this.
Therefore fixed points of f must be
equilibrium prices.
: k kf
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Use Brouwer’s Theorem
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When is p an equilibrium price?
s(j): total supply of good j.
B(i): unique optimal bundle which agent i wants to buy after selling her initial
endowment at prices p.
d(j): total demand of good j.
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When is p an equilibrium price?
s(j): total supply of good j.
B(i): unique optimal bundle which agent i wants to buy after selling her initial
endowment at prices p.
d(j): total demand of good j.
For each good j: s(j) = d(j).
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What if p is not an equilibrium price?
s(j) < d(j) => p(j)
s(j) > d(j) => p(j)
Also ensure kp
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Let
S(j) < d(j) =>
S(j) > d(j) =>
N is s.t.
( )'( )
p jp j
N
'( ) 1j
p j
( ) [ ( ) ( )]'( )
p j d j s jp j
N
( ) 'f p p
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is a cts. fn.
=> is a cts. fn. of p
=> is a cts. fn. of p
=> f is a cts. fn. of p
: ( )i B i
: ( )j d j
: ii u
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is a cts. fn.
=> is a cts. fn. of p
=> is a cts. fn. of p
=> f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist.
: ( )i B i
: ( )j d j
: ii u
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is a cts. fn.
=> is a cts. fn. of p
=> is a cts. fn. of p
=> f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist. q.e.d.!
: ( )i B i
: ( )j d j
: ii u
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Kakutani’s Fixed Point Theorem
convex, compact set
non-empty, convex,
upper hemi-continuous correspondence
s.t.
: 2Sf S
x S ( )x f x
nS R
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Fisher reduces to Arrow-Debreu
Fisher: n buyers, k goods
AD: n+1 agents
first n have money, utility for goods last agent has all goods, utility for money only.