documentj1
TRANSCRIPT
Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Polynomial Time Algorithms
For Market Equilibria
1) History and Basic Notions
Markets
Stock Markets
Internet
Revolution in definition of markets
Revolution in definition of markets
New markets defined byGoogle AmazonYahoo!Ebay
Revolution in definition of markets
Massive computational power available
Revolution in definition of markets
Massive computational power available
Important to find good models and
algorithms for these markets
Adwords Market
Created by search engine companiesGoogleYahoo!MSN
Multi-billion dollar market
Totally revolutionized advertising, especially
by small companies.
New algorithmic and game-theoretic questions
Queries are coming on-line. Instantaneously decide which bidder gets it.
Monika Henzinger, 2004: Find on-line alg.
to maximize Google’s revenue.
New algorithmic and game-theoretic questions
Queries are coming on-line. Instantaneously decide which bidder gets it.
Monika Henzinger, 2004: Find on-line alg.
to maximize Google’s revenue.
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm. Optimal.
How will this market evolve??
The study of market equilibria has occupied
center stage within Mathematical Economics
for over a century.
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.
2). Algorithmic Game Theory
Combinatorial algorithms for
traditional market models
3). New Market Models
Resource Allocation Model of Kelly, 1997
3). New Market Models
Resource Allocation Model of Kelly, 1997
For mathematically modeling
TCP congestion control
Highly successful theory
A Capitalistic Economy
Depends crucially on
pricing mechanisms to ensure:
Stability Efficiency Fairness
Adam Smith
The Wealth of Nations
2 volumes, 1776.
Adam Smith
The Wealth of Nations
2 volumes, 1776.
‘invisible hand’ of the market
Supply-demand curves
Leon Walras, 1874
Pioneered general
equilibrium theory
Irving Fisher, 1891
First fundamental
market model
Fisher’s Model, 1891
milkcheese
winebread
¢¢
$$$$$$$$$$$$$$$$$$
$$
$$$$$$$$
People want to maximize happiness – assume
linear utilities.Find prices s.t. market clears
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
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
Arrow-Debreu Model, 1954Exchange Economy
Second fundamental market model
Celebrated theorem in Mathematical Economics
Kenneth Arrow
Nobel Prize, 1972
Gerard Debreu
Nobel Prize, 1983
Arrow-Debreu Model
n agents, k goods
Arrow-Debreu Model
n agents, k goods
Each agent has: initial endowment of goods,
& a utility function
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
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
Agents: Buyers/sellers
Arrow-Debreu Model
Initial endowment of goods Agents
Goods
Agents
Prices
Goods
= $25 = $15 = $10
Incomes
Goods
Agents
=$25 =$15 =$10
$50
$40
$60
$40
Prices
Goods
Agents1 2: ( , , )i nU x x x R
Maximize utility
$50
$40
$60
$40
=$25 =$15 =$10Prices
Find prices s.t. market clears
Goods
Agents
$50
$40
$60
$40
=$25 =$15 =$10Prices
1: ( , )i nU x x R
Maximize utility
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
Arrow-Debreu Theorem
For continuous, monotonic, strictly concave
utility functions, market clearing prices
exist.
Proof
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
Proof
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
Will illustrate main idea via Brouwer’s Fixed
Point Theorem (buggy proof!!)
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
Brouwer’s Fixed Point Theorem
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
Use Brouwer’s Theorem
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.
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).
What if p is not an equilibrium price?
s(j) < d(j) => p(j)
s(j) > d(j) => p(j)
Also ensure kp
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
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
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
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
Bug??
Boundaries of k
Boundaries of
B(i) is not defined at boundaries!!
k
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
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.
Money