Download Polynomial Time Algorithms For Market Equilibria Vazirani

Algorithmic
Game
Theory
Polynomial Time Algorithms
and Internet Computing
For Market Equilibria
Vijay V. Vazirani
1) History and
Basic Notions
Markets
Stock Markets
Internet

Revolution in definition of markets

Revolution in definition of markets

New markets defined by
 Google
 Amazon
 Yahoo!
 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 companies
 Google
 Yahoo!
 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
$
$$$$$$$$$
¢
wine
bread
cheese

milk
$$$$
People want to maximize happiness – assume
Findutilities.
prices s.t. market clears
linear
Fisher’s Model


n buyers, with specified money, m(i) for buyer i
k goods (unit amount of each good) U  u x

Linear utilities: uij is utility derived by i
on obtaining one unit of j
Total utility of i,
i


u  u x
x [0,1]
i
ij
j
ij
ij
j
ij ij
Fisher’s Model


n buyers, with specified money, m(i)
k goods (each unit amount, w.l.o.g.) U  u x

Linear utilities: uij is utility derived by i
on obtaining one unit of j
Total utility of i,
i


u  u x
i

j
Find prices s.t. market clears, i.e.,
all goods sold, all money spent.
ij
ij
j
ij ij
Arrow-Debreu Model, 1954
Exchange 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 goods
 Buys optimal bundle using this money
 No surplus or deficiency of any good
Utility function of agent i

ui : R  R
k

Continuous, monotonic and strictly concave

For any given prices and money m,
there is a unique utility maximizing bundle
for agent i.
Arrow-Debreu Model
Agents:
Buyers/sellers
Initial endowment of goods
Agents
Goods
Prices
= $25
= $15
= $10
Agents
Goods
Incomes
Agents
$50
$60
Goods
Prices
=$25
$40
$40
=$15
=$10
Maximize utility
U i : ( x1 , x2 ,
xn )  R
Agents
$50
$60
Goods
Prices
=$25
$40
$40
=$15
=$10
Find prices s.t. market clears
Agents
$50
$60
Goods
Prices
=$25
$40
$40
=$15
=$10
Maximize utility
U i : ( x1 , xn )  R

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 : k-1 dimensional
unit simplex
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 S  R be a non-empty, compact, convex set

Continuous function

Then
n
f :S  S
x  S : f ( x)  x
Brouwer’s Fixed Point Theorem
Idea of proof
f : k  k

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.
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
p  k

Let
f ( p)  p '
S(j) < d(j) =>
p( j )  [d ( j )  s( j )]
p '( j ) 
N

S(j) > d(j) =>
p( j )
p '( j ) 
N

N is s.t.

 p '( j )  1
j
i : ui
is a cts. fn.
=>
i : B(i )
is a cts. fn. of p
=>
j : d ( j )
is a cts. fn. of p
=>
f is a cts. fn. of p
i : ui
is a cts. fn.
=>
i : B(i )
is a cts. fn. of p
=>
j : d ( j )
is a cts. fn. of p
=>
f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist.
i : ui
is a cts. fn.
=>
i : B(i )
is a cts. fn. of p
=>
j : d ( j )
is a cts. fn. of p
=>
f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist.
q.e.d.!
Bug??

Boundaries of  k

Boundaries of  k

B(i) is not defined at boundaries!!
Kakutani’s Fixed Point Theorem


S  Rn
convex, compact set
non-empty, convex,
f : S  2S
upper hemi-continuous correspondence
 x  S
s.t.
x  f ( x)
Fisher reduces to Arrow-Debreu

Fisher: n buyers, k goods

AD:
 first
n+1 agents
n have money, utility for goods
 last agent has all goods, utility for money only.
Money