Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Polynomial Time Algorithms

For Market Equilibria

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.

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

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.