Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

Market Equilibrium and

Pricing of Goods

Adam Smith

The Wealth of Nations, 1776.

“It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest.” Each participant in a competitive economy is “led by an invisible hand to promote an end which was no part of his intention.”

What is Economics?

‘‘Economics is the study of the use of

scarce resources which have alternative uses.’’

Lionel Robbins

(1898 – 1984)

How are scarce resources assigned to alternative uses?

How are scarce resources assigned to alternative uses?

Prices!

How are scarce resources assigned to alternative uses?

Prices

Parity between demand and supply

How are scarce resources assigned to alternative uses?

Prices

Parity between demand and supplyequilibrium prices

Leon Walras, 1874

Pioneered general

equilibrium theory

General Equilibrium TheoryOccupied center stage in Mathematical

Economics for over a century

Mathematical ratification!

Central tenet

Markets should operate at equilibrium

Central tenet

Markets should operate at equilibrium

i.e., prices s.t.

Parity between supply and demand

Do markets even admitequilibrium prices?

Do markets even admitequilibrium prices?

Easy if only one good!

Supply-demand curves

Do markets even admitequilibrium prices?

What if there are multiple goods and multiple buyers with diverse desires

and different buying power?

Irving Fisher, 1891

Defined a fundamental

market model

Special case of Walras’

model

utility

Concave utility function

(Of buyer i for good j)

amount of j

( )i ij ijj G

u f x

total utility

For given prices,find optimal bundle of goods

1p 2p3p

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys.

Equilibrium prices

1p 2p3p

Several buyers with different utility functions and moneys.

Show equilibrium prices exist.

1p 2p3p

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.

First Welfare Theorem

Competitive equilibrium =>

Pareto optimal allocation of resources

Pareto optimal = impossible to make

an agent better off without making some

other agent worse off

Second Welfare Theorem

Every Pareto optimal allocation of resources

comes from a competitive equilibrium

(after redistribution of initial endowments).

Kenneth Arrow

Nobel Prize, 1972

Gerard Debreu

Nobel Prize, 1983

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

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, quasi-concave and

satisfying non-satiation.

Given prices and money m,

there is a unique utility maximizing bundle.

: kiu R R

Proof of Arrow-Debreu Theorem

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

. .x s t x f x

Brouwer’s Fixed Point Theorem

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

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

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 '( j) =

p( j) −[s( j) −d( 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

S: compact, convex set in

upper hemi-continuous

: 2Sf S

nR

. . ( )x s t x f x

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.

Pricing of Digital Goods

Music, movies, video games, …

cell phone apps., …, web search results,

… , even ideas!

Pricing of Digital Goods

Music, movies, video games, …

cell phone apps., …, web search results,

… , even ideas!

Once produced, supply is infinite!!

What is Economics?

‘‘Economics is the study of the use of

scarce resources which have alternative uses.’’

Lionel Robbins

(1898 – 1984)

Jain & V., 2010:

Market model for digital goods,

with notion of equilibrium.

Proof of existence of equilibrium.

Idiosyncrasies of Digital Realm

Staggering number of goods available with

great ease, e.g., iTunes has 11 million songs!

Once produced, infinite supply.

Want 2 songs => want 2 different songs,

not 2 copies of same song.

Agents’ rating of songs varies widely.

Game-Theoretic Assumptions

Full rationality, infinite computing power:

not meaningful!

Game-Theoretic Assumptions

Full rationality, infinite computing power:

not meaningful!

e.g., song A for $1.23, song B for $1.56, …

Game-Theoretic Assumptions

Full rationality, infinite computing power:

not meaningful!

e.g., song A for $1.23, song B for $1.56, …

Cannot price songs individually!

Market Model

Uniform pricing of all goods in a category.Assume g categories of digital goods.

Each agent has a total order over all songs

in a category.

Arrow-Debreu-Based Market Model

Assume 1 conventional good: bread.

Each agent has a utility function over

g digital categories and bread.

Optimal bundle for i, given prices p

First, compute i’s optimal bundle, i.e.,

#songs from each digital category

and no. of units of bread.

Next, from each digital category,

i picks her most favorite songs.

Agents are also producers

Feasible production of each agent

is a convex, compact set in

Agent’s earning: no. of units of bread producedno. of copies of each song sold

Agent spends earnings on optimal bundle.

Rg+1

Equilibrium

(p, x, y) s.t.

Each agent, i, gets optimal bundle &

“best” songs in each category. Each agent, k, maximizes earnings,

given p, x, y(-k)

Market clears, i.e., all bread sold &

at least 1 copy of each song sold.

Theorem (Jain & V., 2010):

Equilibrium exists.

(Using Kakutani’s fixed-point theorem)

Arrow-Debreu Theorem, 1954

Celebrated theorem in Mathematical Economics

Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem.

Highly non-constructive!

Leon Walras

Tatonnement process:

Price adjustment process to arrive at equilibrium

Deficient goods: raise pricesExcess goods: lower prices

Leon Walras

Tatonnement process:

Price adjustment process to arrive at equilibrium

Deficient goods: raise pricesExcess goods: lower prices

Does it converge to equilibrium?

GETTING TO ECONOMIC EQUILIBRIUM: A PROBLEM AND ITS HISTORY

For the third International Workshop on Internet and Network Economics

Kenneth J. Arrow

OUTLINE

I. BEFORE THE FORMULATION OF GENERAL EQUILIBRIUM THEORY

II. PARTIAL EQUILIBRIUMIII. WALRAS, PARETO, AND HICKSIV. SOCIALISM AND DECENTRALIZATIONV. SAMUELSON AND SUCCESSORSVI. THE END OF THE PROGRAM

Part VI: THE END OF THE PROGRAM

A. Scarf’s exampleB. Saari-Simon Theorem: For any dynamic system

depending on first-order information (z) only, there is a set of excess demand functions for which stability fails. (In fact, theorem is stronger).

C. Uzawa: Existence of general equilibrium is equivalent to fixed-point theorem

D. Assumptions on individual demand functions do not constrain aggregate demand function (Sonnenschein, Debreu, Mantel)

Several buyers with different utility functions and moneys.

Find equilibrium prices!!

1p 2p3p

The new face of computing

New markets defined by Internet companies, e.g., Microsoft Google eBay Yahoo! Amazon

Massive computing power available.

Need an inherently-algorithmic theory of markets and market equilibria.

Today’s reality

Standard sufficient conditions

on utility functions (in Arrow-Debreu Theorem):

Continuous, quasiconcave,

satisfying non-satiation.

Complexity-theoretic question

For “reasonable” utility fns.,

can market equilibrium be computed in P?

If not, what is its complexity?

Several buyers with different utility functions and moneys.

Find equilibrium prices.

1p 2p3p

“Stock prices have reached what looks like

a permanently high plateau”

“Stock prices have reached what looks like

a permanently high plateau”

Irving Fisher, October 1929

Linear Fisher Market

Assume:Buyer i’s total utility,

mi : money of buyer i.

One unit of each good j.

i ij ijj G

v u x

Eisenberg-Gale Program, 1959

max log

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

m v

s t

i v

j

ij

u xx

x

Eisenberg-Gale Program, 1959

max log

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

m v

s t

i v

j

ij

u xx

x

prices pj

Why remarkable?

Equilibrium simultaneously optimizes

for all agents.

How is this done via a single objective function?

Rational convex program

Always has a rational solution,

using polynomially many bits,

if all parameters are rational.

Eisenberg-Gale program is rational.

KKT Conditions

Generalization of complementary slackness

conditions to convex programs.

Help prove optimal solution to EG program:Gives market equilibriumIs rational

Lagrange relaxation technique

Take constraints into objective with a penalty

Yields dual LP.

min cT . x

s.t. A x =b

L(x, y) = (cT . x−yT (Ax−b))

g(y) =minx {cT . x−yT (Ax−b)}

∀y: g(y) ≤ opt

Best lower bound = maxy g(y)

g(y) =minx {(cT −yTA)x+ yTb}

If (cT −yTA) ≠0, g(y) =−∞

Therefore, maxyT .b

s.t. yT A=cT