Market Equilibrium and
Algorithmic Game Theory
Pricing of Goods
and Internet Computing
Vijay V. Vazirani
Georgia Tech
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 supply
equilibrium prices
Leon Walras, 1874

Pioneered general
equilibrium theory
General Equilibrium Theory
Occupied 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 admit
equilibrium prices?
Do markets even admit
equilibrium prices?
Easy if only one good!
Supply-demand curves
Do markets even admit
equilibrium 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
Concave utility function
(Of buyer i for good j)
utility
amount of j
total utility
ui   fij (xij )
jG
For given prices,
find optimal bundle of goods
p1
p2
p3
Several buyers with
different utility functions and moneys.
Several buyers with
different utility functions and moneys.
Equilibrium prices
p1
p2
p3
Several buyers with
different utility functions and moneys.
Show equilibrium prices exist.
p1
p2
p3
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
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
Ui : ( 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
Ui : ( x1,
xn )  R
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, quasi-concave and
satisfying non-satiation.

Given prices and money m,
there is a unique utility maximizing bundle.
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 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
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 : k-1 dimensional
unit simplex
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.
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)  [s( j)  d( 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:
compact, convex set in R
f : S 2
S
upper hemi-continuous
 x s.t. x  f ( x)
n
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.
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
R
g1
Agent’s earning:
 no.
of units of bread produced
 no. of copies of each song sold

Agent spends earnings on optimal bundle.
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 prices
 Excess goods:
lower prices
Leon Walras

Tatonnement process:
Price adjustment process to arrive at equilibrium
 Deficient
goods: raise prices
 Excess 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.
II.
III.
IV.
V.
VI.
BEFORE THE FORMULATION OF
GENERAL EQUILIBRIUM THEORY
PARTIAL EQUILIBRIUM
WALRAS, PARETO, AND HICKS
SOCIALISM AND
DECENTRALIZATION
SAMUELSON AND SUCCESSORS
THE END OF THE PROGRAM
Part VI: THE END OF THE PROGRAM
A.
B.
C.
D.
Scarf’s example
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).
Uzawa: Existence of general equilibrium is
equivalent to fixed-point theorem
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!!
p1
p2
p3
The new face of computing
Today’s reality

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.
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.
p1
p2
p3
“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,
vi   uij xij
jG

mi : money of buyer i.
 One unit of each good j.
Eisenberg-Gale Program, 1959
max  mi log vi
i
s.t.
i : vi   j u ij x ij
 x 1
ij : x  0
j :
ij
i
ij
Eisenberg-Gale Program, 1959
max  mi log vi
i
s.t.
i : vi   j u ij x ij
 x 1
ij : x  0
j :
ij
i
ij
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 equilibrium
 Is rational
Lagrange relaxation technique

Take constraints into objective with a penalty

Yields dual LP.
T
min c . x
s.t.
Ax  b
L(x, y)  (c . x  y (Ax  b))
T
T
g(y)  min x {c . x  y (Ax  b)}
T
y : g(y)  opt
T
Best lower bound = max y g(y)
g(y)  min x {(c  y A)x  y b}
T
T
T
If (cT  yT A)  0, g(y)   
Therefore,
s.t.
T
max y . b
y Ac
T
T