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Market Equilibrium and Pricing of Goods. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Adam Smith. The Wealth of Nations, 1776. - PowerPoint PPT Presentation
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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