![Page 1: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/1.jpg)
Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Georgia Tech
Market Equilibrium and
Pricing of Goods
![Page 2: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/2.jpg)
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.”
![Page 3: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/3.jpg)
What is Economics?
‘‘Economics is the study of the use of
scarce resources which have alternative uses.’’
Lionel Robbins
(1898 – 1984)
![Page 4: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/4.jpg)
How are scarce resources assigned to alternative uses?
![Page 5: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/5.jpg)
![Page 6: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/6.jpg)
How are scarce resources assigned to alternative uses?
Prices!
![Page 7: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/7.jpg)
![Page 8: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/8.jpg)
How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supply
![Page 9: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/9.jpg)
How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supplyequilibrium prices
![Page 10: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/10.jpg)
Leon Walras, 1874
Pioneered general
equilibrium theory
![Page 11: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/11.jpg)
General Equilibrium TheoryOccupied center stage in Mathematical
Economics for over a century
Mathematical ratification!
![Page 12: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/12.jpg)
Central tenet
Markets should operate at equilibrium
![Page 13: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/13.jpg)
Central tenet
Markets should operate at equilibrium
i.e., prices s.t.
Parity between supply and demand
![Page 14: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/14.jpg)
Do markets even admitequilibrium prices?
![Page 15: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/15.jpg)
Do markets even admitequilibrium prices?
Easy if only one good!
![Page 16: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/16.jpg)
Supply-demand curves
![Page 17: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/17.jpg)
Do markets even admitequilibrium prices?
What if there are multiple goods and multiple buyers with diverse desires
and different buying power?
![Page 18: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/18.jpg)
Irving Fisher, 1891
Defined a fundamental
market model
Special case of Walras’
model
![Page 19: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/19.jpg)
![Page 20: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/20.jpg)
![Page 21: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/21.jpg)
utility
Concave utility function
(Of buyer i for good j)
amount of j
![Page 22: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/22.jpg)
( )i ij ijj G
u f x
total utility
![Page 23: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/23.jpg)
For given prices,find optimal bundle of goods
1p 2p3p
![Page 24: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/24.jpg)
Several buyers with different utility functions and moneys.
![Page 25: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/25.jpg)
Several buyers with different utility functions and moneys.
Equilibrium prices
1p 2p3p
![Page 26: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/26.jpg)
Several buyers with different utility functions and moneys.
Show equilibrium prices exist.
1p 2p3p
![Page 27: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/27.jpg)
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.
![Page 28: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/28.jpg)
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
![Page 29: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/29.jpg)
Second Welfare Theorem
Every Pareto optimal allocation of resources
comes from a competitive equilibrium
(after redistribution of initial endowments).
![Page 30: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/30.jpg)
Kenneth Arrow
Nobel Prize, 1972
![Page 31: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/31.jpg)
Gerard Debreu
Nobel Prize, 1983
![Page 32: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/32.jpg)
Agents: buyers/sellers
Arrow-Debreu Model
![Page 33: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/33.jpg)
Initial endowment of goods Agents
Goods
![Page 34: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/34.jpg)
Agents
Prices
Goods
= $25 = $15 = $10
![Page 35: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/35.jpg)
Incomes
Goods
Agents
=$25 =$15 =$10
$50
$40
$60
$40
Prices
![Page 36: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/36.jpg)
Goods
Agents1 2: ( , , )i nU x x x R
Maximize utility
$50
$40
$60
$40
=$25 =$15 =$10Prices
![Page 37: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/37.jpg)
Find prices s.t. market clears
Goods
Agents
$50
$40
$60
$40
=$25 =$15 =$10Prices
1: ( , )i nU x x R
Maximize utility
![Page 38: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/38.jpg)
Arrow-Debreu Model
n agents, k goods
![Page 39: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/39.jpg)
Arrow-Debreu Model
n agents, k goods
Each agent has: initial endowment of goods,
& a utility function
![Page 40: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/40.jpg)
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
![Page 41: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/41.jpg)
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
![Page 42: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/42.jpg)
Proof of Arrow-Debreu Theorem
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
![Page 43: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/43.jpg)
Proof
Uses Kakutani’s Fixed Point Theorem.Deep theorem in topology
Will illustrate main idea via Brouwer’s Fixed
Point Theorem (buggy proof!!)
![Page 44: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/44.jpg)
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
![Page 45: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/45.jpg)
Brouwer’s Fixed Point Theorem
. .x s t x f x
![Page 46: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/46.jpg)
Brouwer’s Fixed Point Theorem
![Page 47: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/47.jpg)
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
![Page 48: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/48.jpg)
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
![Page 49: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/49.jpg)
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.
![Page 50: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/50.jpg)
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).
![Page 51: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/51.jpg)
What if p is not an equilibrium price?
s(j) < d(j) => p(j)
s(j) > d(j) => p(j)
Also ensure kp
![Page 52: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/52.jpg)
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
![Page 53: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/53.jpg)
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
![Page 54: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/54.jpg)
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
![Page 55: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/55.jpg)
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
![Page 56: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/56.jpg)
Bug??
![Page 57: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/57.jpg)
Boundaries of k
![Page 58: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/58.jpg)
Boundaries of
B(i) is not defined at boundaries!!
k
![Page 59: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/59.jpg)
Kakutani’s fixed point theorem
S: compact, convex set in
upper hemi-continuous
: 2Sf S
nR
. . ( )x s t x f x
![Page 60: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/60.jpg)
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.
![Page 61: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/61.jpg)
Pricing of Digital Goods
Music, movies, video games, …
cell phone apps., …, web search results,
… , even ideas!
![Page 62: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/62.jpg)
Pricing of Digital Goods
Music, movies, video games, …
cell phone apps., …, web search results,
… , even ideas!
Once produced, supply is infinite!!
![Page 63: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/63.jpg)
What is Economics?
‘‘Economics is the study of the use of
scarce resources which have alternative uses.’’
Lionel Robbins
(1898 – 1984)
![Page 64: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/64.jpg)
Jain & V., 2010:
Market model for digital goods,
with notion of equilibrium.
Proof of existence of equilibrium.
![Page 65: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/65.jpg)
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.
![Page 66: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/66.jpg)
Game-Theoretic Assumptions
Full rationality, infinite computing power:
not meaningful!
![Page 67: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/67.jpg)
Game-Theoretic Assumptions
Full rationality, infinite computing power:
not meaningful!
e.g., song A for $1.23, song B for $1.56, …
![Page 68: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/68.jpg)
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!
![Page 69: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/69.jpg)
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.
![Page 70: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/70.jpg)
Arrow-Debreu-Based Market Model
Assume 1 conventional good: bread.
Each agent has a utility function over
g digital categories and bread.
![Page 71: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/71.jpg)
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.
![Page 72: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/72.jpg)
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
![Page 73: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/73.jpg)
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.
![Page 74: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/74.jpg)
Theorem (Jain & V., 2010):
Equilibrium exists.
(Using Kakutani’s fixed-point theorem)
![Page 75: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/75.jpg)
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!
![Page 76: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/76.jpg)
Leon Walras
Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise pricesExcess goods: lower prices
![Page 77: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/77.jpg)
Leon Walras
Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise pricesExcess goods: lower prices
Does it converge to equilibrium?
![Page 78: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/78.jpg)
GETTING TO ECONOMIC EQUILIBRIUM: A PROBLEM AND ITS HISTORY
For the third International Workshop on Internet and Network Economics
Kenneth J. Arrow
![Page 79: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/79.jpg)
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
![Page 80: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/80.jpg)
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)
![Page 81: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/81.jpg)
Several buyers with different utility functions and moneys.
Find equilibrium prices!!
1p 2p3p
![Page 82: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/82.jpg)
The new face of computing
![Page 83: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/83.jpg)
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
![Page 84: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/84.jpg)
Standard sufficient conditions
on utility functions (in Arrow-Debreu Theorem):
Continuous, quasiconcave,
satisfying non-satiation.
![Page 85: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/85.jpg)
Complexity-theoretic question
For “reasonable” utility fns.,
can market equilibrium be computed in P?
If not, what is its complexity?
![Page 86: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/86.jpg)
Several buyers with different utility functions and moneys.
Find equilibrium prices.
1p 2p3p
![Page 87: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/87.jpg)
![Page 88: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/88.jpg)
“Stock prices have reached what looks like
a permanently high plateau”
![Page 89: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/89.jpg)
“Stock prices have reached what looks like
a permanently high plateau”
Irving Fisher, October 1929
![Page 90: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/90.jpg)
![Page 91: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/91.jpg)
![Page 92: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/92.jpg)
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
![Page 93: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/93.jpg)
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
![Page 94: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/94.jpg)
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
![Page 95: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/95.jpg)
Why remarkable?
Equilibrium simultaneously optimizes
for all agents.
How is this done via a single objective function?
![Page 96: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/96.jpg)
Rational convex program
Always has a rational solution,
using polynomially many bits,
if all parameters are rational.
Eisenberg-Gale program is rational.
![Page 97: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/97.jpg)
KKT Conditions
Generalization of complementary slackness
conditions to convex programs.
Help prove optimal solution to EG program:Gives market equilibriumIs rational
![Page 98: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/98.jpg)
Lagrange relaxation technique
Take constraints into objective with a penalty
Yields dual LP.
![Page 99: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/99.jpg)
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
![Page 100: Algorithmic Game Theory and Internet Computing](https://reader030.vdocument.in/reader030/viewer/2022032805/5681324d550346895d98c531/html5/thumbnails/100.jpg)
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