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?

How are scarce resources assigned to alternative uses?
Parity between demand and supply

How are scarce resources assigned to alternative uses?
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

utility
Concave utility function
(Of buyer i for good j ) amount of j

total utility u i
  f x ij ij

For given prices, find optimal bundle of goods p
1 p
2 p
3

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys.
Equilibrium prices p
1 p
2 p
3

Several buyers with different utility functions and moneys.
Show equilibrium prices exist.
p
1 p
2 p
3

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 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
II.
III.
IV.
I.
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!!
p
1 p
2 p
3

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.
p
1 p
2 p
3

“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

Herbert Scarf, 1973
 Approximate fixed point algorithms for market equilibria.
 Not polynomial time.

Curtis Eaves, 1984
 Polynomial time algorithm for
Cobb-Douglas utilities.

utility of i
Linear utility function f ij
( x ij
)
 u ij
.
x ij amount of j

Linear Fisher Market
 Assume:
 Buyer i
’s total utility, v i
  u x ij ij
 m i
: money of buyer i.
 One unit of each good j.

Eisenberg-Gale Program, 1959 max
 i m log v i i
 i : v i
 j :
 i
  j u x ij x ij

1
 ij : x ij

0

Eisenberg-Gale Program, 1959 max
 i m log v i i
 i : v i
 j :
 i
  j u x ij x ij

1
 ij : x ij

0 prices p j

Why remarkable?
 Equilibrium simultaneously optimizes for all agents.
 How is this done via a single objective function?

Rational convex program
 A nonlinear convex program that always has a rational solution, using polynomially many bits, if all parameters are rational.
 Eisenberg-Gale program is rational.

Combinatorial Algorithm for
Linear Case of Fisher’s Model
 Devanur, Papadimitriou, Saberi & V., 2002
By extending the primal-dual paradigm to the setting of convex programs & KKT conditions

Auction for Google’s TV ads
N. Nisan et. al, 2009:
 Used market equilibrium based approach.
 Combinatorial algorithms for linear case provided “inspiration”.

Jain, 2004:
 Rational convex program for Arrow-Debreu market with linear utilities.
 Polynomial time exact algorithm, using ellipsoid method.

utility
Additively separable over goods amount of j

Long-standing open problem
 Complexity of finding an equilibrium for
Fisher and Arrow-Debreu models under separable, plc utilities?

Generalize EG program to piecewise-linear, concave utilities?
utility utility/unit of j u ijk l ijk amount of j

max
 i m log v i i
 i :
 j :
 ijk :
 ijk : v i

  x ij u x

1 ijk x ijk
 l ijk x ijk

0

max
 i m log v i i
 i :
 j :
 ijk :
 ijk : v i

  x ij u x

1 ijk x ijk
 l ijk x ijk

0

 V. & Yannakakis, 2007:
Equilibrium is rational for Fisher and
Arrow-Debreu models under separable, plc utilities.
 In P??

Markets with piecewise-linear, concave utilities
 Chen, Dai, Du, Teng, 2009:
 PPAD-hardness for Arrow-Debreu model

NP-hardness does not apply
 Megiddo, 1988:
 Equilibrium NP-hard => NP = co-NP
 Papadimitriou, 1991: PPAD
 2-player Nash equilibrium is PPAD-complete
 Rational
 Etessami & Yannakakis, 2007: FIXP
 3-player Nash equilibrium is FIXP-complete
 Irrational

Markets with piecewise-linear, concave utilities
 Chen, Dai, Du, Teng, 2009:
 PPAD-hardness for Arrow-Debreu model
 Chen & Teng, 2009:
 PPAD-hardness for Fisher’s model
 V. & Yannakakis, 2009:
 PPAD-hardness for Fisher’s model

Markets with piecewise-linear, concave utilities
V. & Yannakakis, 2009:
 Membership in PPAD for both models.

Algorithmic ratification of the
“invisible hand of the market”
How do we salvage the situation??

Is PPAD really hard??
What is the “right” model??

Price discrimination markets
 Business charges different prices from different customers for essentially same goods or services.
 Goel & V., 2009:
Perfect price discrimination market.
Business charges each consumer what they are willing and able to pay.

plc utilities

 Middleman buys all goods and sells to buyers, charging according to utility accrued.
 Given p , each buyer picks rate for accruing utility.

 Middleman buys all goods and sells to buyers, charging according to utility accrued.
 Given p , each buyer picks rate for accruing utility.
 Equilibrium is captured by a rational convex program!

Generalization of EG program works!
max
 i m log v i i
 i :
 j :
 ijk :
 ijk : v i

  x ij u x

1 ijk x ijk
 l ijk x ijk

0

V., 2010: Generalize to
 Continuously differentiable, quasiconcave
(non-separable) utilities, satisfying non-satiation.

V., 2010: Generalize to
 Continuously differentiable, quasiconcave
(non-separable) utilities, satisfying non-satiation.
 Compare with Arrow-Debreu utilities!!
continuous, quasiconcave, satisfying non-satiation.

Price discrimination market
(plc utilities)
Spending constraint market
EG convex program =
Devanur’s program

Eisenberg-Gale Markets
Jain & V., 2007
(Proportional Fairness)
(Kelly, 1997)
Price disc. market
Nash Bargaining
V., 2008
Spending constraint market
EG convex program =
Devanur’s program

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.

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)