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### Algorithmic Game Theoryand Internet Computing

### GETTING TO ECONOMIC EQUILIBRIUM: A PROBLEM AND ITS HISTORY

The “Invisible Hand of the Market”:

Algorithmic Ratification and

the Digital Economy

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 supplyequilibrium prices

Leon Walras, 1874

- Pioneered general
equilibrium theory

General Equilibrium TheoryOccupied center stage in MathematicalEconomics 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?

Easy if only one good!

Do markets even admitequilibrium prices?

What if there are multiple goods and multiple buyers with diverse desires and different buying power?

Do markets even admitequilibrium prices?

Irving Fisher, 1891

- Defined a fundamental
market model

- Special case of Walras’
model

For given prices,find optimal bundle of goods

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys.Equilibrium prices

Several buyers with different utility functions and moneys.Show equilibrium prices exist.

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?

For the third International Workshop on Internet and Network Economics

Kenneth J. Arrow

OUTLINE

- 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

- 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!!

Today’s reality

- New markets defined by Internet companies, e.g.,
- Microsoft
- 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.

“Stock prices have reached what looks likea permanently high plateau”

“Stock prices have reached what looks likea 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.

Eisenberg-Gale Program, 1959

prices pj

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”.

- Rational convex program for Arrow-Debreu
market with linear utilities.

- Polynomial time exact algorithm, using
ellipsoid method.

Long-standing open problem

- Complexity of finding an equilibrium for
Fisher and Arrow-Debreu models under

separable, plc utilities?

How do we build on solution to the linear case?

Build on combinatorial insights

- 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.

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.

- 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!

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.

market

Price discrimination market

(plc utilities)

EG convex program = Devanur’s program

Jain & V., 2007

(Proportional Fairness)

(Kelly, 1997)

Price disc.

market

Spending constraint

market

Nash Bargaining

V., 2008

EG convex program = Devanur’s program

Triumph of Combinatorial Approach toSolving Convex Programs!!

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)

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!

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

- 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.

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