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Algorithmic Game Theory and Internet Computing

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The “Invisible Hand of the Market”:

Algorithmic Ratification and

the Digital Economy

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

- 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.”

‘‘Economics is the study of the use of

scarce resources which have alternative uses.’’

Lionel Robbins

(1898 – 1984)

Prices!

Prices

Parity between demand and supply

Prices

Parity between demand and supplyequilibrium prices

- Pioneered general
equilibrium theory

Mathematical ratification!

- Markets should operate at equilibrium

- 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?

Do markets even admitequilibrium prices?

- Defined a fundamental
market model

- Special case of Walras’
model

amount ofj

Concave utility function

(Of buyer i for good j)

utility

- Celebrated theorem in Mathematical Economics
- Established existence of market equilibrium under
very general conditions using a deep theorem from

topology - Kakutani fixed point theorem.

- Competitive equilibrium =>
Pareto optimal allocation of resources

- Pareto optimal = impossible to make
an agent better off without making some

other agent worse off

- Every Pareto optimal allocation of resources
comes from a competitive equilibrium

(after redistribution of initial endowments).

- Nobel Prize, 1972

- Nobel Prize, 1983

- 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!

- Tatonnement process:
Price adjustment process to arrive at equilibrium

- Deficient goods: raise prices
- Excess goods: lower prices

- 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

- BEFORE THE FORMULATION OF GENERAL EQUILIBRIUM THEORY
- PARTIAL EQUILIBRIUM
- WALRAS, PARETO, AND HICKS
- SOCIALISM AND DECENTRALIZATION
- SAMUELSON AND SUCCESSORS
- 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)

- 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.

on utility functions (in Arrow-Debreu Theorem):

- Continuous, quasiconcave,
satisfying non-satiation.

- For “reasonable” utility fns.,
can market equilibrium be computed in P?

- If not, what is its complexity?

Irving Fisher, October 1929

Approximate fixed point algorithms

for market equilibria.

Not polynomial time.

Polynomial time algorithm for

Cobb-Douglas utilities.

Linear utility function

utility of i

amount of j

- Assume:
- Buyer i’s total utility,
- mi: money of buyer i.
- One unit of each good j.

prices pj

- Equilibrium simultaneously optimizes
for all agents.

- How is this done via a single objective function?

- A nonlinear convex program that
always has a rational solution,

using polynomially many bits,

if all parameters are rational.

- Eisenberg-Gale program is rational.

- Devanur, Papadimitriou, Saberi & V., 2002
By extending the primal-dual paradigm to the

setting of convex programs & KKT conditions

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.

Piecewise linear, concave

utility

Additively separable over goods

amount ofj

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

separable, plc utilities?

amount ofj

Generalize EG program to piecewise-linear, concave utilities?

utility/unit of j

utility

- V. & Yannakakis, 2007:
Equilibrium is rational for Fisher and

Arrow-Debreu models under separable,

plc utilities.

- In P??

- Chen, Dai, Du, Teng, 2009:
- PPAD-hardness for Arrow-Debreu model

- 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

- 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

V. & Yannakakis, 2009:

- Membership in PPAD for both models.

Algorithmic ratification of the

“invisible hand of the market”

What is the “right” model??

- 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!

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.

Spending constraint

market

Price discrimination market

(plc utilities)

EG convex program = Devanur’s program

Eisenberg-Gale Markets

Jain & V., 2007

(Proportional Fairness)

(Kelly, 1997)

Price disc.

market

Spending constraint

market

Nash Bargaining

V., 2008

EG convex program = Devanur’s program

Music, movies, video games, …

cell phone apps., …, web search results,

… , even ideas!

Music, movies, video games, …

cell phone apps., …, web search results,

… , even ideas!

Once produced, supply is infinite!!

‘‘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.

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.

Full rationality, infinite computing power:

not meaningful!

Full rationality, infinite computing power:

not meaningful!

e.g., song A for $1.23, song B for $1.56, …

Full rationality, infinite computing power:

not meaningful!

e.g., song A for $1.23, song B for $1.56, …

Cannot price songs individually!

- 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.

Assume 1 conventional good: bread.

Each agent has a utility function over

g digital categories and bread.

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.

- 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.

(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)