Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

Market Equilibrium:

The Quest for the “Right” Model

Arrow-Debreu Theorem, 1954

Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

Provides a mathematical ratification of Adam Smith’s “invisible hand of the market”.

Need algorithmic ratification!!

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?

Irving Fisher, 1891

Defined a fundamental

market model

Special case of Walras’

model

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys.

Find equilibrium prices.

1p 2p3p

Linear Fisher Market

DPSV, 2002:

First polynomial time algorithm

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.

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”.

utility

Piecewise linear, concave

amount of j

Additively separable over goods

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?

utility

amount of j

Generalize EG program to piecewise-linear, concave utilities?

ijkl

ijkuutility/unit of j

,

,

max log

. .

:

: 1

:

: 0

i ii

i ijk ijkj k

iji k

ijk ijk

ijk

m v

s t

i v

j

ijk

ijk

u xx

x l

x

Generalization of EG program

,

,

max log

. .

:

: 1

:

: 0

i ii

i ijk ijkj k

iji k

ijk ijk

ijk

m v

s t

i v

j

ijk

ijk

u xx

x l

x

Generalization of EG program

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

Given prices p, are they equilibrium prices?

Build on combinatorial insights

utility

amount of j

Case 1

ijx

fully allocated

partially allocated

utility

amount of j

Case 2: no p.a. segment

ijx

fully allocated

p full & partial segments

Network N(p)

Theorem: p equilibrium prices iff

max-flow in N(p) = unspent money.

Network N(p)

m’(1)

m’(2)

m’(3)

m’(4)

p’(1)

p’(2)

p’(3)

p’(4)

partially allocated segments

st

LP for max-flow in N(p); variables = fe’s

Next, let p be variables!

“Guess” full & partial segments – gives N(p)

Write max-flow LP -- it is still linear! variables = fe’s & pj’s

Rationality proof

If “guess” is correct,

at optimality, pj’s are equilibrium prices.

Hence rational!

Rationality proof

If “guess” is correct,

at optimality, pj’s are equilibrium prices.

Hence rational!

In P??

Markets with piecewise-linear, concave utilities

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

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,

Sufficient condition: non-satiation.

Markets with piecewise-linear, concave utilities

V, & Yannakakis, 2009: Membership in PPAD for both models,

Sufficient condition: non-satiation

Otherwise, for both models, deciding existence of

equilibrium is NP-hard!

Markets with piecewise-linear, concave utilities – same for approx. eq.

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 modelMembership in PPAD for both models,

Sufficient condition: non-satiationOtherwise, deciding existence of eq. is NP-hard

Markets with piecewise-linear, concave utilities

V, & Yannakakis, 2009: Membership in PPAD for both models.

Path-following algorithm??

NP-hardness does not apply

Megiddo, 1988:Equilibrium NP-hard => NP = co-NP

Papadimitriou, 1991: PPAD2-player Nash equilibrium is PPAD-completeRational

Etessami & Yannakakis, 2007: FIXP3-player Nash equilibrium is FIXP-completeIrrational

FIXP = Problems whose solutions are

fixed points of Brouwer functions.

(Given as polynomially computable algebraic

circuits over the basis {+,-,*,/,max,min}.)

Etessami & Yannakakis, 2007:

PPAD = Restriction of FIXP to

piecewise-linear Brouwer functions.

M: market instance

Prove: PPAD alg. for finding equilibrium for M

Geanakoplos, 2003: Brouwer function-based

proof of existence of equilibrium

Piecewise-linear Brouwer function??

Instead …

F: correspondence in Kakutani-based proof.

G: piecewise-linear Brouwer approximation of F

(p*, x*): fixed point of G.

Can be found in PPAD.

Instead …

F: correspondence in Kakutani-based proof.

G: piecewise-linear Brouwer approximation of F

(p*, x*): fixed point of G.

Can be found in PPAD

Yields “guess”

Solve rationality LP to find equilibrium for M!

How do we salvage the situation??

Algorithmic ratification of the

“invisible hand of the market”

Is PPAD really hard??

What is the “right” model??

Linear Fisher Market

DPSV, 2002:

First polynomial time algorithm

Extend to separable, plc utilities??

What makes linear utilities easy?

Weak gross substitutability:

Increasing price of one good cannot

decrease demand of another.

Piecewise-linear, concave utilities do not

satisfy this.

utility

Piecewise linear, concave

amount of j

rate

rate = utility/unit amount of j

amount of j

Differentiate

rate

amount of j

rate = utility/unit amount of j

money spent on j

rate

rate = utility/unit amount of j

money spent on j

Spending constraint utility function

$20 $40 $60

Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability

2). Polynomial time algorithm for computing equilibrium.

An unexpected fallout!!

Has applications to

Google’s AdWords Market!

rate

rate = utility/click

money spent on keyword j

Application to Adwords market

$20 $40 $60

Is there a convex program for this model?

“We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

,

max log

. .

:

:

: 0

ijij

i j j

j iji

ij ij

ij

ub

p

s t

j p b

i b m

ij b

Devanur’s program for linear Fisher

max log

. .

:

:

:

: 0

ijkijk

ijk j

j ijkik

ijk ijk

ijk ijk

ijk

ub

p

s t

j p b

i b m

ijk b l

ijk b

C. P. for spending constraint!

EG convex program = Devanur’s program

Fisher marketwith plc utilities

Spending constraint market

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, there is a well defined rate for each buyer.

Middleman buys all goods and sells to buyers,

charging according to utility accrued.Given p, there is a well defined rate for each buyer.

Equilibrium is captured by a convex program Efficient algorithm for equilibrium

Middleman buys all goods and sells to buyers,

charging according to utility accrued.Given p, there is a well defined rate for each buyer.

Equilibrium is captured by a convex program Efficient algorithm for equilibrium

Market satisfies both welfare theorems!

,

,

max log

. .

:

: 1

:

: 0

i ii

i ijk ijkj k

iji k

ijk ijk

ijk

m v

s t

i v

j

ijk

ijk

u xx

x l

x

Generalization of EG program works!

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.

EG convex program = Devanur’s program

Price discrimination market(plc utilities)

Spending constraint market

EG convex program = Devanur’s program

Price disc. market

Spending constraint market

Nash BargainingV., 2008

Eisenberg-Gale MarketsJain & V., 2007

(Proportional Fairness)(Kelly, 1997)

Triumph of Combinatorial Approach toSolving Convex Programs!!

A new development

Orlin, 2009: Strongly polynomial algorithm

for Fisher’s linear case.

Open: For rest.

AGT&E’s gift to theory of algorithms!

New complexity classes: PPAD, FIXPStudy complexity of total problems

A new algorithmic directionCombinatorial algorithms for convex programs