Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

Combinatorial Approximation

Algorithms for Convex Programs?!

Rational convex program

Always has a rational solution,

using polynomially many bits,

if all parameters are rational.

Some important problems in mathematical

economics and game theory are captured by

rational (nonlinear) convex programs.

A recent development

Combinatorial exact algorithms for

these problems and hence for optimally

solving their convex programs.

General equilibrium theory

A central tenet

Prices are such that demand equals supply, i.e.,

equilibrium prices.

Easy if only one good

Supply-demand curves

Irving Fisher, 1891

Defined a fundamental

market model

utility

Concave utility function

(for good j)

amount of j

( )i ij ijj G

u f x

total utility

For given prices,find optimal bundle of goods

1p 2p3p

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys.

Find equilibrium prices.

1p 2p3p

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual paradigm

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual paradigm

Solves Eisenberg-Gale convex program

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?

Why seek combinatorial algorithms?

Why seek combinatorial algorithms?

Structural insightsHave led to progress on related problemsBetter understanding of solution concept

Useful in applications

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 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

Long-standing open problem

Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?

2009: Both PPAD-complete (using combinatorial insights from [DPSV])

Chen, Dai, Du, TengChen, TengV., Yannakakis

utility

Piecewise linear, concave

amount of j

Additively separable over goods

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.

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!

EG convex program = Devanur’s program

Price discrimination market(plc utilities)

Spending constraint market

Nash bargaining game, 1950

Captures the main idea that both players

gain if they agree on a solution.

Else, they go back to status quo.

Example

Two players, 1 and 2, have vacation homes:

1: in the mountains

2: on the beach

Consider all possible ways of sharing.

Utilities derived jointly

1v

2v

S : convex + compact

feasible set

Disagreement point = status quo utilities

1v

2v

1c

2c

S

Disagreement point = 1 2( , )c c

Nash bargaining problem = (S, c)

1v

2v

1c

2c

S

Disagreement point = 1 2( , )c c

Nash bargaining

Q: Which solution is the “right” one?

Solution must satisfy 4 axioms:

Pareto optimality

Invariance under affine transforms

Symmetry

Independence of irrelevant alternatives

Thm: Unique solution satisfying 4 axioms

1 2( , ) 1 1 2 2( , ) max {( )( )}v v SN S c v c v c

1v

2v

1c

2c

S

Generalizes to n-players

Theorem: Unique solution

1 1( , ) max {( ) ... ( )}v S n nN S c v c v c

Nash bargaining solution is

optimal solution to convex program:

max log( )

. .

i ii

v c

s t v S

Nash bargaining solution is

optimal solution to convex program:

Polynomial time separation oracle

max log( )

. .

i ii

v c

s t v S

Q: Compute solution combinatoriallyin polynomial time?

How should they exchange their goods?

State as a Nash bargaining game

: (.,.,.)

: (.,.,.)

: (.,.,.)

f

b

m

u R

u R

u R

(1, 0,0)

(0, 1,0)

(0, 0,1)

f f

b b

m m

c u

c u

c u

S = utility vectors obtained by distributing goods among players

Special case: linear utility functions

: (.,.,.)

: (.,.,.)

: (.,.,.)

f

b

m

u R

u R

u R

(1, 0,0)

(0, 1,0)

(0, 0,1)

f f

b b

m m

c u

c u

c u

S = utility vectors obtained by distributing goods among players

ADNB

B: n players with disagreement points, ci

G: g goods, unit amount each

S = utility vectors obtained by distributing

goods among players

0i ij ij ijj G

v u x x

Convex program for ADNB

max log( )

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

v c

s t

i v

j

ij

u xx

x

Theorem (V., 2008)

Nash bargaining program is rational.

Theorem (V., 2008)

Nash bargaining solution is rational.

Combinatorial polynomial time algorithm

for finding it.

Game-theoretic properties of NB games -- “stress tests”

Chakrabarty, Goel, V. , Wang & Yu, 2008:

Efficiency (Price of bargaining)Fairness Full competitiveness

An application (Lucent)

“fair” throughput problem on a

wireless channel.

EG convex program = Devanur’s program

Price disc. market

Spending constraint market

ADNB

EG convex program = Devanur’s program

Price disc. market

Spending constraint market

Kelly, 1997: proportional fairness Jain & V., 2007: Eisenberg-Gale markets

ADNB

A new development

Orlin, 2009: Strongly polynomial algorithm

for Fisher’s linear case.

Open: For rest.

AGT’s gift to theory of algorithms!

New complexity classes: PPAD, FIXPStudy complexity of total problems

A new algorithmic directionCombinatorial algorithms for convex programs

Nonlinear programs with rational solutions!

Open

Nonlinear programs with rational solutions!

Solvable combinatorially!!

Open

Primal-Dual Paradigm

Combinatorial Optimization (1960’s & 70’s):

Integral optimal solutions to LP’s

Primal-Dual Paradigm

Combinatorial Optimization (1960’s & 70’s):

Integral optimal solutions to LP’s

Approximation Algorithms (1990’s):

Near-optimal integral solutions to LP’s

Primal-Dual Paradigm

Combinatorial Optimization (1960’s & 70’s):

Integral optimal solutions to LP’s

Approximation Algorithms (1990’s):

Near-optimal integral solutions to LP’s

Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs

Primal-Dual Paradigm

Combinatorial Optimization (1960’s & 70’s):

Integral optimal solutions to LP’s

Approximation Algorithms (1990’s):

Near-optimal integral solutions to LP’s

Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs

Approximation algorithms for convex programs?!

Extending primal-dual paradigm to framework of

convex programs and KKT conditions

Eisenberg-Gale Program, 1959

max ( ) log

. .

:

: 1

: 0

ii

i ij ijj

iji

ij

m i u

s t

i u

j

ij

u xx

x

Main point of departure

Complementary slackness conditions:

involve primal or dual variables, not both.

KKT conditions: involve primal and dual

variables simultaneously.

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

j

j iji

ij i

j

ij ijij jij

j

j p

j p x

u ui j

p m i

u xui j x

p m i

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Only exception: Edmonds, 1965: algorithm

for weight matching.

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Only exception: Edmonds, 1965: algorithm

for weight matching.

Otherwise primal objects go tight and loose.

Difficult to account for these reversals

in the running time.

Our algorithms

Dual variables (prices) are raised greedily

Our algorithms

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

Our algorithms

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

New algorithmic ideas are needed!