algorithmic game theory and internet computing

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

Combinatorial Algorithms for

Convex Programs

(Capturing Market Equilibria and

Nash Bargaining Solutions)

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?


Prices!


Prices

Parity between demand and supply


Prices

Parity between demand and supplyequilibrium prices

Do markets even admitequilibrium prices?

General Equilibrium TheoryOccupied center stage in Mathematical

Economics for over a century


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.


Easy if only one good!

Supply-demand curves


What if there are multiple goods and multiple buyers with diverse desires

and different buying power?

Irving Fisher, 1891

Defined a fundamental

market model

0i ij ij ijj G

v u x x

linear utilities

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

“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

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!

An almost entirely non-algorithmic theory!

General Equilibrium Theory

The new face of computing

New markets defined by Internet companies, e.g., Google eBay Yahoo! Amazon

Massive computing power available.

Need an inherenltly-algorithmic theory of

markets and market equilibria.

Today’s reality

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using the primal-dual paradigm

Combinatorial algorithm

Conducts an efficient search over

a discrete space.

E.g., for LP: simplex algorithm

vs

ellipsoid algorithm or interior point algorithms.

Combinatorial algorithm

Conducts an efficient search over

a discrete space.

E.g., for LP: simplex algorithm

vs

ellipsoid algorithm or interior point algorithms.

Yields deep insights into structure.

No LP’s known for capturing equilibrium allocations for Fisher’s model


Eisenberg-Gale convex program, 1959


Eisenberg-Gale convex program, 1959

Extended primal-dual paradigm to

solving a nonlinear convex program

Linear Fisher Market

B = n buyers, money mi for buyer i

G = g goods, w.l.o.g. unit amount of each good : utility derived by i

on obtaining one unit of j Total utility of i,

Find market clearing prices.

i ij ijj

U u xiju

[0,1]

i ij ijj

ij

v u xx

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


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?

Theorem

If all parameters are rational, Eisenberg-Gale

convex program has a rational solution! Polynomially many bits in size of instance

Theorem



Combinatorial polynomial time algorithm

for finding it.

Theorem



Combinatorial polynomial time algorithm

for finding it.

Discrete space

Idea of algorithm

primal variables: allocations dual variables: prices of goods iterations:

execute primal & dual improvements

Allocations Prices (Money)


Prices

Parity between demand and supply

Yin & Yang

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.

Complete information game.

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:

Paretto 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

1v

2v

1c

2c

v

S

( , ),

& ( , )

v N S c

T S v T v N T c

1v

2v

1c

2c

v

S

T

( , ),

& ( , )

v N S c

T S v T v N T c

Generalizes to n-players

Theorem: Unique solution

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

Linear Nash Bargaining (LNB)

Feasible set is a polytope defined by

linear constraints

Nash bargaining solution is

optimal solution to convex program:

max log( )

. .

i ii

v c

s t

linear constraints

Q: Compute solution combinatoriallyin polynomial time?

Game-theoretic properties of LNB games

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

Fairness Efficiency (Price of bargaining)Monotonicity

Insights into markets

V., 2005: spending constraint utilities (Adwords market)

Megiddo & V., 2007: continuity properties

V. & Yannakakis, 2009: piecewise-linear, concave utilities Nisan, 2009: Google’s auction for TV ads

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

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)

If all parameters are rational,

solution to ADNB is rational! Polynomially many bits in size of instance

Theorem (V., 2008)

If all parameters are rational, solution to ADNB is rational!

Polynomially many bits in size of instance

Combinatorial polynomial time algorithm for finding it.

Flexible budget markets

Natural variant of linear Fisher markets

ADNB flexible budget markets

Primal-dual algorithm for finding an

equilibrium

How is primal-dual paradigm

adapted to nonlinear setting?

Fundamental difference betweenLP’s and convex programs

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 iij

j

j p

j p x

u vi j

p m i

u vi j x

p m i

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( ) ( )

j

j iji

ij i

j

ij ijij jiij

j

j p

j p x

u vi j

p m i

u xu vi j x

p m i m i

Primal-dual algorithms so far(i.e., LP-based)

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Primal-dual algorithms so far



Only exception: Edmonds, 1965: algorithm

for max weight matching.

Primal-dual algorithms so far



Only exception: Edmonds, 1965: algorithm

for max weight matching.

Otherwise primal objects go tight and loose.

Difficult to account for these reversals --

in the running time.

Our algorithm

Dual variables (prices) are raised greedily

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

Our algorithm

Dual variables (prices) are raised greedily

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

New algorithmic ideas needed!

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

Exact Algorithms for Cornerstone Problems in P

Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching




Approximation Algorithms (1990’s):

Near-optimal integral solutions to LP’s





Near-optimal integral solutions to LP’s WGMV1992

Approximation Algorithms

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .






Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs






Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs

Approximation algorithms for convex programs?!

Goel & V., 2009:

ADNB with piecewise-linear, concave utilities

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


max log

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

m v

s t

i v

j

ij

u xx

x

Common generalization

max log( )

. .

:

: 1

: 0

i i ii

i ij ijj

iji

ij

w v c

s t

i v

j

ij

u xx

x


Is it meaningful?

Can it be solved via a combinatorial,

polynomial time algorithm?


Is it meaningful? Nonsymmetric ADNB

Kalai, 1975: Nonsymmetric bargaining games wi : clout of player i.


Is it meaningful? Nonsymmetric ADNB

Kalai, 1975: Nonsymmetric bargaining games wi : clout of player i.

Algorithm

Open

Can Fisher’s linear case

or ADNB

be captured via an LP?

algorithmic game theory and internet computing

Documents

equilibrium allocations

celebrated theorem

alternative uses

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market equilibria

arrowdebreu theorem

centurydo markets

market cle