Combinatorial Algorithms for
Convex Programs
Algorithmic
Game
Theory
(Capturing Market Equilibria and
and
Internet
Computing
Nash Bargaining Solutions)
Vijay V. Vazirani
Georgia Tech
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 supply
How are scarce resources assigned
to alternative uses?
Prices
Parity between demand and supply
equilibrium prices
Do markets even admit
equilibrium prices?
Do markets even admit
equilibrium prices?
General Equilibrium Theory
Occupied 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.
Do markets even admit
equilibrium prices?
Do markets even admit
equilibrium prices?
Easy if only one good!
Supply-demand curves
Do markets even admit
equilibrium prices?
What if there are multiple goods and
multiple buyers with diverse desires
and different buying power?
Irving Fisher, 1891

Defined a fundamental
market model
linear utilities
vi   uij xij
jG
xij  0
For given prices,
find optimal bundle of goods
p1
p2
p3
Several buyers with
different utility functions and moneys.
Several buyers with
different utility functions and moneys.
Find equilibrium prices.
p1
p2
p3
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!
General Equilibrium Theory
An almost entirely
non-algorithmic theory!
The new face of computing
Today’s reality

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.
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

No LP’s known for capturing equilibrium
allocations for Fisher’s model

Eisenberg-Gale convex program, 1959
Eisenberg-Gale Program, 1959
max  mi log vi
i
s.t.
i : vi   j u ij x ij
 x 1
ij : x  0
j :
ij
i
ij
Eisenberg-Gale Program, 1959
max  mi log vi
i
s.t.
i : vi   j u ij x ij
 x 1
ij : x  0
j :
ij
i
ij
prices pj

No LP’s known for capturing equilibrium
allocations for Fisher’s model

Eisenberg-Gale convex program, 1959

Extended primal-dual paradigm to
solving a nonlinear convex program
Theorem

If all parameters are rational, Eisenberg-Gale
convex program has a rational solution!

Polynomially many bits in size of instance
Theorem

If all parameters are rational, Eisenberg-Gale
convex program has a rational solution!


Polynomially many bits in size of instance
Combinatorial polynomial time algorithm
for finding it.
Theorem

If all parameters are rational, Eisenberg-Gale
convex program has a rational solution!


Polynomially many bits in size of instance
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)
How are scarce resources assigned
to alternative uses?
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
v2
S : convex + compact
feasible set
v1
Disagreement point = status quo utilities
v2
S
c2
c1
Disagreement point = (c1 , c2 )
v1
Nash bargaining problem = (S, c)
v2
S
c2
c1
Disagreement point = (c1 , c2 )
v1
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
N ( S , c)  max ( v1 ,v2 )S {(v1  c1 )(v2  c2 )}
v2
S
c2
c1
v1
Generalizes to n-players
 Theorem:
Unique solution
N (S , c)  max vS {(v1  c1 ) ... (vn  cn )}
Linear Nash Bargaining (LNB)

Feasible set is a polytope defined by
linear packing constraints

Nash bargaining solution is
optimal solution to convex program:
max  log(vi  ci )
i
s.t.
packing constraints
Q: Compute solution combinatorially
in polynomial time?
How should they exchange
their goods?
State as a Nash bargaining game
u f : (.,.,.)  R

ub : (.,.,.)  R 
um : (.,.,.)  R

c f  u f (1, 0, 0)
cb  ub (0, 1, 0)
cm  um (0, 0,1)
S = utility vectors obtained by distributing
goods among players
Special case: linear utility functions
u f : (.,.,.)  R

ub : (.,.,.)  R 
um : (.,.,.)  R

c f  u f (1, 0, 0)
cb  ub (0, 1, 0)
cm  um (0, 0,1)
S = utility vectors obtained by distributing
goods among players
Convex program for ADNB
max  log(vi  ci )
i
s.t.
i : vi   j u ij x ij
j :
ij :
 x 1
x 0
i
ij
ij
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 between
LP’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.j : p j  0
2.j : p j  0   i xij  1
uij
vi
3.i, j :

p j m(i )
uij
vi
4.i, j : xij  0 

p j m(i )
KKT conditions
1.j : p j  0
2.j : p j  0   i xij  1
uij
vi
3.i, j :

p j m(i )
uij
vi
4.i, j : xij  0 


p j m(i )

u
x
ij
ij
j
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

Raise dual variables greedily. (Lot of effort spent
on designing more sophisticated dual processes.)
 Only
exception: Edmonds, 1965: algorithm
for max 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 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 loose
 Because
of enhanced KKT conditions
Our algorithm

Dual variables (prices) are raised greedily

Yet, primal objects go tight and loose
 Because

of enhanced KKT conditions
New algorithmic ideas needed!
Open
Nonlinear programs
with rational solutions!
Open
Nonlinear programs
with rational solutions!
Solvable combinatorially!!
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
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
Approximation Algorithms
set cover
Steiner tree
Steiner network
k-MST
scheduling . . .
facility location
k-median
multicut
feedback vertex set
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?!
Convex program for ADNB
max  log(vi  ci )
i
s.t.
i : vi   j u ij x ij
j :
ij :
 x 1
x 0
i
ij
ij
Eisenberg-Gale Program, 1959
max  mi log vi
i
s.t.
i : vi   j u ij x ij
 x 1
ij : x  0
j :
ij
i
ij
Common generalization
max  wi log(vi  ci )
i
s.t.
i : vi   j u ij x ij
j :
ij :
 x 1
x 0
i
ij
ij
Common generalization

Is it meaningful?

Can it be solved via a combinatorial,
polynomial time algorithm?
Common generalization

Is it meaningful?

Kalai, 1975: Nonsymmetric bargaining games

wi : clout of player i.
Nonsymmetric ADNB
Common generalization

Is it meaningful?

Kalai, 1975: Nonsymmetric bargaining games


wi : clout of player i.
Algorithm
Nonsymmetric ADNB