Primal-Dual Algorithms for
Rational Convex Programs II:
Algorithmic Game Theory
Dealing with Infeasibility
and Internet Computing
Vijay V. Vazirani
Georgia Tech
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
v  N (S , c),
T  S & v  T  v  N (T , c)
v2
S
v
c2
c1
v1
v  N (S , c),
T  S & v  T  v  N (T , c)
v2
S
T
v
c2
c1
v1
Thm: Unique solution satisfying 4 axioms
v2
N (S , c)  max(v1 ,v2 )S {(v1  c1 )(v2  c2 )}
S
c2
c1
v1
Generalizes to n-players
 Theorem:
Unique solution
N (S , c)  max vS {(v1  c1 ) ... (vn  cn )}
Linear Nash Bargaining

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.
linear constraints
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
ADNB

Generalize further: assume ci 's
are arbitrary, i.e., not status quo utilities.

Given game is feasible iff
vS s.t. i : vi  ci
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)
 This
convex program is rational.
 Combinatorial
polynomial time algorithm
for determining feasibility,
and if feasible, for solving it.
Game plan

Use KKT conditions to
transform ADNB to determining feasibility &
computing the equilibrium in a certain market.

Design algorithm using primal-dual paradigm.
Several buyers with
different utility functions and moneys.
Find equilibrium prices!
p1
p2
p3
Linear Fisher Market

Assume:
 Buyer
i’s total utility,
vi   uij xij
jG

mi : money of buyer i.
 One unit of each good j.
 n buyers and g goods.

Find equilibrium prices!
Flexible budget market,
only difference:

Buyers don’t spend a fixed amount of money.

Instead, they have a strict lower bound on
the utility they desire.
Flexible budget market,
only difference:

Buyers don’t spend a fixed amount of money.

Instead, they have a strict lower bound on
the utility they desire.

Money spent = f (utility desired, prices of goods)
Most cost-effective goods


At prices p, for buyer i:
Define
 p j 
Si  arg min j  
 uij 
 p j 
cost (i )  min j  
 uij 
Flexible budget market

Agent i wants utility  ci

At prices p, must spend ci . cost (i) to get utility
ci
Flexible budget market

Agent i wants utility  ci

At prices p, must spend ci . cost (i) to get utility

Define

Find market clearing prices.
mi  1  ci . cost (i)
ci
Flexible budget market

Agent i wants utility  ci

At prices p, must spend ci . cost (i) to get utility

Define

Find market clearing prices -- may not exist!!
mi  1  ci . cost (i)
ci
Flexible budget market

Agent i wants utility  ci

At prices p, must spend ci . cost (i) to get utility

Define

Find market clearing prices -- may not exist!!
mi  1  ci . cost (i)
feasible/infeasible
ci
Theorem:
ADNB
reduces to
Equilibrium for flexible budget markets
Theorem:
ADNB
reduces to
Equilibrium for flexible budget markets
(S(u), c)  M(u, c)

(S, c) is feasible iff M is feasible.

If feasible, x is Nash bargaining solution
iff x is equilibrium allocation.
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
prices pj
An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.
An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.

Equilibrium prices are unique!
Max flow in N(p)
p(1)
m(1)
m(2)
p(2)
p(3)
m(3)
m(4)
p(4)
p: equilibrium prices iff both cuts saturated
Weak Domination

p
q if
j : p j  q j  0
q(1)
m(1)
m(2)
t
m(3)
m(4)
q(2)
q(3)
q(4)
Lemma: If M is feasible, p equilibrium prices
and s is a min-cut in N(q) then p
q
s
Small prices: q s.t. s is a min-cut in N(q)
q(1)
m(1)
m(2)
t
m(3)
m(4)
q(2)
q(3)
q(4)
s
Invariant

Throughout algorithm: s is a min-cut in N(p)

i.e., algorithm works with small prices.
Feasible Prices


p: feasible if small and i : surplus(i)  1
Lemma: p feasible  i : vi  ci
Exists iff M is feasible.
Feasible Prices

p: feasible if small and i : surplus(i)  1

Lemma: p feasible  i : vi  ci
Exists iff M is feasible.

Observe: Notion defined via balanced flow!
1-surplus
i  surplus(i) - 1

Because of Invariant, surplus(i)  0

Hence, i   1

p is feasible iff p is small and i : i  0
M is feasible iff


maxvS mini vi ci  0
Write this as an LP
M is feasible iff t > 0
max t
s.t.
i :
u x
jG
j :
x
iB
i, j :
ij ij
ij
 ci  t
 1
xij  0
Proof of infeasibility: dual soln.  0
max t
s.t.
i :
u x
jG
j :
x
iB
i, j :
ij ij
ij
 ci  t
 1
xij  0
Proof of infeasibility

Lemma: If

i
i
 0 and
p
j
j
then M is infeasible.
0
Proof of infeasibility

Lemma: If

i
i
 0 and
p
j
0
j
then M is infeasible.

Observe:
If p  0, then i : surplus(i) = 0, hence i  0
Initialization




i : mi  1
Find equilibrium prices, p, for this linear
Fisher market.
i : mi  1 ci . cost(i)
Invariant holds.
Search
Allocations
Prices (Money)
Infeasible
Feasible
?
Decision
Allocations
Prices (Money)

Feasible:
i : i  0

Infeasible:

i
i
 0 and
p
j
j
0
Network N(p)
N’(I, J)
i  0
i  0
J
I
N - N’
Network N(p)
N’(I, J)
i  0
i  0
J
I
N - N’
Network N(p)
N’(I, J)
i  0
i  0
J
I
N - N’
Best viewed as a tug-of-war
between 2 teams of buyers!
GOOD:
i s.t. i  0
NOT GOOD:
i s.t. i  0
GOOD
NOT GOOD
i  min
rest i  0
i  0
I
J
i  min
rest i  0
i  0
I
J
i  min
rest i  0
i  0
I
J
i  min
rest i  0
i  0
I
J
Decision takes polynomial time

Use balanced flow!


i2
i ( B1  B2 )
1
 Decreases by
fraction in each iteration.
2
n
Search
Allocations
Prices (Money)
Infeasible
Feasible
?
Decision
Allocations
Prices (Money)
Network N(p)
N - N’
I
N’(I, J)
J
p  implies mi 
mi  1 ci . cost(i)

What about surplus(i)?
p  implies mi 
mi  1 ci . cost(i)


What about surplus(i)?
surplus(i) decreases iff
i  0
equilibrium
init
feasible
equilibrium
?
init
feasible
Feasibility for LP’s is easy!

Non-total problems just as easy as total ones.

e.g., max. wt. perfect matching
vs
max. wt. matching.
Theorem: Algorithm runs in polynomial time.
Theorem: Algorithm runs in polynomial time.
Q: Find strongly polynomial algorithm!
Open

Can rational convex programs be solved
in polynomial time using an LP-solver?
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 & Economics:
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 & Economics:
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