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Combinatorial Algorithms for Convex Programs (Capturing Market Equilibria and Nash Bargaining Solutions). Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. What is Economics?. ‘‘Economics is the study of the use of - PowerPoint PPT Presentation
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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?
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 supplyequilibrium prices
Do markets even admitequilibrium prices?
General Equilibrium TheoryOccupied center stage in Mathematical
Economics for over a century
Do markets even admitequilibrium prices?
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 admitequilibrium prices?
Do markets even admitequilibrium prices?
Easy if only one good!
Supply-demand curves
Do markets even admitequilibrium 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
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
No LP’s known for capturing equilibrium allocations for Fisher’s model
Eisenberg-Gale convex program, 1959
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
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
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?
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
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
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 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
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 WGMV1992
Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
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?!
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
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
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
Common generalization
Is it meaningful?
Can it be solved via a combinatorial,
polynomial time algorithm?
Common generalization
Is it meaningful? Nonsymmetric ADNB
Kalai, 1975: Nonsymmetric bargaining games wi : clout of player i.
Common generalization
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?