Algorithmic Game Theory and Internet Computing
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Transcript of Algorithmic Game Theory and Internet Computing
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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)
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What is Economics?
‘‘Economics is the study of the use of
scarce resources which have alternative uses.’’
Lionel Robbins
(1898 – 1984)
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How are scarce resources assigned to alternative uses?
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How are scarce resources assigned to alternative uses?
Prices!
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How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supply
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How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supplyequilibrium prices
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Do markets even admitequilibrium prices?
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General Equilibrium TheoryOccupied center stage in Mathematical
Economics for over a century
Do markets even admitequilibrium prices?
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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.
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Do markets even admitequilibrium prices?
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Do markets even admitequilibrium prices?
Easy if only one good!
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Supply-demand curves
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Do markets even admitequilibrium prices?
What if there are multiple goods and multiple buyers with diverse desires
and different buying power?
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Irving Fisher, 1891
Defined a fundamental
market model
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0i ij ij ijj G
v u x x
linear utilities
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For given prices,find optimal bundle of goods
1p 2p3p
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Several buyers with different utility functions and moneys.
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Several buyers with different utility functions and moneys.
Find equilibrium prices.
1p 2p3p
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“Stock prices have reached what looks like
a permanently high plateau”
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“Stock prices have reached what looks like
a permanently high plateau”
Irving Fisher, October 1929
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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!
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An almost entirely non-algorithmic theory!
General Equilibrium Theory
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The new face of computing
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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
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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using the primal-dual paradigm
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Combinatorial algorithm
Conducts an efficient search over
a discrete space.
E.g., for LP: simplex algorithm
vs
ellipsoid algorithm or interior point algorithms.
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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.
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No LP’s known for capturing equilibrium allocations for Fisher’s model
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No LP’s known for capturing equilibrium allocations for Fisher’s model
Eisenberg-Gale convex program, 1959
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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
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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
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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
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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
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Why remarkable?
Equilibrium simultaneously optimizes
for all agents.
How is this done via a single objective function?
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Theorem
If all parameters are rational, Eisenberg-Gale
convex program has a rational solution! Polynomially many bits in size of instance
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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.
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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
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Idea of algorithm
primal variables: allocations dual variables: prices of goods iterations:
execute primal & dual improvements
Allocations Prices (Money)
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How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supply
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Yin & Yang
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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.
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Example
Two players, 1 and 2, have vacation homes:
1: in the mountains
2: on the beach
Consider all possible ways of sharing.
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Utilities derived jointly
1v
2v
S : convex + compact
feasible set
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Disagreement point = status quo utilities
1v
2v
1c
2c
S
Disagreement point = 1 2( , )c c
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Nash bargaining problem = (S, c)
1v
2v
1c
2c
S
Disagreement point = 1 2( , )c c
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Nash bargaining
Q: Which solution is the “right” one?
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Solution must satisfy 4 axioms:
Paretto optimality
Invariance under affine transforms
Symmetry
Independence of irrelevant alternatives
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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
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1v
2v
1c
2c
v
S
( , ),
& ( , )
v N S c
T S v T v N T c
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1v
2v
1c
2c
v
S
T
( , ),
& ( , )
v N S c
T S v T v N T c
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Generalizes to n-players
Theorem: Unique solution
1 1( , ) max {( ) ... ( )}v S n nN S c v c v c
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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
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Q: Compute solution combinatoriallyin polynomial time?
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Game-theoretic properties of LNB games
Chakrabarty, Goel, V. , Wang & Yu, 2008:
Fairness Efficiency (Price of bargaining)Monotonicity
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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
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How should they exchange their goods?
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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
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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
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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
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Theorem (V., 2008)
If all parameters are rational,
solution to ADNB is rational! Polynomially many bits in size of instance
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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.
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Flexible budget markets
Natural variant of linear Fisher markets
ADNB flexible budget markets
Primal-dual algorithm for finding an
equilibrium
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How is primal-dual paradigm
adapted to nonlinear setting?
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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.
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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
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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
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Primal-dual algorithms so far(i.e., LP-based)
Raise dual variables greedily. (Lot of effort spent
on designing more sophisticated dual processes.)
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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.
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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.
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Our algorithm
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
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Our algorithm
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
New algorithmic ideas needed!
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Nonlinear programs with rational solutions!
Open
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Nonlinear programs with rational solutions!
Solvable combinatorially!!
Open
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Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
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Exact Algorithms for Cornerstone Problems in P
Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching
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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
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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
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Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
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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
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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?!
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Goel & V., 2009:
ADNB with piecewise-linear, concave utilities
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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
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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
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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
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Common generalization
Is it meaningful?
Can it be solved via a combinatorial,
polynomial time algorithm?
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Common generalization
Is it meaningful? Nonsymmetric ADNB
Kalai, 1975: Nonsymmetric bargaining games wi : clout of player i.
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Common generalization
Is it meaningful? Nonsymmetric ADNB
Kalai, 1975: Nonsymmetric bargaining games wi : clout of player i.
Algorithm
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Open
Can Fisher’s linear case
or ADNB
be captured via an LP?