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 Approximation
Algorithms for Convex Programs?!
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Rational convex program
Always has a rational solution,
using polynomially many bits,
if all parameters are rational.
Some important problems in mathematical
economics and game theory are captured by
rational (nonlinear) convex programs.
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A recent development
Combinatorial exact algorithms for
these problems and hence for optimally
solving their convex programs.
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General equilibrium theory
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A central tenet
Prices are such that demand equals supply, i.e.,
equilibrium prices.
Easy if only one good
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Supply-demand curves
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Irving Fisher, 1891
Defined a fundamental
market model
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utility
Concave utility function
(for good j)
amount of j
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( )i ij ijj G
u f x
total utility
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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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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm
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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm
Solves Eisenberg-Gale convex program
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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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Why seek combinatorial algorithms?
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Why seek combinatorial algorithms?
Structural insightsHave led to progress on related problemsBetter understanding of solution concept
Useful in applications
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Auction for Google’s TV ads
N. Nisan et. al, 2009:
Used market equilibrium based approach.
Combinatorial algorithms for linear case
provided “inspiration”.
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utility
Piecewise linear, concave
amount of j
Additively separable over goods
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Long-standing open problem
Complexity of finding an equilibrium for
Fisher and Arrow-Debreu models under
separable, plc utilities?
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How do we build on solution to linear case?
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utility
amount of j
Generalize EG program to piecewise-linear, concave utilities?
ijkl
ijkuutility/unit of j
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,
,
max log
. .
:
: 1
:
: 0
i ii
i ijk ijkj k
iji k
ijk ijk
ijk
m v
s t
i v
j
ijk
ijk
u xx
x l
x
Generalization of EG program
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,
,
max log
. .
:
: 1
:
: 0
i ii
i ijk ijkj k
iji k
ijk ijk
ijk
m v
s t
i v
j
ijk
ijk
u xx
x l
x
Generalization of EG program
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Long-standing open problem
Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?
2009: Both PPAD-complete (using combinatorial insights from [DPSV])
Chen, Dai, Du, TengChen, TengV., Yannakakis
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utility
Piecewise linear, concave
amount of j
Additively separable over goods
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What makes linear utilities easy?
Weak gross substitutability:
Increasing price of one good cannot
decrease demand of another.
Piecewise-linear, concave utilities do not
satisfy this.
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rate
rate = utility/unit amount of j
amount of j
Differentiate
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rate
amount of j
rate = utility/unit amount of j
money spent on j
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rate
rate = utility/unit amount of j
money spent on j
Spending constraint utility function
$20 $40 $60
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Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability
2). Polynomial time algorithm for computing equilibrium.
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An unexpected fallout!!
Has applications to
Google’s AdWords Market!
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rate
rate = utility/click
money spent on keyword j
Application to Adwords market
$20 $40 $60
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Is there a convex program for this model?
“We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”
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,
max log
. .
:
:
: 0
ijij
i j j
j iji
ij ij
ij
ub
p
s t
j p b
i b m
ij b
Devanur’s program for linear Fisher
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max log
. .
:
:
:
: 0
ijkijk
ijk j
j ijkik
ijk ijk
ijk ijk
ijk
ub
p
s t
j p b
i b m
ijk b l
ijk b
C. P. for spending constraint!
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EG convex program = Devanur’s program
Fisher marketwith plc utilities
Spending constraint market
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Price discrimination markets
Business charges different prices from different
customers for essentially same goods or services.
Goel & V., 2009:
Perfect price discrimination market.
Business charges each consumer what
they are willing and able to pay.
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plc utilities
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Middleman buys all goods and sells to buyers,
charging according to utility accrued.Given p, there is a well defined rate for each buyer.
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Middleman buys all goods and sells to buyers,
charging according to utility accrued.Given p, there is a well defined rate for each buyer.
Equilibrium is captured by a convex program Efficient algorithm for equilibrium
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Middleman buys all goods and sells to buyers,
charging according to utility accrued.Given p, there is a well defined rate for each buyer.
Equilibrium is captured by a convex program Efficient algorithm for equilibrium
Market satisfies both welfare theorems!
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,
,
max log
. .
:
: 1
:
: 0
i ii
i ijk ijkj k
iji k
ijk ijk
ijk
m v
s t
i v
j
ijk
ijk
u xx
x l
x
Generalization of EG program works!
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EG convex program = Devanur’s program
Price discrimination market(plc utilities)
Spending constraint market
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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.
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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:
Pareto 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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Generalizes to n-players
Theorem: Unique solution
1 1( , ) max {( ) ... ( )}v S n nN S c v c v c
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Nash bargaining solution is
optimal solution to convex program:
max log( )
. .
i ii
v c
s t v S
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Nash bargaining solution is
optimal solution to convex program:
Polynomial time separation oracle
max log( )
. .
i ii
v c
s t v S
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Q: Compute solution combinatoriallyin polynomial time?
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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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ADNB
B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players
0i ij ij ijj G
v u x x
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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)
Nash bargaining program is rational.
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Theorem (V., 2008)
Nash bargaining solution is rational.
Combinatorial polynomial time algorithm
for finding it.
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Game-theoretic properties of NB games -- “stress tests”
Chakrabarty, Goel, V. , Wang & Yu, 2008:
Efficiency (Price of bargaining)Fairness Full competitiveness
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An application (Lucent)
“fair” throughput problem on a
wireless channel.
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EG convex program = Devanur’s program
Price disc. market
Spending constraint market
ADNB
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EG convex program = Devanur’s program
Price disc. market
Spending constraint market
Kelly, 1997: proportional fairness Jain & V., 2007: Eisenberg-Gale markets
ADNB
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A new development
Orlin, 2009: Strongly polynomial algorithm
for Fisher’s linear case.
Open: For rest.
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AGT’s gift to theory of algorithms!
New complexity classes: PPAD, FIXPStudy complexity of total problems
A new algorithmic directionCombinatorial algorithms for convex programs
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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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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
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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Extending primal-dual paradigm to framework of
convex programs and KKT conditions
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Eisenberg-Gale Program, 1959
max ( ) log
. .
:
: 1
: 0
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i ij ijj
iji
ij
m i u
s t
i u
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u xx
x
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Main point of departure
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( )
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Primal-dual algorithms so far
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 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 weight matching.
Otherwise primal objects go tight and loose.
Difficult to account for these reversals
in the running time.
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Our algorithms
Dual variables (prices) are raised greedily
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Our algorithms
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
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Our algorithms
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
New algorithmic ideas are needed!