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Algorithms for the Linear Case, and Beyond …. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Irving Fisher, 1891. Defined a fundamental market model Special case of Walras’ model. - PowerPoint PPT Presentation
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Georgia Tech
Algorithms for the Linear Case,
and Beyond …
Irving Fisher, 1891
Defined a fundamental
market model
Special case of Walras’
model
Several buyers with different utility functions and moneys.
Find equilibrium prices!!
1p 2p3p
Linear Fisher Market
Assume:Buyer i’s total utility,
mi : money of buyer i.
One unit of each good j.
Find market clearing prices!
i ij ijj G
v u 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
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
Convex programs thatcapture market equilibria
Underly all “efficient” markets (so far).
Rational convex programs for many markets!
Algorithms: combinatorial, for rational (primal-dual) continuous (ellipsoid/interior point)
Combinatorial algorithms for rational convex programs
Natural extension of field of
combinatorial optimization!
Combinatorial algorithms for rational convex programs
Natural extension of field of
combinatorial optimization!
Central aspect of C.O. – efficient algorithms
for solving integral LP’s, e.g. matching, flow.
Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
By extending the primal-dual paradigm to the setting of convex programs & KKT conditions
Combinatorial algorithms
Yield deep structural insights.
Preferable for applications.
Auction for Google’s TV ads
N. Nisan et. al, 2009:
Used market equilibrium based approach.
Combinatorial algorithms for linear case
provided “inspiration”.
Primal-Dual Paradigm
Highly successful algorithm design
technique from exact and
approximation algorithms
Exact Algorithms for Cornerstone Problems in P:
Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching
Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
Yin & Yang
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!
At prices p, buyer i’s most
desirable goods, Si =
Any goods from Si worth m(i)
constitute i’s optimal bundle
arg max ijj
j
u
p
Bang-per-buck
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
For each buyer, most desirable goods, i.e.
arg max iji j
j
uS
p
Network N(p)
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
infinite capacities
st
Max flow in N(p)
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: equilibrium prices iff both cuts saturated
Idea of algorithm
“primal” variables: allocations
“dual” variables: prices of goods
Approach equilibrium prices from below:start with very low prices; buyers have surplus money iteratively keep raising prices and decreasing surplus
An important consideration
The price of a good never exceeds
its equilibrium price
Invariant: s is a min-cut
Invariant: s is a min-cut in N(p)
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: low prices
s
Idea of algorithm
Iterations:
execute primal & dual improvements
Allocations Prices
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!
Key Algorithmic Idea
Dual variables (prices) are raised greedily
Yet, primal objects go tight and looseBecause of enhanced KKT conditions
Balanced Flows: For limiting no. of such events
Max-flow in N
m p
W.r.t. a max-flow f, surplus(i) = m(i) – f(i,t)
i
t s
Max-flow in N
m p
surplus vector = vector of surpluses w.r.t. f
i
Obvious potential function
Total surplus money = l1 norm of surplus vector
Reduce l1 norm of surplus vector by
inverse polynomial fraction in each iteration
Balanced flow
A max-flow that
minimizes l2 norm of surplus vector.
Makes surpluses as equal as possible.
Balanced flow
A max-flow that
minimizes l2 norm of surplus vector.
Makes surpluses as equal as possible.
All balanced flows have same surplus vector.
Our algorithm
Reduces l2 norm of surplus vector by
inverse polynomial fraction in each iteration.
s1
s2
(1, 0)
(0, 1)
Property 1
f: max-flow in N.
R: residual graph w.r.t. f.
If surplus (i) < surplus(j) then there is no
path from i to j in R.
Property 1
i
j
R:
surplus(i) < surplus(j)
Property 1
i
surplus(i) < surplus(j)
j
R:
Property 1
i
Circulation gives a more balanced flow.
j
R:
Property 1
Theorem: A max-flow is balanced iff
it satisfies Property 1.
Construct N’(I, J)
Raise prices in J
New edge enters N
Stop when Invariant is threatened
Algorithm for an iteration
Network N(p)
m p
buyers goods
bang-per-buck edges
Construct N’(I, J)Find a balanced flow in N(p)
Let d = max surplus w.r.t. balanced flowI = buyers with surplus dJ = goods desired by I
Raise prices in J
New edge enters N
Stop when Invariant is threatened
Network N(p)
N’(I, J)I J
N - N’
Network N(p)
N’(I, J)I J
N - N’
Construct N’(I, J)
Raise prices in J N’ is decoupled from N - N’
New edge enters N
Stop when Invariant is threatened
Network N(p)
N’(I, J)I J
N - N’
Network N(p)
N’(I, J)I J
N - N’
By Property 1, this edge did not carry any flow.Hence Invariant is not violated by its removal.
Raise prices in J
proportionately, so that
edges in N’ don’t change.
p . x, for each p in J initialize x = 1raise x
Construct N’(I, J)
Raise prices in J
New edge enters N
Stop when Invariant is threatened
Network N(p)
N’(I, J)I J
N - N’
Construct N’(I, J)
Raise prices in J
New edge enters NRecompute balanced flowBuyers in N - N’ having residual paths to N’
Move to N’
Stop when Invariant is threatened
Network N(p)
N’(I, J)I J
N - N’
Network N(p)
N’(I, J)I J
N - N’
Network N(p)
N’(I, J)I J
N - N’
Construct N’(I, J)
Raise prices in J
New edge enters NRecompute balanced flowBuyers moved to N’ will have
sufficiently large surplus
Stop when Invariant is threatened
Construct N’(I, J)
Raise prices in J
New edge enters N
Stop when Invariant is threatened
Algorithm for an iteration
Tight set: p(S) = m(T)
N’(I, J)
T S
N - N’
Surplus of buyers in T drops to 0
Surplus of buyers in T drops to 0
l1 norm of surplus vector drops by 1/n fraction
after the iteration.
Assume k sub-iterations.
Let d0 = d. At the end of lth sub-iteration,
dl = min {surplus(i) | i is in I}. So, dk = 0.
Network N(p)
N’(I, J)I J
N - N’
Some i in old I will achieve minimum.
Its surplus must drop by at least (dl-1 – dl).
Therefore, decrease in l1 norm
in sub-iteration l is at least (dl-1 – dl)
Therefore, decrease in iteration is at least d
Network N(p)
N’(I, J)I J
N - N’
Assume k sub-iterations.
Let d0 = d. At the end of lth sub-iteration,
dl = min {surplus(i) | i is in I}. So, dk = 0.
Decrease in l1 norm in sub-iteration l
is at least (dl-1 – dl)
Decrease in l22 norm in sub-iteration l
is at least (dl-1 – dl)2
Our algorithm
Reduces l2 norm of surplus vector by
1/n2 fraction in each iteration
Open question
Can define balanced flow without l2 normBalanced flow = lexicographically smallest flow
Q: Can we dispense with l2 norm in proof?
Open question
Can define balanced flow without l2 normBalanced flow = lexicographically smallest flow
Q: Can we dispense with l2 norm in proof?
V, 2008: Family of examples s.t. l1 norm of surplus vector decreases by inverse exponential fraction in an iteration!
s1
s2
(1, 0)
(0, 1)
KKT conditions were relaxed
e(i): money currently spent by i
w.r.t. a balanced flow in N
surplus money of i γi =mi −e(i)
Relaxed 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
e(i)
e(i)
Potential function
2 2 21 2 ... nγ γ γ
Algorithm drops potential by an inverse polynomial
factor in each iteration (strongly polynomial time).
Potential function
2 2 21 2 ... nγ γ γ
Algorithm drops potential by an inverse polynomial
factor in each iteration (strongly polynomial time).
= poly mii∑( )
Second point of departure
KKT conditions are satisfied via a
continuous process Normally: in discrete steps
utility
Piecewise linear, concave
amount of j
Additively separable over goods
Long-standing open problem
Complexity of finding an equilibrium for
Fisher and Arrow-Debreu models under
separable, plc utilities?
How do we build on solution to the linear case?
utility
amount of j
Generalize EG program to
piecewise-linear, concave utilities?
ijkl
ijkuutility/unit of j
,
,
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
,
,
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
V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities.
Given prices p, are they equilibrium prices?
Build on combinatorial insights
utility
amount of j
Case 1
ijx
fully allocated
partially allocated
utility
amount of j
Case 2: no p.a. segment
ijx
fully allocated
p full & partial segments
Network N(p)
Theorem: p equilibrium prices iff
max-flow in N(p) = unspent money.
Network N(p)
partially allocated segments
s
t
q1
q2
q3
q4
m '1
m '2
m '3
m '4
LP for max-flow in N(p); variables = fe’s
Next, let p be variables!
“Guess” full & partial segments – gives N(p)
Write max-flow LP -- it is still linear! variables = fe’s & pj’s
Rationality proof
If “guess” is correct,
at optimality, pj’s are equilibrium prices.
Hence rational!
Rationality proof
If “guess” is correct,
at optimality, pj’s are equilibrium prices.
Hence rational!
In P??
NP-hardness does not apply
Megiddo, 1988:Equilibrium NP-hard => NP = co-NP
Papadimitriou, 1991: PPAD2-player Nash equilibrium is PPAD-completeRational
Etessami & Yannakakis, 2007: FIXP3-player Nash equilibrium is FIXP-completeIrrational
Markets with piecewise-linear, concave utilities
Chen, Dai, Du, Teng, 2009: PPAD-hardness for Arrow-Debreu model
Markets with piecewise-linear, concave utilities
Chen, Dai, Du, Teng, 2009: PPAD-hardness for Arrow-Debreu model
Chen & Teng, 2009: PPAD-hardness for Fisher’s model
V. & Yannakakis, 2009:PPAD-hardness for Fisher’s model
Markets with piecewise-linear, concave utilities
V, & Yannakakis, 2009: Membership in PPAD for both models,
How do we salvage the situation??
Algorithmic ratification of the
“invisible hand of the market”
Is PPAD really hard??
What is the “right” model??
Open
Can Fisher’s linear case
be captured via an LP?