University of Pittsburgh
Market Equilibrium via a
Primal-Dual-Type Algorithm
Written By
Nikhil R. Devanur, Christos H. Papadimitriou,
Amin Saberi, Vijay V. Vazirani
Presented by
Zhouyan Wang
Advised by
Prof. Kirk Pruhs
November 18, 2003
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Outline
Introduction
Basic Algorithm:
--- Basic Idea
--- High Level Idea
Algorithm with Polynomial Time
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Introduction
Definition
• Set A --- n Goods with each price pj
• Set B --- n’ Buyers with amount ei budget
• uij --- the utility derived by i on obtaining a
unit amount of goods j
• Buyer will use all his money to buy the
goods with Max (uij / pj ) since he will get
the maximal satisfaction.
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Introduction
Graph
Buyers
Goods
u11 /p1
e1
u1n
1, 2, …,i,… n’
en’
p1
/pn
1, 2, …,j,… n
un’1/p1
un’n /pn
ei : Budget
pn
pj :Price
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Introduction
Problem
The question is:
--- How to find the market clearing
prices?
--- Namely, find the prices for goods such
that the buyers will use up all money on
hand to buy goods and there are no goods
left in the market?
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Introduction
Literature Review
First defined by Irving Fisher in 1891
No Algorithm with polynomial time has
been found
This paper finds such algorithm by
assuming uij is a linear function of the
amount of goods
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Basic Algorithm
Basic Idea
In the bipartition Graph(A,B), add a source s and
sink t
There is an edge from source to every goods.
Their capacity is pj
For every buyer i and goods j, connect them by an
edge only if goods j can give buyer i the maximal
utility per dollar and its capacity of the edge is infinite
There is an edge from buyer i to sink t and its
capacity is ei
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Basic Algorithm
New Graph
Buyers
1
Goods
1
Infinite
e1
p1
Sink
j
2, …,i,…
en’
ei : Budget
n’
Source
pj
pn
pj :Price
n
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Basic Algorithm
Primal Dual Type Algorithm
In order to find the clear price, solve the max
flow problem by:
Keep the price low such that all goods will
always be sold out: (s, A U B U t) is the min cut
--- Invariant Property.
And then increase the prices gradually such that
all buyers have spent all their money.
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Basic Algorithm
High Level Idea
Define
1. For S  B, Define its money as m(S )  iB ei
2. For S  A, Define its money as m(S )  iB pi
3. For S  A, Define its neighborhood:
N (S )  { j  B i  S , (i, j )  G}
Lemma: For given prices p, graph satisfies the
Invariant IFF
S  A : m( S )  m( N ( S ))
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Basic Algorithm
High Level Idea,Cont.
Define:
1. Tight Set: a unique maximal set S  A, such that
m( S )  m( N ( S ))---
be frozen by the algorithm
2. Active Sub-graph:( A  S , B  N ( S ))
3. Drop the edges (i,j) with i  N ( S ), j  ( A  S )
Lemma: If the Invariant holds and S  A is the tight
set, then each goods j  ( A  S ) has an edge to certain
buyer i  ( B  N ( S ))
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Basic Algorithm
High Level Idea,Cont.
Then gradually raise the prices of all goods in the
active graph by the same pace xp, then:
Event 1: A nonempty set R goes tight. Add it to
the tight set and continue the next iteration.
Event 2: An edge (i,j) with i  ( B  N ( S )), j  S
becomes an equality edge. Suppose j Sl , remove
component (Sl, Tl) into the active subgraph.
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Basic Algorithm
Find Tight Sets
Let x* denote the value at which a nonempty set goes tight
S* denote the tight set at prices x*p
Lemma: For prices (x p)
If x <= x*, then (s, A U B U t) is a min-cut
If x > x*, then (s, A U B U t) is not a min-cut.
Moreover, if (s U A1 U B1, A2 U B2 U t) is a min-cut in
G(x p), then S*  A1
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Basic Algorithm
Find Tight Sets, Cont.
Lemma: x* and S* could be found using n max-flow
computations.
Proof: Let x = m(B)/m(A). Clearly, x >= x*. If (s, A U B U t) is a
min cut in G(x P), then by the above lemma, we get: x = x* and
S* = A.
Otherwise, let (s U A1 U B1, A2 U B2 U t) be the min cut.
By the above lemma again, S*  A1  A. Therefore, it is
sufficient to get S* by at most n recursions on the smaller graph
(A1, N(A1)).
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Basic Algorithm
Basic Algorithm
Partition the whole algorithm into phases, each
phase terminates with the occurrence of Event 1
Each phase is partitioned into iterations which
conclude with a new edge entering the equality
subgraph (Event 2)
Define M be the total money possessed by the buyers
f be the max flow in the graph at prices P
M – f is the surplus money with buyers.
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Basic Algorithm
Basic Algorithm, Cont.
Let U  max iB , jA{uij } and   nU n
Lemma 1: Each phase consists of at most n iterations. At
the termination of a phase, the prices of the goods in the
newly tight set must be rational numbers with
denominator  
Lemma 2: Consider two phases P and P’ , not
necessarily consecutive, such that goods j lies in the
newly tight sets at the end of P as well as P’. Then the
increase in the price of j, going from P to P’ is  1 / 2
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Basic Algorithm
Basic Algorithm, Cont.
Lemma 3: After k phases, f  k / 2
Corollary 4: The algorithm will end with market
clearing price in at most M /(1 / 2 )  M2 phases, and
execute n M2 iterations or n 2 M2 times of computations.
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Advanced Algorithm
Advanced Algorithm
After the each phase in the basic algorithm, instead of
freezing the tight set by prices p, we add one more step:
Let   0 be fixed and ADD  to the price of each
active subgraph and find a min cut in graph G that
maximizes the s-side. Let S '  A be the subset of A on
the s-side of the min cut. Freeze ( S ' , N ( S ' )) .
Clearly, S  S ' and m( N ( S ))  m( S )  S 
The algorithm will continue the next phase from the
prices p, so Invariant will be maintained.
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Advanced Algorithm
Advanced Algorithm,Cont.
However, for the set of goods in the active subgraph
A' :
S  A': m( N (S ))  m(S )  S 
Namely,the next phase must increase f by at least  .
Consider the situation when all goods are frozen and call
it as an epoch.
After the 1st epoch, the total surplus is  n , set
   / e and run the next epoch…
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Advanced Algorithm
Advanced Algorithm,Cont.
Consider the epoch that   1 /( n ) .At the end
2
of this epoch, the surplus f  1 /  , and we will set   0
for the next epoch.
1st
2
By Lemma 2, the next epoch will be the final epoch
and we will get the market clearing prices.
Lemma: The number of total phases is: O(ln( M2 ))
and the total iterations are O(n(ln( M )))
2
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