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2) Combinatorial Algorithms for Traditional Market Models
Vijay V. Vazirani
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Arrow-Debreu Theorem: Equilibria exist.
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Arrow-Debreu Theorem: Equilibria exist.
Do markets operate at equilibria?
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Arrow-Debreu Theorem: Equilibria exist.
Do markets operate at equilibria?
Can equilibria be computed efficiently?
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Arrow-Debreu is highly non-constructive
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Arrow-Debreu is highly non-constructive
“Invisible hand” of the market: Adam Smith
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Arrow-Debreu is highly non-constructive
“Invisible hand” of the market: Adam Smith
Scarf, 1973: approximate fixed point algs.
Convex programs: Fisher: Eisenberg & Gale, 1957Arrow-Debreu: Newman and Primak, 1992
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Used for deciding tax policies, price of new
products etc.
New markets on the Internet
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Algorithmic Game Theory
Use powerful techniques from modern algorithmic theory and notions from game theory to address issues raised by Internet.
Combinatorial algorithms for finding market equilibria.
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Two Fundamental Models
Fisher’s model
Arrow-Debreu model,
also known as exchange model
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Combinatorial Algorithms
Primal-dual schema based algorithms Devanur, Papadimitriou, Saberi & V., 2002
Combinatorial algorithm for Fisher’s model
Auction-based algorithmsGarg & Kapoor, 2004
Approximation algorithms.
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Approximation
Find prices s.t. all goods clear
Each buyer get goods providing
at least optimal utility.(1 )
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Primal-Dual Schema
Highly successful algorithm design
technique from exact and
approximation algorithms
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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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Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
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Main new idea
Previous: problems captured via
linear programs
DPSV: nonlinear convex program
Eisenberg-Gale Convex Program, 1959
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Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i
on obtaining one unit of j Total utility of i,
i ij ijj
U u xiju
]1,0[
x
xuuij
ijj iji
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Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i
on obtaining one unit of j Total utility of i,
Find prices s.t. market clears
i ij ijj
U u xiju
xuu ijj iji
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Can equilibrium allocations be captured via an LP?
Set of feasible allocations:
1
, 0
iji
ij
j x
i j x
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Does equilibrium optimize a global objective function?
Guess 1: Maximize sum of utilities, i.e.,
Problem: and
are equivalent utility functions.
max ( ) maxi ij iji i j
u x u x
2 ( )iu x( )iu x
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However,
1i 1
1i 1
Maximize ( ) ( ) does not necessarily
maximize 2 ( ) ( )
1
, 0
i
i
iji
ij
u x u x
u x u x
j x
i j x
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Guess 2: Product of utilities.
11
11
Maximize ( ) ( )
maximizes 2 ( ) ( )
1
, 0
ii
ii
iji
ij
u x u x
u x u x
j x
i j x
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However, suppose a buyer with $200 is
split into two buyers with $100 each
And same utility function.
Clearly, equilibrium should not change.
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However,
11
21
1
Maximize ( ) ( ) does not necessarily
maximize ( ) ( )
1
, 0
ii
ii
iji
ij
u x u x
u x u x
j x
i j x
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Money of buyers is relevant.
Assume a utility function is written on
each dollar in market
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Guess 3: Product of utilities over all dollars
( )Max ( )
1
, 0
m ii
i
iji
ij
u x
j x
i j x
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Eisenberg-Gale Program, 1959
max ( ) log
. .
:
: 1
: 0
ii
i ij ijj
iji
ij
m i u
s t
i u
j
ij
u xx
x
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Via KKT Conditions can establish:
Optimal solution gives equilibrium
allocations
Lagrange variables give prices of goods
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DPSV Algorithm
“primal” variables: allocations of goods
“dual” variables: prices
algorithm: primal & dual improvements
Allocations Prices
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Buyer i’s optimization program:
Global Constraint:
Market Equilibrium
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People Goods
$100
$60
$20
$140
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Prices and utilities
$100
$60
$20
$140
$20
$40
$10
$60
10
20
4
2
utilities
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Bang per buck
$100
$60
$20
$140
$20
$40
$10
$60
10
20
4
2
10/20
20/40
4/10
2/60
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Bang per buck
Utility of $1 worth of goods
Buyers will only buy goods providing
maximum bang per buck
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Equality subgraph
$100
$60
$20
$140
$20
$40
$10
$60
10
20
4
2
10/20
20/40
4/10
2/60
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Equality subgraph
$100
$60
$20
$140
$20
$40
$10
$60
Most desirable goods for each buyer
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Any goods sold in equality subgraph make agents happiest
How do we maximize sales in equality subgraph?
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Any goods sold in equality subgraph make agents happiest
How do we maximize sales in equality subgraph?
Use max-flow!
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Max flow
100
60
20
140
20
40
10
60
infinite capacities
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Idea of Algorithm
Invariant: source edges form min-cut
(agents have surplus)
Iterations: gradually raise prices,
decrease surplus
Terminate: when surplus = 0, i.e.,
sink edges also form a min-cut
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Ensuring Invariant initially
Set each price to 1/n
Assume buyers’ money integral
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How to raise prices? Ensure equality edges retained
i
j
l
ij il
j l
u u
p p
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How to raise prices? Ensure equality edges retained
i
j
l
ij il
j l
u u
p p
• Raise prices proportionatelyj ij
l il
p u
p u
ij il
j l
u u
p p
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100
60
20
140
20x
40x
10x
60x
initialize: x = 1
x
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100
60
20
140
20x
40x
10x
60x
x = 2: another min-cut
x>2: Invariant violated
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100
60
20
140
40x
80x
20
120
active
frozenreinitialize: x = 1
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100
60
20
140
50
100
20
120
active
frozen x = 1.25
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100
60
20
140
50
100
20
120
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100
60
20
140
50
100
20
120
unfreeze
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100
60
20
140
50x
100x
20x
120x
x = 1, x
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m
buyers goods
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m p
buyers goods
ensure Invariant
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m p
buyers goodsequality
subgraph ensure Invariant
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m px
x = 1, x
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}{ S( )S
( ) ( ( ))x p S m S
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}{ S( )S
( ) ( ( ))x p S m S freeze S
tight set
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}{ S( )S
prices in S are market clearing
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x = 1, x
S( )S
active
frozen
px
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x = 1, x
S( )S
active
frozen
px
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x = 1, x
S( )S
active
frozen
px
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new edge enters equality subgraph
S( )S
active
frozen
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unfreeze component
active
frozen
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• All goods frozen => terminate
(market clears)
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• All goods frozen => terminate
(market clears)
• When does a new set go tight?
•Solve as parametric cut problem
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Termination Prices in S* have denominators
Terminates in max-flows.
,nnU
max { }ij ijU u
2 2Mn
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Polynomial time? Problem: very little price increase
between freezings
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Polynomial time? Problem: very little price increase
between freezings
Solution: work with buyers having
large surplus
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Max flow
100
60
20
140
20
40
10
60
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100
60
20
140
20
40
10
60
20
0
10
60
40
0
Max flow
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surplus(i) = m(i) – f(i)
100
60
20
140
20
40
10
60
20
0
10
60
40
0
40
60
20
70
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surplus(i) = m(i) – f(i)
100
60
20
140
20
40
10
60
20
0
10
60
40
0
40
60
20
70
Surplus vector = (40, 60, 20, 70)
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Balanced flow
A max-flow that minimizes l2 norm of
surplus vector
tries to make surpluses as equal as possible
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Algorithm
Compute balanced flow
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active
frozen
Active subgraph: Buyers with maximum surplus
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active
frozen
x = 1, x
px
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active
frozen
new edge enters equality subgraph
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active
frozen
Unfreeze buyers having residual path to
active subgraph
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active
frozen
Unfreeze buyers having residual path to
active subgraph
Do they have large surplus?
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f: balanced flow
R(f): residual graph
Theorem: If R(f) has a path from i to j then
surplus(i) > surplus(j)
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active
frozen
New set tight
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active
frozen
New set tight: freeze
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Theorem: After each freezing, l2 norm of
surplus vector drops by (1 - 1/n2 ) factor.
Two reasons: total surplus decreasesflow becomes more balanced
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Idea of Algorithm
algorithm: primal & dual improvements
measure of progress: l2-norm of surplus vector
Allocations Prices
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Weak gross substitutability
Increasing price of one good cannot decrease
demand for another good.
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Weak gross substitutability
Increasing price of one good cannot decrease
demand for another good.
=> never need to decrease
prices (dual variables).
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Weak gross substitutability
Increasing price of one good cannot decrease
demand for another good.
=> never need to decrease
prices (dual variables).
Almost all primal-dual algs work this way.
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Polynomial time
2 2( ( log log ))O n n U MnTheorem:
max-flow computations suffice.
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Arrow-Debreu Model
Approximate equilibrium algorithms:
Jain, Mahdian & Saberi, 2003:
Use DPSV as black box.
Devanur & V., 2003: More efficient, by
opening DPSV.
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Garg & Kapoor, 2004Auction-based algorithm
Start with very low prices
Keep increasing price of good that is in demand
B has excess money. Favorite good: g Currently at price p and owned by B’
B outbids B’
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(1 )p
p
B 'B
p(1 )p
Outbid
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Auction-based algorithm Go in rounds:
In each round, total surplus decreases by factor
Hence iterations suffice, M= total moneytotal money
1 2, ,... nB B B
(1 )
(1 )log M
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Arrow-Debreu Model
Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money
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Arrow-Debreu Model
Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money
Any good with price >1 is fully sold
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Arrow-Debreu Model
Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money
Any good with price >1 is fully sold
Eventually every good will have price >1
maxmax
min minij
ij
uprice
price u
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Garg, Kapoor & V., 2004:
Auction-based algorithms for
additively separable concave utilities
satisfying weak gross substitutability
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Kapoor, Mehta & V., 2005:
Auction-based algorithm for
a (restricted) production model
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Q: Distributed algorithm for equilibria?
Appropriate model?
Primal-dual schema operates via
local improvements
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