Download - How Intractable is the ‘‘Invisible Hand’’: Polynomial Time Algorithms for Market Equilibria
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How Intractable is the ‘‘Invisible Hand’’:Polynomial Time Algorithms for
Market Equilibria
Vijay V. Vazirani
Georgia Tech
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Market Equilibrium
cheese milk
winebread
¢¢
$$$$$$$$$$$$$$$$$$
$$
$$$$$$$$
•People want to maximize happiness
•Find prices s.t. market clears
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Walras, 1874
• Pioneered mathematical theory of
general economic equilibrium
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Arrow-Debreu Theorem, 1954
• Celebrated theorem in Mathematical Economics
• Shows existence of equilibrium prices
using Kakutani’s fixed point theorem
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Arrow-Debreu Theorem is highly non-constructive
• How do markets find equilibria?
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Arrow-Debreu Theorem is highly non-constructive
• How do markets find equilibria?
– “Invisible hand” of the market: Adam Smith
Wealth of Nations, 1776
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Arrow-Debreu Theorem is highly non-constructive
• How do markets find equilibria?
– “Invisible hand” of the market: Adam Smith
Wealth of Nations, 1776
– Scarf, 1973: approximate fixed point algorithms
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Arrow-Debreu Theorem is highly non-constructive
• How do markets find equilibria?
– “Invisible hand” of the market: Adam Smith
Wealth of Nations, 1776
– Scarf, 1973: approximate fixed point algorithms
– Use techniques from modern theory of algorithms
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Arrow-Debreu Theorem is highly non-constructive
• How do markets find equilibria?
– “Invisible hand” of the market: Adam Smith
Wealth of Nations, 1776
– Scarf, 1973: approximate fixed point algorithms
– Use techniques from modern theory of algorithms
Deng, Papadimitriou & Safra, 2002: linear case in P?
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Market Equilibrium
cheese milk
winebread
$$$$
$$
$$
$$
•People want to maximize happiness
•Find prices s.t. market clears
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History
• Irving Fisher 1891 (concave functions)– Hydraulic apparatus for calculating equilibrium
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History
• Irving Fisher 1891 (concave functions)– Hydraulic apparatus for calculating equilibrium
• Eisenberg & Gale 1959– (unique) equilibrium exists
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History
• Irving Fisher 1891 (concave functions)– Hydraulic apparatus for calculating equilibrium
• Eisenberg & Gale 1959– (unique) equilibrium exists
• Devanur, Papadimitriou, Saberi & V. 2002– poly time alg for linear case
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History
• Irving Fisher 1891 (concave functions)– Hydraulic apparatus for calculating equilibrium
• Eisenberg & Gale 1959– (unique) equilibrium exists
• Devanur, Papadimitriou, Saberi & V. 2002– poly time alg for linear case
• V. 2002: alg for generalization of linear case
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Market Equilibrium• n buyers, with specified money,• m goods (unit amount)• Linear utilities: utility derived by i
on obtaining one unit of j
iju
i ij ijj
U u x
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Market Equilibrium• n buyers, with specified money,• m goods (unit amount)• Linear utilities: utility derived by i
on obtaining one unit of j
• Find prices s.t. market clears
iju
i ij ijj
U u x
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People Goods
$100
$60
$20
$140
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Bang per buck
$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
$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
}/{max jijji pu
Given prices , each i picks goods to maximize
her bang per buck, i.e.,
jp
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Equality subgraph
$100
$60
$20
$140
$20
$40
$10
$60
for all i: most desirable j’s
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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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Max flow
100
60
20
140
20
40
10
60
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Idea of Algorithm
Invariant: source edges form min-cut
(agents have surplus)
Want: prices s.t. sink edges also form min-cut
Gradually raise prices, decrease surplus,
until 0
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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
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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 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?
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}{ S( )S
* ( ( ) ): min
( )S A
m Sx
p S
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Try S = A (all goods)
Let
Clearly,
If s is min-cut, x=x* and S* = A.
( ).
( )
m Bx
p A
*x x
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m px
Otherwise, x > x* .
s
S*( *)S
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Sufficient to recurse on smaller graph
s
S*( *)S
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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
• Pre-emptively freeze sets that have small
surplus (at most ).
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m s
S*( *)S
*x p
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m s
S*( *)S
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m s
S*( *)S }freeze
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m
add to prices and find new min-cut
s
S*( *)S
*x p
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• Next freezing: prices must increase
• Problem: at end, surplus
.
0.
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• Next freezing: prices must increase
• Problem: at end, surplus
• But, surplus
.
0.
.n
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initial surplus
M
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ε
2
M
n
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final surplus2
Mn
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Polynomial time
2 2( ( log log ))O n n U MnTheorem:
max-flow computations suffice.
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Question
• Is main algorithm, i.e., without
pre-emptive freezing, polynomial time?
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Question
• Is main algorithm, i.e., without
pre-emptive freezing, polynomial time?
• Strongly polynomial?
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Post-mortem
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Primal-Dual Schema
Highly successful algorithm design
technique from exact and
approximation algorithms
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Central Idea Behind Primal-Dual Schema
Two processes
making local improvements
(relative to each other) and
achieving global objective
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Post-mortem
• “primal” variables: flow in equality subgraph
• “dual” variables: prices
• algorithm: primal & dual improvements
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Post-mortem
• “primal” variables: flow in equality subgraph
• “dual” variables: prices
• algorithm: primal & dual improvements
• nonzero flow from j to i iijj up /
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Post-mortem
• “primal” variables: flow in equality subgraph
• “dual” variables: prices
• algorithm: primal & dual improvements
• nonzero flow from j to i
“complementary slackness condition”
iijj up /
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Post-mortem
• “primal” variables: flow in equality subgraph
• “dual” variables: prices
• algorithm: primal & dual improvements
• algorithm inspired by Kuhn’s
primal-dual algorithm for bipartite matching
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‘‘Primal-Dual-Type’’ Algorithms
• Formal mathematical setting?
• Analogous setting for primal-dual
algorithms: LP-Duality Theory
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Concave utilities
• Buyers get satiated by goods
• Fix prices => each buyer has unique
optimal bundle
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Concave utilities
• Buyers get satiated by goods
• Fix prices => each buyer has unique optimal bundle
Economy of communication!
Distributed market clearing
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utility
Concave utility function
amount of j
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utility
Piece-wise linear, concave
amount of j
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utility
PTAS for concave fn.
amount of j
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utility
Piece-wise linear, concave
amount of j
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rate
rate = utility/amount of j
amount of j
Differentiate
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$20 $70 $100
for buyer i, good j
rate
rate = utility/unit amount of j
ijf
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0
( )( )
x ij
j
f yu x dy
p
Fix price of j, then utility/$ given by
ij
j
f
p
utility derived,
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$20 $70 $100
utility
fix price of j
piecewise-linear, concave
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V. 2002:
Rate at which i derives happiness depends on fraction of budget spent on j.
Spending constraint model
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Theorem: Equilibrium price is unique,
and can be computed in polynomial time
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Theorem: Equilibrium price is unique,
and can be computed in polynomial time
(Not unique in traditional model,
even for piece-wise linear case)
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$20 $40 $100
Can generalize notion of ‘‘market clearing’’ --assume that buyers have utility for money.
rate
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Does the spending constraint
model measure up to
traditional theory?
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$100
for buyer i, good j
rate
continuous, decreasing rate function
ijf
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0
( )( )
x ij
j
f yu x dy
putility derived,
Strictly concave function.
Each buyer has unique optimal bundle.
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Devanur & V., 2003
Equilibrium exists for cts., decreasing rate fns.
(Proof uses Brauwer’s fixed point theorem),
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Devanur & V., 2003
Equilibrium exists for cts., decreasing rate fns.
(Proof uses Brauwer’s fixed point theorem),
and prices are unique.
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Devanur & V., 2003
Equilibrium exists for cts., decreasing rate fns.
(Proof uses Brauwer’s fixed point theorem),
and prices are unique.
PTAS for computing equilibrium.
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Devanur & V., 2003
Extend model to Arrow-Debreu setting.
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Devanur & V., 2003
Extend model to Arrow-Debreu setting.
Equilibrium exists.
(Proof uses Kakutani’s fixed point theorem.)
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Algorithmic Game Theory
• Mechanism design: find equilibria that
ensure truthful, fair functioning of agents,
and are efficiently computable.
• Approximations: deal with NP-hardness,
stringent game-theoretic notions
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Q: Distributed algorithm for equilibria?
• Appropriate model?
• Primal-dual schema operates via
local improvements
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• Global optimality via local improvement
• Exploit in distributed setting
Kelly, Low, Lapsley, Doyle, Paganini …
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TCP congestion control
primal process: packet rates at sources
dual process: packet drop at links
AIMD + RED solves utility maximization
problem in limit
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Kelly, `97: charging, rate control and routing for elastic traffic
Kelly & V. 2002:
It is essentially a market equilibrium question, and generalizes Fisher’s problem!
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Kelly, `97: charging, rate control and routing for elastic traffic
Kelly & V. 2002:
It is essentially a market equilibrium question, and generalizes Fisher’s problem!
Q: Polynomial time alg?
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Develop an algorithmic theory
of market equilibria,
via polynomial time exact and
approximation algorithms
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• w.r.t. prices p, i sorts segments according
to utility/$, and partitions into classes
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• w.r.t. prices p, i sorts segments according
to utility/$, and partitions into classes
$50 $30 $40 $40 $60
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• w.r.t. prices p, i sorts segments according
to utility/$, and partitions into classes
$50 $30 $40 $40 $60
Assume i has $100
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• w.r.t. prices p, i sorts segments according
to utility/$, and partitions into classes
$50 $30 $40 $40 $60
forcedflexible
undesirable
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
– as prices change, allocs may become undesirable
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
– as prices change, allocs may become undesirable
Deallocate
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
– as prices change, allocs may become undesirable
Deallocate - exponential time??
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
– as prices change, allocs may become undesirable
Deallocate - exponential time??
Reduce prices
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• Invariant 1: Approach eq. from below
• Invariant 2: Forced allocs follow sorted order
– simultaneously, for all i
– as prices change, allocs may become undesirable
Deallocate - exponential time??
Reduce prices - measure of progress??
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Maintains both Invariants
+ deallocations
+ monotonicity of prices
Algorithm
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flexibleforced undesirable
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flexibleforced undesirable
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flexibleforced undesirable
Can ensure prices