Leontief Exchange Markets Can Solve
Multivariate Polynomial Equations,
Yielding FIXP and ETR Hardness
Ruta Mehta
Joint work with Jugal Garg, Vijay V. Vazirani
and
Sadra
Yazdanbod
Joint works with
Ruta
Mehta, Milind
Sohoni and Vijay Vazirani
Exchange Market
Agent :
Endowment
Utility function
At Equilibrium:
Agents sell goods, and buy
utility maximizing bundles.
Market clears.
Utility Functions
Concave
PLC = piecewiselinear, concave
Leontief
Leontief Functions

i.e. needed in a fixed proportion
#

Eg: Utility = min{
, #butter}

Goods are Complements
Wassily Leontief
Nobel prize (1973)
Equilibrium Computation

Fisher model




Linear Utilities: Devanur, Papadimitriou, Saberi and Vazirani’02,
Orlin’10
Spending constraint: Devanur and Vazirani’04, Vegh’12
Leontief, CES: Eisenberg’61 + Ellipsoid
Exchange model


Linear: Jain’07, ye’08, Duan and Mehlhorn’13
CES with 1
1: Jain and Varadrajan’06,
And the list goes on …
Complexity of Equilibrium Computation

Two central solution concepts in Econ: Nash & Market

Existence of a solution guaranteed.

Complexity classes: PPAD (P’94) & FIXP (EY’07)

PPAD: Approximate Fixed-points (Sperner’s Lemma)


Solutions are rational
FIXP: Fixed-points of an algebraic function.

Solutions are irrational but algebraic
max
+
Perspective Through Dichotomies
2-Nash
k-Nash,
Nature of
solution
Rational
Algebraic;
Irrational e.g.:
Nash’51
Complexity
PPAD-complete
DGP’06, CD’06
Practical
algorithm
Lemke-Howson
algorithm
FIXP-complete
EY’07
Utility Functions
Concave
PLC = piecewiselinear, concave
Leontief
Separable
PLC
Perspective Through Dichotomies
Separable PLC
PLC
Nature of
solution
Rational
Algebraic;
Irrational e.g.:
Mas-Colell’75
Complexity
PPAD-complete
CDT’09,VY’09
Practical
algorithm
Lemke-based
algorithm (GM.SV’12)
In FIXP (GM.V’14)
FIXP-complete?
Perspective: What we know
Fisher model is in P (Convex
formulation by Eisenberg’61)
Concave
Pairing is PPAD-complete
(CSVY’05)
PLC
Irrational solutions (M-C’75)
VY’09: FIXP-complete?
In FIXP
(GM.V’14)
In FIXP (Yannakakis’13)
Leontief
Separable
PLC
Rational solutions.
PPAD-Complete (CDT’07, VY’09)
What we show

Given a system of polynomials


0, ∀ , where
∈
,
,
,
0
We construct a Leontief exchange market

is captured through price of a good

Theorem. Solutions of 
Corollary. Leontief markets are FIXP-hard!

Equilibrium prices of
3-Nash (FIXP-complete) → System
→ Leontief market
Some Classical Results
Sonnenschein (Econometrica’72):

Given , ∃ an exchange market with concave utilities such that
excess demand functions of its goods are
.
Exchange Market
Agent :
Endowment
Utility function
,…,
:
→
Agents sell goods, and buy
utility maximizing bundles
Excess demand of a good:
total demand – total supply
Some Classical Results
Sonnenschein (Econometrica’72):

Given , ∃ an exchange market with concave utilities such that
excess demand functions of its goods are
.
Sonnenschein–Mantel–Debreu Theorem. “Anything goes”

s need not be polynomials.
Tatonnement:
Walras (1874)


Excess demand? → increase price
Excess supply? → decrease price
Zeros of s are equilibrium
Non-convergence of Tatonnement
Some Classical Results
Sonnenschein (Econometrica’72):

Given , ∃ an exchange market with concave utilities such that
excess demand functions of its goods are
.
Sonnenschein–Mantel–Debreu Theorem. “Anything goes”


s need not be polynomials.
Our result differs: 1.Excess demand functions
2.Leontief utilities
Reduction
Basic Operations

implification):


is one of
,
LIN.
QD.
=0


:

:
,
:
, ,
0
Basic Operations

implification):





LIN.
QD.
is one of
,
, ,
0
is captured as a price
. At equilibrium, prices
Have to ensure s
Are bounded because
are bounded
Market Construction
Step 2: A market gadget for LIN and QD
 Same variable in many
gadgets have common
goods

Difficult to analyze flow of goods among these gadgets
Closed Submarket
Endowment
& Utility
=( , )
=(
,
)
: Submarket of
: Closed if at every equilibrium of ,
is locally at
’s demand equals
’s supply
equilibrium
The Idea
A closed submarket for each
 At equilibrium, no flow of goods among submarkets
even though there are common goods!

Helps simulate fan-out (same variable in multiple
s)
Equilibrium in Leontief Market
Agent :
Endowment
Utility
,…,
min
,…,
At equilibrium
For each agent i:
- Buys ∗ units of good j
for an
0
- Spending = Earning
For each good j:
- Demand Supply
Example
W
1: 1
1: 1
Earnings
Demand
3
1.5, 0, 1.5
2
0, 2/3, 2/3
Not an Equilibrium
Example
W
Earnings
Demand
3
1, 0, 1
3
0, 1, 1
1: 1
1: 1
Equilibrium!
Example
W
1: 1
1: 1
Earnings
Demand
a+b
1, 0, 1
a+b
0, 1, 1
Equilibrium!
Closed Submarket for
W
Earnings
Spend
c, 0, c
1: 1
0, d, d
1: 1
Exclusive ⇒
Demand
2
1
At Equilibrium
⇒
∎
LIN
Similar to
Theorem.
a closed submarket that enforces
at equilibrium.
QD
(amount, price)
High-Level Idea:
 For some good , establish its

& price
:1
consumes all of , and brings a unit of



Supply =
,
Set price of ′
Earning
= Spending
)
1,
QD
(amount, price)
High-Level Idea:
 For some good , establish its

& price
′: 1
consumes all of , and brings a unit of



Supply
Supply ==
,
Set price of ′
Earning
= Spending
)
1,
Gadget: Price to Supply
1,
2
,1
Issue:
is an unknown ⇒ variable amount
Gadget: Price to Supply
1,
1,
:
2
,1
, )
Suppose:
Prices are explicitly
set using ‘=‘ gadget
is known ⇒ constant amount
Gadget: Price to Supply
1,
1,
1,
)
∶
2
:1
,
,1
,
hastoearn
has to spend as much
Issue:
is an unknown ⇒ variable amount
Gadgets
,1
,
,
2
1,
,
)
,
: variable
)
,
)
,
)
Closed Submarket for
,
,
1,
,
1
~ 2
2
,
1,
,
We get
Theorem. Leontief exchange markets can solve a systems
of multivariate polynomial equations, where each variable
for some
Consequences

3-Nash is FIXP-complete (Etessami-Yannakakis’07)




Reduces to system of polynomials
FIXP-hardness of Leontief exchange markets
In FIXP (Yannakakis’13)
Decision 3-Nash: Is there a NE in 0.5-ball?


Schaefer-Stefankovic’12: ETR-complete (Existential Theory of Reals)
ETR-complete: Is there an equilibrium in a Leontief market?
Open

Fisher markets with Leontief utilities are in P


Complexity of Fisher markets with PLC utilities



Eisenberg’s convex program
In FIXP (Garg-M.-Vazirani’14)
Is it FIXP-hard?
Exchange markets with CES

→
∞ is Leontief.
?
Open

Strongly polynomial-time algorithm for linear exchange
markets.

Rational Convex Program (Devanur, Garg and Vegh’13)

Convergence of Tatonnement type dynamics in exchange
markets.

Average case analysis of Lemke-based algorithms.

For both Nash and market equilibrium
Thanks!