Finding Equilibrium Points
Alejandro Jofré, U. Chile, Santiago
R.T. Rockafellar, U. Washington, Seattle
Roger J-B Wets, U. California, Davis
July 14, 2006
Eco.Math.School 2006
1
Collaborators - Contributors
Adib Bagh, Univ. California, Davis
Sergio Lucero, Univ. California, Davis
Michael Ferris, Univ. of Wisconsin


S. Robinson, J.-S. Pang, D. Ralph, …
T. Munson, S. Dirkse, …
Hedy Attouch, Univ. de Montpellier
July 14, 2006
Eco.Math.School 2006
2
Further readings
Jofré, A. & R. Wets, Variational convergence of bivariate functions: theoretical
foundations. To appear in Mathematical Programming (2006).
Jofré, A., R.T. Rockafellar R.T & R. Wets. Variational Inequalities and economic
equilibrium. To appear in Math. Operation Research (2006?)
Jofré, A., RT. Rockafellar & R. Wets, A variational inequality scheme for
determining an economic equilibrium of classical or extended type. In
“Variational analysis and applications”, 553--577, Nonconvex Optim. Appl., 79,
Springer, New York, 2005.
Jofré, A. & R. Wets, Continuity properties of Walras equilibrium points.
Stochastic equilibrium problems in economics and game theory. Annals of
Operations Research, 114 (2002), 229--243.
S. P. Dirkse and M. C. Ferris. The PATH solver: A non-monotone stabilization
scheme for mixed complementarity problems. Optimization Methods and
Software, 5:123-156, 1995.
July 14, 2006
Eco.Math.School 2006
3
I. Walras Equilibrium Model
Deterministic
Pure Exchange model
July 14, 2006
Eco.Math.School 2006
4
Pure Exchange: Walras
agent’s problem: Agents: a ! A | A| finite "large"
ca !arg max ua (c) so that p, c " p, ea , c !Ca
ea : endowment of agent a, ea ! int Ca
ua : utility of agent a, concave, usc
Ca " !, Ca # ! n (survival set) convex
market clearing: s( p) =
equilibrium price:
#
a"A
(ea ! ca ) excess supply
p !" such that s( p) # 0
! unit simplex
July 14, 2006
Eco.Math.School 2006
5
The Walrasian
W ( p, q) = q, s( p) , W : ! " ! # !
p equilibirum price
! p "arg max p (infq W ( p, q)) & s( p) # 0
Properties of W :
continuous in p (ea "int Ca , 'a-inf-compact')
linear in q, $ compact
convex
usc
W ( p, p) # 0, %p "$
i.e., W is a Ky Fan function
July 14, 2006
Eco.Math.School 2006
6
Ky Fan functions & inequality
K : B ! B " ! is a Ky Fan function if
(a) #y : x " K(x, y) usc
(b) #x : y " K(x, y) convex
Theorem. K Ky Fan fcn, dom K = B ! B, B compact
" argmax-inf K # $
if K(x, x) % 0 on dom K, x & argmax-inf K
" infy K(x, y) % 0.
The Walrasian is a Ky Fan function
yields existence of equilibrium price.
July 14, 2006
Eco.Math.School 2006
7
II. Numerical Strategies
- Augmented Lagrangian
- Path Solver
July 14, 2006
Eco.Math.School 2006
8
Augmented Walrasian
Augmented Walrasian:
p ! argmax-inf W
! saddle point ( p, q ) of W! r
{
}
W! r ( p, q)= infz W ( p, z) z ! q " r ,
i an appropriate norm (|i|# e.g.)
July 14, 2006
Eco.Math.School 2006
9
from Variational Analysis
Properties: W! r is usc in p, convex, lsc in (q, r)
non-increasing in r
(
)
W ( p, q) =
Saddle-points: sup inf
q!"
sup
p!"
(
)
p!"
inf W" r ( p, q) =
q!",r !! +
#
&
"
inf % sup Wr ( p, q)(
q!",r !! + $ p!"
'
argument similar to augmented Lagrangian
July 14, 2006
Eco.Math.School 2006
10
Iterations
W ( p, q) = q, s( p) on ! " !
{
W! r ( p, q) = infz W ( p, z) z # q $ r
q k +1 = arg min &' min z z, s( p k )
q%!
}
z # q $ rk ()
minimizing a linear form on a ball
reduces to finding the smallest element of s(pk)
p k +1 = arg max &' min z z, s( p)
p%!
z # q k +1 $ rk +1 ()
as rk " *, p k + p (Walras equilibrium point)
July 14, 2006
Eco.Math.School 2006
11
Test: Demand functions
Cobb-Douglas utility function (usc):
ua (c) = ! a #
n
"al
l =1 l
c
with
budget constraint:
$
!
l
l
c
(
p)
=
(
!
demand: a
a / pl )
n
l =1
" = 1, " % 0
l
a
l
a
l
p
c
"
p
e
!l l a
l l l
(
)
k
p
e
" k l a , l = 1,…, n
experiments: 10 agents, 150 goods (blink!)
July 14, 2006
Eco.Math.School 2006
12
Variational Inequality I
max ua (c) so that p, c ! p, ea , c "Ca . a " A
c
! a (ea " ca ) = s( p) # 0.
!
e "int ! a Ca
a a
KKT-Optimality Conditions & Market Clearing:
ca !Ca optimal " # $a % 0 (linear constraint)
(a) p, ea & ca % 0 (feasibility)
(b) $a ( p, ea & ca
)=0
(c) 'ua ( ca ) & $a p = 0
(d) ( a ( ea & ca ) = 0
July 14, 2006
(compl. slackness)
(ea ! int Ca )
(market clearing)
Eco.Math.School 2006
13
Variational Inequality II
max ua (c) so that p,c ! p, ea , c "Ca , #a
c
! a (ea " ca ) = s( p) # 0.
{
N D (z )
z
}
N D (z ) = v v, z !!z e"a "int
0, #z!$D
C
a
a a
D
G( p,(ca ),(!a )) = %& $ a (ea " ca ); ( !a p " #ua (ca )); p, ea " ca '(
D =!"
(# C ) " (# ! )
a
a
a
+
!G( p,(ca ),("a )) #N D ( p,(ca ),("a ))
D (unfortunately) is unbounded
July 14, 2006
Eco.Math.School 2006
14
bounding D: “solvable” V.I.
from D to D̂ bounded with
explicit bounds derived via duality (finite # goods)
(global bound for Ca , ! a depends on 'var'(ua ))
!G( p,(ca ),("a )) #N D̂ ( p,(ca ),("a ))
D̂ = ! "
(# Ĉ ) " (# [0,$ ])
a
a
a
a
Polyhedral case: efficient algorithmic procedures
July 14, 2006
Eco.Math.School 2006
15
Path Solver .. (by Ferris & all)
!G(z ) "N D (z ), z = ( p,(ca ),(#a ))
D =!"
(# C ) " (# ! ) = {z Az $ b}
a
a
a
+
Complementarity problem:
!G(z) = AT y, y " 0, Az ! b # y
with K = ! N ! ! M+ :
(z, y) "K, H (z, y) "#K $ , (z, y) % H (z, y)
&G(z) + AT y ) , 0 /
H (z, y) = (
+ # . b1
' Az
* - 0
July 14, 2006
Eco.Math.School 2006
16
Equivalent non-smooth mapping
0 = H (prjK (z, y)) + (z, y) ! prjK (z, y)
with simplified K = ! N+ :
(CP) 0 ! x " F(x) # 0 Complementarity Problem
(NS) 0 = F(x+ ) + x $ x+ Nonlinear system
x sol'n (CP) ! x! sol'n (NS):
x!i = xi if Fi (x ) = 0, x!i = "Fi (xi ) if Fi (x ) > 0
x! sol'n (NS) ! x! + sol'n (CP):
x! + # 0, F( x! + ) = x! + " x! # 0 & x! + $ x! + " x!
July 14, 2006
Eco.Math.School 2006
17
PATH Solver:
x = (z, y), x+ = prjK (x, y)
PATH: Newton method based on
non-smooth normal mapping:
H (x+ ) + x ! x+
Newton point: solution of piecewise
linearization:
H (x+k ) + !H (x+k ), x+ " x+k + x " x+ = 0
July 14, 2006
Eco.Math.School 2006
18
The “Newton” step
July 14, 2006
Eco.Math.School 2006
19
with PATH Solver (experimental)
Economy: (5 goods)



Skilled & unskilled workers
Businesses: Basic goods & leisure
Banker: bonds (riskless), 2 stocks
2-stages, solved under # of scenarios
utilities: CSE-functions (gen. Cobb-Douglas)


Utility in stage 2 assigned to financial instruments
only used for transfer in stage 1
so far: mostly calibration
numerically: `blink’
July 14, 2006
Eco.Math.School 2006
(5000 iterations).
20
III. Continuity, Stability Issues
equilibrium points
solutions of V.I., …
July 14, 2006
Eco.Math.School 2006
21
Variational Convergence
solutions of optimization problems

arg min f ! " arg min f : epi-convergence

arg max f ! " arg max f : hypo-convergence
stability of saddle points

saddle pts K ! " saddle pts K : epi/hypo-convergence
stability of maxinf points

max inf K ! " max inf K : lopsided convergence (tightly)
July 14, 2006
Eco.Math.School 2006
22
Walras Equilibrium points
{
!a " A: ca ( p) "arg max C ua (ca ) p,ca # p,ea
a
s( p) = " a (ea ! d a (ca )) excess supply
}
find p !" (unit simplex) so that s( p) # 0
Walrasian: W ( p,q) = q, s( p)
Ky Fan fcn
p !arg max"inf W # s( p) $ 0
conditions: ea !int dom ua , "globally compact"
!
!
u
"
u
,
e
Convergence: a hypo a a " ea #
W ! converge lopsided tightly to W
July 14, 2006
Eco.Math.School 2006
23
Variational Inequalities
N C (u)
C ! ! n non-empty, convex
G : C ! ! n continuous
u
C
find u !C such that " G(u ) ! N C (u )
where v ! N C (u ) " v,u # u $ 0, %u !C
with K(u,v) = G(u),v ! u on dom K = C " C
K is a Ky Fan function, K(u,u) ≥ 0.
Find
July 14, 2006
u !arg max"inf K(i,i) so that K(u ,i) # 0
Eco.Math.School 2006
24
Lopsided convergence: definition
! y" #D"
$ y #D
• (x, y)
C!D
!
!
•
(x , y )
!
C "D
lim sup! K ! (x! , y! ) " K(x, y) when x #C
K ! (x! , y! ) $ % & when x 'C
!
! x" #C " $ x
"
! y #D
$y
"
•(x, y)
C!D
•
!
(x! , y )
lim inf! K ! (x! , y! ) " K(x, y) when y #D
K ! (x! , y! ) $ % when x &D
C ! " D!
! x" #C " $ x #C
July 14, 2006
Eco.Math.School 2006
25
Lopsided tightly
K C! ! " D!
#
lop $tightly
K C " D if K C! ! " D! # K C " D &
lop
(b) %x &C,' x! # x, % y! & D! and y! # y :
lim inf K ! (x! , y! ) ( K(x, y) if y & D
K ! (x! , y! ) # ) if y * D
but also %+ >0, ' B+ compact (depends on x! # x) :
inf B
+
, D!
K ! (x! ,i) - inf D! K ! (x! ,i) + + , %! ( ! +
THM. K C! ! " D! # K C " D lopsided tightly, x cluster point of
{x! $arg max%inf K C! ! " D! }! $! & x $arg max%inf K C " D
July 14, 2006
Eco.Math.School 2006
26
Proof ….
K C! ! " D!
#
lop $tightly
KC " D
Let g ! = inf y%D! K ! (i, y), g = inf y%D K(i, y).
{
}
(C ! = x %C ! g ! (x) > $'
* g
!
& g # g when )
,hypo
*+ Cg = x %C g(x) > $'
{
}
then apply
!
gC! ! " gC , x! #arg max C! g ! , x k " x #C $ x #arg max C g
hypo
July 14, 2006
Eco.Math.School 2006
27
Extending Ky Fan’s inequality
K ! " K lopsided
K ! Ky Fan # K Ky Fan
& when arg max!inf K " # $
if x % cluster-pts {arg max!inf K " }
& x %arg max!inf K & K(x ,i) ' 0
Ky Fan fcns closed under tight-lopsided
saddle fcns closed under hypo/epi-convergence
usc fcns closed under hypo-convergence
July 14, 2006
Eco.Math.School 2006
28