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NUFFIELD.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME NUFFIELD
* Problem :
* *********
* A problem from economics.
* Maximize a 2-D integral representing consumer surplus subject to
* linear and quadratic constraints representing incentive compatibility
* Let v( . , . ) : R^2 -> R, Omega = [a,a+1] x [a,a+1], and
* the corners A, B, C, D be as follows:
* (a+1,a+1)
* A *-----* B
* | |
* | |
* D *-----* C
* (a,a)
* The problem is to maximize
* (a+1) line integral_{AB U BC} v(w)dw
* - a line integral_{CD U DA} v(w)dw
* - 3 volume integral_{Omega} v(w)dw
* subject to v being symmetric (i.e., v(x,y) = v(y,x))
* v(a,a) = 0
* nabla_w v(w) >= 0
* < e, nabla_w v(w) > <= 1
* and nabla_ww v(w) positive definite
* this last constraint is guaranteed by ensuring that
* d^2 v/dx^2 >= 0
* d^2 v/dy^2 >= 0
* ( d^2 v/dx^2 )( d^2 v/dy^2 ) >= ( d^2 v/dxdy )^2
* Symmetry is ensured by only considering v(x,y) for x <= y
* Here v(x,y) is the consumer surplus. that is if the consumer values good
* 1 at x pounds and good 2 at y pounds then they will have a utility
* equivalent to v(x,y) pounds after being faced with the optimal monopoly
* pricing strategy. (Apparently, from this we can infer what the optimal
* pricing strategy was... ).
* More background is available from
* "Optimal Selling Strategies: When to haggle, when to hold firm",
* Riley and Zeckhauser. The Quarterly Journal of Economics, 1983, and
* "Multidimensional Incentive Compatibility and Mechanism Design",
* McAfee and McMillan. The Journal of Economic Theory, 1988.
* Source: John Thanassoulis <[email protected]>
* Standard finite-differences are used to ap[proximate derivatives, and
* 1- and 2-D trapezoidal rules to approximate integrals
* SIF input: Nick Gould, February 2001
* classification LQR2-AN-V-V
* The parameter a
RE A 5.0 $-PARAMETER
* Number of nodes in each direction
*IE N 10 $-PARAMETER
*IE N 20 $-PARAMETER
*IE N 30 $-PARAMETER
IE N 40 $-PARAMETER
*IE N 100 $-PARAMETER
* Other useful parameters
IE 0 0
IE 1 1
IE 2 2
IA N-1 N -1
RI RN N
RD H RN 1.0
R= 1/H RN
RM -1/H 1/H -1.0
R* H**2 H H
R* 1/H**2 1/H 1/H
RM -2/H**2 1/H**2 -2.0
R* 1/H**4 1/H**2 1/H**2
RA A+1 A 1.0
RM -A-1 A+1 -1.0
RM C2 H 3.0
RM C3 C2 0.5
R+ C4 C3 A
R+ C1 C3 -A-1
RA C5 C3 -1.0
RM C5 C5 0.5
RM C6 C3 0.5
R+ C6 C6 -A-1
RM C6 C6 0.5
R* C1 C1 H
R* C2 C2 H
R* C3 C3 H
R* C4 C4 H
R* C5 C5 H
R* C6 C6 H
VARIABLES
* V(I,J) gives the value of v(a+I/N,a+J/N)
DO I 0 N
DO J 0 I
X V(I,J)
ND
GROUPS
* objective function terms (for minimization!)
DO J 1 N-1
ZN OBJ V(N,J) C1
ND
DO I 2 N-1
IA I-1 I -1
DO J 1 I-1
ZN OBJ V(I,J) C2
ND
DO I 1 N-1
ZN OBJ V(I,I) C3
ND
DO I 1 N-1
ZN OBJ V(I,0) C4
ND
ZN OBJ V(N,0) C5
ZN OBJ V(N,N) C6
* positive gradient
DO I 0 N-1
IA I+1 I 1
DO J 0 I
ZG VX(I,J) V(I+1,J) 1/H
ZG VX(I,J) V(I,J) -1/H
ZL VV(I,J) V(I+1,J) 1/H
ZL VV(I,J) V(I,J) -1/H
ND
DO J 0 N-1
ZG VX(N,J) V(N,J) 1/H
ZG VX(N,J) V(N-1,J) -1/H
ZL VV(N,J) V(N,J) 1/H
ZL VV(N,J) V(N-1,J) -1/H
ND
ZG VX(N,N) V(N,N) 1/H
ZG VX(N,N) V(N,N-1) -1/H
ZL VV(N,N) V(N,N) 1/H
ZL VV(N,N) V(N,N-1) -1/H
DO I 1 N
IA I-1 I -1
DO J 0 I-1
IA J+1 J 1
ZG VY(I,J) V(I,J+1) 1/H
ZG VY(I,J) V(I,J) -1/H
ZL VV(I,J) V(I,J+1) 1/H
ZL VV(I,J) V(I,J) -1/H
ND
DO I 1 N-1
IA I+1 I 1
ZG VY(I,I) V(I+1,I) 1/H
ZG VY(I,I) V(I,I) -1/H
ZL VV(I,I) V(I+1,I) 1/H
ZL VV(I,I) V(I,I) -1/H
ND
ZG VY(N,N) V(N,N) 1/H
ZG VY(N,N) V(N,N-1) -1/H
ZL VV(N,N) V(N,N) 1/H
ZL VV(N,N) V(N,N-1) -1/H
* positive curvature
DO I 1 N-1
IA I-1 I -1
IA I+1 I 1
DO J 0 I-1
ZG VXX(I,J) V(I+1,J) 1/H**2
ZG VXX(I,J) V(I,J) -2/H**2
ZG VXX(I,J) V(I-1,J) 1/H**2
OD J
ND
DO I 2 N
IA I-1 I -1
DO J 1 I-1
IA J-1 J -1
IA J+1 J 1
ZG VYY(I,J) V(I,J+1) 1/H**2
ZG VYY(I,J) V(I,J) -2/H**2
ZG VYY(I,J) V(I,J-1) 1/H**2
OD J
ND
DO I 1 N-1
IA I-1 I -1
IA I+1 I 1
ZG VXX(I,I) V(I+1,I) 1/H**2
ZG VXX(I,I) V(I,I) -2/H**2
ZG VXX(I,I) V(I,I-1) 1/H**2
ZG VYY(I,I) V(I+1,I) 1/H**2
ZG VYY(I,I) V(I,I) -2/H**2
ZG VYY(I,I) V(I,I-1) 1/H**2
ND
DO I 1 N-1
DO J 1 I
XG C(I,J)
ND
CONSTANTS
DO I 0 N
DO J 0 I
X NUFFIELD VV(I,J) 1.0
ND
BOUNDS
XR NUFFIELD 'DEFAULT'
XX NUFFIELD V(0,0) 0.0
ELEMENT TYPE
EV CONVEX VIP1J
EV CONVEX VIJP1
EV CONVEX VIJ
EV CONVEX VIM1J
EV CONVEX VIJM1
EV CONVEX VIPJP
EV CONVEX VIPJM
EV CONVEX VIMJM
EV CONVEX VIMJP
IV CONVEX VXX
IV CONVEX VYY
IV CONVEX VXY
ELEMENT USES
* positive curvature
DO I 1 N-1
IA I+1 I 1
IA I-1 I -1
DO J 1 I
IA J+1 J 1
IA J-1 J -1
XT C(I,J) CONVEX
ZV C(I,J) VIP1J V(I+1,J)
ZV C(I,J) VIM1J V(I-1,J)
ZV C(I,J) VIJP1 V(I,J+1)
ZV C(I,J) VIJM1 V(I,J-1)
ZV C(I,J) VIJ V(I,J)
ZV C(I,J) VIPJP V(I+1,J+1)
ZV C(I,J) VIMJM V(I-1,J-1)
ZV C(I,J) VIPJM V(I+1,J-1)
ZV C(I,J) VIMJP V(I-1,J+1)
ND
GROUP USES
DO I 1 N-1
DO J 1 I
ZE C(I,J) C(I,J) 1/H**4
ND
OBJECT BOUND
* Solutions (may be local!)
*LO SOLTN -2.512312500 $ (n=10)
*LO SOLTN -2.512359371 $ (n=20)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS NUFFIELD
INDIVIDUALS
T CONVEX
R VXX VIP1J 1.0 VIJ -2.0
R VXX VIM1J 1.0
R VYY VIJP1 1.0 VIJ -2.0
R VYY VIJM1 1.0
R VXY VIPJP 0.25 VIMJM 0.25
R VXY VIMJP -0.25 VIPJM -0.25
F VXX * VYY - VXY * VXY
G VXX VYY
G VYY VXX
G VXY - 2.0 * VXY
H VXX VYY 1.0
H VXY VXY -2.0
ENDATA