#!/usr/bin/python

# Figures 6.25 and 6.26, page 342.
# Consumer preference analysis.

import pylab
from cvxopt import solvers
from cvxopt.base import matrix, sqrt
from cvxopt.modeling import variable, op
solvers.options['show_progress'] = 0

def utility(x, y): 
    return (1.1 * sqrt(x) + 0.8 * sqrt(y)) / 1.9

B = matrix([ 
    4.5e-01,  9.6e-01,
    2.1e-01,  3.4e-01,
    2.8e-01,  8.7e-01,
    9.6e-01,  3.0e-02,
    8.0e-02,  9.2e-01,
    2.0e-02,  2.2e-01,
    0.0e+00,  3.9e-01,
    2.6e-01,  6.4e-01,
    3.5e-01,  9.7e-01,
    9.1e-01,  7.8e-01,
    1.2e-01,  1.4e-01,
    5.8e-01,  8.4e-01,
    3.2e-01,  7.3e-01,
    4.9e-01,  2.7e-01,
    7.0e-02,  8.0e-01,
    9.3e-01,  8.7e-01,
    4.4e-01,  8.6e-01,
    3.3e-01,  4.2e-01,
    8.9e-01,  9.0e-01,
    4.9e-01,  7.0e-02,
    9.5e-01,  3.3e-01,
    6.6e-01,  2.6e-01,
    9.5e-01,  7.3e-01,
    4.2e-01,  9.1e-01,
    6.8e-01,  2.0e-01,
    8.7e-01,  1.7e-01,
    5.2e-01,  6.2e-01,
    7.7e-01,  6.3e-01,
    2.0e-02,  2.9e-01,
    9.8e-01,  2.0e-02,
    5.0e-02,  7.9e-01,
    7.9e-01,  1.9e-01,
    6.2e-01,  6.0e-02, 
    6.9e-01,  1.0e-01,
    6.9e-01,  3.7e-01,
    0.0e+00,  7.2e-01,
    6.3e-01,  4.0e-02,
    4.0e-02,  4.6e-01,
    3.6e-01,  9.5e-01,
    8.2e-01,  6.7e-01 ], (2, 40)) 
m = B.size[1]

# Plot some contour lines.
nopts = 200
a = (1.0/nopts)*matrix(range(nopts), tc='d')
X, Y = a[:,nopts*[0]].T,  a[:,nopts*[0]]
pylab.figure(1, facecolor='w')
pylab.plot(B[0,:], B[1,:], 'wo', markeredgecolor='b')
pylab.contour(pylab.array(X), pylab.array(Y), pylab.array(utility(X,Y)),
    [.1*(k+1) for k in xrange(9)], colors='k')
pylab.xlabel('x1')
pylab.ylabel('x2')
pylab.title('Goods baskets and utility function (fig. 6.25)')
#print "Close figure to start analysis."
#pylab.show()


# P are basket indices in order of increasing preference 
l = zip(utility(B[0,:], B[1,:]), range(m))
l.sort()
P = [ e[1] for e in l ]

# baskets with known preference relations 
u = variable(m)    
gx = variable(m)  
gy = variable(m)  

# comparison basket at (.5, .5) has utility 0
gxc = variable(1)
gyc = variable(1)

monotonicity = [ gx >= 0, gy >= 0, gxc >= 0, gyc >= 0 ]
preferences = [ u[P[j+1]] >= u[P[j]] + 1.0 for j in xrange(m-1) ]
concavity = [ u[j] <= u[i] + gx[i] * ( B[0,j] - B[0,i] ) + 
    gy[i] * ( B[1,j] - B[1,i] ) for i in xrange(m) for j in 
    xrange(m) ] 
concavity += [ 0 <= u[i] + gx[i] * ( 0.5 - B[0,i] ) + 
    gy[i] * ( 0.5 - B[1,i] ) for i in xrange(m) ]  
concavity += [ u[j] <= gxc * ( B[0,j] - 0.5 ) + 
    gyc * ( B[1,j] - 0.5 ) for j in xrange(m) ]  

preferred, rejected, neutral = [], [], []
for k in xrange(m):
    p = op(-u[k], monotonicity + preferences + concavity)
    p.solve()
    if p.status == 'optimal' and p.objective.value()[0] > 0:
        rejected += [k]
        print "Basket (%1.2f, %1.2f) rejected." %(B[0,k],B[1,k])
    else: 
        p = op(u[k], monotonicity + preferences + concavity)
        p.solve()
        if p.status == 'optimal' and p.objective.value()[0] > 0: 
            print "Basket (%1.2f, %1.2f) preferred." %(B[0,k],B[1,k])
            preferred += [k]
        else:
            print "No conclusion about basket (%1.2f, %1.2f)." \
                %(B[0,k],B[1,k])
            neutral += [k]

pylab.figure(1, facecolor='w')
pylab.plot(B[0,:], B[1,:], 'wo', markeredgecolor='b')
pylab.contour(pylab.array(X), pylab.array(Y), pylab.array(utility(X,Y)),
    [.1*(k+1) for k in xrange(9)], colors='k')
pylab.xlabel('x1')
pylab.ylabel('x2')
pylab.title('Goods baskets and utility function (fig. 6.25)')

pylab.figure(2, facecolor='w')
pylab.plot(B[0,preferred], B[1,preferred], 'go')
pylab.plot(B[0,rejected], B[1,rejected], 'ro')
pylab.plot(B[0,neutral], B[1,neutral], 'ys')
pylab.plot([0.5], [0.5], '+')
pylab.plot([0.5, 0.5], [0,1], ':', [0,1], [0.5,0.5], ':')
pylab.axis([0,1,0,1])
pylab.contour(pylab.array(X), pylab.array(Y), pylab.array(utility(X,Y)),
    [utility(0.5,0.5)], colors='k')
pylab.xlabel('x1')
pylab.ylabel('x2')
pylab.title('Result of preference analysis (fig. 6.26)')
pylab.show()
