Example 6.9: Bounding consumer preference

Jump to:     Source code     Text output      Plots     Library index

% Section 6.5.5, Figures 6.25-6.26
% Boyd & Vandenberghe "Convex Optimization"
% Original by Lieven Vandenberghe
% Adapted for CVX Argyris Zymnis - 11/30/2005
%
% We are given a set of consumer preference data for bundles
% of two goods x_1 and x_2. These points are generated by
% taking 40 random points and comparing them using the
% utility function: u(x_1,x_2) = (1.1*sqrt(x_1)+0.8*sqrt(x_2))/1.9
% Then, if we have u(i) >= u(j) we say that (i,j) is in Pweak.
%
% Given this, we wish to compare the point (0.5,0.5) to each
% of the bundles in the given dataset. I.e. for each point k in the
% dataset, we wish to decide wether u(k) >= u(0) or u(k) <= u(0),
% or both, in which case we cannot make any conclusions about
% consumer preferences.
%
% To do this, we have to solve two LPs for each point:
%       minimize     u(k) - u(0)
%       subject to   g_i  >= 0
%                    u(j) <= u(i) + g_i^T(a_j - a_i), for all i,j
%                    u(i) >= u(j), for all (i,j) in Pweak
%
% and:
%       maximize     u(k) - u(0)
%       subject to   g_i  >= 0
%                    u(j) <= u(i) + g_i^T(a_j - a_i), for all i,j
%                    u(i) >= u(j), for all (i,j) in Pweak
%
% If the second LP has a strictly negative solution, we can deduce that
% u(k) < u(0). If on the other hand the second LP has a nonnegative
% solution and the first LP has a strictly positive solution, we can
% deduce that u(k) > u(0). Finally if none of the two previous cases
% holds, we cannot make a decision between the two bundles.
%
% NOTE: This file requires the auxilliary function utilfun.m to run.

data= [...
   4.5e-01   9.6e-01
   2.1e-01   3.4e-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
   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
   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
   2.8e-01   8.7e-01
   6.9e-01   1.0e-01
   6.9e-01   3.7e-01
   0.0e+00   7.2e-01
   8.7e-01   1.7e-01
   6.3e-01   4.0e-02
   3.2e-01   7.3e-01
   4.0e-02   4.6e-01
   3.6e-01   9.5e-01
   8.2e-01   6.7e-01 ];

% objective point
obj=[0.5,0.5];

figure(1);
% display the utility function's level sets on some data points.

plot(data(:,1),data(:,2),'o');
hold on;

[X,Y] = meshgrid(0:.01:1,0:.01:1);
Z=(1.1*X.^(1/2)+0.8*Y.^(1/2))/1.9;

[C,h] = contour(X,Y,Z,[.1,.2,.3,.4,.5,.6,.7,.8,.9],'--');
clear X Y Z C
hold off;
xlabel('x_1');
ylabel('x_2');
hold off

m = size(data,1);  % number of baskets, including 0,1

% add preference data
Pweak = zeros(m+1,m+1);
for i=1:m,
   for j=1:m
      if (i~=j) & (1.1*data(i,1).^(1/2)+0.8*data(i,2).^(1/2))/1.9 >= ...
             (1.1*data(j,1).^(1/2)+0.8*data(j,2).^(1/2))/1.9,
         Pweak(i,j) = 1;
      end;
   end;
end;

% Find consumer preferences
data = [data; 0.5 0.5];
bounds = zeros(m,2);
for k = 1:m
    fprintf(1,'Deciding on bundle %d of %d: ',k,m);

    % Check for u(k) >= u(0.5,0.5)
    cvx_begin quiet
        variables u(m+1) g_x(m+1) g_y(m+1)
        minimize(u(k)-u(m+1))
        subject to
            g_x >= 0;
            g_y >= 0;
            ones(m+1,1)*u' <= u*ones(1,m+1)+(g_x*ones(1,m+1)).*...
              (ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+...
              (g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
            (u*ones(1,m+1)).*Pweak >= (ones(m+1,1)*u').*Pweak;
    cvx_end
    bounds(k,1) = cvx_optval;
    fprintf( 1,'%g', round(cvx_optval) );

    % Check for u(0.5,0.5) >= u(k)
    cvx_begin quiet
        variables u(m+1) g_x(m+1) g_y(m+1)
        maximize(u(k)-u(m+1))
        subject to
            g_x >= 0;
            g_y >= 0;
            ones(m+1,1)*u' <= u*ones(1,m+1) + (g_x*ones(1,m+1)).*...
              (ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+...
              (g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
            (u*ones(1,m+1)).*Pweak >= (ones(m+1,1)*u').*Pweak;
    cvx_end
    bounds(k,2) = cvx_optval;
    fprintf( 1,' %g\n', round(cvx_optval) );

end

figure(2);
hold off

% plot data pt and contour line through it
val = 1.1*sqrt(0.5)+ 0.8*sqrt(.5);   % value at center
t = linspace(((val-.8)/1.1)^2, 1, 1000);
y = ( (val - 1.1*(t.^(1/2)))/.8 ).^2;
plot(t,y,'--', [.5 .5], [0 1], ':', [0 1], [.5 .5], ':');
axis([0 1 0 1]);
hold on

for k=1:m
   if bounds(k,2) < 1e-5,  % preferred over (.5,.5)
      dot = plot(data(k,1),data(k,2),'o');
      %'MarkerSize',8);
   elseif bounds(k,1) > -1e-5,  % rejected in favor of (.5,.5)
      dot = plot(data(k,1),data(k,2),'o','MarkerFaceColor',[0 0 0]);
   else % no conclusion
      dot = plot(data(k,1),data(k,2),'square', 'LineWidth',1.0,...
           'MarkerSize',10);
   end;
end;
xlabel('x_1');  ylabel('x_2');

Deciding on bundle 1 of 40: 0 Inf
Deciding on bundle 2 of 40: -Inf -0
Deciding on bundle 3 of 40: -Inf -0
Deciding on bundle 4 of 40: -Inf -0
Deciding on bundle 5 of 40: -Inf -0
Deciding on bundle 6 of 40: -Inf 0
Deciding on bundle 7 of 40: -Inf -0
Deciding on bundle 8 of 40: 0 Inf
Deciding on bundle 9 of 40: 0 Inf
Deciding on bundle 10 of 40: -Inf -0
Deciding on bundle 11 of 40: 0 Inf
Deciding on bundle 12 of 40: -Inf -0
Deciding on bundle 13 of 40: -Inf -0
Deciding on bundle 14 of 40: 0 Inf
Deciding on bundle 15 of 40: 0 Inf
Deciding on bundle 16 of 40: -Inf -0
Deciding on bundle 17 of 40: 0 Inf
Deciding on bundle 18 of 40: -Inf -0
Deciding on bundle 19 of 40: 0 Inf
Deciding on bundle 20 of 40: -Inf -0
Deciding on bundle 21 of 40: 0 Inf
Deciding on bundle 22 of 40: 0 Inf
Deciding on bundle 23 of 40: -Inf -0
Deciding on bundle 24 of 40: 0 Inf
Deciding on bundle 25 of 40: 0 Inf
Deciding on bundle 26 of 40: -Inf -0
Deciding on bundle 27 of 40: -Inf -0
Deciding on bundle 28 of 40: -Inf -0
Deciding on bundle 29 of 40: -Inf Inf
Deciding on bundle 30 of 40: -Inf -0
Deciding on bundle 31 of 40: -Inf Inf
Deciding on bundle 32 of 40: -Inf -0
Deciding on bundle 33 of 40: -Inf Inf
Deciding on bundle 34 of 40: -Inf -0
Deciding on bundle 35 of 40: -Inf Inf
Deciding on bundle 36 of 40: -Inf -0
Deciding on bundle 37 of 40: -Inf Inf
Deciding on bundle 38 of 40: -Inf -0
Deciding on bundle 39 of 40: 0 Inf
Deciding on bundle 40 of 40: 0 Inf