Vpasolve can not find a solution that I know it exist

clear
clc
%% Parameters and Assumptions
syms x11 x12 x21 x22 positive % Actions
syms a11 a12 a21 a22 positive % Marginal effects
syms c1 c2 positive % Costs
syms W1 W2 positive % Budget constraint
syms epsilon1 epsilon2 positive % Inital voters (can be seen as partisian voters)
syms b1 b2 positive % Total effects
syms V positive % Prize of winning
assume(a11 > 0 & a11 < 1)
assume(a12 > 0 & a12 < 1)
assume(a21 > 0 & a21 < 1)
assume(a22 > 0 & a22 < 1)
%% Theoretical Model
% Votes gain functions
f1 = b1*(x11^a11)*(x12^a12);
f2 = b2*(x21^a21)*(x22^a22);
F1 = f1 + epsilon1;
F2 = f2 + epsilon2;
% Payoff - Probability of winning.
P1 = (F1/(F1 + F2))*V;
P2 = (F2/(F1 + F2))*V;
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
isAlways(DP11 > 0)

ans = logical
   1

isAlways(DP21 > 0)

ans = logical
   1

% The probability of winning is strictly increasing in activity one (and also two).
% Therefore candidates spend all budget.
% Budget constraints
BC1 = W1 == x11*c1 + x12*c2;
BC2 = W2 == x21*c1 + x22*c2;
%% Theoretical Solution
BC1 = isolate(BC1,x12);
BC2 = isolate(BC2,x22);
P1 = subs(P1, lhs(BC1), rhs(BC1));
P2 = subs(P2, lhs(BC2), rhs(BC2));
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
eqns1 = [DP11 == 0];
vars1 = [x11];
S11 = solve(eqns1,vars1,'Real',true);
eqns2 = [DP21 == 0];
vars2 = [x21];
S21 = solve(eqns2,vars2,'Real',true);
% S11 and S21 are the equilibrium efforts of activity one for candidates one and two.
S12 = (W1 - S11*c1)/c2;
S22 = (W2 - S21*c1)/c2;
S12 = simplify(S12);
S22 = simplify(S22);
%% Equations
% First order conditions
Eq1 = ((a11*W1)/(c1*(a11+a12))) - x11 == 0; 
Eq2 = ((a12*W2)/(c2*(a21+a22))) - x12 == 0; 
% Election result (it is number of votes)
Eq3 = f1 - 1189313 == 0;
Eq4 = f2 - 1152271 == 0;
% Parameter restrictions
Eq5 = a11 + a12 - 1 == 0;
Eq6 = a21 + a22 - 1 == 0;
%% Substitution
% Candidate number one is LLP and candidate number two is DM.
Data = [28.752/c1,0.911/c2,29.664,31.012/c1,2.490/c2,33.502,1022013,1143681,1,1];
% The budget is in millions of UYU, epsilon1 and epsilon2 is number of inicial voters.
% Since do not have enough equations for parameters, assume b1 = b2 = 1.
% Last two digits of the vector "Data" reflect that.
E1 = subs(Eq1,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data);
E2 = subs(Eq2,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data);
E3 = subs(Eq3,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data);
E4 = subs(Eq4,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data);
E5 = a11 + a12 -1 == 0;
E6 = a21 + a22 -1 == 0;
%% Calibration
X0 = [0 1; 0 1; 0 1; 0 1; 0 Inf; 0 Inf];
S = vpasolve([E1,E2,E3,E4,E5,E6],[a11, a12, a21, a22, c1, c2],X0)

S = struct with fields:
    a11: [0x1 sym]
    a12: [0x1 sym]
    a21: [0x1 sym]
    a22: [0x1 sym]
     c1: [0x1 sym]
     c2: [0x1 sym]

I know there is a solution to this system of equations but Matlab does not find it and I can see what is wrong on the code.

Answer 1

[Torsten]

Maybe it must read

Eq2 = ((a21*W2)/(c2*(a21+a22))) - x12 == 0; 

instead of

Eq2 = ((a12*W2)/(c2*(a21+a22))) - x12 == 0; 

?

clear
clc
%% Parameters and Assumptions
syms x11 x12 x21 x22 positive % Actions
syms a11 a12 a21 a22 positive % Marginal effects
syms c1 c2 positive % Costs
syms W1 W2 positive % Budget constraint
syms epsilon1 epsilon2 positive % Inital voters (can be seen as partisian voters)
syms b1 b2 positive % Total effects
syms V positive % Prize of winning
assume(a11 > 0 & a11 < 1)
assume(a12 > 0 & a12 < 1)
assume(a21 > 0 & a21 < 1)
assume(a22 > 0 & a22 < 1)
%% Theoretical Model
% Votes gain functions
f1 = b1*(x11^a11)*(x12^a12);
f2 = b2*(x21^a21)*(x22^a22);
F1 = f1 + epsilon1;
F2 = f2 + epsilon2;
% Payoff - Probability of winning.
P1 = (F1/(F1 + F2))*V;
P2 = (F2/(F1 + F2))*V;
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
% The probability of winning is strictly increasing in activity one (and also two).
% Therefore candidates spend all budget.
% Budget constraints
BC1 = W1 == x11*c1 + x12*c2;
BC2 = W2 == x21*c1 + x22*c2;
%% Theoretical Solution
BC1 = isolate(BC1,x12);
BC2 = isolate(BC2,x22);
P1 = subs(P1, lhs(BC1), rhs(BC1));
P2 = subs(P2, lhs(BC2), rhs(BC2));
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
eqns1 = [DP11 == 0];
vars1 = [x11];
S11 = solve(eqns1,vars1,'Real',true);
eqns2 = [DP21 == 0];
vars2 = [x21];
S21 = solve(eqns2,vars2,'Real',true);
% S11 and S21 are the equilibrium efforts of activity one for candidates one and two.
S12 = (W1 - S11*c1)/c2;
S22 = (W2 - S21*c1)/c2;
S12 = simplify(S12);
S22 = simplify(S22);
%% Equations
% First order conditions
Eq1 = ((a11*W1)/(c1*(a11+a12))) - x11 == 0; 
Eq2 = ((a21*W2)/(c2*(a21+a22))) - x12 == 0; 
% Election result (it is number of votes)
Eq3 = f1 - 1189313 == 0;
Eq4 = f2 - 1152271 == 0;
% Parameter restrictions
Eq5 = a11 + a12 - 1 == 0;
Eq6 = a21 + a22 - 1 == 0;
%% Substitution
% Candidate number one is LLP and candidate number two is DM.
Data = [28.752/c1,0.911/c2,29.664,31.012/c1,2.490/c2,33.502,1022013,1143681,1,1];
% The budget is in millions of UYU, epsilon1 and epsilon2 is number of inicial voters.
% Since do not have enough equations for parameters, assume b1 = b2 = 1.
% Last two digits of the vector "Data" reflect that.
E1 = subs(Eq1,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E1 = 

E2 = subs(Eq2,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E2 = 

E3 = subs(Eq3,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E3 = 

E4 = subs(Eq4,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E4 = 

E5 = a11 + a12 -1 == 0

E5 = 

E6 = a21 + a22 -1 == 0

E6 = 

syms c1s c2s
sola = solve([E1,E2,E5,E6],[a11 a12 a21 a22]);
E3 = subs(E3,[a11 a12 a21 a22],[sola.a11 sola.a12 sola.a21 sola.a22]);
E4 = subs(E4,[a11 a12 a21 a22],[sola.a11 sola.a12 sola.a21 sola.a22]);
E3 = E3 + 1189313;
E3 = simplify(log(lhs(E3))) == log(rhs(E3));
E3 = subs(E3,[log(c1) log(c2)],[c1s c2s]);
E4 = E4 + 1152271;
E4 = simplify(log(lhs(E4))) == log(rhs(E4));
E4 = subs(E4,[log(c1) log(c2)],[c1s c2s]);
solcs = solve([E3 E4],[c1s c2s]);
a11 = vpa(sola.a11)

a11 = 

0.96925566343042071197411003236246

a12 = vpa(sola.a12)

a12 = 

0.03074433656957928802588996763754

a21 = vpa(sola.a21)

a21 = 

0.027192406423497104650468628738583

a22 = vpa(sola.a22)

a22 = 

0.97280759357650289534953137126142

c1 = vpa(exp(solcs.c1s))

c1 = 

0.00002339002395546612021028972551146

c2 = vpa(exp(solcs.c2s))

c2 = 

0.0000021694431095803284252266058790349

[Roque Ivan Acosta Bellizzi]

Hi, thanks for fast response. You are right.

Answer 2

[Walter Roberson]

There is no solution in the reals.

clear
clc
%% Parameters and Assumptions
syms x11 x12 x21 x22 positive % Actions
syms a11 a12 a21 a22 positive % Marginal effects
syms c1 c2 positive % Costs
syms W1 W2 positive % Budget constraint
syms epsilon1 epsilon2 positive % Inital voters (can be seen as partisian voters)
syms b1 b2 positive % Total effects
syms V positive % Prize of winning
assume(a11 > 0 & a11 < 1)
assume(a12 > 0 & a12 < 1)
assume(a21 > 0 & a21 < 1)
assume(a22 > 0 & a22 < 1)
%% Theoretical Model
% Votes gain functions
f1 = b1*(x11^a11)*(x12^a12);
f2 = b2*(x21^a21)*(x22^a22);
F1 = f1 + epsilon1;
F2 = f2 + epsilon2;
% Payoff - Probability of winning.
P1 = (F1/(F1 + F2))*V;
P2 = (F2/(F1 + F2))*V;
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
isAlways(DP11 > 0)

ans = logical
   1

isAlways(DP21 > 0)

ans = logical
   1

% The probability of winning is strictly increasing in activity one (and also two).
% Therefore candidates spend all budget.
% Budget constraints
BC1 = W1 == x11*c1 + x12*c2;
BC2 = W2 == x21*c1 + x22*c2;
%% Theoretical Solution
BC1 = isolate(BC1,x12);
BC2 = isolate(BC2,x22);
P1 = subs(P1, lhs(BC1), rhs(BC1));
P2 = subs(P2, lhs(BC2), rhs(BC2));
DP11 = diff(P1,x11);
DP21 = diff(P2,x21);
eqns1 = [DP11 == 0];
vars1 = [x11];
S11 = solve(eqns1,vars1,'Real',true);
eqns2 = [DP21 == 0];
vars2 = [x21];
S21 = solve(eqns2,vars2,'Real',true);
% S11 and S21 are the equilibrium efforts of activity one for candidates one and two.
S12 = (W1 - S11*c1)/c2;
S22 = (W2 - S21*c1)/c2;
S12 = simplify(S12);
S22 = simplify(S22);
%% Equations
% First order conditions
Eq1 = ((a11*W1)/(c1*(a11+a12))) - x11 == 0; 
Eq2 = ((a12*W2)/(c2*(a21+a22))) - x12 == 0; 
% Election result (it is number of votes)
Eq3 = f1 - 1189313 == 0;
Eq4 = f2 - 1152271 == 0;
% Parameter restrictions
Eq5 = a11 + a12 - 1 == 0;
Eq6 = a21 + a22 - 1 == 0;
%% Substitution
% Candidate number one is LLP and candidate number two is DM.
Data = [28.752/c1,0.911/c2,29.664,31.012/c1,2.490/c2,33.502,1022013,1143681,1,1];
% The budget is in millions of UYU, epsilon1 and epsilon2 is number of inicial voters.
% Since do not have enough equations for parameters, assume b1 = b2 = 1.
% Last two digits of the vector "Data" reflect that.
E1 = subs(Eq1,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E1 = 

E2 = subs(Eq2,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E2 = 

E3 = subs(Eq3,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E3 = 

E4 = subs(Eq4,[x11,x12,W1,x21,x22,W2,epsilon1,epsilon2,b1,b2],Data)

E4 = 

E5 = a11 + a12 -1 == 0

E5 = 

E6 = a21 + a22 -1 == 0

E6 = 

eqns = [E1, E2, E3, E4, E5, E6].';
partial_a11 = solve(eqns(1), a11, 'returnconditions', true)

partial_a11 = struct with fields:
           a11: (599*a12)/19
    parameters: [1×0 sym]
    conditions: a12 < 19/599

eqns2 = subs(eqns(2:end), a11, partial_a11.a11)

eqns2 = 

partial_a12 = solve(eqns2(4), a12)

partial_a12 = 

eqns3 = subs(eqns2([1 2 3 5]), a12, partial_a12)

eqns3 = 

partial_a21 = solve(eqns3(4), a21)

partial_a21 = 

eqns4 = subs(eqns3([1 2 3]), a21, partial_a21)

eqns4 = 

%partial_c2 = Inf;    %by inspection

%at this point we have something divided by c2, raised to the power of a22.
%With c2 being inf to satisfy 3677/(30900*c2) == 0, normally
%(249/(100*c2))^a22 would give 0^a22 which would give 0, and 0 times something minus something cannot give 0.
% But if a22 is 0 then it would be 0^0 which would give 1, and we survive
% anther step
partial_a22 = 0;     %by inspection
eqns5 = subs(eqns4(2:end), a22, partial_a22)

eqns5 = 

partial_c1 = solve(eqns5(2), c1)

partial_c1 = 

eqns6 = subs(eqns5(1), c1, partial_c1)

eqns6 = 

partial_c2 = solve(eqns6, c2)

partial_c2 = 

subs(eqns4(1), c2, partial_c2)

ans = 

[Roque Ivan Acosta Bellizzi]

Hi, you are right. Like @Torsten suggests, there is a mistake in the code. Thanks.