Reduce, Piecewise function gives result that does not make any sense given condition constraint

Question

Here is my code:

N1 = 1;
N2 = 1;
 X1 = N2/(1 - \[Sigma]);


   c = 1
surpluspayoff10011010 = (N1 + x + y)/((N1 + x + y) + N2)* X1 - 
   c y - (((N1 + x)/((N1 + x) + N2))*X1);
yargmax10011010 = Simplify[y /. NSolve[\!\(
\*SubscriptBox[\(\[PartialD]\), \(y\)]\((surpluspayoff10011010)\)\) ==
        0, y]][[2]];
yargmax10011010 = 
  Simplify[Piecewise[{{0, 
      Simplify@
       Reduce[{Rationalize@yargmax10011010 < 0  && 0 <= \[Sigma] < 1 && 
          ymax > 1}, x]}, {yargmax10011010, 
      Simplify@
       Reduce[{Rationalize@yargmax10011010 <= ymax && 
          0 <= \[Sigma] < 1 && ymax > 1}, x]}, {ymax, 
      Simplify@
       Reduce[{Rationalize@yargmax10011010 > ymax && 
          0 <= \[Sigma] < 1 && ymax > 1}, x]  }}]];
surpluspayoff10011010 = 
  Simplify[(N1 + yargmax10011010 + 
        x)/((N2) + (N1 + x + yargmax10011010))* X1 - 
    yargmax10011010 - (((N1 + x)/((N1 + x) + N2))*X1) ];
yargmax10011010 = 
  Simplify[Piecewise[{{0, 
      Simplify@
       Reduce[{Rationalize@surpluspayoff10011010 < 0 && 
          0 <= \[Sigma] < 1  && ymax > 1}, x]}, {yargmax10011010, 
      Simplify@
       Reduce[{Rationalize@surpluspayoff10011010 >= 0 && 
          0 <= \[Sigma] < 1 && ymax > 1}, x]}}]];

I cannot put the result here, since it will look very ugly here. But the thing is I already gave the condition for 0<\sigma<1 , 1< ymax, but the result for yargmax10011010 is something that violated these conditions. Basicly,look at the result of yargmax10011010 when y=0, we have conditions such as ymax =<1, and such

Answer 1

I used simpler variable names (e.g., y1 in place of yargmax10011010) and rather than redefining them created new ones (this last because when you are tracking down a problem it often is better not to redefine variables).

I believe that MarcoB is correct, adding assumptions to Simplify removes the offending inequalities.

N1 = 1;
N2 = 1;
X1 = N2/(1 - σ);
c = 1;

s1 = (N1 + x + y)/((N1 + x + y) + N2)*X1 - 
  c y - (((N1 + x)/((N1 + x) + N2))*X1)
(* -y - (1 + x)/((2 + x) (1 - σ)) + 
 (1 + x + y)/((2 + x + y) (1 - σ)) *)

The first pass

y1 = Simplify[y /. NSolve[D[s1, y] == 0, y]][[2]]
(* (2. + x + Sqrt[1. - 1. σ] - 2. σ - 
 1. x σ)/(-1. + σ) *)

The second pass

y2 = Simplify[
  Piecewise[{
    {0, Simplify@
      Reduce[{Rationalize@y1 < 0 && 0 <= σ < 1 && ymax > 1}, 
       x]}, {y1, 
     Simplify@
      Reduce[{Rationalize@y1 <= ymax && 0 <= σ < 1 && 
         ymax > 1}, x]}, {ymax, 
     Simplify@
      Reduce[{Rationalize@y1 > ymax && 0 <= σ < 1 && ymax > 1},
        x]}}
   ], 
  ymax > 1
  ]

I will have to use a snapshot for the result

Use y2 for new version of s1

s2 = Simplify[(N1 + y2 + x)/((N2) + (N1 + x + y2))*X1 - 
      y2 - (((N1 + x)/((N1 + x) + N2))*X1), ymax > 1]

and finally

y3 = Simplify[
  Piecewise[{
    {0, Simplify@
      Reduce[{Rationalize@s2 < 0 && 0 <= \[Sigma] < 1 && ymax > 1}, 
       x]}, {y2, 
     Simplify@
      Reduce[{Rationalize@s2 >= 0 && 0 <= \[Sigma] < 1 && ymax > 1}, 
       x]}
    }],
  ymax > 1
  ]