System of inequalities with Max command

Question

I'm trying to use both Max and Reduce function but Mathematica seems not able to find any solution. I put the code below and I hope one can tell me what's wrong in my coding. Thanks in advance for your propositions!

Reduce[{a^2/(
    4 (b + c1 + s θ1)) + β ((a^2 (1 - p))/(
       4 (b + c1 + s θ1)) + 
       p Max[-F3 + a^2/(4 (b + c3 + s θ3)), a^2/(
         4 (b + c1 + s θ1))]) > -F2 + a^2/(
    4 (b + c2 + s θ2)) + β ((a^2 (1 - p))/(
       4 (b + c2 + s θ2)) + 
       p Max[-F3 + a^2/(4 (b + c3 + s θ3)), a^2/(
         4 (b + c2 + s θ2))]), a > 0, 1 > p > 0, s > 0, 
  1 > b > 0, 1 > θ1 > θ2 > θ3 > 0, c1 > 0, 
  c2 > 0, c3 > 0, F2 > 0, F3 > 0, 
  0 < F2 < (a^2 c1 - a^2 c2 + a^2 c1 β - a^2 c2 β + 
      a^2 s θ1 + a^2 s β θ1 - a^2 s θ2 - 
      a^2 s β θ2)/(4 b^2 + 4 b c1 + 4 b c2 + 4 c1 c2 + 
      4 b s θ1 + 4 c2 s θ1 + 4 b s θ2 + 
      4 c1 s θ2 + 4 s^2 θ1 θ2)}, {c1, c2, c3, p, 
  F2, F3, F3, s}]

Answer 1

Even 8 unknowns are too much for Reduce, saying nothing about 6 parameters and nonlinear inequalities with Max. My best is

a = 1; b = 1/Pi; β = 3; θ1 = 1/3; θ2 = 1/4; θ3 = 1/5;
FindInstance[{a^2/(4 (b + c1 + 
     s θ1)) + β ((a^2 (1 - p))/(4 (b + c1 + 
        s θ1)) + 
   p* Max[-F3 + a^2/(4 (b + c3 + s θ3)), 
     a^2/(4 (b + c1 + s θ1))]) > -F2 + 
a^2/(4 (b + c2 + 
     s θ2)) + β ((a^2 (1 - p))/(4 (b + c2 + 
        s θ2)) + 
   p Max[-F3 + a^2/(4 (b + c3 + s θ3)), 
     a^2/(4 (b + c2 + s θ2))]), a > 0, 1 > p > 0, s > 0, 
  1 > b > 0, 1 > θ1 > θ2 > θ3 > 0, c1 > 0, 
  c2 > 0, c3 > 0, F2 > 0, F3 > 0,   0 < F2 < (a^2 c1 - a^2 c2 + 
a^2 c1 β - a^2 c2 β + 
  a^2 s θ1 + a^2 s β θ1 - a^2 s θ2 - 
  a^2 s β θ2)/(4 b^2 + 4 b c1 + 4 b c2 + 4 c1 c2 + 
  4 b s θ1 + 4 c2 s θ1 + 4 b s θ2 + 
  4 c1 s θ2 + 4 s^2 θ1 θ2)}, {c1, c2, c3, p, F2, F3, F3, s}, Reals]
(*{{c1 -> 5/64, c2 -> 5/64, c3 -> 7/128, p -> 3/4, F2 -> 9/64,  F3 -> 9/128, s -> 1}}*)

If a=1 is replaced by a=1.0 in the above, then the result is

(*{{c1 -> 0.0238732, c2 -> 0.0238732, c3 -> 0.0155176, p -> 0.5,  F2 -> 0.0658013, F3 -> 0.014157, s -> 0.143239}}*)