Plot[{Abs[x], Sign[x]},  {x, -10, 10}]

Sign'[x_] := 0;

Abs'[x_] := Sign[x];

\(\(\( (*\ 
      2 - Parameter\ ICE\ Preference\ \(Maps : \ 
            Functional\ DEFINITION\)\ *) \)\(
\)\)\)

\(\(\(P[x_, 
          y_, β_, γ_] := \ \(\((x\^2 + 
                  y\^2)\)\^\(\((β - γ)\)/2\)\) Sign[x - y] 
          Abs[\ x - y]\^γ;\)\(
\)
    \)\)

Derivative[1, 0, 0, 0][P][x, y, β, γ]

\(\(\(
\)\( (*\ NOTE : \ Sign[]\^2\ is\ almost\ always\ 1, \ 
      but\ this\ cannot\ be\ declared\ because\ "Tag Power is Protected."\ *) \)\(
\)\)\)

FullSimplify[Derivative[1, 0, 0, 0][P][x, 
        y, β, γ]]

dPdx[x_, 
          y_, β_, γ_] :=  +((((x^2) + (y^2)))^(-1 + (β - γ)/2)) Abs[
                x - y]^(-1 + γ) (β (x^2) + (γ - β) xy + γ (y^2));

Derivative[0, 1, 0, 0][P][x, y, β, γ]

FullSimplify[Derivative[0, 1, 0, 0][P][x, 
        y, β, γ]]

dPdy[x_, 
          y_, β_, γ_] :=  -((((x^2) + (y^2)))^(-1 + (β - γ)/2)) Abs[
                x - y]^(-1 + γ) (γ (x^2) + (γ - β) x y + β (y^2));

ContourPlot[P[x, y, 0.2, 2], {x, -10, 10}, {y, -10, 10},  
      PlotPoints -> 250, Contours -> 41,  
      ContourShading -> False]

Plot[dPdy[2, y, 0.2, 2], {y, -10, 10}]

ContourPlot[
      dPdy[x, y, 0.2, 2], {x, -10, 10}, {y, -10, 10},  
      PlotPoints -> 250, 
      Contours -> 41]

Solve[γ (x^2) + (γ - β) 
            x y + β (y^2) == 0, x]

Solve[(γ^2) - 6  γ β + (β^2) == 
        0, γ]

N[3 + 2 (Sqrt[ 2]), 25]

(* Note that lower bound of 3 - 
        2 (Sqrt[ 2]) 
          is also the reciprocal of the upper bound. *)

N[3 - 2 (Sqrt[ 2]), 25]

\(\(\( (*\ 
      This\ notebook\ explores\ properties\ of\ the\ most\ "directional"\ ICE\ preference\ *) \)\(
\)\( (*\ 
      maps\ that\ satisfy\ the\ axiom\ of\ Cartesian\ \(\(Monotonicity\ [
            CM]\)\(.\)\)\ *) \)\(
\)\( (*	
      These\ extreme\ maps\ are\ called\ "ICE-Omega"\ preference\ maps\ because\ their\ *) \)\(
\)\( (*\ 
      power - parameter - ratio, \ γ/β, \ 
      equals\ the\ CM\ upper\ limit\ of\ Ω\  = \ \((1\  + \ \@2)\)\^2, \ *) \)\(
\)\( (*\ 
      which\ is\ approximately\ 5.828\ *) \)\)\)

P[x_, 
          y_, β_, Ω_] :=  ((((x^2) + 
                  (y^2)))^(β (1 - Ω)/2)) 
          Sign[x - y] 
          Abs[ x - y]^(β Ω);

FullSimplify[{1 - (((1 + (Sqrt[2])))^2)}/2]

\(\(\( (*\ 
      The\ ContourPlot\ below\ depicts\ an\ ICE - Omega\ map\ which, \ 
      due\ to\ β\  = \ 0.2\  < \ 1, \ *) \)\(\ \)\(
\)\( (*\ 
      corresponds\ to\ decreasing\ returns - 
        to - \(\(sacle\)\(.\)\)\ *) \)\)\)

ContourPlot[
      P[x, y, 0.2, (((1 + (Sqrt[2])))^2)], {x, -10, 10}, {y, -10, 10},  
      PlotPoints -> 250, Contours -> 41,  
      ContourShading -> False]

ContourPlot[
      P[x, y, 1, (((1 + (Sqrt[2])))^2)], {x, -10, 10}, {y, -10, 10},  
      PlotPoints -> 250, Contours -> 41,  
      ContourShading -> False]

(* Willingness is a function of only the std .  
            ICE ratio ρ  =  
        x/y when γ/β  ==  Ω. *)

Willingness[ρ_, Ω_] := (1 + (Ω - 
                    1) ρ + Ω (ρ^2)) / (Ω  + (Ω - 
                    1) ρ +  (ρ^2));

Solve[Willingness[ρ, (((1 + (Sqrt[2])))^2)] == 0, ρ]

Solve[Derivative[1, 0][Willingness][ρ, (((1 + (Sqrt[2])))^2)] == 
        0, ρ]

N[Solve[Derivative[1, 0][
              Willingness][ρ, (((1 + (Sqrt[2])))^2)] == 
          0, ρ]]

Plot[{Willingness[ρ, (((1 + (Sqrt[2])))^2)]},  {ρ, -5, 
        5}]

N[Willingness[-1 - (Sqrt[2]), (((1 + (Sqrt[2])))^2)]]

N[Willingness[1 - (Sqrt[2]), (((1 + (Sqrt[2])))^2)]]

\(\(\( (*	\(Unlike\ other\ 2\ parameter\ ICE\ preference\ maps\ possessing\ Monotonicity\)\(,\)\ *) \)\(\ \)\(
\)\( (*\ 
      Willingness\ within\ ICE - 
        Omega\ maps\ ranges\ from\ 0\ to\  + ∞ . 		In\ maps\ with\ a\ *) \)\(
\)\( (*\ 
      power - parameter - 
          ratio\  = \ γ/β\ that\ is\ strictly\ between\ 1/Ω\ and\ Ω, \ the\ minimum\ *) \)\(
\)\( (*\ 
      Willingness\ is\ strictly\ positive, \ 
      and\ the\ maximum\ Willingness\ is\ \(\(bounded\)\(.\)\)\ *) \)\)\)

Solve[Derivative[1, 0][Willingness][ρ, η] == 
        0, ρ]

FullSimplify[
      Willingness[((1 - (Sqrt[η]))/(1 + (Sqrt[η]))), η]]

Plot[-1 + ((4 (Sqrt[η]))/(-1 + 2 (Sqrt[η]) + η)),  {η, 0.3, 6}]

FullSimplify[
      Willingness[((1 + (Sqrt[η]))/(1 - (Sqrt[η]))), η]]

Plot[-1 + ((4 (Sqrt[η]))/(1 + 2 (Sqrt[η]) - η)),  {η, 
        0, 5}]

Ψ[
          r_, θ_, β_, Ω_] :=  (r^β) 
          Sign[Cos[θ]] 
          Abs[ Cos[θ]]^(β Ω);

ContourPlot[Ψ[r, θ, 
        0.2, (((1 + (Sqrt[2])))^2)], {θ, -π/4, π/4}, {r, 0, 
        10},  PlotPoints -> 250, Contours -> 41,  
      ContourShading -> False]

\(\(\( (*\ When\ ICE\ radius\ is\ restricted, \ 
      such\ as\ "r at most 10"\ in\ the\ above\ graph, \ 
      some\ *) \)\(
\)\( (*\ 
      relatively\ large\ values\ of\ ICE\ preference\ can\ be\ achived\ within\ the\ SE\ quadrant\ *) \)\(
\)\( (*\ 
      that\ cannot\ be\ achieved\ within\ any\ other\ \(\(quadrant\)\(.\)\)\ *) \)\)\)

Plot[Cos[θ],  {θ, -π/4, π/4}]

\(\(\( (*\ 
      Along\ the\ θ\  = \ π/
            4\ boundary\ between\ the\ SE\ and\ NE\ quadrants, \ 
      preference\ within\ *) \)\(
\)\( (*\ 
      ICE - Omega\ maps\ can\ be\ made\ arbitrarily\ large\ simply\ by\ increasing\ the\ ICE\ *) \)\(
\)\( (*\ radius, \ r, \ 
      as\ long\ as\ one\ restricts\ β\ to\ be\ strictly\ \(\(positive\)\(.\)\)\ *) \)\)\)

FullSimplify[Ψ[
        r, π/4, β, (((1 + (Sqrt[2])))^2)]]

Plot[Ψ[r, π/4, 0.2, (((1 + (Sqrt[2])))^2)],  {r, 0,  
        10}]

Plot[Ψ[r, π/4, 2, (((1 + (Sqrt[2])))^2)],  {r, 0,  
        10}]

\(\(Will2[θ_, Ω_] := \((1 + \((Ω - 
                    1)\) Tan[θ - π/
                      4] + Ω\ Tan[θ - π/4]\^2)\)\ /\ 
		\((Ω\  + \((Ω - 1)\) 
                Tan[θ - π/4] + 
              Tan[θ - π/4]\^2)\);\)\)

Solve[Will2[θ, (((1 + (Sqrt[2])))^2)] == 0, θ]

N[(1/4) (π + 4 ArcTan[1 - (Sqrt[2])])]

N[180* 0.392699/π]

N[Sin[(1/4) (π + 4 ArcTan[1 - (Sqrt[2])])]]

N[1/Sin[(1/4) (π + 4 ArcTan[1 - (Sqrt[2])])]]

N[Will2[π/8, (((1 + (Sqrt[2])))^2)]]

N[Will2[-π/8, (((1 + (Sqrt[2])))^2)]]

P[x_, y_, β_, η_, γ_,  ρ_] := 
        If[(x^2) + 
              (y^2) < (ρ^2), (((((x^2) + (y^2))/(ρ^2)))^((β - γ)/2)) Sign[x - y] 
            Abs[x - y]^γ,  (((((x^2) + (y^2))/(ρ^2)))^((η - γ)/2)) Sign[x - y] 
            Abs[x - y]^γ];

 (* Inside the circle of radius ρ,  
      one rescaling of the 2 - 
        parameter (β, γ) 
          map applies. *) 
 (* 
      Outside the circle of radius ρ,  
      a different rescaling of the 2 - 
        parameter (η, γ) 
          map applies. *) 
 (* 
      On the circle of radius ρ,  
      differences between β and η are unimportant because 1 to any power is 1.  *)

ContourPlot[
      P[x, y, 0.2,  0.5, 1,  5], {x, -10, 10}, {y, -10, 10},  
      PlotPoints -> 250, Contours -> 41,  
      ContourShading -> False]