Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -(r-g)(α l+cs)+(1-e)β(α l+cs)

D[Profitv,e]

Solve[%==0,e]

e:=((l-l α+θ)/(2 k))

Profiti

Solve[%==0,r]

r:=((β (2 k-l+l α-θ))/(2 k))

Profits

Simplify[-cs-l α-(cs+l α) (-g+((β (2 k-l+l α-θ))/(2 k)))+((((1+d-p) p-l (1-α)) (l-l α+θ))/(2 k))+(cs+l α) β (1-((l-l α+θ)/(2 k)))]

D[%,l]

Solve[%==0,l]

D[(1/(2 k))(2 cs (-1+g) k-(l^2) (((-1+α))^2)+(1+d-p) p θ+l (-(1+d) p (-1+α)+(p^2) (-1+α)+2 (-1+g) k α+(-1+α) θ)),p]

Solve[%==0,p]

l:=((p+d p-(p^2)-2 k α+2 g k α-p α-d p α+(p^2) α-θ+α θ)/(2 (((-1+α))^2)))
p:=((1+d)/2)
l

Simplify[(1/(2 (((-1+α))^2)))(((1+d)/2)+(1/2) d (1+d)-(1/4) (((1+d))^2)-(1/2) (1+d) α-(1/2) d (1+d) α+(1/4) (((1+d))^2) α-2 k α+2 g k α-θ+α θ)]

e

Simplify[%18]

r

Simplify[%20]

Solve[1-((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 k (-1+α)))==0,α]

{{α->((1+2 d+(d^2)-16 k+4 θ)/(1+2 d+(d^2)-8 k-8 g k+4 θ))}}
Solve[((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 k (-1+α)))==0,α]

1. When the contract launch price is the optimal price for SA:

We solve this game by backward induction. By checking the Hessian matrix, we find that πSA  is jointly concave in p and l, when α<(((((1+d))^2)+4 θ)/((((1+d))^2)+8 k(1-g)+4 θ)).in this paper, we consider the case with  (((((1+d))^2)-16k+4 θ)/((((1+d))^2)-8 k(1+g)+4 θ))<α<(((((1+d))^2)+4 θ)/((((1+d))^2)+8 k(1-g)+4 θ)), which is the common situation because it means launch successfully propability is above zero and below one.
e = ((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α(1-g)-4 θ+4 α θ)/(16 k (-1+α))), that (ψ/(16 k (1-α))), where  ψ=1-2 d (-1+α)-(d^2) (-1+α)-α-8 k α+4 θ-4 α θ=(1-α)(((1-d))^2)-8αk(1-g)+(1-α)4θ

r = β(1-(ψ/(16 k (1-α)))),
p =(( 1+d)/( 2)),
l = ((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1+8 (-1+g) k+4 θ))/(8 (((-1+α))^2))) = ((ψ-8θ(1-α))/(8 (((1-α))^2)))

Profitv

Simplify[%8]

Profits

Simplify[%10]

CS:=(1/8) (((1+d))^2)

SW:=Profits+Profitv+CS

SW

Simplify[%14]

Simplify[+(1/8) (((1+d))^2)+(1/(32 k (((-1+α))^3)))(1-α) (((1+d)/2)+(1/2) d (1+d)-(1/4) (((1+d))^2)-(1/2) (1+d) α-(1/2) d (1+d) α+(1/4) (((1+d))^2) α-2 k α+2 g k α-θ+α θ) (-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))-((((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ)))^2)/(256 k (((-1+α))^2)))+(1/(16 k (-1+α)))(-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ)) ((1/2) (1+d) (1+(1/2) (-1-d)+d)-(1/(2 (((-1+α))^2)))(1-α) (((1+d)/2)+(1/2) d (1+d)-(1/4) (((1+d))^2)-(1/2) (1+d) α-(1/2) d (1+d) α+(1/4) (((1+d))^2) α-2 k α+2 g k α-θ+α θ))-(cs+(1/(2 (((-1+α))^2)))α (((1+d)/2)+(1/2) d (1+d)-(1/4) (((1+d))^2)-(1/2) (1+d) α-(1/2) d (1+d) α+(1/4) (((1+d))^2) α-2 k α+2 g k α-θ+α θ)) (-g+((β (-2 d (-1+α)-(d^2) (-1+α)+8 k (-2+α+g α)-(-1+α) (1+4 θ)))/(16 k (-1+α))))-θ (1-((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 k (-1+α))))+β (cs+(1/(2 (((-1+α))^2)))α (((1+d)/2)+(1/2) d (1+d)-(1/4) (((1+d))^2)-(1/2) (1+d) α-(1/2) d (1+d) α+(1/4) (((1+d))^2) α-2 k α+2 g k α-θ+α θ)) (1-((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 k (-1+α))))]

Derivation of the sensitivity analysis for Model NG. How d,k,θ  and  will affect these optimal decisions:

e:=((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 k (-1+α)))
r:=((β (-2 d (-1+α)-(d^2) (-1+α)+8 k (-2+α+g α)-(-1+α) (1+4 θ)))/(16 k (-1+α)))
l:=((p+d p-(p^2)-2 k α+2 g k α-p α-d p α+(p^2) α-θ+α θ)/(2 (((-1+α))^2)))
p:=((1+d)/2)
Profitv:=(1/(256 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+16 g k α+(α^2)-16 k (α^2)-16 g k (α^2)-192 (k^2) (α^2)+128 g (k^2) (α^2)+64 (g^2) (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+64 g k α θ+8 (α^2) θ-64 k (α^2) θ-64 g k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 (1+g) k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 (1+g) k-4 θ)+4 θ))
Profits:=(1/(128 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)+128 cs (-1+g) k (((-1+α))^2)-2 α-16 k α+16 g k α+(α^2)+16 k (α^2)-16 g k (α^2)+64 (k^2) (α^2)-128 g (k^2) (α^2)+64 (g^2) (k^2) (α^2)+8 θ-16 α θ+64 k α θ-64 g k α θ+8 (α^2) θ-64 k (α^2) θ+64 g k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 (-1+g) k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 (-1+g) k-4 θ)+4 θ))
CS:=(1/8) (((1+d))^2)
SW:=(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)+64 (-1-2 g+3 (g^2)) (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)+16 k (-1+α) (-2-16 cv (-1+α)+16 cs (-1+g) (-1+α)+3 α-3 g α+16 θ-12 α θ+4 g α θ)-4 d (-1+α) (8 k (2+3 (-1+g) α)-3 (-1+α) (1+4 θ))-2 (d^2) (-1+α) (8 k (2+3 (-1+g) α)-3 (-1+α) (3+4 θ)))

d

D[e,d]

Simplify[((2 (-1+α)+2 d (-1+α))/(16 k (-1+α)))]

D[r,d]

Simplify[(((-2 (-1+α)-2 d (-1+α)) β)/(16 k (-1+α)))]

D[l,d]

Simplify[(((1/2)+(1/2) (-1-d)+(d/2)+((1+d)/2)-(α/2)-((d α)/2))/(2 (((-1+α))^2)))]

D[p,d]

D[Profits,d]

Simplify[%23]

(((1+d) (-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ)))/(32 k (-1+α)))

D[Profitv,d]

Simplify[%25]

(((1+d) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 (1+g) k α-4 θ+4 α θ))/(64 k (-1+α)))

D[CS,d]

D[SW,d]

Simplify[%28]

(1/(64 k (-1+α)))(1+d) (6 d (-1+α)+3 (d^2) (-1+α)-8 k (2+3 (-1+g) α)+3 (-1+α) (1+4 θ))

k

D[e,k]

Simplify[(((8-8 g) α)/(16 k (-1+α)))-((-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 k-8 g k+4 θ))/(16 (k^2) (-1+α)))]

-((1+2 d+(d^2)+4 θ)/(16 (k^2)))
D[r,k]

Simplify[(((-2+α+g α) β)/(2 k (-1+α)))-((β (-2 d (-1+α)-(d^2) (-1+α)+8 k (-2+α+g α)-(-1+α) (1+4 θ)))/(16 (k^2) (-1+α)))]

((β (1+2 d+(d^2)+4 θ))/(16 (k^2)))

D[l,k]

Simplify[((-2 α+2 g α)/(2 (((-1+α))^2)))]

(((-1+g) α)/(((-1+α))^2))

D[p,k]

D[Profits,k]

Simplify[%38]

-(1/(128 (k^2) (((-1+α))^2)))(4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)+4 d (((-1+α))^2) (1+4 θ)+(((1+4 θ))^2)-2 α (((1+4 θ))^2)+2 (d^2) (((-1+α))^2) (3+4 θ)+(α^2) (-64 (((-1+g))^2) (k^2)+(((1+4 θ))^2)))

y2:=((((1-α)(((1+d))^2) -8 k α(1-g)+4 θ-4 α θ))^2)+16α k(1-g)((1-α)(((1+d))^2) -8 k α(1-g)+4 θ-4 α θ)-(4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)+4 d (((-1+α))^2) (1+4 θ)+(((1+4 θ))^2)-2 α (((1+4 θ))^2)+2 (d^2) (((-1+α))^2) (3+4 θ)+(α^2) (-64 (((-1+g))^2) (k^2)+(((1+4 θ))^2)))
y2

Simplify[%23]

D[Profitv,k]

Simplify[%40]

y1:=((((1-α)(((1+d))^2) -8 k α(1-g)+4 θ-4 α θ))^2)+16α k(1-g)((1-α)(((1+d))^2) -8 k α(1-g)+4 θ-4 α θ+16α k)-(4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-64 (-3+2 g+(g^2)) (k^2) (α^2)+4 d (((-1+α))^2) (1+4 θ)+(((1+4 θ))^2)-2 α (((1+4 θ))^2)+2 (d^2) (((-1+α))^2) (3+4 θ)+(((α+4 α θ))^2))
y1

Simplify[%20]

D[CS,k]

D[SW,k]

Simplify[%43]

θ

D[e,θ]

Simplify[((-4+4 α)/(16 k (-1+α)))]

D[r,θ]

D[l,θ]

D[p,θ]

D[Profits,θ]

Simplify[%50]

D[CS,θ]

D[SW,θ]

Simplify[%53]

g

D[e,g]

D[r,g]

D[l,g]

D[p,g]

D[Profits,g]

Simplify[(1/(128 k (((-1+α))^2)))(128 cs k (((-1+α))^2)+16 k α-32 d k (-1+α) α-16 (d^2) k (-1+α) α-16 k (α^2)-128 (k^2) (α^2)+128 g (k^2) (α^2)-64 k α θ+64 k (α^2) θ)]

D[Profitv,g]

Simplify[(1/(256 k (((-1+α))^2)))(16 k α-32 d k (-1+α) α-16 (d^2) k (-1+α) α-16 k (α^2)+128 (k^2) (α^2)+128 g (k^2) (α^2)+64 k α θ-64 k (α^2) θ)]

D[SW,g]

Simplify[(1/(256 k (((-1+α))^2)))(-96 d k (-1+α) α-48 (d^2) k (-1+α) α+64 (-2+6 g) (k^2) (α^2)+16 k (-1+α) (16 cs (-1+α)-3 α+4 α θ))]

.2 When the contract launch price is the acceptal price for VM:

Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -(r-g)(α l+cs)+(1-e)β(α l+cs)

e:=((l-l α+θ)/(2 k))
r:=((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/((Sqrt[k]) (-1+α)))

Solve[Profitv==0,l]

Simplify[((-2 α-(θ/k)+((α θ)/k)+((2 (Sqrt[cv-2 cv α+cv (α^2)+k (α^2)+θ-α θ]))/(Sqrt[k])))/((1/k)-((2 α)/k)+((α^2)/k)))]

l:=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))

e

Simplify[((θ+((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))-((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2)))/(2 k))]

e:=(((Sqrt[k]) α-(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/((Sqrt[k]) (-1+α)))

Profitv

Simplify[%14]

Profits

Simplify[%8]

Simplify[%16]

D[-cs+((α (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-(1/((Sqrt[k]) (((-1+α))^2)))(-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) ((1+d) p (-1+α)-(p^2) (-1+α)-2 k α-θ+α θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))+(1/((Sqrt[k]) (((-1+α))^3)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-(-g+r) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))),p]

Solve[%==0,p]

p:=((1+d)/2)

Profits

Simplify[%11]

Simplify[(1/((Sqrt[k]) (((-1+α))^3)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-(-g+((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/((Sqrt[k]) (-1+α)))) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2)))]

y

Derivation of the sensitivity analysis for Model NG. How d,k,θ  and  will affect these optimal decisions:

l:=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))
p:=((1+d)/2)
e:=(((Sqrt[k]) α-(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/((Sqrt[k]) (-1+α)))
r:=((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/((Sqrt[k]) (-1+α)))
Profits:=-cs+((α (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-(1/(4 (Sqrt[k]) (((-1+α))^2)))(-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 k α-4 θ+4 α θ+8 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))+(1/((Sqrt[k]) (((-1+α))^3)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-(-g+((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/((Sqrt[k]) (-1+α)))) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2)))
CS:=(1/8) (((1+d))^2)
SW:=(1/(8 (Sqrt[k]) (((-1+α))^2)))(-16 (1+g) (k^(3/2)) (α^2)+16 (1+g) k α (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])-2 (-1+α) (1+2 d+(d^2)+4 θ) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])+(Sqrt[k]) (-1+α) (-1-2 d-(d^2)-16 cv (-1+α)+8 cs (-1+g) (-1+α)+3 α+6 d α+3 (d^2) α+16 θ+8 g α θ))

d

D[e,d]

D[Profits,d]

Simplify[-(((2 (-1+α)+2 d (-1+α)) (-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(4 (Sqrt[k]) (((-1+α))^2)))]

k

D[e,k]

Simplify[(((α/(2 (Sqrt[k])))-((α^2)/(2 (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))))/((Sqrt[k]) (-1+α)))-(((Sqrt[k]) α-(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(2 (k^(3/2)) (-1+α)))]

D[Profits,k]

Simplify[%49]

Solve[%==0,g]

Simplify[((cv/(8 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((cv d (-1+α))/(4 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((cv (d^2) (-1+α))/(8 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((cv α)/(8 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((cv α)/((Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((2 (Sqrt[k]) (α^3))/((((-1+α))^2) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))+((cv θ)/(2 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((cv α θ)/(2 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))+((α θ)/((Sqrt[k]) (-1+α) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))+((θ (1+2 d+(d^2)+4 θ))/(8 (k^(3/2)) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))))/(-((2 (α^2))/(((-1+α))^2))+((cv α)/((Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))+((2 (Sqrt[k]) (α^3))/((((-1+α))^2) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-((α θ)/((Sqrt[k]) (-1+α) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))))]

θ

D[Profits,θ]

Simplify[%54]

g

D[Profits,g]

Simplify[cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))]