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

D[Profitv,e]

Solve[%==0,e]

e:=((l-l α+θ)/(2 kb))

Profiti

Solve[%==0,r]

r:=((β (2 kb-l+l α-θ))/(2 kb))

Profits

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

D[%,l]

Solve[%==0,l]

D[(1/(2 kb))(-2 cs kb-2 csb kb-2 kb l α+(cs+l α) β (2 kb+l (-1+α)-θ)+((1+b+d-p) p+l (-1+α)) (l-l α+θ)+(cs+l α) (2 g kb+β (-2 kb+l-l α+θ))),p]

Solve[%==0,p]

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

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

l

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

l:=-((-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α+4 θ-4 α θ)/(8 (((-1+α))^2)))

e

Simplify[%21]

r

Simplify[%23]

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+b))^2)+4 θ)/((((1+d+b))^2)+8 kb+4 θ)).in this paper, we consider the case with  (((((1+d+b))^2)-16kb+4 θ)/((((1+d+b))^2)-8 kb+4 θ))<α<(((((1+d))^2)+4 θ)/((((1+d))^2)+8 kb+4 θ)), which is the common situation because it means launch successfully propability is above zero and below one.
e = ((-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α-4 θ+4 α θ)/(16 kb (-1+α))), that (λ/(16 kb (1-α))), where  λ=1-2 d (-1+α)-(d^2) (-1+α)-(b^2) (-1+α)-2 b (1+d) (-1+α)-α-8 kb α+4 θ-4 α θ=(1-α)(((1+d+b))^2)-8αkb(1-g)+(1-α)4θ

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

e:=((-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α-4 θ+4 α θ)/(16 kb (-1+α)))
r:=(1/(16 kb (-1+α)))β (1-16 kb-(b^2) (-1+α)-2 d (-1+α)-(d^2) (-1+α)-2 b (1+d) (-1+α)-α+8 kb α+8 g kb α+4 θ-4 α θ)
l:=-(1/(8 (((-1+α))^2)))(-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α+4 θ-4 α θ)
p:=(1/2) (1+b+d)
Profits

Simplify[%13]

Profits:=(1/(128 kb (((-1+α))^2)))(1-128 cs kb-128 csb kb+128 cs g kb+(b^4) (((-1+α))^2)+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)+4 (b^3) (1+d) (((-1+α))^2)-2 α-16 kb α+256 cs kb α+256 csb kb α+16 g kb α-256 cs g kb α+(α^2)+16 kb (α^2)-128 cs kb (α^2)-128 csb kb (α^2)-16 g kb (α^2)+128 cs g kb (α^2)+64 (kb^2) (α^2)-128 g (kb^2) (α^2)+64 (g^2) (kb^2) (α^2)+8 θ-16 α θ+64 kb α θ-64 g kb α θ+8 (α^2) θ-64 kb (α^2) θ+64 g kb (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) kb-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 (-1+g) kb-4 θ)+4 θ)+4 b (1+d) (-1+α) (-1+2 d (-1+α)+(d^2) (-1+α)-4 θ+α (1+8 kb-8 g kb+4 θ))+2 (b^2) (-1+α) (-3+6 d (-1+α)+3 (d^2) (-1+α)-4 θ+α (3+8 kb-8 g kb+4 θ)))

Profitv

Simplify[%15]

Profitv:=(1/256) (-256 cv-256 cvb-(1/(((-1+α))^2))32 α (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α+4 θ-4 α θ)+(1/(kb (-1+α)))16 θ (-1+16 kb+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α-8 kb α-8 g kb α-4 θ+4 α θ)+(1/(kb (((-1+α))^2)))2 (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α+4 θ-4 α θ) (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α-4 θ+4 α θ)-(1/(kb (((-1+α))^2)))(((-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α-4 θ+4 α θ))^2))

CS:=(1/2)(((1+d+b-p))^2)

CS

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

SW:=Profits+Profitv+CS

SW

Simplify[%21]

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

d

D[e,d]

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

D[r,d]

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

D[l,d]

Simplify[-((2 (-1+α)+2 b (-1+α)+2 d (-1+α))/(8 (((-1+α))^2)))]

D[p,d]

D[Profits,d]

Simplify[%30]

D[Profitv,d]

Simplify[%32]

D[CS,d]

Simplify[(1/2) (1+b+(1/2) (-1-b-d)+d)]

D[SW,d]

Simplify[%36]

kb

D[e,kb]

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

D[r,kb]

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

D[l,kb]

Simplify[-((8 α-8 g α)/(8 (((-1+α))^2)))]

D[p,kb]

D[Profits,kb]

Simplify[%47]

D[Profitv,kb]

Simplify[%49]

D[CS,kb]

D[SW,kb]

Simplify[%52]

θ

D[e,θ]

Simplify[((-4+4 α)/(16 kb (-1+α)))]

D[r,θ]

Simplify[(((4-4 α) β)/(16 kb (-1+α)))]

D[l,θ]

Simplify[-((4-4 α)/(8 (((-1+α))^2)))]

D[p,θ]

D[Profits,θ]

Simplify[%61]

D[Profitv,θ]

Simplify[%63]

D[CS,θ]

D[SW,θ]

Simplify[%66]

g

D[e,g]

D[r,g]

D[l,g]

D[Profitv,g]

Simplify[(1/256) (((256 kb (α^2))/(((-1+α))^2))-((128 α θ)/(-1+α))-((16 α (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α-8 g kb α+4 θ-4 α θ))/(((-1+α))^2)))]

D[Profits,g]

Simplify[%15]

D[SW,g]

Simplify[%17]

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

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

e:=((l-l α+θ)/(2 kb))
r:=((β (2 kb-l+l α-θ))/(2 kb))

Solve[Profitv==0,l]

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

l:=((-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/(((-1+α))^2))
e:=(((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/((Sqrt[kb]) (-1+α)))
r:=((β (-(Sqrt[kb])+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/((Sqrt[kb]) (-1+α)))
p:=((1+d+b)/2)

Profits

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

Simplify[(1/((Sqrt[kb]) (-1+α)))((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])) ((1/2) (1+b+d) (1+b+(1/2) (-1-b-d)+d)-(((1-α) (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))]

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

l:=((-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/(((-1+α))^2))
p:=((1+d+b)/2)
e:=(((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/((Sqrt[kb]) (-1+α)))
r:=((β (-(Sqrt[kb])+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/((Sqrt[kb]) (-1+α)))
Profits:=-cs-csb-((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2))+(1/((Sqrt[kb]) (-1+α)))((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])) ((1/2) (1+b+d) (1+b+(1/2) (-1-b-d)+d)-(((1-α) (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))+β (1-(((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/((Sqrt[kb]) (-1+α)))) (cs+((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))-(-g+((β (-(Sqrt[kb])+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/((Sqrt[kb]) (-1+α)))) (cs+((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))
CS:=(1/8) (((1+d+b))^2)
SW:=-cs-csb-((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2))+(1/((Sqrt[kb]) (-1+α)))((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])) ((1/2) (1+b+d) (1+b+(1/2) (-1-b-d)+d)-(((1-α) (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))+β (1-(((Sqrt[kb]) α-(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))/((Sqrt[kb]) (-1+α)))) (cs+((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))-(-g+((β (-(Sqrt[kb])+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/((Sqrt[kb]) (-1+α)))) (cs+((α (-2 kb α-θ+α θ+2 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])))/(((-1+α))^2)))+(1/8) (((1+d+b))^2)

d

D[e,d]

D[Profits,d]

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

kb

D[e,kb]

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

D[Profits,kb]

Simplify[%28]

y:=-θ-2 b θ-(b^2) θ-2 d θ-2 b d θ-(d^2) θ+2 α θ+4 b α θ+2 (b^2) α θ+4 d α θ+4 b d α θ+2 (d^2) α θ-(α^2) θ-2 b (α^2) θ-(b^2) (α^2) θ-2 d (α^2) θ-2 b d (α^2) θ-(d^2) (α^2) θ+(((1+d+b))^2)(((1-α))^2) θ

y

Simplify[-θ-2 b θ-(b^2) θ-2 d θ-2 b d θ-(d^2) θ+(((1+b+d))^2) (((1-α))^2) θ+2 α θ+4 b α θ+2 (b^2) α θ+4 d α θ+4 b d α θ+2 (d^2) α θ-(α^2) θ-2 b (α^2) θ-(b^2) (α^2) θ-2 d (α^2) θ-2 b d (α^2) θ-(d^2) (α^2) θ]

y1:=16 (kb^2) (α^3)+16 g (kb^2) (α^3)-θ-2 b θ-(b^2) θ-2 d θ-2 b d θ-(d^2) θ+2 α θ+4 b α θ+2 (b^2) α θ+4 d α θ+4 b d α θ+2 (d^2) α θ+8 kb α θ+8 g kb α θ-(α^2) θ-2 b (α^2) θ-(b^2) (α^2) θ-2 d (α^2) θ-2 b d (α^2) θ-(d^2) (α^2) θ-8 kb (α^2) θ-8 g kb (α^2) θ-4 (θ^2)+8 α (θ^2)-4 (α^2) (θ^2)-16 (kb^(3/2)) (α^2) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])-16 g (kb^(3/2)) (α^2) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])+cv (((-1+α))^2) (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α+8 g kb α-4 θ+4 α θ)+cvb (((-1+α))^2) (-1+(b^2) (-1+α)+2 d (-1+α)+(d^2) (-1+α)+2 b (1+d) (-1+α)+α+8 kb α+8 g kb α-4 θ+4 α θ)+(((1+d+b))^2)(((1-α))^2) θ

y1

Simplify[%47]

θ

D[e,θ]

D[Profits,θ]

Simplify[%50]

y:=-1-8 cvb-2 d-(d^2)+(b^2) (-1+α)+2 b (1+d) (-1+α)-8 cv (((-1+α))^2)+α+16 cvb α+2 d α+(d^2) α-8 kb α-8 g kb α-8 cvb (α^2)-8 kb (α^2)-12 θ+12 α θ+16 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])+8 g (Sqrt[kb]) α (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ])+(1-α)(((1+b+d))^2)

y

Simplify[%53]

g

D[e,g]

D[Profits,g]

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