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(α l+cs)+(1-e)β(α l+cs)-csb

D[Profitv,e]

Solve[%==0,e]

e:=((l-l α+θ)/(2 kb))

Profiti

Simplify[r (cs+l α)-(cs+l α) β (1-((l-l α+θ)/(2 kb)))]

Solve[%==0,r]

r:=((β (2 kb-l+l α-θ))/(2 kb))

Profits

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

Solve[%==0,p]

{{p->(1/2) (1+b+d)}}
p:=(1/2) (1+b+d)

l:=((p+b p+d p-(p^2)-2 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-θ+α θ)]

e

Simplify[%22]

r

Simplify[%24]

Solve[1-(1/(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-4 θ+4 α θ)==0,α]

Solve[(1/(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-4 θ+4 α θ)==0,α]

{{α->((1+2 b+(b^2)+2 d+2 b d+(d^2)+4 θ)/(1+2 b+(b^2)+2 d+2 b d+(d^2)+8 kb+4 θ))}}
η

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-4 θ+4 α θ))/(16 kb (-1+α))), that (η/(16 kb (1-α))), where  η=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-α)4θ

r = β(1-(η/(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-4 θ+4 α θ))/(8 (((-1+α))^2)))=((η-8θ(1-α))/(8 (((1-α))^2)))

Profits

Simplify[%9]

Profitv

Simplify[%11]

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

SW:=Profits+Profitv+CS

SW

Simplify[%22]

e:=(1/(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-4 θ+4 α θ)
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+4 θ-4 α θ)
p:=(1/2) (1+b+d)
l:=((p+b p+d p-(p^2)-2 kb α-p α-b p α-d p α+(p^2) α-θ+α θ)/(2 (((-1+α))^2)))
Profits:=(1/(128 kb (((-1+α))^2)))(1-128 cs kb-128 csb 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+(α^2)+16 kb (α^2)-128 cs kb (α^2)-128 csb kb (α^2)+64 (kb^2) (α^2)+8 θ-16 α θ+64 kb α θ+8 (α^2) θ-64 kb (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 kb α-4 θ+4 α θ)+4 b (1+d) (-1+α) (-1+2 d (-1+α)+(d^2) (-1+α)+α+8 kb α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 kb+4 θ))+2 (b^2) (-1+α) (-3+6 d (-1+α)+3 (d^2) (-1+α)-4 θ+α (3+8 kb+4 θ)))
Profitv:=(1/(256 kb (((-1+α))^2)))(1-256 cv kb-256 cvb 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+512 cv kb α+512 cvb kb α+(α^2)-16 kb (α^2)-256 cv kb (α^2)-256 cvb kb (α^2)-192 (kb^2) (α^2)+8 θ-256 kb θ-16 α θ+320 kb α θ+8 (α^2) θ-64 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 kb-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 kb-4 θ)+4 θ)+4 b (1+d) (-1+α) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 kb α-4 θ+4 α θ)+2 (b^2) (-1+α) (-3+6 d (-1+α)+3 (d^2) (-1+α)-4 θ+α (3-8 kb+4 θ)))
CS:=(1/8) (((1+b+d))^2)
SW:=(1/(256 kb (((-1+α))^2)))(3+32 kb-256 cs kb-256 csb kb-256 cv kb-256 cvb kb+3 (b^4) (((-1+α))^2)+12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)+12 (b^3) (1+d) (((-1+α))^2)-6 α-80 kb α+512 cs kb α+512 csb kb α+512 cv kb α+512 cvb kb α+3 (α^2)+48 kb (α^2)-256 cs kb (α^2)-256 csb kb (α^2)-256 cv kb (α^2)-256 cvb kb (α^2)-64 (kb^2) (α^2)+24 θ-256 kb θ-48 α θ+448 kb α θ+24 (α^2) θ-192 kb (α^2) θ+48 (θ^2)-96 α (θ^2)+48 (α^2) (θ^2)+4 d (-1+α) (8 kb (-2+3 α)+3 (-1+α) (1+4 θ))+4 b (1+d) (-1+α) (6 d (-1+α)+3 (d^2) (-1+α)+8 kb (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 kb (-2+3 α)+3 (-1+α) (3+4 θ))+2 (b^2) (-1+α) (18 d (-1+α)+9 (d^2) (-1+α)+8 kb (-2+3 α)+3 (-1+α) (3+4 θ)))

d

D[e,d]

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

((1+b+d)/(8 kb))

D[r,d]

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

D[l,d]

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

D[p,d]

D[Profits,d]

Simplify[%39]

D[Profitv,d]

Simplify[%41]

D[CS,d]

D[SW,d]

Simplify[%44]

kb

D[e,kb]

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

D[r,kb]

Simplify[(((-16+8 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+4 θ-4 α θ)]

D[l,kb]

D[p,kb]

D[Profits,kb]

Simplify[%53]

D[Profitv,kb]

Simplify[%55]

y1:=((((1-α)(((1+d+b))^2) -8 kb α+4 θ-4 α θ))^2)+16α kb((1-α)(((1+d+b))^2) -8 kb α+4 θ-4 α θ+16α kb)-(1+(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+(α^2)+192 (kb^2) (α^2)+8 θ-16 α θ+8 (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (((-1+α))^2) (1+4 θ)+2 (d^2) (((-1+α))^2) (3+4 θ)+4 b (1+d) (((-1+α))^2) (1+2 d+(d^2)+4 θ)+2 (b^2) (((-1+α))^2) (3+6 d+3 (d^2)+4 θ))
y1

Simplify[%2]

D[CS,kb]

D[SW,kb]

Simplify[%58]

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]

D[p,θ]

D[Profits,θ]

Simplify[(1/(128 kb (((-1+α))^2)))(8-16 α+64 kb α+8 (α^2)-64 kb (α^2)+2 (b^2) (-1+α) (-4+4 α)+4 d (-1+α) (-4+4 α)+2 (d^2) (-1+α) (-4+4 α)+4 b (1+d) (-1+α) (-4+4 α)+32 θ-64 α θ+32 (α^2) θ)]

D[Profitv,θ]

Simplify[(1/(256 kb (((-1+α))^2)))(8-256 kb-4 d (4-4 α) (-1+α)-2 (d^2) (4-4 α) (-1+α)-16 α+320 kb α+8 (α^2)-64 kb (α^2)+2 (b^2) (-1+α) (-4+4 α)+4 b (1+d) (-1+α) (-4+4 α)+32 θ-64 α θ+32 (α^2) θ)]

D[CS,θ]

D[SW,θ]

Simplify[(1/(256 kb (((-1+α))^2)))(24-256 kb+24 (b^2) (((-1+α))^2)+48 d (((-1+α))^2)+24 (d^2) (((-1+α))^2)+48 b (1+d) (((-1+α))^2)-48 α+448 kb α+24 (α^2)-192 kb (α^2)+96 θ-192 α θ+96 (α^2) θ)]

.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(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

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

r

Simplify[(1/(2 kb))β (2 kb-θ-(1/(((-1+α))^2))(-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/(((-1+α))^2))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])))]

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[%17]

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

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+(1/(((-1+α))^2))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/(4 (Sqrt[kb]) (((-1+α))^2)))(-(Sqrt[kb]) α+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-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-4 θ+4 α θ+8 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))
CS:=(1/8) (((1+d+b))^2)
SW:=-cs-csb+(1/(((-1+α))^2))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/(4 (Sqrt[kb]) (((-1+α))^2)))(-(Sqrt[kb]) α+(Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-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-4 θ+4 α θ+8 (Sqrt[kb]) (Sqrt[cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ]))+(1/8) (((1+d+b))^2)

d

D[e,d]

D[p,d]

D[Profits,d]

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

D[CS,d]

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[r,kb]

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

D[l,kb]

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

D[Profits,kb]

Simplify[%29]

θ

D[e,θ]

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

D[r,θ]

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

D[l,θ]

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

D[Profits,θ]

Simplify[%37]