VM provide Insurance Model
in this version, if the launch is failed, the VM will compensate the prepay price and a part of the satellite

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv +rv cs-(1-e) β cs
Profits:=-α p+e(f-(1-α)p)-cs -rv cs+(1-e)β cs

The VM’s Effort Under Insurance Model

D[Profitv,e]

Solve[%==0,e]

We now solve the Stackelberg game  by using backward induction. First, given any contingent price p , by considering the first-order condition of Profitv, the vehicle manufacture’s best response is given as

e:=((p-p α+cs β+θ)/(2 k))

Profitv

Simplify[-cv+cs rv+p α+((p (1-α) (p-p α+cs β+θ))/(2 k))-((((p-p α+cs β+θ))^2)/(4 k))-cs β (1-((p-p α+cs β+θ)/(2 k)))-θ (1-((p-p α+cs β+θ)/(2 k)))]

Solve[%==0,p]

Hence the vehicle manufacture’s participation condition, i.e., Profitv≥0, can be written as 
p ≥ ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)-(((1-α))^2)k rv Cs+(1-α)kβCs]) -2k α -(1-α)(θ+β Cs))/(((1-α))^2))

The SO’s Price

Profits:=-α p+e(f-(1-α)p)-cs -rv cs+(1-e)β cs 
e:=((p-p α+cs β+θ)/(2 k))

Profits

Simplify[-cs-cs rv-p α+(((f-p (1-α)) (p-p α+cs β+θ))/(2 k))+cs β (1-((p-p α+cs β+θ)/(2 k)))]

D[%,p]

Solve[%==0,p]

for the satellite owner, her unconstrained optimal contract (p,α) can be written as p=(((f-θ)(1-α)-2 k α -(1-α)2β Cs)/(2 (((1-α))^2)))
s.t.  p ≥ ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)-(((1-α))^2)k rv Cs+(1-α)kβCs]) -2k α -(1-α)(θ+β Cs))/(((1-α))^2))( the level of price that vehicle manufacture will accept)

Δp:=((f-f α-2 k α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))-(1/(1-2 α+(α^2)))(-2 k α-cs β+cs α β-θ+α θ+2 √(cv k-cs k rv-2 cv k α+2 cs k rv α+cv k (α^2)+(k^2) (α^2)-cs k rv (α^2)+cs k β-cs k α β+k θ-k α θ))

Δp

Simplify[((f-f α-2 k α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))-(1/(1-2 α+(α^2)))(-2 k α-cs β+cs α β-θ+α θ+2 √(cv k-cs k rv-2 cv k α+2 cs k rv α+cv k (α^2)+(k^2) (α^2)-cs k rv (α^2)+cs k β-cs k α β+k θ-k α θ))]

Solve[%==0,f]

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

The profit

1.  If the (p^S)>(p^VA),  which satisfied with F≥((-2 k α-(1-α) θ+4 (Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α )-cs (-1+α) (rv (-1+α)+β)])))/(1-α)) (the dark region) then the effort is e=(1/4) (((f+θ)/k)-((2 α)/(1-α))+(θ/k)), the contract [p,α] satellite owner offered is p=(((f-θ)(1-α)-2 k α-2 cs β(1- α ))/(2 (((1-α))^2))). The profit of vehicle manufacture is 
(1/(16 k (((-1+α))^2)))(16 cs k rv+(f^2) (((-1+α))^2)-16 cv k (((-1+α))^2)-32 cs k rv α-12 (k^2) (α^2)+16 cs k rv (α^2)-16 cs k β+16 cs k α β-16 k θ+20 k α θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)-2 f (-1+α) (2 k α+θ-α θ))
which can be written as (((((F+θ))^2)(((1-α))^2) + 4k [ F (1-α)α- 3(α^2)k - θ (1- α)(4-α)  ]-16(1-α)k β Cs)/(16k (((1-α))^2)))-Cv+rv Cs
The profit of satellite owner is 
(1/(8 k (((-1+α))^2)))((f^2) (((-1+α))^2)-8 cs k (-1+α) (-1+rv (-1+α)+α+β)+2 f (-1+α) (2 k α+(-1+α) θ)+(((2 k α+θ-α θ))^2))
wich can be written as (((((F+θ))^2)(((1-α))^2)+4kα [αk +(θ -F)(1-α) ]+8(1-α)k β Cs)/(8k (((1-α))^2)))-(1+rv)Cs

the whole chain payoff is ((3(((F+θ))^2)(((1-α))^2) - 4(α^2)(k^2)+ 4k (1- α)(-α F-4θ + 3αθ )  )/(16k (((1-α))^2)))-Cv-Cs, the same.
Compared with the insurance model, the whole supply chain efficiency doesn’t change, but for SO it is fitable for the SO to choose the insurance from the vm.

2.If the (p^S)<(p^VA), which satisfied with F<((-2 k α-(1-α) θ+4 (Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α )-cs (-1+α) (rv (-1+α)+β)])))/(1-α))=((-2 k α-(1-α) θ+4 E)/(1-α))

the effort is e=((-k α+(Sqrt[k (cv (((1-α))^2)+k (α^2) +cs (1-α) (β-rv (1-α))+θ(1-α))]))/(k (1-α)))=((-k α+E)/(k (1-α)))

  the price is ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)-(((1-α))^2)k rv Cs+(1-α)kβCs]) -2k α -(1-α)(θ+β Cs))/(((1-α))^2))=((2E-2k α -(1-α)(θ+ β Cs))/(((1-α))^2))
  The profit of satellite owner is -(1/(k (((-1+α))^2)))(2 cv k+cs k (((-1+α))^2)-4 cv k α+f k α+2 cv k (α^2)-f k (α^2)+2 (k^2) (α^2)+2 k θ-2 k α θ-f (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+f α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-2 k α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+α θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))
(1/(k (((-1+α))^2)))(-cs k-2 cv k+cs k rv+2 cs k α+4 cv k α-f k α-2 cs k rv α-cs k (α^2)-2 cv k (α^2)+f k (α^2)-2 (k^2) (α^2)+cs k rv (α^2)-cs k β+cs k α β-2 k θ+2 k α θ+f (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])-f α (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])+2 k α (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])+θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])-α θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)]))
   which can be written as 
   (([(F+θ)(1-α)+2αk]*(Sqrt[k[((Cv(1-α))^2)+((kα)^2)+θ(1-α)]-(((1-α))^2)k rv Cs+(1-α)kCs])- k (1-α) (α F+2θ ) - 2 (k^2)(α^2) -(1-α)β k Cs)/((k(1-α))^2))
- (Cs+2Cv)
   令 E=(Sqrt[k[((Cv(1-α))^2)+((kα)^2)+θ(1-α)]-(((1-α))^2)k rv Cs+(1-α)kCs])   > A
     (([(F+θ)(1-α)+2αk]*E- k (1-α) (α F+2θ ) - 2 (k^2)(α^2)- (1-α)β k Cs )/((k(1-α))^2)) - (Cs+2Cv)
   the profit of vehicle manufacture is 0
  the whole chain payoff is (([(F+θ)(1-α)+2αk]*E- k (1-α) (α F+2θ ) - 2 (k^2)(α^2)- (1-α)β k Cs )/((k(1-α))^2)) - (Cs+2Cv)

Proof
1. ps>pvA

e:=((p-p α+cs β+θ)/(2 k))
p:=((f-f α-2 k α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv +rv cs-(1-e) β cs
Profits:=-α p+e(f-(1-α)p)-cs -rv cs+(1-e)β cs

e

Simplify[((cs β+θ+((f-f α-2 k α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))-((α (f-f α-2 k α-2 cs β+2 cs α β-θ+α θ))/(2 (((-1+α))^2))))/(2 k))]

(1/4) ((f/k)+((2 α)/(-1+α))+(θ/k))

Profitv

Simplify[%35]

(1/(16 k (((-1+α))^2)))(16 cs k rv+(f^2) (((-1+α))^2)-16 cv k (((-1+α))^2)-32 cs k rv α-12 (k^2) (α^2)+16 cs k rv (α^2)-16 cs k β+16 cs k α β-16 k θ+20 k α θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)-2 f (-1+α) (2 k α+θ-α θ))

Profits

Simplify[%37]

(1/(8 k (((-1+α))^2)))((f^2) (((-1+α))^2)-8 cs k (-1+α) (-1+rv (-1+α)+α+β)+2 f (-1+α) (2 k α+(-1+α) θ)+(((2 k α+θ-α θ))^2))

Profit=(1/(16 k (((-1+α))^2)))(16 cs k rv+(f^2) (((-1+α))^2)-16 cv k (((-1+α))^2)-32 cs k rv α-12 (k^2) (α^2)+16 cs k rv (α^2)-16 cs k β+16 cs k α β-16 k θ+20 k α θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)-2 f (-1+α) (2 k α+θ-α θ)+2(f^2) (((-1+α))^2)-16 cs k (-1+α) (-1+rv (-1+α)+α+β)+4 f (-1+α) (2 k α+(-1+α) θ)+2(((2 k α+θ-α θ))^2))

Simplify[Profit]

2. ps<pvA

e:=((p-p α+cs β+θ)/(2 k))
p:=(1/(1-2 α+(α^2)))(-2 k α-cs β+cs α β-θ+α θ+2 √(cv k-cs k rv-2 cv k α+2 cs k rv α+cv k (α^2)+(k^2) (α^2)-cs k rv (α^2)+cs k β-cs k α β+k θ-k α θ))
Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv +rv cs-(1-e) β cs
Profits:=-α p+e(f-(1-α)p)-cs -rv cs+(1-e)β cs

e

Simplify[%5]

Profitv

Simplify[%45]

Profits

Simplify[%47]

(1/(k (((-1+α))^2)))(-cs k-2 cv k+cs k rv+2 cs k α+4 cv k α-f k α-2 cs k rv α-cs k (α^2)-2 cv k (α^2)+f k (α^2)-2 (k^2) (α^2)+cs k rv (α^2)-cs k β+cs k α β-2 k θ+2 k α θ+f (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])-f α (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])+2 k α (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])+θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)])-α θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)-cs (-1+α) (rv (-1+α)+β)+θ-α θ)]))