Blockchain Model
The VM’s Effort Under Insurance Model

Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb+rvb cs-(1-eb) β cs
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rvb cs+(1-eb)β cs -csb

D[Profitvb,eb]

Solve[%==0,eb]

We now solve the Stackelberg game  by using backward induction. First, given any contingent price (p^B) , by considering the first-order condition of Profitv, the vehicle manufacture’s best response is given as
e^B=(((1-α)p+θ+β Cs)/(2(k^B)))

eb:=((pb-pb α+cs β+θ)/(2 kb))

Profitvb

Simplify[-cv-cvb+cs rvb+pb α+((pb (1-α) (pb-pb α+cs β+θ))/(2 kb))-((((pb-pb α+cs β+θ))^2)/(4 kb))-cs β (1-((pb-pb α+cs β+θ)/(2 kb)))-θ (1-((pb-pb α+cs β+θ)/(2 kb)))]

Solve[%==0,pb]

{{pb->(1/(1-2 α+(α^2)))(-2 kb α-cs β+cs α β-θ+α θ-2 √(cv kb+cvb kb-cs kb rvb-2 cv kb α-2 cvb kb α+2 cs kb rvb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)-cs kb rvb (α^2)+cs kb β-cs kb α β+kb θ-kb α θ))},{pb->(1/(1-2 α+(α^2)))(-2 kb α-cs β+cs α β-θ+α θ+2 √(cv kb+cvb kb-cs kb rvb-2 cv kb α-2 cvb kb α+2 cs kb rvb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)-cs kb rvb (α^2)+cs kb β-cs kb α β+kb θ-kb α θ))}}

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

The SO’s Optimal Contract Under Blockchain Model

eb:=((pb-pb α+θ+cs β)/(2 kb))
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rvb cs+(1-eb)β cs -csb

Profitsb

Simplify[-cs-csb-cs rvb-pb α+(((f-pb (1-α)) (pb-pb α+cs β+θ))/(2 kb))+cs β (1-((pb-pb α+cs β+θ)/(2 kb)))]

D[%,pb]

Simplify[(((f+pb (-1+α)) (1-α))/(2 kb))-α-((cs (1-α) β)/(2 kb))+(((-1+α) (pb-pb α+cs β+θ))/(2 kb))]

Solve[%==0,pb]

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

Δp:=((f-f α-2 kb α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))-(1/(1-2 α+(α^2)))(-2 kb α-cs β+cs α β-θ+α θ+2 √(cv kb+cvb kb-cs kb rvb-2 cv kb α-2 cvb kb α+2 cs kb rvb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)-cs kb rvb (α^2)+cs kb β-cs kb α β+kb θ-kb α θ))

Δp

Simplify[%21]

Solve[%==0,f]

令  ：G=√(kb(cv+cvb-rvb cs) (((1-α))^2)+kb (α^2)+(θ+β cs)(1-α))

1.  If the (p^S)>(p^VA),  which satisfied with F≥((-2 kb α-(1-α)θ+4 (Sqrt[kb(cv+cvb-rvb cs) (((1-α))^2)+kb (α^2)+(θ+β cs)(1-α)])))/(1-α)) =((-2 (k^B) α-(1-α) θ+4 G)/(1-α))(the dark region) then the effort is e=(1/4) (((f+θ+β cs)/(k^B))-((2 α)/(1-α))), the contract [p,α] satellite owner offered is p=(((f-θ-2cs β)(1-α)-2 (k^B) α )/(2 (((1-α))^2))). The profit of vehicle manufacture is 
(1/(16 kb (((-1+α))^2)))(-16 cvb kb+16 cs kb rvb+(f^2) (((-1+α))^2)-16 cv kb (((-1+α))^2)+32 cvb kb α-32 cs kb rvb α-16 cvb kb (α^2)-12 (kb^2) (α^2)+16 cs kb rvb (α^2)-16 cs kb β+16 cs kb α β-16 kb θ+20 kb α θ-4 kb (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)-2 f (-1+α) (2 kb α+θ-α θ))

which can be written as (((((F+θ))^2)(((1-α))^2) + 4(k^B) [ F (1-α)α- 3(α^2)((k )^B)- θ (1- α)(4-α)  ]-16(1-α)k β Cs)/(16(k^B) (((1-α))^2)))-Cv+rv Cs-Cvb
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))
(1/(8 kb (((-1+α))^2)))(-8 csb kb+(f^2) (((-1+α))^2)+16 csb kb α-8 csb kb (α^2)+4 (kb^2) (α^2)-8 cs kb (-1+α) (-1+rvb (-1+α)+α+β)+4 kb α θ-4 kb (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)+2 f (-1+α) (2 kb α+(-1+α) θ))
wich can be written as (((((F+θ))^2)(((1-α))^2)+4(k^B)α [((αk)^B) +(θ -F)(1-α) ]+8(1-α)((k )^B)β Cs)/(8(k^B) (((1-α))^2)))-(1+rv)Cs-Csb

the whole chain payoff is ((3(((F+θ))^2)(((1-α))^2) - 4(α^2)((k^B)^2)+ 4(k^B) (1- α)(-α F-4θ + 3αθ )  )/(16(k^B) (((1-α))^2)))-Cv-Cs-Cvb-Csb, 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 kb α-(1-α)θ+4 (Sqrt[kb(cv+cvb-rvb cs) (((1-α))^2)+kb (α^2)+(θ+β cs)(1-α)])))/(1-α))=((-2 (k^B) α-(1-α) θ+4 G)/(1-α))

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

  the price is ((2 (Sqrt[(k^B)(cv +cvb-rvb cs)(((1-α))^2)+(α^2)((k^B)^2) +(θ+β cs)(k^B) (1-α )]) -2 (k^B) α-(1-α)( θ+β cs))/( (((1-α))^2)))=((2G-2k α -(1-α)(θ+ β Cs))/(((1-α))^2))
  The profit of satellite owner is 
-(1/(kb (((-1+α))^2)))(csb kb+2 cv kb+2 cvb kb-2 csb kb α-4 cv kb α-4 cvb kb α+f kb α+csb kb (α^2)+2 cv kb (α^2)+2 cvb kb (α^2)-f kb (α^2)+2 (kb^2) (α^2)-cs kb (-1+α) (1+rvb (-1+α)-α+β)+2 kb θ-2 kb α θ-f (Sqrt[kb (-cs rvb+cv (((-1+α))^2)+cvb (((-1+α))^2)+2 cs rvb α+kb (α^2)-cs rvb (α^2)+cs β-cs α β+θ-α θ)])+f α (Sqrt[kb (-cs rvb+cv (((-1+α))^2)+cvb (((-1+α))^2)+2 cs rvb α+kb (α^2)-cs rvb (α^2)+cs β-cs α β+θ-α θ)])-2 kb α (Sqrt[kb (-cs rvb+cv (((-1+α))^2)+cvb (((-1+α))^2)+2 cs rvb α+kb (α^2)-cs rvb (α^2)+cs β-cs α β+θ-α θ)])-θ (Sqrt[kb (-cs rvb+cv (((-1+α))^2)+cvb (((-1+α))^2)+2 cs rvb α+kb (α^2)-cs rvb (α^2)+cs β-cs α β+θ-α θ)])+α θ (Sqrt[kb (-cs rvb+cv (((-1+α))^2)+cvb (((-1+α))^2)+2 cs rvb α+kb (α^2)-cs rvb (α^2)+cs β-cs α β+θ-α θ)]))
   which can be written as 
   (([(F+θ)(1-α)+2((αk)^B)]*G- (k^B) (1-α) (α F+2θ ) - 2 ((k^B)^2)(α^2) -(1-α)β (k^B) Cs)/(((k^B)(1-α))^2))
- (Cs+2Cv+2Cvb+Csb)+rvb Cs
  
   the profit of vehicle manufacture is 0
  the whole chain payoff is (([(F+θ)(1-α)+2((αk)^B)]*G- (k^B) (1-α) (α F+2θ ) - 2 ((k^B)^2)(α^2) -(1-α)β (k^B) Cs)/(((k^B)(1-α))^2))
- (Cs+2Cv+2Cvb+Csb)+rvb Cs

Proof

eb:=((pb-pb α+θ+cs β)/(2 kb))
pb:=((f-f α-2 kb α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))
Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb+rvb cs-(1-eb) β cs
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rvb cs+(1-eb)β cs -csb

Profitvb

Simplify[%28]

Profitsb

Simplify[%30]

Profitc:=Profitsb+Profitvb

Profitc

Simplify[%33]

Proof ps <pva

eb:=((pb-pb α+θ+cs β)/(2 kb))
pb:=(1/(1-2 α+(α^2)))(-2 kb α-cs β+cs α β-θ+α θ+2 √(cv kb+cvb kb-cs kb rvb-2 cv kb α-2 cvb kb α+2 cs kb rvb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)-cs kb rvb (α^2)+cs kb β-cs kb α β+kb θ-kb α θ))

Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb+rvb cs-(1-eb) β cs
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rvb cs+(1-eb)β cs -csb
eb

Simplify[%15]

Profitsb

Simplify[%11]