Blockchain Model
The VM’s Effort Under Insurance Model

Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb
Profitib:=rb(α pb+cs+f)-(1-eb)(α pb+cs+f)
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-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)/(2(k^B)))

eb:=((pb-pb α+θ)/(2 kb))

Profitvb

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

Solve[%==0,pb]

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

The IC’s Premium Rate Under Blockchain Model
Observing the contract price p selected by the satellite owner, the insurance company can anticipate the vehicle manufacture’s effort e as given above. Operating in a competitive insurance market, the insurance company sets its premium rate r to breakeven in expectation. In other words, under the premium rate r that it offers, the insurance company’s expected profit,r(α p+cs+f), equals the expected coverage 
(1-e) (α p+cs+f).
Substituting e given ine:=((p(1- α)+θ)/(2 (k^B))) , the insurance company breakeven condition can be satisfied if and only if p ≥ (p^IA) ≡((2 k^B (1-r)- θ)/((1-α) )) ,which we refer as the insurance company’s underwriting constraint.  And the premium rate satisfied with r=((2 (k^B)-p(1- α)-θ)/(2 (k^B)))

Profitib:=rb(α pb+cs+f)-(1-eb)(α pb+cs+f)

Solve[Profitib==0,rb]

The SO’s Optimal Contract Under Blockchain Model

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

Profitsb

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

D[%,pb]

Solve[%==0,pb]

{{pb->((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))}}

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

Δp:=((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))-(1/(1-2 α+(α^2)))(-2 kb α-θ+α θ+2 (Sqrt[cv kb+cvb kb-2 cv kb α-2 cvb kb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)+kb θ-kb α θ]))
Δp

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

Solve[%==0,f]

kb:=60
k:=100
cv:=30
cvb:=10
θ:=80

f1:=((-((kb α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))
pf1=Plot[f1,{α,0,1},GridLines->{{1},{}}]

f2:=1800
pf2=Plot[f2,{α,0,1}]

Plot[{f1,f2,f3},{α,0,1},Filling->{1->{3}},GridLines->{{1},{}}]

Show[%48,AxesLabel->{HoldForm[α],HoldForm[F]},PlotLabel->HoldForm[F-α],LabelStyle->{GrayLevel[0]}]

k:=100

f3:=((-((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)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))

pf3=Plot[{f2,f3},{α,0,1},Filling->{1->
{2}},GridLines->{{1},{}}]

Show[pf3,AxesLabel->{HoldForm[α],None},PlotLabel->HoldForm[F-α],LabelStyle->{GrayLevel[0]}]

Show[pf3,Background->White]

The Profit

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


The profit of satellite owner is -cs-csb+((α (f (-1+α)+2 kb α+θ-α θ))/(2 (((-1+α))^2)))-(((f (-1+α)+2 kb α+(-1+α) θ) (f-f α+2 kb α+θ-α θ))/(8 kb (((-1+α))^2)))
wich can be written as (((((F+θ))^2)(((1-α))^2)-4(k^B) α[αk + (θ - F) (1-a) ])/(8(k^B) (((1-α))^2)))-Cs-Csb,
and the difference is (((k-(k^B))[(((1-α))^2)(((F+θ))^2)-4k (k^B)(α^2)])/(8 k (k^B) (((1-α))^2)))-Csb
The premium rate is r = 1+(1/4) (((2 α)/(1-α))-((f+θ)/(k^B)))

2. If the (p^S)<(p^VA), which satisfied with F<((-2 (k^B) α-(1-α) θ+4 (Sqrt[(k^B) (cv (((1-α))^2)+(k^B) (α^2)+θ(1-α )])))/(1-α)),
  the effort is e=((-(k^B) α+(Sqrt[(k^B) (cv (((1-α))^2)+(k^B) (α^2)+θ(1-α) )]))/((k^B) (1-α)))
  The premium rate is r=(((k^B)-(Sqrt[(k^B) (cv (((1-α))^2)+(k^B) (α^2)+θ(1-α ))]))/((k^B) (1-α)))
  the price is ((2(Sqrt[(k^B) cv·(((1-α))^2) + (α^2)((k^B)^2) + θ (k^B)(1-α)]) -2(k^B) α -(1-α)θ)/(((1-α))^2))
  The profit of satellite owner is -(1/(kb (((-1+α))^2)))(cs kb+csb kb+2 cv kb+2 cvb kb-2 cs kb α-2 csb kb α-4 cv kb α-4 cvb kb α+f kb α+cs kb (α^2)+csb kb (α^2)+2 cv kb (α^2)+2 cvb kb (α^2)-f kb (α^2)+2 (kb^2) (α^2)+2 kb θ-2 kb α θ-f (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+f α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-2 kb α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+α θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)]))
   which can be written as 
   (([(F+θ)(1-α)+2((αk)^B)]*(Sqrt[(k^B)[((Cv(1-α))^2)+(k^B)(α^2)+θ(1-α)]])- (k^B) (1-α)(α F+2θ )  - 2 ((k^B)^2)(α^2) )/(((k^B)(1-α))^2))
- (Cs+2Cv+Csb+2Cvb)
   
   (([(F+θ)(1-α)+2((αk)^B)]*B- (k^B) (1-α)(α F+2θ )  - 2 ((k^B)^2)(α^2) )/(((k^B)(1-α))^2))
- (Cs+2Cv+2Csb+Cvb)
   
   and the difference is  (1/(k (k^B) (((-1+α))^2)))(2k (k^B)(α^2)(k-(k^B))+(F+θ)(1-α)(k(Sqrt[(k^B) (cv (((1-α))^2)+cvb (((1-α))^2)+(k^B) (α^2)+θ(1-α ))])-(k^B)(Sqrt[k (cv (((1-α))^2)+cvb (((1-α))^2)+k (α^2)+θ(1-α ))]))+2((αkk)^B)((Sqrt[(k^B) (cv (((1-α))^2)+cvb (((1-α))^2)+(k^B) (α^2)+θ(1-α ))])-(Sqrt[k (cv (((1-α))^2)+cvb (((1-α))^2)+k (α^2)+θ(1-α ))]))
    令  A=(Sqrt[k (cv (((1-α))^2)+((kα)^2)+θ(1-α ))])
         B=(Sqrt[(k^B) (cv (((1-α))^2)+(k^B)(α^2)+θ(1-α ))])
         C=(Sqrt[k (cv (((1-α))^2)+cvb (((1-α))^2)+k (α^2)+θ(1-α ))])
         D=(Sqrt[(k^B) (cv (((1-α))^2)+cvb (((1-α))^2)+(k^B) (α^2)+θ(1-α ))])
         A<B<C<D
         (((2k (k^B)(α^2)(k-(k^B))+(F+θ)(1-α)(k*D-(k^B)C)+2((αkk)^B)(D-C)))/(k (k^B) (((1-α))^2)))
-(Cs+Csb+2Cv+2Cvb)
   the profit of vehicle manufacture is 0

Proof1:

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

eb

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

(1/4) ((f/kb)+((2 α)/(-1+α))+(θ/kb))
rb

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

((f+2 kb (-2+α)-f α+θ-α θ)/(4 kb (-1+α)))

rb:=1+(1/4) (((2 α)/(1-α))-((f+θ)/kb))

eb:=(1/4) (((f+θ)/kb)-((2 α)/(1-α)))
pb:=((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-csb

Profitsb

-cs-csb-((α (f-f α-2 kb α-θ+α θ))/(2 (((-1+α))^2)))+(1/4) (-((2 α)/(1-α))+((f+θ)/kb)) (f-(((1-α) (f-f α-2 kb α-θ+α θ))/(2 (((-1+α))^2))))

Simplify[%26]

-cs-csb+((α (f (-1+α)+2 kb α+θ-α θ))/(2 (((-1+α))^2)))-(((f (-1+α)+2 kb α+(-1+α) θ) (f-f α+2 kb α+θ-α θ))/(8 kb (((-1+α))^2)))

ΔProfitsb:=-cs-csb+((α (f (-1+α)+2 kb α+θ-α θ))/(2 (((-1+α))^2)))-(((f (-1+α)+2 kb α+(-1+α) θ) (f-f α+2 kb α+θ-α θ))/(8 kb (((-1+α))^2))) -(-cs-((α (f-f α-2 k α-θ+α θ))/(2 (((-1+α))^2)))+(1/4) (-((2 α)/(1-α))+((f+θ)/k)) (f-(((1-α) (f-f α-2 k α-θ+α θ))/(2 (((-1+α))^2)))))
ΔProfitsb

Simplify[%36]

(1/(8 k kb (((-1+α))^2)))((f^2) (k-kb) (((-1+α))^2)-8 csb k kb (((-1+α))^2)-4 (k^2) kb (α^2)+4 k (kb^2) (α^2)+2 f (k-kb) (((-1+α))^2) θ+k (θ^2)-kb (θ^2)-2 k α (θ^2)+2 kb α (θ^2)+k (α^2) (θ^2)-kb (α^2) (θ^2))

Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb

Profitvb

Simplify[-cv-cvb+((α (f-f α-2 kb α-θ+α θ))/(2 (((-1+α))^2)))+(((1-α) (f-f α-2 kb α-θ+α θ) (-((2 α)/(1-α))+((f+θ)/kb)))/(8 (((-1+α))^2)))-(1/16) kb (((-((2 α)/(1-α))+((f+θ)/kb)))^2)-θ (1+(1/4) (((2 α)/(1-α))-((f+θ)/kb)))]

(1/(16 kb (((-1+α))^2)))(-16 cvb kb+(f^2) (((-1+α))^2)-16 cv kb (((-1+α))^2)+32 cvb kb α-16 cvb kb (α^2)-12 (kb^2) (α^2)-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))

eb:=(1/4) (((f+θ)/kb)-((2 α)/(1-α)))
pb:=((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))
Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv-cvb
Profitvb

Simplify[-cv-cvb+((α (f-f α-2 kb α-θ+α θ))/(2 (((-1+α))^2)))+(((1-α) (f-f α-2 kb α-θ+α θ) (-((2 α)/(1-α))+((f+θ)/kb)))/(8 (((-1+α))^2)))-(1/16) kb (((-((2 α)/(1-α))+((f+θ)/kb)))^2)-θ (1+(1/4) (((2 α)/(1-α))-((f+θ)/kb)))]

(1/(16 kb (((-1+α))^2)))(-16 cvb kb+(f^2) (((-1+α))^2)-16 cv kb (((-1+α))^2)+32 cvb kb α-16 cvb kb (α^2)-12 (kb^2) (α^2)-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))

ΔProfitvb:=(1/(16 kb (((-1+α))^2)))(-16 cvb kb+(f^2) (((-1+α))^2)-16 cv kb (((-1+α))^2)+32 cvb kb α-16 cvb kb (α^2)-12 (kb^2) (α^2)-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))-
(1/(16 (((α-1))^2) k))((α^2) (θ^2)-2 α (θ^2)-16 (((α-1))^2) cv k+(((α-1))^2) (f^2)-2 (α-1) f (-(α θ)+θ+2 α k)+(θ^2)-12 (α^2) (k^2)-4 (α^2) θ k+20 α θ k-16 θ k)
ΔProfitvb

Simplify[%11]

Proof2

eb:=((pb-pb α+θ)/(2 kb))
rb:=1-eb
pb:=(1/(1-2 α+(α^2)))(-2 kb α-θ+α θ+2 (Sqrt[cv kb+cvb kb-2 cv kb α-2 cvb kb α+cv kb (α^2)+cvb kb (α^2)+(kb^2) (α^2)+kb θ-kb α θ]))

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

eb

Simplify[%9]

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

D[eb,kb]

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

eb 对kb求一阶导数，小于0

Profitsb

Simplify[%7]

-(1/(kb (((-1+α))^2)))(cs kb+csb kb+2 cv kb+2 cvb kb-2 cs kb α-2 csb kb α-4 cv kb α-4 cvb kb α+f kb α+cs kb (α^2)+csb kb (α^2)+2 cv kb (α^2)+2 cvb kb (α^2)-f kb (α^2)+2 (kb^2) (α^2)+2 kb θ-2 kb α θ-f (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+f α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-2 kb α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+α θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)]))

ΔProfitsb2:=-(1/(kb (((-1+α))^2)))(cs kb+csb kb+2 cv kb+2 cvb kb-2 cs kb α-2 csb kb α-4 cv kb α-4 cvb kb α+f kb α+cs kb (α^2)+csb kb (α^2)+2 cv kb (α^2)+2 cvb kb (α^2)-f kb (α^2)+2 (kb^2) (α^2)+2 kb θ-2 kb α θ-f (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+f α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-2 kb α (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])-θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)])+α θ (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)]))-(-(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)])))

ΔProfitsb2

Simplify[%12]