base model
the SO search for a VM and insurance company by himself, and contract with the VM directly which is timingly and complex.

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv
Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)
Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)-T

if the SO purchase the launch service from LS seviser instead of the VM, and the LS will help to contact an insurer whose premium rate is lower than above insurance company because of the communication of LS.

Profitv:=α p+e (1-α)p-(1-e)θ- k*(e*e)-cv
Profiti:=β r(α s+cs+f)-(1-e)(α s+cs+f)
Profitl:=α s+e(1-α)s-α p-e (1-α)p-τ s*s
Profits:=-α s+e(f-(1-α)s)-cs -β r(α s+cs+f)+(1-e)(α s+cs+f)

Solve: model 2

D[Profitv,e]

Solve[%==0,e]

e:=((p-p α+θ)/(2 k))

Profiti

Solve[%==0,r]

{{r->((2 k-p+p α-θ)/(2 k β))}}
r:=((2 k-p+p α-θ)/(2 k β))

Profitl

Simplify[-p α+s α-((p (1-α) (p-p α+θ))/(2 k))+((s (1-α) (p-p α+θ))/(2 k))-(s^2) τ]

D[%,s]

Solve[%==0,s]

s:=((p+2 k α-2 p α+p (α^2)+θ-α θ)/(4 k τ))

Profits

Simplify[%13]

D[%,p]

Solve[%==0,p]

p:=((2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ)/(((-1+α))^3))

e

Simplify[((θ+((2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ)/(((-1+α))^3))-((α (2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ))/(((-1+α))^3)))/(2 k))]

((-α+(α^2)+f τ)/(((-1+α))^2))

r

Simplify[((2 k-θ-((2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ)/(((-1+α))^3))+((α (2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ))/(((-1+α))^3)))/(2 k β))]

-((-1+α+f τ)/((((-1+α))^2) β))

s

Simplify[((2 k α+θ-α θ+((2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ)/(((-1+α))^3))-((2 α (2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ))/(((-1+α))^3))+(((α^2) (2 k α-2 k (α^2)+θ-2 α θ+(α^2) θ-2 f k τ))/(((-1+α))^3)))/(4 k τ))]

(f/(2-2 α))