Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)

p:=((-2 e k+θ)/(-1+α))
r:=((2 k-p+p α-θ)/(2 k))

D[Profits,e]

Simplify[%8]

Solve[%==0,e]

Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)

e:=((p-p α+θ)/(2 k))

r:=1-((p-p α+θ)/(2 k))

Profits

D[%,p]

Solve[%==0,p]

P:=((f-f α-2 k α-θ+α θ)/(2 (((-1+α))^2)))

P:=((-2 e k+θ)/(-1+α))

Solve[((-2 e k+θ)/(-1+α))-((f-f α-2 k α-θ+α θ)/(2 (((-1+α))^2)))==0,e]

e0:=((-f+f α+2 k α-θ+α θ)/(4 k (-1+α)))-((α/(-1+α))+((Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])/((1-α)k)))

e0

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

α:=0.2
k:=100
θ:=50
cs:=50

Solve[f-f α+2 k α+θ-α θ-4 (Sqrt[k (cv-2 cv α+cv (α^2)+k (α^2)+θ-α θ)])==0,f]

f:=((2 k α+θ-α θ-4 (Sqrt[k (cv-2 cv α+cv (α^2)+k (α^2)+θ-α θ)]))/(-1+α))

f

f1:=2k-θ

f1

f2:=(Sqrt[(cs+cv+θ) 4 k ])-θ

f2

pf2=Plot[f2,{cv,0,20}]

pf1=Plot[f1,{cv,0,20}]

pf:=Plot[f,{cv,0,10}]
pf

Show[pf,pf1,pf2]

y:=(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,f]

Solve[y==0,cv]

Solve[y==0,]

α:=0.2
k:=100
θ:=50
cv:=50
f:=2 (1-α) (((k α)/(((1-α))^2))+(θ/(2 (1-α)))+((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2))))
f1:=2k-θ
f2:=(Sqrt[(cs+cv+θ) 4 k ])-θ

pf2=Plot[f2,{cs,0,10}]

pf=Plot[f,{cs,0,10}]

pf1=Plot[f1,{cs,0,10}]

Show[pf,pf1,pf2]