ClearAll["Global`*"]
(*cap=(1/2);*)

(* 
Case A : q_L(π|ω_H)=q_H(π|ω_L)
<=π_L=q_L(π|ω_H)
<=π_H<=q_H(π|ω_H)
*)
ClearAll[
hpreAccH,hpreAccL,lpreAccH,lpreAccL,
hDAH,hDAL,lDAH,lDAL,
qhH,qhL,qlH,qlL,
UhpreAccH,UhpreAccH,UlpreAccH,UlsigAccL,
UhDAH,UhDAL,UlDAH,UlDAL,
uh,ul,ph,pl,
v,c,r]
c = 1/2;

hpreAccH=(1) r Integrate[1,{x,ph,1}]// FullSimplify;
hpreAccL=(c) r Integrate[1,{x,ph,1}]// FullSimplify;
lpreAccH=(1-(1)) r Integrate[1,{x,pl,1}]+(1)r Integrate[1,{x,pl,ph}]// FullSimplify;
lpreAccL=(1-(c)) r Integrate[1,{x,pl,1}]+(c) r Integrate[1,{x,pl,ph}]// FullSimplify;

hDAH=(1)(1-r)Integrate[1,{x,qhH,1}]// FullSimplify;
hDAL=(c)(1-r)Integrate[1,{x,qhL,1}]+(1-(c))(1-r)Integrate[1,{x,qhL,qlL}]+(c)r Integrate[1,{x,qhL,pl}]+(1-(c))r Integrate[1,{x,qhL,pl}]// FullSimplify;
lDAH=(1-(1))(1-r)Integrate[1,{x,qlH,1}]+(1)(1-r)Integrate[1,{x,qlH,qhH}]+(1-(1))r Integrate[1,{x,qlH,pl}]+(1)r Integrate[1,{x,qlH,pl}]// FullSimplify;
lDAL=(1-(c))(1-r)Integrate[1,{x,qlL,1}]// FullSimplify;

qhH=qhH/.Solve[cap==hpreAccH+hDAH,qhH]// FullSimplify;
qhL=qhL/.Solve[cap==hpreAccL+hDAL,qhL]// FullSimplify;
qlH=qlH/.Solve[cap==lpreAccH+lDAH,qlH]// FullSimplify;
qlL=qlL/.Solve[cap==lpreAccL+lDAL,qlL]// FullSimplify;

UhpreAccH=(1) r Integrate[x,{x,ph,1}]// FullSimplify;
UhpreAccL=(c) r Integrate[x,{x,ph,1}]// FullSimplify;
UlpreAccH=(1-c) r Integrate[x,{x,pl,1}]+(1)r Integrate[x,{x,pl,ph}]// FullSimplify;
UlsigAccL=(1-1) r Integrate[x,{x,pl,1}]+(c) r Integrate[x,{x,pl,ph}]// FullSimplify;

UhDAH=(1)(1-r)Integrate[x,{x,qhH,1}]// FullSimplify;
UhDAL=(c)(1-r)Integrate[x,{x,qhL,1}]+(1-(c))(1-r)Integrate[x,{x,qhL,qlL}]+(c)r Integrate[x,{x,qhL,pl}]+(1-(c))r Integrate[x,{x,qhL,pl}]// FullSimplify;
UlDAH=(1-(1))(1-r)Integrate[x,{x,qlH,1}]+(1)(1-r)Integrate[x,{x,qlH,qhH}]+(1-(1))r Integrate[x,{x,qlH,pl}]+(1)r Integrate[x,{x,qlH,pl}]// FullSimplify;
UlDAL=(1-(c))(1-r)Integrate[x,{x,qlL,1}]// FullSimplify;

uh=1/2(UhpreAccH + UhDAH)+(1-1/2)(UhpreAccL + UhDAL) // FullSimplify;
ul=1/2(UlpreAccH + UlDAH)+(1-1/2)(UlsigAccL + UlDAL)// FullSimplify;

ph = ph/.Solve[D[uh, ph] ==0,ph]// FullSimplify;
pl = pl/.Solve[D[ul, pl] ==0,pl]// FullSimplify;

qsSA=qhH//FullSimplify
qsLA=qhL//FullSimplify
qlSA=qlH//FullSimplify
qlLA=qlL//FullSimplify
psA = ph//FullSimplify
plA = pl//FullSimplify
usA = uh//FullSimplify
ulA =ul//FullSimplify

(* 
Case A deviation: 
q_L(π|ω_H)=q_L(π|ω_L)
<=q_H(π|ω_L)
<=π_L
<=π_H
<q_H(π|ω_H)
*)
ClearAll[
hpreAccH,hpreAccL,lpreAccH,lpreAccL,
hDAH,hDAL,lDAH,lDAL,
qhH,qhL,qlH,qlL,
UhpreAccH,UhpreAccH,UlpreAccH,UlsigAccL,
UhDAH,UhDAL,UlDAH,UlDAL,
uh,ul,ph,pl,
v,c,r]
c = 1/2;

hpreAccH=(1) r Integrate[1,{x,ph,1}]// FullSimplify;
hpreAccL=(c) r Integrate[1,{x,ph,1}]// FullSimplify;
lpreAccH=(1-(1)) r Integrate[1,{x,pl,1}]+(1)r Integrate[1,{x,pl,ph}]// FullSimplify;
lpreAccL=(1-(c)) r Integrate[1,{x,pl,1}]+(c) r Integrate[1,{x,pl,ph}]// FullSimplify;

hDAH=(1)(1-r)Integrate[1,{x,qhH,1}]// FullSimplify;
hDAL=(c)(1-r)Integrate[1,{x,qhL,1}]+(c)r Integrate[1,{x,qhL,pl}]// FullSimplify;
lDAH=(1-(1))(1-r)Integrate[1,{x,qlH,1}]+(1)(1-r)Integrate[1,{x,qlH,qhH}]+(1-(1))r Integrate[1,{x,qlH,pl}]+(1)r Integrate[1,{x,qlH,pl}]// FullSimplify;
lDAL=(1-(c))(1-r)Integrate[1,{x,qlL,1}]+(c)(1-r)Integrate[1,{x,qlL,qhL}]+(1-(c))r Integrate[1,{x,qlL,pl}]+(c)r Integrate[1,{x,qlL,qhL}]// FullSimplify;

qhH=qhH/.Solve[cap==hpreAccH+hDAH,qhH]// FullSimplify;
qhL=qhL/.Solve[cap==hpreAccL+hDAL,qhL]// FullSimplify;
qlH=qlH/.Solve[cap==lpreAccH+lDAH,qlH]// FullSimplify;
qlL=qlL/.Solve[cap==lpreAccL+lDAL,qlL]// FullSimplify;

UhpreAccH=(1) r Integrate[x,{x,ph,1}]// FullSimplify;
UhpreAccL=(c) r Integrate[x,{x,ph,1}]// FullSimplify;
UlpreAccH=(1-c) r Integrate[x,{x,pl,1}]+(1)r Integrate[x,{x,pl,ph}]// FullSimplify;
UlsigAccL=(1-1) r Integrate[x,{x,pl,1}]+(c) r Integrate[x,{x,pl,ph}]// FullSimplify;

UhDAH=(1)(1-r)Integrate[x,{x,qhH,1}]// FullSimplify;
UhDAL=(c)(1-r)Integrate[x,{x,qhL,1}]+(c)r Integrate[x,{x,qhL,pl}]// FullSimplify;
UlDAH=(1-(1))(1-r)Integrate[x,{x,qlH,1}]+(1)(1-r)Integrate[x,{x,qlH,qhH}]+(1-(1))r Integrate[x,{x,qlH,pl}]+(1)r Integrate[x,{x,qlH,pl}]// FullSimplify;
UlDAL=(1-(c))(1-r)Integrate[x,{x,qlL,1}]+(c)(1-r)Integrate[x,{x,qlL,qhL}]+(1-(c))r Integrate[x,{x,qlL,pl}]+(c)r Integrate[x,{x,qlL,qhL}]// FullSimplify;

uh=1/2(UhpreAccH + UhDAH)+(1-1/2)(UhpreAccL + UhDAL) // FullSimplify;
ul=1/2(UlpreAccH + UlDAH)+(1-1/2)(UlsigAccL + UlDAL)// FullSimplify;

ph = ph/.Solve[D[uh, ph] ==0,ph]// FullSimplify;
pl = pl/.Solve[D[ul, pl] ==0,pl]// FullSimplify;

qsSADEV1=qhH//FullSimplify
qsLADEV1=qhL//FullSimplify
qlSADEV1=qlH//FullSimplify
qlLADEV1=qlL//FullSimplify
psADEV1 = ph//FullSimplify
plADEV1 = pl//FullSimplify
usADEV1 = uh//FullSimplify
ulADEV1 =ul//FullSimplify

usOG = usA /. r ->0;
hpreAccH= usA/usOG
welfareLostB= usB/usOG

usOG = usA /. r ->0;
hpreAccH= usA;
welfareLostB= usB;

Reduce[D[hpreAccH, r] < 0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]

(*Reduce[D[hpreAccH, r] < 0];
FullSimplify[%, Assumptions->0<=v<=1-c&&0<=r<=1&&1/2<=c<=1&&0<cap <= 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]*)

Reduce[D[welfareLostB, r] < 0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]

(*Reduce[D[welfareLostB, r] < 0];
FullSimplify[%, Assumptions->0<=v<=1-c&&0<=r<=1&&1/2<=c<=1&&0<cap <= 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]*)

qs = qsLA /. r->0;
Reduce[qs <=0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2]

Reduce[D[hpreAccH, c]<0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<c]

Reduce[D[welfareLostB,c]<0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c]

cap = 3/10;
v = 15/100;
c = 75/100;

rBreak2[[2]] // N
rBreakUltra //N

rBreakB
Show[
Plot[psA,{r, 0, rBreak}, PlotStyle->{Red}],
Plot[qsSA,{r, 0, rBreak}, PlotStyle->{Orange}],
Plot[qsLA,{r, 0, rBreak}, PlotStyle->{Orange}],
Plot[plA,{r, 0, rBreak}, PlotStyle->{Blue}],
Plot[qlSA,{r, 0, rBreak}, PlotStyle->{Cyan}],
Plot[qlLA,{r, 0, rBreak}, PlotStyle->{Cyan}],

Plot[qSAlt,{r, 0, 1}, PlotStyle->{Black}],
Plot[qLAlt,{r, 0, 1}, PlotStyle->{Black}],

Plot[psB,{r, rBreak, 1}, PlotStyle->{Red}],
Plot[qsSB,{r, rBreak, 1}, PlotStyle->{Orange}],
Plot[qsLB,{r, rBreak, 1}, PlotStyle->{Orange}],
Plot[plB,{r, rBreak, 1}, PlotStyle->{Blue}],
Plot[qlSB,{r, rBreak, 1}, PlotStyle->{Cyan}],
Plot[qlLB,{r, rBreak, 1}, PlotStyle->{Cyan}],
PlotRange-> {{0,1},{0,1}},
AxesLabel->{HoldForm[r],None},PlotLabel->HoldForm[Responsiveness Varying],LabelStyle->{GrayLevel[0]}
]
Show[
Plot[usA,{r, 0, rBreak}, PlotStyle->{Red}],
Plot[usB,{r, rBreak, rBreak2[[2]]}, PlotStyle->{Orange}],
PlotRange-> {{0,1},{0.225, 0.24}}
]
usA /. r ->0.5

qLAlt /. {r -> x}


ClearAll[c, cap, r,v]

qSAlt = 1-cap/(r(1));
qLAlt = 1-cap/(r(c));
usAlt = cap((1/2) ((1+qSAlt)/2)+(1/2)((1+qLAlt)/2));

rBreak2= r /.Solve[usAlt == usB,r];
rBreak2= FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];

Reduce[1>=rBreak2[[2]] >= 0];
 FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];
Reduce[rBreak2[[2]] >= rBreak];
 FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c]

qSBlt = 1-cap/(r(1-(1)));
qLBlt = 1-cap/(r(1-(c)));
usBlt = cap((1/2) ((1+qSBlt)/2)+(1/2)((1+qLBlt)/2))

Solve[usAlt == usB,r];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c]

qhH = qsSA /.r -> 0
qhL = qsLA /.r -> 0
Reduce[D[qsSA-qhH, r]>0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]

Reduce[D[qsSB-qhH, r]>0];
FullSimplify[%, Assumptions->0<v<1-c&&0<r<1&&1/2<c<1&&0<cap < 1/2&&cap<=c];
FullSimplify[%, Assumptions->cap<=c]

cap = 2/10;
v = 10/100;
c = 75/100;
r = 5/10;

vBreak
cBreak // N
rBreak

Show[
Plot[psA, {v, 0, vBreak}, PlotStyle->{Red}],
Plot[qsSA, {v, 0, vBreak}, PlotStyle->{Orange}],
Plot[qsLA, {v, 0, vBreak}, PlotStyle->{Orange}],
Plot[plA, {v,0, vBreak}, PlotStyle->{Blue}],
Plot[qlSA, {v,0, vBreak}, PlotStyle->{Cyan}],
Plot[qlLA, {v,0, vBreak}, PlotStyle->{Cyan}],

Plot[psB, {v, vBreak,1-c}, PlotStyle->{Red}],
Plot[qsSB, {v, vBreak, 1-c}, PlotStyle->{Orange}],
Plot[qsLB, {v, vBreak, 1-c}, PlotStyle->{Orange}],
Plot[plB, {v, vBreak, 1-c}, PlotStyle->{Blue}],
Plot[qlSB, {v, vBreak, 1-c}, PlotStyle->{Cyan}],
Plot[qlLB, {v, vBreak, 1-c}, PlotStyle->{Cyan}],
PlotRange-> {{0,1-c},{0,1}},
AxesLabel->{HoldForm[v],None},PlotLabel->HoldForm[Volatility Varying],LabelStyle->{GrayLevel[0]}
]

ClearAll[c, cap, r,v]

ClearAll[
hpreAccH,hpreAccL,lpreAccH,lpreAccL,
hDAH,hDAL,lDAH,lDAL,
qhH,qhL,qlH,qlL,
UhpreAccH,UhpreAccH,UlpreAccH,UlsigAccL,
UhDAH,UhDAL,UlDAH,UlDAL,
uh,ul,ph,pl,
v,c,r]

hpreAccH=(1) r Integrate[1,{x,ph,1}]+(1-c) r Integrate[1,{x,ph,pl}]// FullSimplify;
hpreAccL=(c) r Integrate[1,{x,ph,1}]+(1-1) r Integrate[1,{x,ph,pl}]// FullSimplify;
lpreAccH=(1-c) r Integrate[1,{x,pl,1}]// FullSimplify;
lpreAccL=(1-1) r Integrate[1,{x,pl,1}]// FullSimplify;

hDAH=(1)(1-r)Integrate[1,{x,qhH,1}]// FullSimplify;
hDAL=(c)(1-r)Integrate[1,{x,qhL,1}]+(1-1)(1-r)Integrate[1,{x,qhL,qlL}]+(1)r Integrate[1,{x,qhL,pl}]+(1-c)r Integrate[1,{x,qhL,pl}]// FullSimplify;
lDAH=(1-c)(1-r)Integrate[1,{x,qlH,1}]+(1)(1-r)Integrate[1,{x,qlH,qhH}]+(1-c)r Integrate[1,{x,qlH,ph}]+(1)r Integrate[1,{x,qlH,ph}]// FullSimplify;
lDAL=(1-1)(1-r)Integrate[1,{x,qlL,1}]// FullSimplify;

qhH=qhH/.Solve[cap==hpreAccH+hDAH,qhH]// FullSimplify;
qhL=qhL/.Solve[cap==hpreAccL+hDAL,qhL]// FullSimplify;
qlH=qlH/.Solve[cap==lpreAccH+lDAH,qlH]// FullSimplify;
qlL=qlL/.Solve[cap==lpreAccL+lDAL,qlL]// FullSimplify;

UhpreAccH=(1) r Integrate[x,{x,ph,1}]// FullSimplify;
UhpreAccL=(c) r Integrate[x,{x,ph,1}]// FullSimplify;
UlpreAccH=(1-c) r Integrate[x,{x,pl,1}]+(1)r Integrate[x,{x,pl,ph}]// FullSimplify;
UlsigAccL=(1-1) r Integrate[x,{x,pl,1}]+(c) r Integrate[x,{x,pl,ph}]// FullSimplify;

UhDAH=(1)(1-r)Integrate[x,{x,qhH,1}]// FullSimplify;
UhDAL=(c)(1-r)Integrate[x,{x,qhL,1}]+(1-1)(1-r)Integrate[x,{x,qhL,qlL}]+(1)r Integrate[x,{x,qhL,pl}]+(1-c)r Integrate[x,{x,qhL,pl}]// FullSimplify;
UlDAH=(1-c)(1-r)Integrate[x,{x,qlH,1}]+(1)(1-r)Integrate[x,{x,qlH,qhH}]+(1-c)r Integrate[x,{x,qlH,pl}]+(1)r Integrate[x,{x,qlH,pl}]// FullSimplify;
UlDAL=(1-1)(1-r)Integrate[x,{x,qlL,1}]// FullSimplify;

uh=1/2(UhpreAccH + UhDAH)+(1-1/2)(UhpreAccL + UhDAL) // FullSimplify;
ul=1/2(UlpreAccH + UlDAH)+(1-1/2)(UlsigAccL + UlDAL)// FullSimplify;

ph = ph/.Solve[D[uh, ph] ==0,ph]// FullSimplify
pl = pl/.Solve[D[ul, pl] ==0,pl]// FullSimplify

qsSC=qhH//FullSimplify;
qsLC=qhL//FullSimplify;
qlSC=qlH//FullSimplify;
qlLC=qlL//FullSimplify;
psC = ph//FullSimplify;
plC = pl//FullSimplify;
usC = uh//FullSimplify;
ulC =ul//FullSimplify;