with(plots):
#t1:=textplot([1,1.5,"Figure 1.5"],align={BELOW,LEFT});
y3:=1-y1-y2:
sets:=inequal( {-2*y1+2*y2-y3<1,3*y1+y3<1,y1>=0,y2>=0,y1+y2<=1}, y1=-0.5..0.75, y2=-0.25..1,
optionsfeasible=(color=red),
optionsopen=(color=black,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white) );
#goal1:=implicitplot(-4*y+2*x=-6,x=-1..3,y=-1..4,color=black);
#goal2:=implicitplot(-4*y+2*x=-8,x=-1..3,y=-1..4,color=blue,thickness=2);
#goal3:=implicitplot(-4*y+2*x=-10,x=-1..3,y=-1..4,color=blue,thickness=3);
display(sets);
#figure 2.3
restart:with(plots):
#t1:=textplot([1.5,3.5,"Decreasing Cost"],align={BELOW,LEFT});
sets:=inequal( { x+y>=1/2, x-y<=1,y+2*x<=3,x>=0,y>=0}, x=-0.5..2, y=0..3.5,
optionsfeasible=(color=grey),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=3),
optionsexcluded=(color=white) );
#z:=animate(implicitplot,[-3*y+2*x=k,x=0..5,y=0..4],k=-15..10,thickness=3);
goal1:=implicitplot(-4*y+2*x=-6,x=-1..3,y=-1..4,color=blue);
goal2:=implicitplot(-4*y+2*x=-8,x=-1..3,y=-1..4,color=blue,thickness=2);
goal3:=implicitplot(-4*y+2*x=-10,x=-1..3,y=-1..4,color=blue,thickness=3);
display(sets,goal1,goal2,goal3,labels=[``,``]);
restart:
with(plots):with(plottools):
with(LinearAlgebra):
A:=Matrix([[1,-1],[2,-3]]);B:=Matrix([[1,2],[-1,-3]]);
f:=(x,y)->expand(Transpose(<x,1-x>).A.<y,1-y>);
expand(f(x,y));
g:=(x,y)->expand(Transpose(<x,1-x>).B.<y,1-y>);
points:={seq(seq([f(x,y),g(x,y)],x=0..1,0.05),y=0..1,0.05)}:
pure:=([[1,1],[-1,2],[-3,-3],[2,-1]]);
pp:=pointplot(points);
pq:=polygon(pure,color=yellow);
display(pq,pp,title="Payoffs (E[I},E[II]) in Chicken Game");
restart:
with(Statistics):
X := RandomVariable(Normal(0.5,0.2)):
PDF(X, u);
PDF(X, .5);
Mean(X);
Variance(X);
plot(1-CDF(X,u)-u*PDF(X,u),u=0..1,color=black,thickness=2);
fsolve(1-CDF(X,u)-u*PDF(X,u),u);
restart:with(plots):
p:=N->(1/(N+1))^(1/N);
plot(p(N),N=0..50,color=black,thickness=2);
restart:with(plots):
sets:=inequal( {1<=x1,2>=x1,2<=x2,3>=x2,x1+x2<=5,x1+x2>=4}, x1=0..6, x2=0..6,
optionsfeasible=(color=gray),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white) );
display(sets);
restart:with(plots):with(plottools):
inequal( { x<=8/15,y<=11/15,x+y >=7/15, x+y<=1,x>=0,y>=0}, x=-.25..1, y=-.25..1,
optionsfeasible=(color=gray),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white),labels=[`x1`,`x2`] );
restart:
cnsts:={-x1<=z,-x2<=z,-(5/2-x1-x2)<=z,2-x1-x2<=z,
1-x2-(5/2-x1-x2)<=z,-x1-(5/2-x1-x2)<=z};
Core:=subs(z=0,cnsts);
with(plots):
inequal(Core,x1=0..2,x2=0..3,optionsfeasible=(color=gray),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white));
animate(inequal,[cnsts, x1=0..2,x2=0..3,
optionsfeasible=(color=red),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black,thickness=3),
optionsexcluded=(color=white)],
z=-1..0,frames=50);
restart:
with(plots):
cnsts:={-x1<=z,-x2<=z,-(1-x1-x2)<=z,4/5-x1-x2<=z,2/5-x1-(1-x1-x2)<=z,1/5-x2-(1-x1-x2)<=z};
animate(inequal,[cnsts,x1=0..1,x2=0..1,optionsfeasible=(color=red),optionsopen=(color=blue,thickness=2),optionsclosed=(color=black,thickness=3),optionsexcluded=(color=white)],z=-1..0,frames=100);
restart:with(plots):with(plottools):with(LinearAlgebra):
A:=Matrix([[2,-10],[-19,17]]);B:=Matrix([[17,-22],[-7,2]]);
f:=(x,y)->expand(Transpose(<x,1-x>).A.<y,1-y>);
expand(f(x,y));
g:=(x,y)->expand(Transpose(<x,1-x>).B.<y,1-y>);
points:={seq(seq([f(x,y),g(x,y)],y=0..1,0.05),x=0..1,0.05)}:
pure:=([[2,17],[17,2],[-10,-22],[-19,-7]]);
pp:=pointplot(points,symbol=point);
pq:=polygon(pure,color=white);
l3:=line([-5/2,-13/4],[22,-13/4],color=black,linestyle=SOLID,thickness=3):
l4:=line([-5/2,-13/4],[-5/2,19],color=black,linestyle=SOLID,thickness=3):
#t5:=textplot([-18,-4,"Safety point(-13/4,-5/2)"],align={BELOW,RIGHT}):
display(pq,pp,title="Payoffs with and without cooperation");
display(l3,l4,pq);
display(%,%%);
restart:
f:=(x,y)->(x+13/4)*(y+5/2);
gcnst:={x >=-13/4,y>=-5/2,y<=-x+19,
y<=24/21*x+14.71,y>=24/27*x-15.22};
with(plots):with(plottools):
cp:=contourplot(f(x,y),x=0..25,y=0..25,
axes=normal,thickness=2,contours=4):
ineq:=inequal( gcnst,x=-4..25, y=-3..25,
optionsfeasible=(color=gray),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white) ):
pointp:=pointplot([73/8,79/8],thickness=5,symbol=circle):
t1:=textplot([16,13,"(73/8,79/8)"],align={BELOW,LEFT});
display3d(cp,ineq,pointp,
labels=[``,``] );
restart:
with(plots):with(plottools):
cnst:={u>=-1/4,v>=-1/3,0<=10+5*u-5*v,0<=10+u+3*v,0<=5-4*u+3*v,0<=5-2*u-v};
ineq:=inequal( cnst,u=-1..3, v=-1..4,
optionsfeasible=(color=gray),
optionsopen=(color=blue,thickness=2),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white) ):
pointp:=pointplot([29/24,31/12],thickness=5,symbol=circle):
display(ineq,pointp);
restart:with(plots):with(plottools):
mypoints:=[[-1, -1], [-2, 2], [1, 1], [2, -2], [-1, -1]];
constr:={0 <=x,-3*x-y<=4, x+3*y<=4, 3*x+y<=4, -x-3*y<=4, 1<=y};
z:=(x+0)*(y-1);
iplot2:=inequal(constr,x=-0.5..2, y=-0.5..2,
optionsfeasible=(color=gray),
optionsclosed=(color=black, thickness=2),
optionsexcluded=(color=white),
title="Feasible Set with (0,1) security"):
pol:=plots[polygonplot](mypoints, color=white):
cp:=plots[contourplot](z, x=-2..3,y=-2..3,
contours=30, axes=normal,thickness=2,color=gray):
pointp:=pointplot([1/2,7/6],thickness=5,symbol=circle):
display(iplot2, pointp,pol,cp,labels=[``,``]);
restart:a:=-1:b:=-2:
ode:=diff(p1(t),t)=p1(t)*(a*p1(t)-a*p1(t)^2-b*p2(t)^2),
diff(p2(t),t)=p2(t)*(b*p2(t)-a*p1(t)^2-b*p2(t)^2);
with(DEtools):
:
DEplot([ode],[p1(t),p2(t)],t=0..20,
stepsize=.05,p1=0..1,p2=0..1,
scene=[p1(t),p2(t)],arrows=large,
title="a<0,b<0");
DEplot([ode],[p1(t),p2(t)],t=0..20,
[[p1(0)=.8,p2(0)=.2],[p1(0)=.4,p2(0)=.6],[p1(0)=.6,p2(0)=.4],[p1(0)=.75,p2(0)=.25],[p1(0)=.1,p2(0)=.9],[p1(0)=2/3,p2(0)=1/3]],
stepsize=.01,p1=0..1,p2=0..1,
scene=[t,p2(t)],arrows=large,linecolor=black,
title="a>0,b>0");
restart:
ode:=diff(x(t),t)=x(t)*(1-x(t))*(2*x(t)-1);
with(DEtools):
:
DEplot([ode],[x(t)],t=0..20,
stepsize=.05,x=0..1,[x(0)=1/2,x(0)=0.6,x(0)=0.3],
scene=[t,x(t)],arrows=large,linecolour=black,
title="Asymptotic Solution");
restart:with(LinearAlgebra):
with(Optimization):
A:=Matrix([[0,-2,1],[1,0,1],[1,3,0]]);
X:=<0,1/4,3/4>; Y:=<y1,y2,y3>;
uYY:=expand(Transpose(Y).A.Y); uYX:=expand(Transpose(Y).A.X);
Maximize(uYX,{y1+y2+y3=1,y1>=0,y2>=0,y3>=0});
uXY:=Transpose(X).A.Y;
w:=subs(y1=0,y2=y,y3=1-y,uYY); v:=subs(y1=0,y2=y,y3=1-y,uXY);
plot(w-v,y=0..1,color=black,thickness=2);
restart:with(DEtools):with(plots):with(LinearAlgebra):
A:=Matrix([[0,1,-1],[-1,0,1],[1,-1,0]]);
X:=<x1,x2,x3>;
Transpose(X).A.X;
s:=expand(%);
L:=A.X;
L[1]-s;L[2]-s;L[3]-s;
ode:={D(p1)(t)=p1(t)*(p2(t)-p3(t)),
D(p2)(t)=p2(t)*(-p1(t)+p3(t)),
D(p3)(t)=p3(t)*(p1(t)-p2(t))};
subs(p3(t)=1-p1(t)-p2(t),ode);
G:=simplify(%);
DEplot3d({G[1],G[2]},{p1(t),p2(t)},
t=0..20,
[[p1(0)=0.1,p2(0)=0.2],[p1(0)=0.6,p2(0)=.2],[p1(0)=1/3,p2(0)=1/3]],
scene=[t,p1(t),p2(t)],
stepsize=.1,linecolor=black,title="Cycle around the steady state");
restart:with(DEtools):with(plots):with(LinearAlgebra):
A:=Matrix([[0,-1,1],[1,0,-1],[-1,1,0]]);
X:=<x1,x2,x3>;
Transpose(X).A.X;
s:=expand(%);
L:=A.X;
L[1]-s;L[2]-s;L[3]-s;
ode:={D(p1)(t)=p1(t)*(-p2(t)+p3(t)),
D(p2)(t)=p2(t)*(p1(t)-p3(t)),
D(p3)(t)=p3(t)*(-p1(t)+p2(t))};
subs(p3(t)=1-p1(t)-p2(t),ode);
G:=simplify(%);
DEplot3d({G[1],G[2]},{p1(t),p2(t)},
t=0..20,
[[p1(0)=0.1,p2(0)=0.2],[p1(0)=0.6,p2(0)=.2],[p1(0)=1/3,p2(0)=1/3]],
scene=[t,p1(t),p2(t)],
stepsize=.1,linecolor=black,title="Cycle around the steady state");
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