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Qyw+SSViZXRhRzYiLkYkIiIiPkkoZXBzaWxvbkdGJS5GKUYnPkkiQUdGJS1JJ21hdHJpeEdGJTYlIiIkRjA3JTclRiciIiFGJDclRiRGJ0YzNyVGM0YkRidGJz5JIkJHRiUtRi42JUYwRjA3JTclLCRGJCEiIkYnRjM3JUYzRjxGJzclRidGM0Y8Ric+SSNBdEdGJS1JKnRyYW5zcG9zZUdGJTYjRixGJw==
SSViZXRhRzYi
SShlcHNpbG9uRzYi
PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJiIiITYkRiciIiNJJWJldGFHRiM2JEYmRidGLTYkRidGJ0YmNiRGJkYsRio2JEYsRiZGLTYkRixGLEYmNiRGLEYnRio2JEYmRiZGJg==
PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJkYmNiRGJyIiIyIiITYkRiZGJ0YsNiRGJ0YnLCRJJWJldGFHRiMhIiI2JEYmRitGJjYkRitGJkYsNiRGK0YrRi82JEYrRidGJjYkRiZGJkYv
PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJkklYmV0YUdGIzYkRiciIiMiIiE2JEYmRidGLTYkRidGJ0YmNiRGJkYsRio2JEYsRiZGLTYkRixGLEYmNiRGLEYnRio2JEYmRiZGJg==
fyi: Start on V_1 : this is the line in Sigma^B where player A is indifferent between strategy 2 and 3 (it goes from R^B_{23} to Q^B_{23}. Since we want to look at the problem in the projective sense, we will take p^B in \134partial \134Sigma^B and for simplicity we take p^A in the mid-point. The flow during the ith-leg is denoted by pAi,pBi, i=1,2,3,4. During 1st leg aim is p^A_1,P^B_1, during 2nd leg P^A_1, P^B_2, 3rd leg aim is P^A_2, P^B_2 and 4th leg is P^A_2,P_1^B.
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LCYqJiwmIiIiRiVJInRHNiIhIiJGJT1GJzYjO0YlIiIkRVxbbCRGJSNGJUYsIiIjRi5GLEYuRiVGJSomRiZGJT1GJ0YqRVxbbCRGJUYlRi8iIiFGLEYzRiVGJQ==
PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
p^A starting in a point on boundary Sigma^A, just to the right of R^A_{12}
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PTYiNiM7IiIiIiIkRVxbbCRGJiomLCgqJkklYmV0YUdGI0YmSShlcHNpbG9uR0YjRiZGJkYmRiZGLUYmRiYsJiIiI0YmRixGJiEiIkYvLCQqJiwmRjBGJkYtRiZGJkYuRjBGMEYnLCQqKEYuRjAsKEYmRiZGLEYmRi1GJkYmRixGJkYw
Hit R^B_{13}.
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KiZJKGVwc2lsb25HNiIiIiIsKEYlRiVJJWJldGFHRiRGJUYjRiUhIiI=
fyi: these are the positions at the end of leg one:
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fyi: these are the flows during the 2nd leg PkkkcEEyRzYiLCYqJiwmIiIiRihJInRHRiQhIiJGKEklcEF0MUdGJEYoRigqJkYpRigtSSd2ZWN0b3JHRiQ2IzclRigiIiFGMUYoRig=
LCYqJiwmIiIiRiVJInRHNiIhIiJGJUklcEF0MUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiVGJSIiIyIiIUYuRjFGJUYl
PkkkcEIyRzYiLCYqJiwmIiIiRihJInRHRiQhIiJGKEklcEJ0MUdGJEYoRigqJkYpRigtSSd2ZWN0b3JHRiQ2IzclIiIhRihGMUYoRig=
LCYqJiwmIiIiRiVJInRHNiIhIiJGJUklcEJ0MUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUiIiEiIiNGJUYuRjBGJUYl
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fyi: these are the positions when player B becomes indifferent, so pAt2 intersects indifference line (connecting MidA to R^A_{12})
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QyY+SSRwQTRHNiIsJiomLCYiIiJGKUkidEdGJSEiIkYpSSVwQXQzR0YlRilGKSomRipGKS1JJ3ZlY3RvckdGJTYjNyUiIiFGKUYyRilGKUYpPkkkcEI0R0YlLCYqJkYoRilJJXBCdDNHRiVGKUYpKiZGKkYpLUYvNiM3JUYpRjJGMkYpRilGKQ==
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Target in Sigma_A resp Sigma_B: this is R^A_{12} resp. Q^B_{13}
Compute TA which is Q^A_{12} x:=vector([t, 1-t, 0]);utilBx:=evalm(x.B);solve(utilBx[1]=utilBx[2],t);
PTYiNiM7IiIiIiIkRVxbbCRGJkkidEdGIyIiIywmRiZGJkYpISIiRiciIiE=
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiZJInRHRiNGJkklYmV0YUdGI0YmISIiIiIjLCZGK0YmKiYsJkYmRiZGK0YtRiZGLEYmRi1GJ0Yx
KiZJJWJldGFHNiIiIiIsJkYjIiIjRiVGJSEiIg==
QyQ+SSNUQUc2Ii1JJmV2YWxtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJiwmSSViZXRhR0YlIiIjIiIiRjAhIiItSSd2ZWN0b3JHRiU2IzclRi4sJkYwRjBGLkYwIiIhRjBGMA==
PTYiNiM7IiIiIiIkRVxbbCRGJiomSSViZXRhR0YjRiYsJkYqIiIjRiZGJiEiIkYsKiZGK0YtLCZGJkYmRipGJkYmRiciIiE=
Compute TB which is R^B_{12}.
x:=vector([t,1-t,0]);utilAx:=evalm(x . At);solve(utilAx[1]=utilAx[2],t);
QyQ+SSNUQkc2Ii1JJmV2YWxtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJiwmIiIjIiIiSSViZXRhR0YlISIiRjEtSSd2ZWN0b3JHRiU2IzclRi8sJkYvRi9GMEYxIiIhRi9GLw==
PTYiNiM7IiIiIiIkRVxbbCRGJkkidEdGIyIiIywmRiZGJkYpISIiRiciIiE=
PTYiNiM7IiIiIiIkRVxbbCRGJkkidEdGIyIiIywoKiZGKUYmSSViZXRhR0YjRiZGJkYmRiZGKSEiIkYnKiYsJkYmRiZGKUYuRiZGLUYm
LCQqJCwmISIjIiIiSSViZXRhRzYiRiYhIiJGKQ==
PTYiNiM7IiIiIiIkRVxbbCRGJiokLCYiIiNGJkklYmV0YUdGIyEiIkYtRisqJkYqRi0sJkYmRiZGLEYtRiZGJyIiIQ==
V1 is plane in Sigma^B determined by U^B := pB * At is indifferent between 2 and 3
(where we ignore whether 1 is larger or not). First take line through the targets to compute the cone over these 'base points' and then find intersection with V1 .
conepAt1:=(1-tau)* TA + tau * evalm(pAt1);conepBt1:=(1-tau)*TB+tau*evalm(pBt1);conepAt2:=(1-tau)* TA + tau * evalm(pAt2);conepBt2:=(1-tau)*TB+tau*evalm(pBt2);conepAt3:=(1-tau)* TA + tau * evalm(pAt3);conepBt3:=(1-tau)*TB+tau*evalm(pBt3);conepAt4:=(1-tau)* TA + tau * evalm(pAt4);conepBt4:=(1-tau)*TB+tau*evalm(pBt4);
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUqJCwmIiIjRiVJJWJldGFHRidGJUYoRjIiIiFGLiomLCZGJUYlRjNGJUYlRjFGKEYlRiU=
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQkdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUsJComLChGJUYlSSViZXRhR0YnRiVJKGVwc2lsb25HRidGLkYlLChGJUYlRjNGJUY0RiVGKCNGJUYuIiIjLCQqJiwmRiVGJUYzRiVGJUY1RihGNkYuRjhGJUYl
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUqKCwsRihGJUklYmV0YUdGJ0YoSShlcHNpbG9uR0YnISIkKiZGMkYlRjNGJUYlKiZGM0YlRjIiIiNGJUYlLCpGKEYlRjJGKEYzISIjRjVGJUYoLCZGN0YlRjJGJUYoRjciIiFGLiwkKiosKEYlRiVGMkYlRjNGJUYlLCZGJUYlRjJGJUYlRjhGKEY6RihGKEYlRiU=
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQkdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUsJComLChGJUYlSSViZXRhR0YnRiVJKGVwc2lsb25HRidGLkYlLCpGKEYlRjNGKEY0ISIjKiZGM0YlRjRGJUYlRigjRihGLiIiIywkKiYsKkYoRiVGM0YoRjQhIiRGN0YuRiVGNUYoI0YlRi5GLiwkKiYsJkYlRiVGM0YlRiVGNUYoRjhGJUYl
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUqKiwsRihGJUklYmV0YUdGJ0YoSShlcHNpbG9uR0YnISIkKiZGMkYlRjNGJUYlKiZGM0YlRjIiIiNGJUYlRjJGJSwmRjdGJUYyRiVGKCwsRjZGNyokRjJGN0YoRjJGKEY1ISIjRjNGKEYoRjcqKCwmRjpGJUYoRiVGJUYzRiVGOUYoRi4sJCosLCZGJUYlRjJGJUYlLChGJUYlRjJGJUYzRiVGJUYyRiVGOEYoRjlGKEYoRiVGJQ==
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQkdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUsJCooLChGJUYlSSViZXRhR0YnRiVJKGVwc2lsb25HRidGLkYlRjNGJSwsKiZGNEYlRjMiIiNGNyokRjNGN0YoRjNGKComRjNGJUY0RiUhIiNGNEYoRigjRihGLkY3LCQqJiwsRjNGKEY4RihGOSEiJEY2IiInRjRGP0YlRjVGKCNGJUYuRi4sJCooLCZGJUYlRjNGJUYlRjNGJUY1RihGO0YlRiU=
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQUdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUqKkklYmV0YUdGJ0YlLCxGKEYlRjFGKEkoZXBzaWxvbkdGJyEiJComRjFGJUYzRiVGJSomRjNGJUYxIiIjRiVGJSwmRjdGJUYxRiVGKCwsRjZGNyokRjFGN0YoRjVGNEYxRihGMyEiI0YoRjcqKCwoRjpGJUY7RiVGMUYoRiVGM0YlRjlGKEYuLCQqLEYxRiUsKEYlRiVGMUYlRjNGJUYlLCZGJUYlRjFGJUYlRjhGKEY5RihGKEYlRiU=
LCYqJiwmIiIiRiVJJHRhdUc2IiEiIkYlSSNUQkdGJ0YlRiUqJkYmRiU9Ric2IztGJSIiJEVcW2wkRiUsJComLCpJJWJldGFHRidGJSokRjMiIiNGJSomRjNGJUkoZXBzaWxvbkdGJ0YlIiInRjdGLkYlLCwqJkY3RiVGM0Y1RjVGNEYoRjYhIiRGM0YoRjchIiNGKCNGKEYuRjUsJComLCxGM0YoRjRGKEY2RjtGOkY4RjdGO0YlRjlGKCNGJUYuRi4sJCooRjNGJSwmRiVGJUYzRiVGJUY5RihGPUYlRiU=
intersect with V1 (line in Sigma^B determined by UtilA[2]=UtilA[3]
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
PTYiNiM7IiIiIiIkRVxbbCRGJioqLDQqJkkoZXBzaWxvbkdGI0YmSSViZXRhR0YjIiIjISImKiZGLEYmRi0iIiVGJiomRi1GJkYsRiZGJiokRi1GMUYuKiRGLUYnRiYqJkYsRiZGLUYnRi5GLUYuRiZGJkYsRiZGJiwsRjIhIiRGNEYmRitGLkYmRiZGLEYmISIiLCZGLUYuRiZGJkY4LCZGLkYmRi1GJkY4Ri4qLkYtRiZGLEYmLCYhIiNGJkYtRiZGJiwmRiZGJkYtRiZGJkY2RjhGOUY4RicqLCwoRi1GOCokRi1GLkYmRiZGJkYmRjZGOCwoRiZGJkYtRiZGLEYmRiZGPkYmRjpGOA==
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiYsLComSSViZXRhR0YjRiZJKGVwc2lsb25HRiNGJiEiJyokRi1GJ0YmKiZGLkYmRi0iIiNGJ0YmRiZGLkYnRiYsLEYsISIkRjBGJkYxRjJGJkYmRi5GJiEiIiNGJkYnRjIsJComLCpGLEY0RjFGJ0YwRiZGJkYmRiZGM0Y1RjZGJywkKigsKEYtRjUqJEYtRjJGJkYmRiZGJkYzRjUsJkYmRiZGLUYmRiZGNg==
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKigsLComSShlcHNpbG9uR0YjRiZJJWJldGFHRiMiIiNGJkYuISIjKiZGLkYmRi1GJkYvRi0hIiQhIiJGJkYmLCZGL0YmRi5GJkYzLCZGLkYvRiZGJkYzRjNGLyooRi1GJiwmRjBGJkYuRiZGJkY1RjNGJyomRjRGMywoRiZGJkYuRiZGLUYmRiY=
PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiwsMiokSSViZXRhR0YjRichIiMqJkkoZXBzaWxvbkdGI0YmRi1GJ0YnKiRGLSIiI0YmKiZGMEYmRi1GMiEiIiomRi1GJkYwRiYhIiRGLUY0RjBGJkY0RiZGJkYtRiYsJkYyRiZGLUYmRjQsJkYtRjJGJkYmRjQsLkYsRiZGMUY0RjNGLkY1RidGLUYmRjBGNEY0RjRGMiwkKiosKkYsRidGMUYuRi1GNEYmRiZGJkYwRiZGOEY0RjlGNEY0RicqLEYtRiYsKEYtRjRGMUYmRiZGJkYmLChGJkYmRi1GJkYwRiZGJkY3RjRGOUY0
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiYsLiokSSViZXRhR0YjRidGJiokRi0iIiMhIiIqJkkoZXBzaWxvbkdGI0YmRi1GLyEiJComRi1GJkYyRiYiIidGLUYmRjJGM0YmLC5GLEYmRi5GMEYxISIjRjRGJ0YtRiZGMkYwRjAjRiZGJ0YvLCQqKEYtRiYsLEYuRiZGLUYwRjRGM0YmRiZGMkYnRiZGNkYwRjhGJywkKihGLUYmLChGLUYwRi5GJkYmRiZGJkY2RjBGOA==
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiYsJkkoZXBzaWxvbkdGI0YmISIiRiZGJiwmIiIjRiZJJWJldGFHRiNGJkYtRi1GLyIiIUYnKiZGLkYtLChGJkYmRjBGJkYsRiZGJg==
PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
Now take intersection with V2 which is the line where player B is indifferent between strategies 1 and 3. That is, this is line in Sigma^A connecting R^A_{13} and Q^A_{13}.
UtilBt1 := evalm(conepAt1*B); tau1 := solve(UtilBt1[3] = UtilBt1[1], tau); xV2t1 := simplify(evalm(subs(tau = tau1, conepAt1))); yV2t1 := simplify(evalm(subs(tau = tau1, conepBt1))); UtilBt2 := evalm(conepAt2*B); tau1 := solve(UtilBt2[3] = UtilBt2[1], tau); xV2t2 := simplify(evalm(subs(tau = tau1, conepAt2))); yV2t2 := simplify(evalm(subs(tau = tau1, conepBt2))); UtilBt3 := evalm(conepAt3*B); tau1 := solve(UtilBt3[3] = UtilBt3[1], tau); xV2t3 := simplify(evalm(subs(tau = tau1, conepAt3))); yV2t3 := simplify(evalm(subs(tau = tau1, conepBt3))); UtilBt4 := evalm(conepAt4*B); tau1 := solve(UtilBt4[3] = UtilBt4[1], tau); xV2t4 := simplify(evalm(subs(tau = tau1, conepAt4))); yV2t4 := simplify(evalm(subs(tau = tau1, conepBt4)));
PTYiNiM7IiIiIiIkRVxbbCRGJiwmKiYsJiooLCZGJkYmSSR0YXVHRiMhIiJGJkklYmV0YUdGI0YmLCZGMCIiI0YmRiZGL0YmKiZGLkYmLCZGMkYmRjBGJkYvRiZGJkYwRiZGLyooRi5GJiwmRiZGJkYwRiZGJkY0Ri9GJkYyLChGLEYmRjNGJioqRi1GJkYxRi9GNkYmRjBGJkYvRicsJiooRi1GJkYxRi9GNkYmRiYqKkYuRiZGNkYmRjRGL0YwRiZGLw==
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PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiosMEklYmV0YUdGIyEjOCokRiwiIiMhIiYqJkYsRiZJKGVwc2lsb25HRiNGJiEiJyEiKEYmRjIhIzoqJEYsRidGJiomRjJGJkYsRi9GJ0YmLCZGJkYmRixGJiEiIiwmISIjRiZGLEYmRjksKEYmRiZGLEYmRjJGJkY5I0YmIiIqRi8sJCoqLDBGLCEjNUYuIiIlRjEhIiRGNiIiKEY3IiInRjRGJkYyRkRGJkY4RjlGOkY5RjxGOUY9RicsJComLCZGL0YmRixGJkYmRjxGOUY9
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LCQqKiwoIiIiRiVJJWJldGFHNiJGJSokRiYiIiNGJUYlLDYiIiRGJSomSShlcHNpbG9uR0YnRiVGJkYrIiIlKiZGLUYlRiZGKUYrKiZGLUYlRiZGLkYlRi0iIiZGKCIjN0YmIiIqKiZGJkYlRi1GJUYxKiRGJkYuRisqJEYmRitGMyEiIiwqRjdGJUYmRjdGLSEiI0Y0RiVGJSwmRilGJUYmRiVGJUY3
PTYiNiM7IiIiIiIkRVxbbCRGJiomLCwqJEklYmV0YUdGI0YnRiYqJEYsIiIjRi5GLEYuRiZGJkkoZXBzaWxvbkdGI0YnRiYsMComRi9GJkYsRidGJkYrRicqJkYvRiZGLEYuRidGLSIiJ0YsRjNGJ0YmRi8iIiYhIiJGLiomLDJGK0YmRjFGJkYtRi5GMkYuRixGLiomRixGJkYvRiZGNUYmRiZGL0YmRiZGMEY1RicqKCwoRiZGJkYsRiZGL0YmRiYsKEYmRiZGLEYmRi1GJkYmRjBGNQ==
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PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
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PTYiNiM7IiIiIiIkRVxbbCRGJiwmKiYsJiooLCZGJkYmSSR0YXVHRiMhIiJGJkklYmV0YUdGI0YmLCZGMCIiI0YmRiZGL0YmKixGLkYmRjBGJiwsRi9GJkYwRi9JKGVwc2lsb25HRiMhIiQqJkYwRiZGNUYmRiYqJkY1RiZGMEYyRiZGJiwmRjJGJkYwRiZGLywsRjhGMiokRjBGMkYvRjdGNkYwRi9GNSEiI0YvRiZGJkYwRiZGLyouRi5GJkYwRiYsKEYmRiZGMEYmRjVGJkYmLCZGJkYmRjBGJkYmRjlGL0Y6Ri9GL0YyLChGLEYmRjNGJiomLCYqKEYtRiZGMUYvRj9GJkYmKipGLkYmLChGO0YmRjxGJkYwRi9GJkY1RiZGOkYvRiZGJkYwRiZGL0YnLChGQ0YmRkRGJiouRi5GJkYwRjJGPkYmRj9GJkY5Ri9GOkYvRiY=
LCQqLCwoIiIiRiVJJWJldGFHNiJGJSokRiYiIiNGJUYlRiYhIiIsNComSShlcHNpbG9uR0YnRiVGJkYpIiIpKiZGJkYlRi1GJSIiJiomRi1GJUYmIiIkIiIlRiYiIipGMkYlRi1GJSokRiZGM0YyKiRGJkYyRjRGKCIjN0YqLCZGKUYlRiZGJUYlLCxGLEYpRihGKkYvISIkRiZGKkYtISIjRiVGKg==
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Now take intersection of xV1projti points with the boundary of Sigma^A \134times Sigma^B, which means taking the 2nd coordinate of the xV1projti equal to zero : 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 JSFH
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LCYqJiwmIiIiRiVJJHJob0c2IiEiIkYlPUYnNiM7RiUiIiRFXFtsJEYlI0YlRiwiIiNGLkYsRi5GJUYlKiZGJkYlSSZ5VjF0MkdGJ0YlRiU=
LCYqJiwmIiIiRiVJJHJob0c2IiEiIkYlPUYnNiM7RiUiIiRFXFtsJEYlI0YlRiwiIiNGLkYsRi5GJUYlKiZGJkYlSSZ4VjF0M0dGJ0YlRiU=
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Now for V1 projected on the boundary of Sigma^A\134times Sigma^B intersect these lines with the boundary, i.e. with the 2nd coordinate of p^A = 0
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rho1 := simplify(solve(evalm(xV1projt2)[2] = 0, rho)); xV1Projt2 := simplify(subs(rho = rho1, evalm(xV1projt2))); yV1Projt2 := simplify(subs(rho = rho1, evalm(yV1projt2)));
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PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
rho1:= simplify(solve( evalm(xV1projt3) [2] = 0, rho));xV1Projt3 := simplify(subs(rho = rho1, evalm(xV1projt3))); yV1Projt3 := simplify(subs(rho = rho1, evalm(yV1projt3)));
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rho1 := simplify(solve(evalm(xV1projt4)[2] = 0, rho)); xV1Projt4 := simplify(subs(rho = rho1, evalm(xV1projt4))); yV1Projt4 := simplify(subs(rho = rho1, evalm(yV1projt4)));
IiIi
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PTYiNiM7IiIiIiIkRVxbbCRGJiNGJkYnIiIjRilGJ0Yp
Now see how these points vary with epsilon. For t0,t2 and t4 the p^B is in the midpoint, so just consider p^A and these are on the side of the triangle Sigma^B
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JSFH
CV1xt2 := evalm(xV1Projt2[1]); DV1R1t2:=simplify(subs(epsilon=0,simplify(diff(CV1xt2, epsilon))));fff2:=simplify(%);ff2:=beta->fff2;
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For t1 and t3 the points p^A are in R^A_{12} and p^B is on the line V_0. It is easiest to consider the l_1 norm (sum of absolute speed in all directions), or just consider the positive terms (because the sum of speeds is zero this is half of the previous) QzY+SSlDVjF5dDFkMUc2Ii1JJmV2YWxtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMmSSp5VjFQcm9qdDFHRiU2IyIiIiEiIi1JJWRpZmZHRik2JEYkSShlcHNpbG9uR0YlRjA+SShEVjF0MWQxR0YlLUkpc2ltcGxpZnlHRig2Iy1JJXN1YnNHRik2JC9GNCIiIUkiJUdGJUYvPkkpQ1YxeXQxZDJHRiUtRic2IyZGLTYjIiIjRjAtRjI2JEZBRjRGMD5JKERWMXQxZDJHRiVGN0YvPkkpQ1YxeXQxZDNHRiUtRic2IyZGLTYjIiIkRjAtRjI2JEZMRjRGMD5JKERWMXQxZDNHRiVGN0YvPkkkZmYxR0YlZio2I0klYmV0YUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlRjZGJUYlRiVGLw== The first one is positive, the 2nd negative, and third one is negative for \134beta\134in (0,1/2) and positve for \134beta\134in (1/2,1). So the sum norm is twice the first one when beta\134in (0,1/2) and twice the 2nd one if beta\134in (1/2,1)
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IiIh
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The first term is negative, the 2nd positive, and the third one is positive when \134beta\134in (0,1/2) and negative otherwise. So the sum norm is twice the first one if beta\134in (0,1/2) and twice the 2nd if beta\134in (1/2,1)
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Now consider the intersections with V2 (which is the where the utility of B is indifferent between 1 and 3), which we computed before, but now project these xV2 and yV2 onto the boundary Sigma\134times Sigma which means (because orbit is close to legs, i.e. yV2
is close to Midpoint) that the 3rdd coordinate of yV2projti must be zero: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZfci1JI21pR0YkNiVRJyZyaG87MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmc29sdmVGJ0YvRjItRlM2JC1GIzYpLUYsNiVRJmV2YWxtRidGL0YyLUZTNiQtRiM2Iy1GLDYlUSp5VjJwcm9qdDFGJ0YvRjJGOS1GUzYmLUYjNiMtSSNtbkdGJDYkUSIzRidGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZnbzYkUSIwRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJiZyaG87RicvRjBGPUY5RjlGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmpwL0ZORltxRmVxLUYsNiVRKnhWMlByb2p0MUYnRi9GMkY1Rk8tRlM2JC1GIzYkLUYsNiVRJXN1YnNGJ0YvRjItRlM2JC1GIzYoRl5xRmBwLUYsNiVRJyZyaG87MUYnRi9GMkZmcEZobi1GUzYkLUYjNiMtRiw2JVEqeFYycHJvanQxRidGL0YyRjlGOUY5LUY2Ni1RIjtGJ0Y5RjtGaXBGQEZCRkRGRkZIRmpwRk0tRiw2JVEqeVYyUHJvanQxRidGL0YyRjVGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuRltvRjlGOUZhc0ZnckY1Rk8tRlM2JC1GIzYkRlctRlM2JC1GIzYpRmhuLUZTNiQtRiM2Iy1GLDYlUSp5VjJwcm9qdDJGJ0YvRjJGOUZib0ZgcEZjcEZmcEZecUY5RjlGYnFGZXFGZXEtRiw2JVEqeFYyUHJvanQyRidGL0YyRjVGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuLUZTNiQtRiM2Iy1GLDYlUSp4VjJwcm9qdDJGJ0YvRjJGOUY5RjlGYXMtRiw2JVEqeVYyUHJvanQyRidGL0YyRjVGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuRmd0RjlGOUZhc0ZnckY1Rk8tRlM2JC1GIzYkRlctRlM2JC1GIzYpRmhuLUZTNiQtRiM2Iy1GLDYlUSp5VjJwcm9qdDNGJ0YvRjJGOUZib0ZgcEZjcEZmcEZecUY5RjlGYnFGZXFGZXEtRiw2JVEqeFYyUHJvanQzRidGL0YyRjVGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuLUZTNiQtRiM2Iy1GLDYlUSp4VjJwcm9qdDNGJ0YvRjJGOUY5RjlGYXMtRiw2JVEqeVYyUHJvanQzRidGL0YyRjVGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuRmN3RjlGOUZhc0ZnckY1Rk8tRlM2JC1GIzYkRlctRlM2JC1GIzYpRmhuLUZTNiQtRiM2Iy1GLDYlUSp5VjJwcm9qdDRGJ0YvRjJGOUZib0ZgcEZjcEZmcEZecUY5RjlGYnFGZXFGZXEtRiw2JVEqeFYyUHJvanQ0RidGL0YyRjVGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuLUZTNiQtRiM2Iy1GLDYlUSp4VjFwcm9qdDRGJ0YvRjJGOUY5RjlGYXMtRiw2JVEqeVYyUHJvanQ0RidGL0YyRjVGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGZXFGTy1GUzYkLUYjNiRGYHItRlM2JC1GIzYoRl5xRmBwRmdyRmZwRmhuRl96RjlGOUZhc0Zgci1GUzYkLUYjNigtRiw2JVEnJmJldGE7RidGYXFGOUZgcC1GZ282JFEiMUYnRjlGZnAtRiw2JVEkZmYxRidGL0YyLUZTNiQtRiM2I0ZnXGxGOUY5RmFz
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IiIh
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IiIh
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JSFH