memo = {o20_$, is, the, payoff, modulo, the, parameters, and, sxak0_set, 
     while, o21_$, is, the, same, modulo, the . more . refined, sxakfr1_set, 
     soln}
 
o20 = {0.015067057346705312, 0.23249224164958188}
  
o21 = {0.014690208304944125571163983998871653746434979595`29.6604, 
     0.251393749004404228122786542555419659559836367043`29.511}
 
sxak0 = {0.0040914, 0.0576848, 0.0488584, 0.3200085, 0.3220528, 0.0905741, 
     0.0763074, 0, 0, 0, 0, 0, 0, 0, 0}
 
sxakfr1 = {0.004395518250090528441156231498555909344533270368`30, 
     0.065208065388218100240021472178606583269734014438`30, 
     0.051926232258494780106621008271103446888771214018`30, 
     0.3256777484428379487329332858157846568753992`30, 
     0.325677748442837948732933285815784656843246`30, 
     0.089986415430917070989709296586619627827393314245`30, 
     0.091749928504781434991499125829726004345923442812`30, 0, 0, 0, 0, 0, 0, 
     0, 0}
 
memo2 = {these,represent,a,vector,separating,payoff,through,stage,2,coalitions,
     and,grand,coalition,payoff,through,stage,3,and,both,are,for,all,three,players}
 
payv = {(-3*x1*(3 - 6*x1 + 4*x1^2)*(1/20 + (x2*(-1 + x3))/20 - x3/20))/
      (10*(-1/20 - (57*x1)/10 + (57*x1^2)/5 - (38*x1^3)/5)*
       (1/20 - (19*x2*(-1 + x3))/20 + (19*x3)/20)), 
     (2*x1*(3 - 6*x1 + 4*x1^2)*(x2*(-1 + x3) - x3))/
      ((-1/20 - (57*x1)/10 + (57*x1^2)/5 - (38*x1^3)/5)*
       (1/20 - (19*x2*(-1 + x3))/20 + (19*x3)/20))}
 
payvo = {(-2*aa*(3 - 6*aa + 4*aa^2)*b*(e5 + ab*(-1 + ac)*e5 - ac*e5))/
      ((6*aa*(-1 + e4) - 12*aa^2*(-1 + e4) + 8*aa^3*(-1 + e4) - e4)*
       (ac + ab*(-1 + ac)*(-1 + e5) + e5 - ac*e5)), 
     (2*aa*(3 - 6*aa + 4*aa^2)*(ab*(-1 + ac) - ac))/
      ((6*aa*(-1 + e4) - 12*aa^2*(-1 + e4) + 8*aa^3*(-1 + e4) - e4)*
       (ac + ab*(-1 + ac)*(-1 + e5) + e5 - ac*e5))}
 
st2pay = (-2*aa*(3 - 6*aa + 4*aa^2)*b*(e5 + ab*(-1 + ac)*e5 - ac*e5))/
     ((6*aa*(-1 + e4) - 12*aa^2*(-1 + e4) + 8*aa^3*(-1 + e4) - e4)*
      (ac + ab*(-1 + ac)*(-1 + e5) + e5 - ac*e5))
 
st3pay = (2*aa*(3 - 6*aa + 4*aa^2)*(ab*(-1 + ac) - ac))/
     ((6*aa*(-1 + e4) - 12*aa^2*(-1 + e4) + 8*aa^3*(-1 + e4) - e4)*
      (ac + ab*(-1 + ac)*(-1 + e5) + e5 - ac*e5))