Let x stand for the point at which the first player is indiffent between standing on one turn or taking another. Let z be the score on the first spin. The probability of winning by taking another turn is the integral from z to 1 of t2 which equals (1-z3)/3. By equating this with the probability of winning by not taking another turn, z2, you find the indifference point: (1-z3)/3= z2. So the indifference point is x where x3+x2-1=0, x =~ 0.53208889 .
The probability of winning is the integral from 0 to x of (1-z3)/3 plus the integral from x to 1 of z2. After integration this answer is (x - x4/4)/3 + (1 -x 3)/3 = (-x4/4 -x3 + x + 1)/3 =~ .45380187 .
Michael Shackleford, A.S.A.