Uniform-offer stopping recursion
= Uniform-offer stopping recursion
{title2=$v_j=(1+v_{j-1}^2)/2,\quad v_1=1/2$}
With independent <uniform distributions> on $[0,1]$ as sequential offers and a finite number of rounds, the continuation value with $j-1$ offers remaining is $v_{j-1}$. Accept the current offer when it reaches that threshold. Integrating the maximum of the offer and continuation value gives the recursion, by the <Snell envelope> principle.