# The last step of the Longstaff-Schwartz method

I'm reading An analysis of the Longstaff-Schwartz algorithm for American option pricing, by Clement, Lamberton and Protter.

They define the stopping times (top of page 4)

$$\tau_j^{[m]} = \begin{cases} L & \text{ if } j = L\\ j \mathbf{1}_{\left\{ Z_j \ge P_j^m(Z_{\tau_{j+1}^{[m]}})\right\}} + \tau_{j+1}^{[m]} \mathbf{1}_{\left\{ Z_j < P_j^m(Z_{\tau_{j+1}^{[m]}})\right\}} & \text{ if } j \le L-1 \end{cases}$$

And then obtain the approximate value function (equation 2.1) $$U_0^{[m]} = \max \left( Z_0, \mathbb{E} Z_{\tau_1^{[m]}} \right)$$

My question is, why didn't they take the more natural $$U_0^{[m]} = \mathbb{E} Z_{\tau_0^{[m]}}$$ instead?

When implementing Longstaff-Schwartz (see equation 2.2 and above), this choice actually makes a difference, as they introduce some high-bias (Jensen) $$U_0^{m, N} = \max \left(Z_0, \frac{1}{N} \sum_{n=1}^N Z^{(n)}_{\tau_1^{n, m, N}}\right)$$ instead of $$\frac{1}{N} \sum_{n=1}^N Z^{(n)}_{\tau_0^{n, m, N}} = \frac{1}{N} \sum_{n=1}^N \max \left(Z_0, Z^{(n)}_{\tau_1^{n, m, N}}\right)$$