Prove or disprove NP-Completeness: An optimal ordering problem
I think your question is better written as follows (to avoid ambiguity):
Find an order $\mathbf{s}=(s_1,s_2,...s_n)$ (that is, a permutation of $(1,...,n)$) such that
$$
E_{\mathbf{s}}(\mathbf{v},\mathbf{p})=p_{s_1}w_1v_{s_1}+(1-p_{s_1})p_{s_2}w_2v_{s_2}+\cdots+\left[\prod_{j=1}^{n-1}(1-p_{s_j})\right]p_{s_n}w_nv_{s_n}.
$$
is maximized, where $w_i$ is the positive weight for the expected achievable value at the $i^{th}$ stage, $\mathbf{v}=\{v_1,v_2,\ldots,v_n\}$ and $\mathbf{p}=\{p_1,p_2,\ldots,p_n\}$ are the positive profits and the probabilities associated with the $n$ receivers. If $w_i$ is a linearly decreasing function, say, if $w_i=1-i\tau$, the objective function becomes:

$$
E_{\mathbf{s}}(\mathbf{v},\mathbf{p})=p_{s_1}(1-\tau)v_{s_1}+(1-p_{s_1})p_{s_2}(1-2\tau)v_{s_2}+\cdots+\left[\prod_{j=1}^{n-1}(1-p_{s_j})\right]p_{s_n}(1-n\tau)v_{s_n}.
$$ 

Now, let us consider the following two orders which differ only in the $(i-1)^{th}$ and the $i^{th}$ positions: $\mathbf{s}=\{\ldots,\underset{i-1}{1},\underset{i}{2},\ldots\}$ and $\mathbf{\tilde{s}}=\{\ldots,\underset{i-1}{2},\underset{i}{1},\ldots\}$. It is easy to show that

$$
E_\mathbf{s}&gt;E_\mathbf{\tilde{s}}\Leftrightarrow\frac{v_1}{\frac{\tau}{\theta_1}+w_i }&gt;\frac{v_2}{\frac{\tau}{\theta_2}+w_i }.
$$
which (I believe) implies that the problem cannot be solved by a simple index policy (i.e., a sorting algorithm), because whether $\mathbf{s}=\{\ldots,\underset{i-1}{1},\underset{i}{2},\ldots\}$ or $\mathbf{\tilde{s}}=\{\ldots,\underset{i-1}{2},\underset{i}{1},\ldots\}$ is better really depends on $w_i$ (or the stage number $i$).

Here is a paper Optimal wideband spectrum sensing order based on decision-making tree in cognitive radio, which gives an exact algorithm assuming $w_i=1-i\tau$ and they claim that it runs in polynomial time $O(n^3)$, although the space complexity seems to be exponential.

I believe their algorithm holds for general monotonically decreasing $w_i$ values (with respect to $i$).

However, it seems that they solve the problem by finding the optimum from a set of candidate sequences. The running time is assumed to be proportional to the size of this candidate set. In order to find this candidate set, a branch and bound approach is adopted, but the time needed to compute this candidate set is ignored. So I double if their running time claim is valid.. I would really appeciate it if someone can convince me that their $O(n^3)$ claim is valid (and so it only takes at most $O(n^3)$ time to find the candidate set)? or if the problem (with $w_i=1-i\tau$) is actually np-hard? Thank you!