This question relates closely to other posts. See note at the bottom.
Problem:
Suppose that a committee with $n$ members needs to vote on whether to accept a proposition.
Each member in the committee can cast a yes/no vote ($q_i\in\{1,0\}$ for $i\in \{1,2,...,n \}$), and each member's vote has a different weight ($w_i\in[0,1]$ for $i\in \{1,2,...,n \}$).
The committee rules stipulate that the proposition is to be accepted if the weighted average of votes is more than 50%, namely, if $$\bar{q}(n)=\sum_{i=1}^{n} w_i q_i>0.5$$
Suppose that (i) voting is sequential, (ii) each vote is a random draw from a Bernoulli distribution with unknown mean $p$, and (iii) the order of voting is independent of voting weights.
I would like to establish a numerical algorithm, such that a decision (accept vs reject) is made with only the first $k$ votes, and yet be statistically confident that the decision made is not far (up to some arbitrary degree) to the one that would have been made if all members had voted.
What I currently have: I am interested in a numerical approach to solve this question. A related question enquiring about the analytical bayesian approach (posted here) suggests there is no tractable exact treatment. For the case in which $w_i=w$, however, the bayesian approach (proposed in another related question here) does have a tractable exact solution.