A countably infinite number of voters, each in possession of some private information, make a decision between two choices, here denoted by \(A\) and \(C\), from the running example of a jury deciding to Acquit or Convict a defendant. The group decision is \(C\) only if the voters unanimously vote \(C\). From the pool of voters, the first \(n\) cast votes. The author provides a necessary condition under which the decision converges to the correct decision, that made under perfect information, as the number of voters \(n\) tends to infinity. The author notes that his condition is equivalent to a convergence result of \textit{P. R. Milgrom} for single-price auctions [Econometrica 47, 679-688 (1979; Zbl 0415.90100)]. Under certain continuity assumptions, the author shows that this condition is also sufficient. The author also gives an example with detailed computations to illustrate the rate of convergence.