ILLUSTRATION 1 --- It suffices to show how to jointly determine one coinflip, for by repetition this yields enough randomness to determine an election's outcome. To jointly determine one coinflip, each Candidate, prior to election, privately records a sequence of 1000 coinflips, sealing this record in an envelope to be stored in the public trust. (Practically, to save an hour of tedium, a trusted computer-saavy team member does this by computer.) After election, each Candidate announces their sequence, and the joint coinflip is be "HEADS" if within the Candidates' announced sequences together there are an odd number of HEADS --- counting only those announcements that, as checked against public unsealing of the envelopes, were faithful.
ANALYSIS OF ILLUSTRATION 1 --- The two key properties are (A) that any one candidate's choice of sequence can arbitrarily change the joint outcome and (B) that these choices, and also the voting choices of the People, are not affected by the choice-of-sequence of other Candidates. What guarantees (B) is that the choices were made privately before the election, using the envelopes as a "commitment scheme" (one also must guarantee that each Candidate's choice of whether or not to faithfully announce their sequence is not influenced by other Candidates' announcements, a minor problem within the realm of problems already solved by current techniques to multiply supervise ballot-counting). Even if Candidates conspire, so long as some Candidate is not in the conspiracy, then the latter's fair randomness makes the joint sequence fairly random. Moreover, if a Candidate uses any even-somewhat-predictable sequence-generation method then a conspiracy of competing Candidates could exploit that predictability to the former Candidates' detriment, so it behooves the former Candidate to use actual fair randomness. The system is therefore robust to large-but-not-unanimous conspiracies. The analysis is similar for the case where each Candidate belongs to some conspiracy but no conspiracy contains all Candidates. Except by a unanimous conspiracy of candidates, this system produces fair randomness.
ILLUSTRATION 2 --- we may refine the method to be more to-the-publicly-verifiably-fair as follows. Have each Candidate publicly label their envelope with some "digest" of the private sequence within. A "digest" is a message related to the private sequence that does not reveal the private sequence, but that together with a purported private sequence can be used to check whether the latter is authentic. This is akin to the 17th century habit of scientists to announce anagrams of latin sentences characterizing their discoveries; this way, they could claim priority without revealing their ongoing work, for when a priority dispute arises they could reveal the scrambled sentence. For many decades we have a plentiful supply of digest-methods built on the combination of many simple arithmetic operations into an intractable-to-invert mess. So by publicly announcing digests, one allows each member of the public --- if they have pen and paper and a spare weekend, or if they have a computer and a spare minute --- to verify whether an announced sequence was faithful.