I think you're asking about the field of research known as
End-to-end auditable or end-to-end voter verifiable (E2E) systems are voting systems with stringent integrity properties and strong tamper resistance. E2E systems often employ cryptographic methods to craft receipts that allow voters to verify that their votes were counted as cast, without revealing which candidates were voted for. As such, these systems are sometimes referred to as receipt-based systems.
As MS Research set of slides is a bit more clear/concise than Wikipedia's article... the problem is defined (going from blurb to more precise statements) as:
Technology exists that enables any inaccuracies and
tampering of election tallies to be detected …
… not just by election officials, but also by any
candidate, media outlet, voter, or other observer …
… and not just external tampering, but corruption by
election officials, equipment vendors, and others.
This is known as End-to-End (E2E) Verifiability.
An election is end-to-end verifiable if
- Voters can verify that their own selections have been correctly recorded.
- Anyone can verify that the recorded votes have been correctly tallied.
The real issue/challenge is that when you also want to enforce vote privacy/secrecy i.e.:
Voters must be unable to disclose their votes to others.
This is what you called "incomplete information". Without this last requirement (typical of democracies in the past 100 years or so, but not so much before), you can have an open ballot where everyone sees how everyone else votes (i.e. "complete information" to use your terms) as a fairly simple, low tech solution to just the first two constraints.
A relatively simple solution/idea when the 3rd constraint (vote secrecy aka "incomplete information" is added) is (thus) homomorphic tallying. Basically
a form of encryption allowing one to perform calculations on encrypted data without decrypting it first. The result of the computation is in an encrypted form, when decrypted the output is the same as if the operations had been performed on the unencrypted data
The MSR slides have an example.
If you're willing to make the voter do some extra work at voting time, then non-cryptographic solutions are possible e.g. Rivest's ThreeBallot.
Wikipedia mentions a couple of small scale tests/elections when such (cryptographic) E2E-V techniques were used. For a recent paper on a UK experiment, see Hao et al. "End-to-end Verifiable E-voting Trial for Polling Station Voting". For the earlier, more technical/theoretical paper from the same authors see "Every Vote Counts: Ensuring Integrity in Large-Scale Electronic Voting". In particular, research from this group attempted to eliminate any use of key-issuing Tallying Authorities.
we combine [a cancellation formula first introduced
in 2006 in the design of an anonymous veto protocol [Hao and Zielinski 2006] with the conventional homomorphic encryption to build a self-enforcing
e-voting protocol. Compared with the existing mix-net or homomorphic aggregation based tallying methods, the new method has the distinctive feature of not requiring any secret keys (hence no TAs).
Wikipedia also has a brief article on AV-nets, which underlies "the other half" of this technique.