The most misunderstood math theorem
The usual interpretation of the Gibbard–Satterthwaite theorem is that preferential voting systems which always give result are either manipulation or dictatorship. We hear it every single time a voting reform is suggested. And there are huge problems with that interpretation.
The red flag is the silent part. The "which always give result" is usually omitted, or mentally skipped over. And exactly this is which tells us a very important thing: voting is just a part of the social decision process. When deliberation is not enough, voting won't magically fill up the gaps. So the right interpretation is:
If the voting system cannot signal that more deliberation is needed, it can lead to manipulation and dictatorship.
To understand how it works, let's take a look at the only major voting system which does not yield result in all cases: Condorcet. When there are intransitive preferences, there is no Condorcet winner. What does is actually mean?
The Condorcet loop is often illustrated with the three city problem: there are three cities, each with a given distance from each other, and with a given population. People vote to choose a capital. Everyone's first choice is their own city, and second choice is the closest one. If the numbers are constructed the right way, there will be a Condorcet loop. Here we assume that the overriding need of the voters are minimal travel, and they are voting in full awareness of their needs. Well, if the minimal travel is such an overriding need, then the obvious way to minimize Bayesian regret is to build a new capital in the center of mass (in respect to population count) of the area. Put it on the ballot, and you break the Condorcet cycle. The right choice was missing from the ballot, and a bit of deliberation would have uncovered it.
A real-world example of a Condorcet cycle is related to Brexit. ( https://blogs.lse.ac.uk/brexit/2019/01/10/deal-remain-no-deal-deal-brexit-and-the-condorcet-paradox/ )
There was a condorcet loop between Deal, Remain and No Deal. Brexit is a famous example where voters were not initially aware of the consequences of their vote. Some deliberation would have helped them to get the full picture.