Here is a small challenge:
In a research community (e.g., uni faculty, conferences) consisting of researchers (e.g., professors, etc), every researcher knows each other. There are good researchers and a corrupt one. Each researcher knows about some other researchers and whether each of them is good or corrupt, but s/he doesn't know whether her/himself is corrupt or not. One day, a queen who has the power to know everything about all communities, came to the research community and told that "there is one corrupt researcher in this community. You should not exchange with each other what you already know about the corruption. I ask any of you to leave in the midnight of the day once you know that yourself is corrupt."
Edit: Readers may find the definition of 'corrupt': https://www.dictionary.com/browse/corrupt
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Corruption detection in Distributed Network
In computer science, if a 'good entity' doesn't act under the rule nor communicate their knowledge, it is said malfunctioned, compromised or corrupted. Theoretically, those entities actually become no different from the corrupt ones who actually targets the network. Mathematically, if many such 'good' entities existed, the whole network is compromised, it can no longer distinguish what is good or not. When the network comes to that state, it is irreversible. Detected corruption is as important as the knowledge, and sharing detected corruption must be part of the rules.
Only computer science is given in this example, readers may get their own intuition in the matters they are concerned.
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Conjecture 1:
Given: there are rules (law) for every good entity to follow, and assume they all follow.
Conjecture: If all good entities still act (do, follow, obey) based on the common rules (law) and share knowledge (truth, facts) via communication, then corruption can be uncovered if not dominant.
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Truth, Majority, Transparency and Education
For any sample of population and any person in the sample has an equal chance to access to or deduce the truth, then majority is likely to get closer to the truth than the remainder. In practice, the chances vary and truths sometimes are restricted to only a small portion. That's why majority may not work in such setting. Transparency and education help.
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Summary of discussions
Knowledge and communication may be not sufficient to stop corruption. It needs rules and transparency.
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Version 2: In a committee of n researchers, each researcher interacts with exactly k other researchers each day and finds out whether any of the k is corrupt. The researcher then gossips the new finding with 1 other research on that day. Note that, the corrupt researcher can also gossip, but his/her message can be true/wrong each time. If the queen comes and tells that there is one corrupt researcher, can the committee spot it out? in how many days? if there is no such queen, can the committee still find it out?