Each night, the carnivore may strike deadly to eat one of the herbivores. Each night, the herbivore's instinct makes them investigate (exactly one) element in their surroundings to discover a threat. Unfortunately they don't know what type they are, as they posess no long-term memory (more than 5 days).
* One of the species is notorious for being fearful of everything. We'll call them PARANOIDS because after they investigate another animal (even one of the same species) their brains marks the target as a carnivore threat (even if it is herbivore).
* The second species is notorious for being too trustful. We'll call them NAIVE because all their investigations lead to them trusting the targeted creature
* The third species is confused, to say the least, INSANE to be more precise. They always trust carnivores upon inspection and distrust herbivores (for no apparent reason).
* The fourth species is SANE. Their advanced senses make their results spot on. Always.
Each day, they can all share their results and discuss. Each day, all alive creatures must cast a vote for one of them to be sacrificed as a possible threat to the group. Votes can be changed once they are cast. If one of them reaches plurality (largest number of votes from total number of possible votes, not necessarily a majority of votes) at the end of the day he is sacrificed by the others and all will learn if they've sacrificed a herbivore or a carnivore before a new night begins.
1) Prove (or disprove) that there is a strategy that can be agreed beforehand (during first night) so that the herbivores win (win = no carnivore is alive in the group).
2) Prove (or disprove) that there is a strategy that can be agreed beforehand (during first night) so that the herbivores win if abstaining is allowed.
3) Prove (or disprove) that there is a strategy that can be agreed beforehand (during first night) so that the herbivores win if changing of votes is NOT allowed.
Or in a simpler setting:
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